This online resource summarizes all empirical ground-motion prediction equations (GMPEs), to estimate earthquake peak ground acceleration (PGA) and elastic response spectral ordinates, published between 1964 and late 2019 (inclusive). This resource replaces: the Imperial College London reports of Douglas (2001a), Douglas (2002) and Douglas (2004a), which provide a summary of all GMPEs from 1964 until the end of 2003; the BRGM report of Douglas (2006), which summarizes all GMPEs from 2004 to 2006 (plus some earlier models); the report of Douglas (2008), concerning GMPEs published in 2007 and 2008 (plus some earlier models); and the report of Douglas (2011), which superseded all these reports and covered the period up to 2010. It is planned to continually update this website when new GMPEs are published or errors/omissions are discovered. In addition, this resource lists published GMPEs derived from simulations, although details are not given since the focus here is on empirical models. Studies that only present graphs are only listed, as are those nonparametric formulations that provide predictions for different combinations of distance and magnitude because these are more difficult to use for seismic hazard analysis than those which give a single formula. Equations for single earthquakes or for earthquakes of approximately the same size are excluded due to their limited usefulness. Those relations based on conversions from macroseismic intensity are only listed.

This website summarizes, in total, the characteristics of 462 empirical GMPEs for the prediction of PGA and 299 empirical models for the prediction of elastic response spectral ordinates. In addition, 82 simulation-based models to estimate PGA and elastic response spectral ordinates are listed but no details are given. 50 complete stochastic models, 41 GMPEs derived in other ways, 31 non-parametric models and 15 backbone (G. M. Atkinson, Bommer, and Abrahamson 2014; Douglas 2018b) models are also listed. Finally, the table provided by Douglas (2012) is expanded and updated to include the general characteristics of empirical GMPEs for the prediction of: Arias intensity (31 models), cumulative absolute velocity (9 models), Fourier spectral amplitudes (18 models), maximum absolute unit elastic input energy (6 models), inelastic response spectral ordinates (5 models), Japanese Meterological Agency seismic intensity (4 models), macroseismic intensity (52 models, commonly called intensity prediction equations), mean period (6 models), peak ground velocity (134 models), peak ground displacement (34 models), relative significant duration (16 models) and vertical-to-horizontal response spectral ratio (11 models). This report will be updated roughly once every six months.

It should be noted that the size of this resource means that it may contain some errors or omissions. The boundaries between empirical, simulation-based and non-parametric ground-motion models are not always clear so I may classify a study differently than expected. No discussion of the merits, ranges of applicability or limitations of any of the relationships is included herein except those mentioned by the authors or inherent in the data used. This compendium is *not* a critical review of the models.

This compilation was made when I was employed at: Imperial College London, University of Iceland, BRGM and University of Strathclyde. I thank: my current and former employers for their support, many people for references, suggestions and encouragement while producing this resource, and the developers of LaTeXand associated packages, without whom this report would never have been written.

If required, you can cite this resource in the following way:

Douglas, J. (2019), Ground motion prediction equations 1964â2019, http://www.gmpe.org.uk.

ESEE Report 01-1 âA comprehensive worldwide summary of strong-motion attenuation relationships for peak ground acceleration and spectral ordinates (1969 to 2000)â (Douglas 2001a) was completed and released in January 2001. A report detailing errata of this report and additional studies was released in October 2002 (Douglas 2002). These two reports were used by Douglas (2003) as a basis for a review of previous ground-motion prediction equations (GMPEs). Following the release of these two reports, some further minor errors were found in the text and tables of the original two reports, and additional studies were found in the literature that were not included in ESEE 01-1 or the follow-on report. Also some new studies were published. Rather than produce another report listing errata and additions it was decided to produce a new report that included details on all the studies listed in the first two reports (with the corrections made) and also information on the additional studies. This report was published as a research report of Imperial College London at the beginning of 2004 (Douglas 2004a). At the end of 2006 a BRGM report was published (Douglas 2006) detailing studies published in 2004â2006 plus a few earlier models that had been missed in previous reports. Finally, at the end of 2008 another BRGM report was published (Douglas 2008) containing summaries of GMPEs from 2007 and 2008 and some additional earlier models that had been recently uncovered.

Because of the large number of new GMPEs published in 2009 and 2010 and the discovery of some additional earlier studies and various errors in the previous reports, it was decided to publish a new comprehensive report to replace the previous reports (Douglas 2001a, 2002, 2004a, 2006, 2008) containing all previous reports plus additional material rather than publish yet another addendum to the 2004 report. It was also decided that, for completeness and due to the lack of another comprehensive and public source for this information, to include a list of GMPEs developed using other methods than regression of strong-motion data, e.g. simulation-based models (e.g. Douglas and Aochi 2008). However, due to the complexity of briefly summarizing these models it was decided not to provide details but only references. This report was published as Douglas (2011).

In order to make the compendium easier to use and to update in the future it was decided to port the entire report to `html`

using the LaTeXTeXÂ 4ht package as well as add models from 2011 to the middle of 2019 and some older GMPEs that were recently found. Finally, GMPEs for intensity measures other than PGA and elastic response spectral ordinates are listed but details are not given (although some of these correspond to models for PGA and elastic spectral ordinates and hence they are summarized elsewhere in this compendium).

This report summarizes, in total, the characteristics of 462 empirical GMPEs for the prediction of peak ground acceleration (PGA) and 299 models for the prediction of elastic response spectral ordinates as well as 31 models for the prediction of Arias intensity, 9 models for cumulative absolute velocity, 18 models for Fourier spectral amplitudes, 6 models for maximum absolute unit elastic input energy, 5 models for inelastic response spectral ordinates, 4 models for Japanese Meterological Agency seismic intensity, 52 models^{1} (intensity prediction equations) for macroseismic intensity, 6 models for mean period, 134 for peak ground velocity, 34 for peak ground displacement, 16 for relative significant duration and 11 models for vertical-to-horizontal response spectral ratio. With this many GMPEs available it is important to have criteria available for the selection of appropriate models for seismic hazard assessment in a given region â Cotton et al. (2006) and, more recently, Bommer et al. (2010) suggest selection requirements for the choice of models. For the selection of GMPEs routinely applicable to state-of-the-art hazard analyses of ground motions from shallow crustal earthquakes Bommer et al. (2010) summarize their criteria thus.

Model is derived for an inappropriate tectonic environment (such as subduction-zone earthquakes or volcanic regions).

Model not published in a Thomson Reuters ISI-listed peer-reviewed journal (although an exception can be made for an update to a model that did meet this criterion).

The dataset used to derive the model is not presented in an accessible format; the minimum requirement would be a table listing the earthquakes and their characteristics, together with the number of records from each event.

The model has been superseded by a more recent publication.

The model does not provide spectral predictions for an adequate range of response periods, chosen here to be from \(0\) to \(2\,\mathrm{s}\).

The functional form lacks either non-linear magnitude dependence or magnitude-dependent decay with distance.

The coefficients of the model were not determined with a method that accounts for inter-event and intra-event components of variability; in other words, models must be derived using one- or two-stage maximum likelihood approaches or the random effects approach.

Model uses inappropriate definitions for explanatory variables, such as \(M_L\) or \(r_{epi}\), or models site effects without consideration of \(V_{s,30}\).

The range of applicability of the model is too small to be useful for the extrapolations generally required in PSHA: \(M_{\min}>5\), \(M_{\max}<7\), \(R_{\max}<80\,\mathrm{km}\).

Model constrained with insufficiently large dataset: fewer than 10 earthquakes per unit of magnitude or fewer than 100 records per \(100\,\mathrm{km}\) of distance.

Similar criteria could be developed for other types of earthquakes (e.g. subduction). For example, the reader is referred to Stewart et al. (2015) for a discussion of the selection of GMPEs for hazard assessments for the three principal tectonic regimes. Application of such criteria would lead to a much reduced set of models. The aim of this report, however, is not to apply these, or any other, criteria but simply to summarize all models that have been published. Bommer et al. (2010) also note that: â[i]f one accepts the general approach presented in this paper, then it becomes inappropriate to develop and publish GMPEs that would subsequently be excluded from use in PSHA [probabilistic seismic hazard analysis] on the basis of not satisfying one or more of the requirements embodied in the criteria.â

Predictions of median ground motions from GMPEs show great dispersion (Douglas 2010a, 2010b, 2012) demonstrating the large epistemic uncertainties involved in the estimation of earthquake shaking. This uncertainty should be accounted for within seismic hazard assessments by, for example, logic trees (e.g. Bommer and Scherbaum 2008).

A number of reviews of GMPEs have been made in the past that provide a good summary of the methods used, the results obtained and the problems associated with such relations. Trifunac and Brady (1975a, 1976) provide a brief summary and comparison of published relations. McGuire (1976) lists numerous early relations. Idriss (1978) presents a comprehensive review of published attenuation relations up until 1978, including a number which are not easily available elsewhere. Hays (1980) presents a good summary of ground-motion estimation procedures up to 1980. Boore and Joyner (1982) provide a review of attenuation studies published in 1981 and they comment on empirical prediction of strong ground motion in general. Campbell (1985) contains a full survey of attenuation equations up until 1985. Joyner and Boore (1988) give an excellent analysis of ground motion prediction methodology in general, and attenuation relations in particular; Joyner and Boore (1996) update this by including more recent studies. N. N. Ambraseys and Bommer (1995) provide an overview of relations that are used for seismic design in Europe although they do not provide details about methods used. Recent reviews include those by Campbell (2003c, 2003a) and Bozorgnia and Campbell (2004a), which provide the coefficients for a number of commonly-used equations for peak ground acceleration and spectral ordinates, and Douglas (2003). Bommer (2006) discusses some pressing problems in the field of empirical ground-motion estimation. The International Institute of Seismology and Earthquake Engineering provides a useful online resource

http://iisee.kenken.go.jp/eqflow/reference/Start.htm summarising a number of GMPEs (particularly those from Japan) and providing coefficients and an Excel spreadsheet for their evaluation. A recent discussion of current and future trends in ground-motion prediction is provided by Douglas and Edwards (2016).

Summaries and reviews of published ground-motion models for the estimation of strong-motion parameters other than PGA and elastic response spectral ordinates are available^{2}. For example: Bommer and MartÄ±Ìnez-Pereira (1999), AlarcÃ³n (2007) and Bommer, Stafford, and AlarcÃ³n (2009) review predictive equations for strong-motion duration; Tromans (2004) summarizes equations for the prediction of PGV and displacement (PGD); Bommer and AlarcÃ³n (2006) provide a more recent review of GMPEs for PGV; Hancock and Bommer (2005) discuss available equations for estimating number of effective cycles; Stafford, Berrill, and Pettinga (2009) briefly review GMPEs for Arias intensity; Rathje et al. (2004) summarize the few equations published for the prediction of frequency-content parameters (e.g. predominant frequency); and Cua et al. (2010) review various intensity prediction equations.

Equations for single earthquakes (e.g. Bozorgnia, Niazi, and Campbell 1995) or for earthquakes of approximately the same size (e.g. Seed et al. 1976; K. Sadigh, Youngs, and Power 1978) are excluded because they lack a magnitude-scaling term and, hence, are of limited use. Also excluded are those originally developed to yield the magnitude of an earthquake (e.g. Espinosa 1980), i.e. the regression is performed the other way round, which should not be used for the prediction of ground motion at a site. The model of Kim and Shin (2017) is not included because it is based on the ratio of the magnitude of the mainshock to an aftershock rather than the magnitude directly. The model of J. X. Zhao and Gerstenberger (2010) is not summarised since it uses recorded motions to estimate motions at sites without observations, within a rapid-response system. Models such as that by Olszewska (2006) and Golik and Mendecki (2012), who use âsource energy logarithmsâ to characterize mining-induced events, have been excluded because such a characterization of event size is rare in standard seismic hazard assessments. Similarly, equations derived using data from nuclear tests, such as those reported by Mickey (1971; Hays 1980), are not included.

Those based on simulated ground motions from stochastic source models (e.g G. M. Atkinson and Boore 1990) and other types of simulations (e.g. Megawati, Pan, and Koketsu 2005), those derived using the hybrid empirical technique (e.g Campbell 2003b; Douglas, Bungum, and Scherbaum 2006), those relations based on intensity measurements (e.g. Battis 1981) and backbone models (G. M. Atkinson, Bommer, and Abrahamson 2014; Douglas 2018b) are listed in ChapterÂ 6 but no details are given because the focus here is on empirical models derived from ground-motion data. Studies using simulation techniques other than the classic stochastic method and which do not provide a closed-form GMPE (e.g. Medel-Vera and Ji 2016) are not listed as they are often difficult to use. Studies which provide graphs to give predictions (e.g. Schnabel and Seed 1973) are only listed and not summarized as are those nonparametric formulations that give predictions for different combinations of distance and magnitude (e.g. Anderson 1997), both of which are generally more difficult to use for seismic hazard analysis than those which report a single formula. For similar reasons, models derived using neural networks (e.g. GÃ¼llÃ¼ and ErÃ§elebi 2007) are only listed.

GMPEs for the prediction of PGA are summarized in ChaptersÂ 2 andÂ 3 and those for spectral ordinates are summarized in ChaptersÂ 4 andÂ 5. ChapterÂ 6 lists other ground-motion models that are not detailed in the previous chapters. The final chapter (ChapterÂ 7) provides the general characteristics of GMPEs for intensity measures other than PGA and elastic spectral ordinates. All the studies that present the same GMPE are mentioned at the top of the section and in the tables of general characteristics (IllustrationsÂ [tab:pga]Â &Â [tab:speccomp]). The information contained within each section, and within tables, is the sum of information contained within each of the publications, i.e. not all the information may be from a single source. Note that GMPEs are ordered in chronological order both in the section titles and the order of the sections. Therefore, a well-known model presented in a journal article may not be listed where expected since it had previously been published in a conference proceedings or technical report. To find a given model it is recommended to examine the table of content carefully or apply a keyword search to the PDF. Some models (e.g. Abrahamson and Silva 1997) provide GMPEs for spectral accelerations up to high frequencies (e.g. \(100\,\mathrm{Hz}\)) but do not explicitly state that these equations can be used for the prediction of PGA. Therefore, they are only listed in the chapters dealing with GMPEs for the prediction of spectral ordinates (ChaptersÂ 4 andÂ 5) and their coefficients are not given. This should be considered when searching for a particular model.

To make it easier to understand the functional form of each GMPE the equations are given with variable names replacing actual coefficients and the derived coefficients and the standard deviation, \(\sigma\), are given separately (for PGA equations). These coefficients are given only for completeness and if an equation is to be used then the original reference should be consulted. If a coefficient is assumed before the analysis is performed then the number is included directly in the formula.

Obviously all the details from each publication cannot be included in this report because of lack of space but the most important details of the methods and data used are retained. The style is telegraphic and hence phrases such as âNote that â¦â should be read âThe authors [of the original model] note that â¦â. The number of records within each site and source mechanism category are given if this information was reported by the authors of the study. Sometimes these totals were found by counting the numbers in each category using the tables listing the data used and, therefore, they may be inaccurate.

This report contains details of all studies for PGA and response spectra that could be found in the literature (journals, conference proceedings, technical reports and some Ph.D. theses) although some may have been inadvertently missed^{3}. Some of the studies included here have not been seen but are reported in other publications and hence the information given here may not be complete or correct. Since this resource has been written in many distinct periods over almost two decades (2000â2019), the amount of information given for each model varies, as does the style.

In the equations unless otherwise stated, \(D\), \(d\), \(R\), \(r\), \(X\), \(\Delta\) or similar are distance and \(M\) or similar is magnitude and all other independent variables are stated. PGA is peak ground acceleration, PGV is peak ground velocity and PSV is relative pseudo-velocity.

In TablesÂ [tab:pga], [tab:speccomp] and [tab:gmpes] the gross characteristics of the data used and equation obtained are only given for the main equation in each study. The reader should refer to the section on a particular publication or the original reference for information on other equations derived in the study.

In earlier reports the name âattenuation relation(ships)â is used for the models reported. The current *de facto* standard is to refer to such models as âground motion prediction equationsâ (GMPEs) and, therefore, this terminology is adopted here. However, as discussed by Boore and Atkinson (2007 Appendix A) there is some debate over the best name for these models (e.g. âground-motion modelâ or âground motion estimation equationsâ) and some people disagree with the use of the word âpredictionâ in this context.

No discussion of the merits, ranges of applicability or limitations of any of the relationships is included herein except those mentioned by the authors or inherent in the data used. This report is *not* a critical review of the models. The ground-motion models are generally reported in the form given in the original references. The boundaries between empirical, simulation-based and non-parametric ground-motion models are not always clear so I may classify a study differently than expected. Note that the size of this report means that it may contain some errors or omissions â the reader is encouraged to consult the original reference if a model is to be used.

Ground-motion model is: \[a=c\exp(\alpha M) R^{-\beta}\] where \(a\) is in \(\,\mathrm{cm/s^2}\), \(c=2000\), \(\alpha=0.8\) and \(\beta=2\) (\(\sigma\) is not given).

Ground-motion model is: \[\begin{aligned} a&=&\frac{a_1}{\sqrt{T_G}} 10^{a_2 M-P \log_{10} R+Q}\\ P&=&a_3+a_4/R\\ Q&=&a_5+a_6/R\end{aligned}\] where \(a\) is in \(\,\mathrm{cm/s^2}\), \(a_1=5\), \(a_2=0.61\), \(a_3=1.66\), \(a_4=3.60\), \(a_5=0.167\) and \(a_6=-1.83\) (\(\sigma\) is not given).

\(T_G\) is the fundamental period of the site.

Ground-motion model is: \[A=\frac{a_1 \mathrm{e}^{a_2 M}}{a_3 \mathrm{e}^{a_4 M} +\Delta^2}\] where \(A\) is in percentage of \(\,\mathrm{g}\), \(a_1=0.69\), \(a_2=1.64\), \(a_3=1.1\) and \(a_4=1.10\) (\(\sigma\) not given).

Use data from Esteva and Rosenblueth (1964).

Ground-motion model is: \[a=c_1 \mathrm{e}^{c_2 M} (R+c_3)^{-c_4}\] where \(a\) is in \(\,\mathrm{cm/s^2}\), \(c_1=1230\), \(c_2=0.8\), \(c_3=25\), \(c_4=2\) and \(\sigma=1.02\) (in terms of natural logarithm).

Records from soils comparable to stiff clay or compact conglomerate.

Records from earthquakes of moderate duration.

Ground-motion model is: \[\log Y=b_1 + b_2 M+b_3 \log R\] where \(Y\) is in \(\,\mathrm{g}\), \(b_1=-0.2\), \(b_2=0.2\) and \(b_3=-1.1\) (\(\sigma\) not given).

Records from near dam on recent unconsolidated lake sediments which are \(\geq 50\,\mathrm{m}\) thick.

Note need for more points and large uncertainty in \(b_1\), \(b_2\) and \(b_3\).

Ground-motion model is: \[A=\alpha \mathrm{e}^{\beta m}R^{-\gamma}\] where \(A\) is in \(\,\mathrm{g}\), \(\alpha=0.279\), \(\beta=0.80\), \(\gamma=1.64\) and \(\sigma=0.74\) (in terms of natural logarithms).

Ground-motion model is: \[\log Y_a=a_1+a_2 M_L+b_3 \log R\] where \(Y_a\) is in \(\,\mathrm{cm/s^2}\), \(a_1=2.91\), \(a_2=0.32\) and \(a_3=-1.45\) (\(\sigma\) is not given).

Use records from Yonki station (20 records) which is on \(50\,\mathrm{m}\) of recent alluvium and from Paguna station (5 records) which is on unconsolidated volcanic rock.

Question validity of combining data at the two sites because of differences in geological foundations.

Note large standard errors associated with coefficients preclude accurate predictions of ground motions.

Also derive equation for Yonki station separately.

Ground-motion model is: \[y=b_1 \mathrm{e}^{b_2 M} (R+25)^{-b_3}\] where \(y\) is in \(\,\mathrm{gal}\), \(b_1=1080\), \(b_2=0.5\), \(b_3=1.32\) and \(\sigma=0.71\). \(25\) adopted from Esteva (1970).

214 (\(32\%\)) records from San Fernando (9/2/1971) earthquake and \(53\%\) of records with PGA less than \(0.5\,\mathrm{m/s^2}\).

Considers portions of data and finds magnitude dependence increases with increasing distance from source and more small accelerations increase magnitude dependence. Thus magnitude and distance cannot be considered independent variables.

Ground-motion model is: \[Y_c=b_1 \mathrm{e}^{b_2 M} (R+b_4)^{-b_3}\] where \(Y_c\) is in \(\,\mathrm{cm/s^2}\), \(b_1=5600\), \(b_2=0.8\), \(b_3=2\), \(b_4=40\) and \(\sigma=0.64\) (in terms of natural logarithm).

Ground-motion model is: \[\log A=c_1+c_2 \log(R+c_3)+c_4 M\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(c_1=2.308\), \(c_2=-1.637\), \(c_3=30\) and \(c_4=0.411\) (\(\sigma\) not reported

^{4}). Also derive equation using \(r_{epi}\): \(c_1=0.982\), \(c_2=-0.129\), \(c_3=0\) and \(c_4=0.466\) (\(\sigma\) not reported^{5}).

Ground-motion model is: \[E[v]=a10^{bM}(R+25)^{-c}\] where \(E\) indicates expectation, \(v\) is in \(\,\mathrm{gal}\), \(a=472\), \(b=0.278\), \(c=1.301\).

Excludes records for which significant soil amplification established but makes no distinction between rock and soil sites.

Focal depths between \(9\) and \(70\,\mathrm{km}\) with most about \(10\,\mathrm{km}\). Most records from earthquakes with magnitudes about \(6.5\) and most distances less than \(50\,\mathrm{km}\). Uses records from 21 different sites.

Notes that physical laws governing ground motion near the source are different than those governing motion at greater distances therefore excludes records with epicentral distance or distance to fault rupture smaller than one-half of estimated length of rupture.

Examines correlation among the records but find negligible effect.

Ground-motion model is: \[A=\lambda 10^{\alpha M} R^{\beta}\] where \(A\) is in \(\,\mathrm{g}\), \(\lambda=6.6\times 10^{-2}\), \(\alpha=0.40\), \(\beta=-1.39\) and \(\sigma=1.99\) (this is multiplication factor).

Use 113 records with distances between \(15\) to \(350\,\mathrm{km}\) from San Fernando earthquake to find distance dependence, \(\beta\).

Use 27 records of Wiggins (1964) from El Centro and Ferndale areas, with magnitudes between \(4.1\) and \(7.0\) and distances between \(17\) and \(94\,\mathrm{km}\) (assuming focal depth of \(15\,\mathrm{km}\)), to compute magnitude dependent terms assuming distance dependence is same as for San Fernando.

Ground-motion model is: \[A=c_1 \exp(c_2 M) (R+c_3)^{-c_4}\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(c_1=1230\), \(c_2=0.8\), \(c_3=13\) and \(c_4=-2\) (\(\sigma\) is not reported).

Ground-motion model is: \[\log Y=b_1 +b_2 M_L+b_3 \log R\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(b_1=0.46\), \(b_2=0.63\), \(b_3=-1.10\) and \(\sigma=0.32\)

^{6}N. N. Ambraseys and Bommer (1995) state that uses earthquakes with maximum focal depth of \(15\,\mathrm{km}\).

Ground-motion model is: \[A=c_1 \exp(c_2 M) (R+c_3)^{-c_4}\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(c_1=5000\), \(c_2=0.8\), \(c_3=40\) and \(c_4=-2\) (\(\sigma\) is not reported).

Ground-motion model is: \[\begin{aligned} \log_{10} a_{\max}&=&M+\log_{10} A_0 (R) -\log_{10} a_0(M,p,s,v)\\ \log_{10} a_0(M,p,s,v)&=&\left\{ \begin{array}{r} ap+bM+c+ds+ev+fM^2-f(M-M_{\max})^2\\ \quad \mbox{for}\quad M\geq M_{\max}\\ ap+bM+c+ds+ev+fM^2\\ \quad \mbox{for}\quad M_{\max}\geq M \geq M_{\min}\\ ap+bM_{\min}+c+ds+ev+fM^2_{\min}\\ \quad \mbox{for}\quad M\leq M_{\min}\\ \end{array} \right.\end{aligned}\] where \(a_{\max}\) is in \(\,\mathrm{cm/s^2}\), \(\log_{10} A_0 (R)\) is an empirically determined attenuation function from Richter (1958) used for calculation of \(M_L\), \(p\) is confidence level and \(v\) is component direction (\(v=0\) for horizontal and \(1\) for vertical). Coefficients are: \(a=-0.898\), \(b=-1.789\), \(c=6.217\), \(d=0.060\), \(e=0.331\), \(f=0.186\), \(M_{\min}=4.80\) and \(M_{\max}=7.50\) (\(\log_{10} A_0(R)\) not given here due to lack of space).

Use three site categories:

Alluvium or other low velocity âsoftâ deposits: \(63\%\) of records.

âIntermediateâ type rock: \(23\%\) of records.

Solid âhardâ basement rock: \(8\%\) of records.

Exclude records from tall buildings.

Do not use data from other regions because attenuation varies with geological province and magnitude determination is different in other countries.

Records baseline and instrument corrected. Accelerations thought to be accurate between \(0.07\) and \(25\,\mathrm{Hz}\) or between \(0.125\) and \(25\,\mathrm{Hz}\) for San Fernando records.

Most records (\(71\%\)) from earthquakes with magnitudes between \(6.0\)â\(6.9\), \(22\%\) are from \(5.0\)â\(5.9\), \(3\%\) are from \(4.0\)â\(4.9\) and \(3\%\) are from \(7.0\)â\(7.7\) (note barely adequate data from these two magnitude ranges). \(63\%\) of data from San Fernando earthquake.

Note that for large earthquakes, i.e. long faults, \(\log_{10} A_0(R)\) would have a tendency to flatten out for small epicentral distances and for low magnitude shocks curve would probably have a large negative slope. Due to lack of data \(\lesssim 20\,\mathrm{km}\) this is impossible to check.

Note difficulty in incorporating anelastic attenuation because representative frequency content of peak amplitudes change with distance and because relative contribution of digitization noise varies with frequency and distance.

Note that \(\log_{10} A_0(R)\) may be unreliable for epicentral distances less than \(10\,\mathrm{km}\) because of lack of data.

Change of slope in \(\log_{10} A_0(R)\) at \(R=75\,\mathrm{km}\) because for greater distances main contribution to strong shaking from surface waves, which are attenuated less rapidly (\(\sim 1/R^{1/2}\)) than near-field and intermediate-field (\(\sim 1/R^{2-4}\)), or far-field body waves (\(\sim 1/R\)).

Note lack of data to reliably characterise \(\log_{10} a_0(M,p,s,v)\) over a sufficiently broad range of their arguments. Also note high proportion of San Fernando data may bias results.

Firstly partition data into four magnitude dependent groups: \(4.0\)â\(4.9\), \(5.0\)â\(5.9\), \(6.0\)â\(6.9\) and \(7.0\)â\(7.9\). Subdivide each group into three site condition subgroups (for \(s=0\), \(1\) and \(2\)). Divide each subgroup into two component categories (for \(v=0\) and \(1\)). Calculate \(\log_{10} a_0(M,p,s,v)=M+\log_{10} A_0(R)-\log_{10} a_{\max}\) within each of the \(24\) parts. Arrange each set of \(n\) \(\log_{10} a_0\) values into decreasing order with increasing \(n\). Then \(m\)th data point (where \(m\) equals integer part of \(pn\)) is estimate for upper bound of \(\log_{10} a_0\) for \(p \%\) confidence level. Then fit results using least squares to find \(a\), â¦\(f\).

Check number of PGA values less than confidence level for \(p=0.1\), â¦, \(0.9\) to verify adequacy of bound. Find simplifying assumptions are acceptable for derivation of approximate bounds.

Ground-motion model is: \[a=b_1 \mathrm{e}^{b_2 M_L} (R+25)^{-b_3}\] where \(a\) is in \(\,\mathrm{gal}\), for \(M_L \leq 6\frac{1}{2}\) \(b_1=0.318\times 29^{1.14\bar{b}}\), \(b_2=1.03\), \(b_3=1.14 \bar{b}\) and \(\sigma=0.930\) (in terms of natural logarithm) and for \(M_L>6\frac{1}{2}\) \(b_1=26.0 \times 29^{1.22\bar{b}}\), \(b_2=0.432\), \(b_3=1.22\bar{b}\) and \(\sigma=0.592\) (in terms of natural logarithm).

Assumes all earthquakes have focal depth of \(8\,\mathrm{km}\).

Makes no distinction for site conditions in first stage where uses only earthquake records.

Studies effects of PGA cutoff (no cutoff, \(0.01\), \(0.02\) and \(0.05\,\mathrm{m/s^2}\)), distance cutoff (no cutoff and \(<150\,\mathrm{km}\)) and magnitude cutoff (all, \(\geq 5\frac{1}{2}\), \(\geq 6\), \(\geq 6 \frac{1}{2}\), \(\geq 6 \frac{3}{4}\) and \(\leq 6\frac{1}{2}\)).

Selects \(6\frac{1}{2}\) as optimum magnitude cutoff but uses all data to derive equation for \(M_L \leq 6\frac{1}{2}\) because not much difference and dispersion is slightly lower (in terms of \(\pm 1\) standard deviation have \(2.53\) and \(2.61\)).

In second stage uses only records from underground nuclear explosions, consistent with natural earthquake records, to derive site factor.

Uses 1911 alluvium and 802 rock records and derive PGA ratio of alluvium to rock assuming their PGAs equal at \(4\,\mathrm{km}\).

Finds site impedance \(\rho V_s\), where \(\rho\) is density and \(V_s\) is shear-wave velocity under site, is best measure of site condition. Use \(2000\,\mathrm{fps}\) (\(600\,\mathrm{m/s}\)) as shear-wave velocity of alluvium stations.

Multiplies equation (after taking logarithms) by \(\bar{b}=\frac{1}{2} \log_{10} (\rho V_s)\) and normalise to \(4\,\mathrm{km}\).

Notes may not be a good model for other regions.

Ground-motion model is: \[\mathrm{ACC}=a_1 \mathrm{e}^{a_2M} R^{a_3}\] where \(\mathrm{ACC}\) is in \(\,\mathrm{g}\), \(a_1=0.04\), \(a_2=1.00\) and \(a_3=-1.4\).

Ground-motion model is: \[\log A=c_1+c_2M+c_3 \log(\Delta)\] where \(A\) is in \(\,\mathrm{cm/s^2}\), for class I: \(c_1=1.455\), \(c_2=0.207\) and \(c_3=-0.598\); for class II: \(c_1=1.121\), \(c_2=0.330\) and \(c_3=-0.806\); for class III: \(c_1=1.507\),

\(c_2=0.254\) and \(c_3=-0.757\); for class IV: \(c_1=0.811\), \(c_2=0.430\) and \(c_3=-0.977\); and for all sites: \(c_1=1.265\), \(c_2=0.302\) and \(c_3=-0.800\). \(\sigma\) not reported.

Use 4 site classes:

29 records

74 records

127 records

68 records

Ground-motion model is: \[\bar{a}=a_1\bar{R}^{a_2}\exp(a_3\bar{M})\] where \(\bar{a}\) is in \(\,\mathrm{cm/s^2}\), \(a_1=1.31\), \(a_2=-0.92\) and \(a_3=1.455\) (\(\sigma\) is not given).

Uses data from former USSR, former Yugoslavia, Portugal, Italy, Iran, Greece and Pakistan.

Peak ground accelerations have either been taken from true-to-scale accelerograms or have been supplied by local networks. Records have not been high- or low-pass filtered because it was found not to work with short records.

Believes body-wave or local magnitude are the appropriate magnitude scales because interested in the high-frequency range of spectra, which are seen and sampled by strong-motion instruments, and most engineering structures have high natural frequencies.

Most of the magnitudes were recalculated using P-waves of periods of not more than \(1.2\,\mathrm{s}\) because it was found that the magnitude was dependent on the period of the P-waves used for its determination.

Groups data into intervals of \(0.5\) magnitude units by \(10\,\mathrm{km}\) in which the mean and standard deviations of the PGAs is calculated. This grouping minimises distance and magnitude-dependent effects. Notes that the number of observations is barely sufficient to allow a statistical treatment of the data and hence only test general trend. Notes that scatter is significant and decreases with increasing magnitude.

Ground-motion model is: \[\begin{aligned} y&=&b_1 \mathrm{e}^{b_2 M} (R+25)^{-b_3}\\ \mbox{where }\quad b_1&=&c_1 R^{-c_2}\\ b_2&=&d_1+d_2 \log R\\ b_3&=&e_1+e_2 \log R\end{aligned}\] where \(y\) is in \(\,\mathrm{gal}\), \(c_1=2,154,000\), \(c_2=2.10\), \(d_1=0.046\), \(d_2=0.445\), \(e_1=2.515\), \(e_2=-0.486\), for \(y=0.01\,\mathrm{g}\) \(\sigma=0.5\), for \(y=0.05\,\mathrm{g}\) \(\sigma=0.48\), for \(y=0.10\,\mathrm{g}\) \(\sigma=0.46\) and for \(y=0.15\,\mathrm{g}\) \(\sigma=0.41\) (in terms of natural logarithm).

Use \(25\) because assume energy centre of Californian earthquakes to be at depth \(5\,\mathrm{km}\).

Consider two site conditions but do not model:

Rock: (21 records)

Stiff soil: (38 records)

\(32\%\) of records from San Fernando (9/2/1971) but verifies that relationship is not significantly biased by this data.

Most records within \(50\,\mathrm{km}\) and most from earthquakes with magnitudes of about \(6.5\).

Recognises that magnitude and distance are not independent variables.

Find \(b_1\), \(b_2\) and \(b_3\) by dividing data according to distance and computing \(b\) parameters for each set using least squares. Find a distinct trend with little scatter.

Ground-motion model is: \[y=a 10^{b M} (R+25)^{-c}\] where \(y\) is in \(\,\mathrm{gal}\), \(a=108.60\), \(b=0.265\), \(c=0.808\) and \(\sigma=0.236\) (in terms of logarithm to base \(10\)).

Records from sites underlain by cohesive or cohesionless soils with shear-wave velocities less than about \(100\,\mathrm{m/s}\) and/or standard penetration resistance \(N\leq10\) in uppermost \(10\,\mathrm{m}\) with layers of considerably stiffer materials either immediately below or at depths not exceeding a few tens of metres.

Focal depths between \(9\) and \(100\,\mathrm{km}\).

Free-field accelerograms, to minimize soil-structure interaction.

Excludes records with \(\mathrm{PGA}<0.4\,\mathrm{m/s^2}\).

21 Japanese records processed with frequency cutoffs of bandpass filter, for baseline correction, adjusted so as to account for length and mean sampling rate of records and response characteristics of SMAC-2. 4 of remaining 7 records processed in same way.

Ground-motion model is: \[\log A=c_1+c_2M+c_3 \log(\Delta+30)\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(c_1=2.610\), \(c_2=0.160\) and \(c_3=-0.752\) (\(\sigma\) not reported

^{7}).Data from alluvial sites

All PGAs \(>50\,\mathrm{gal}\).

Ground-motion model is: \[\ln x=b_1+b_2 M +b_3 \ln R +b_4 Y_s\] where \(x\) is in \(\,\mathrm{cm/s^2}\), \(b_1=3.40\), \(b_2=0.89\), \(b_3=-1.17\), \(b_4=-0.20\) and \(\sigma=0.62\).

Uses two site categories:

Rock: sedimentary or basement rock or soil less than \(10\,\mathrm{m}\) thick, 11 records.

Soil: alluvium or other soft material greater than \(10\,\mathrm{m}\) thick, 59 records.

Uses records from basement of buildings or from âfree-fieldâ. Uses no more than seven records from same earthquake and no more than nine from a single site to minimize underestimation of calculated variance. Retains records which give a large distance and magnitude range.

Notes that near-field ground motion governed by different physical laws than intermediate and far field so excludes near-field data, for example El Centro (19/5/1940) and Cholame-2, from Parkfield earthquake (28/6/1966)

Considers a distance dependent site term but not statistically significant. Also uses a magnitude dependent site term and although it was statistically significant it did not reduce the scatter and also since largest magnitude for a rock site is \(6.5\), result may be biased.

Ground-motion model is: \[\ln y=\ln A+B M_s +E \ln [R+d\exp(fM_s)]\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(d=0.864\) and \(f=0.463\) and for path A (rock): \(A=157\) (for median), \(A=186\) (for mean), \(B=1.04\) and \(E=-1.90\), for path A (stiff soil): \(A=191\) (for median), \(A=224\) (for mean), \(B=0.823\) and \(E=-1.56\) and for path B (stiff soil): \(A=284\) (for median), \(A=363\) (for mean), \(B=0.587\) and \(E=-1.05\) (\(\sigma\) not given).

Separate equations for two types of path:

Shallow focus earthquakes (California, Japan, Nicaragua and India), 63 records.

Subduction (Benioff) zone earthquakes (Japan and South America), 23 earthquakes, \(5.3\leq M_s \leq 7.8\), 32 records.

Use two site categories for path A earthquakes for which derive separate equations:

Rock: 21 records.

Stiff soil: 42 records.

Use only stiff soil records for deriving subduction zone equation.

Most earthquakes for path A have \(5\leq M_s \leq 6.7\).

All data corrected. PGA for corrected Japanese and South American records much higher than uncorrected PGA.

Ground-motion model is: \[\ln A_p=a+b M_L+c \ln (R+25)\] where \(A_p\) is in \(\,\mathrm{cm/s^2}\), \(a=6.74\), \(b=0.859\), \(c=-1.80\) and \(\sigma=0.57\).

No more than 7 records from one earthquake to avoid biasing results.

Records from basements of buildings or free-field.

Ground-motion model is: \[\log y=b_1+b_2M+b_3 \log(R+25)\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(b_1=0.44\), \(b_2=0.33\), \(b_3=-2.66\) and \(\sigma=0.12\).

Uses data from three sedimentary rock sites (Somplago, San Rocco and Robic) because aim of study to provide zoning criteria as free as possible from influence of local conditions.

Compares predictions and observations and find close fit, possibly because of restricted distance range.

Note that use of simple functional form and \(r_{hypo}\) acceptable approximation because of short rupture lengths.

Ground-motion model is: \[\log y=b_1+b_2 M+b_3 \log(R+\alpha)\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(b_1=1.59\pm 0.69\), \(b_2=0.25\pm 0.03\), \(b_3=-0.79\pm 0.12\), \(\alpha=0\) and \(\sigma=0.25\) for horizontal PGA and \(b_1=1.38\pm 1.89\), \(b_2=0.24\pm 0.09\), \(b_3=-0.78\pm 0.25\) and \(\sigma=0.25\) for vertical PGA.

Use two site classes:

Includes alluvium and moraine deposits of varying thicknesses and characteristics.

Includes limestone, dolomite, flysch and cemented conglomerates, even if heavily fractured, overlain by not more than \(4\)â\(5\,\mathrm{m}\) of alluvium.

Use published and unpublished material for classification.

Focal depths between \(6\) and \(11\,\mathrm{km}\).

Use data from Friuli 1976 mainshock and subsequent earthquakes from four networks including temporary stations (ENEL, CNEN, IZIIS and CEA/DSN). Data from ENEL, CNEN and IZIIS from RFT-250 and SMA-1 instruments and data from CEA/DSN from short-period seismographs. Some records not available in digital form so used reported PGAs.

Almost all records from free-field stations.

58 PGAs from \(r_{hypo}\leq 20\,\mathrm{km}\).

\(13\,\mathrm{cm/s^2}\leq PGA \leq 515\,\mathrm{cm/s}\) with \(93\%\) above \(30\,\mathrm{cm/s^2}\).

Best-recorded earthquake (mainshock) contributed 24 PGAs.

One station contributed 17 PGAs.

Also regresses just using data from mainshock.

\(\alpha\) is either \(0\) or \(25\) in regression. Prefer results with \(\alpha=0\) because smaller standard errors in \(b_3\).

Statistical tests show \(b_2\) and \(b_3\) are significantly different than \(0\).

Also present coefficients for rock-like stations only and soil stations only. Find that effect of selection by site class does not greatly affect coefficients.

Process a smaller set of records available in digitized form (76 horizontal components) using high-pass filter (cut-off and roll-off of \(0.4\)â\(0.8\,\mathrm{Hz}\)) based on digitization noise. Note difficulty in standard processing due to high-frequency content and short durations. Use sampling rate of \(100\,\mathrm{Hz}\). Find that corrected horizontal PGAs are on average \(6\%\) lower than uncorrected PGAs and \(15\%\) show difference larger than \(10\%\). For vertical PGAs average difference is \(12\%\). Develop equations based on this subset (for horizontal PGA \(b_1=1.51\pm 0.77\), \(b_2=0.24\pm 0.04\), \(b_3=0.70 \pm 0.21\) and \(\sigma=0.24\)). Note similarity to results for uncorrected PGAs.

Also derive equation using only 39 PGAs from \(r_{hypo}\leq 20\,\mathrm{km}\) and note weak magnitude and distance dependence. Compare to data from shallow soil sites at Forgaria-Cornino and Breginj and note that local site conditions can significantly modify bedrock motions even at close distances.

Ground-motion model is: \[\log A_e=a_1 M+a_2 \log R+a_3\] where \(A_e\) is in \(\,\mathrm{cm/s^2}\), for \(A_e \geq 160\,\mathrm{cm/s^2}\) \(a_1=0.28\), \(a_2=-0.8\) and \(a_3=1.70\) and for \(A_e< 160\,\mathrm{cm/s^2}\) \(a_1=0.80\), \(a_2=-2.3\) and \(a_3=0.80\) (\(\sigma\) not given).

As a rule, PGA corresponds to S-wave.

Use five source mechanism categories (about 70 records, 59 earthquakes from W. N. America including Hawaii, Guatemala, Nicaragua, Chile, Peru, Argentina, Italy, Greece, Romania, central Asia, India and Japan):

Contraction faulting (uplift and thrust), about 16 earthquakes.

Contraction faulting with strike-slip component, about 6 earthquakes.

Strike-slip, about 17 earthquakes.

Strike-slip with dip-slip component, about 6 earthquakes.

Dip-slip, about 9 earthquakes.

Use these approximately 70 records to derive ratios of mean measured, \(A_0\), to predicted PGA, \(A_e\), \(\log(A_0/A_e)\), and for ratios of mean horizontal to vertical PGA, \(\log A_h/A_v\), for each type of faulting. Use every earthquake with equal weight independent of number of records for each earthquake.

Results are:

Category 1 Category 2 Category 3 Category 4 Category 5 \(\log A_0/A_e\) \(0.35\pm 0.13\) (16) \(0.11\pm 0.17\) (5) \(0.22\pm 0.08\) (17) \(0.06\pm 0.13\) (6) \(-0.06 \pm 0.20\) (9) \(\log A_h/A_v\) \(0.32\pm 0.13\) (12) \(0.32\pm 0.08\) (5) \(0.27\pm 0.07\) (12) \(0.18\pm 0.10\) (5) \(0.17 \pm 0.11\) (5) where \(\pm\) gives \(0.7\) confidence intervals and number in brackets is number of earthquakes used.

Also calculate mean envelope increasing speed for P-wave amplitudes, \(A\), obtained at teleseismic distances: \(n=\mathrm{d} \ln A/ \mathrm{d} t\), where \(t\) is time for P-wave arrival and try to relate to ratios for each type of faulting.

Ground-motion model is: \[a=b_1 \mathrm{e}^{b_2 M} (R+k)^{-b_3}\] where \(a\) is in \(\,\mathrm{gal}\), for method using distance partitioning \(b_1=18.4\), \(b_2=0.941\), \(b_3=1.27\) and \(k=25\) and for ordinary one-stage method \(b_1=102\), \(b_2=0.970\), \(b_3=1.68\) and \(k=25\) (\(\sigma\) not given).

Does not use PGA cutoff because PGA is, by itself, a poor index of damage in most cases.

Mean magnitude is \(5.4\) and mean distance is \(84.4\,\mathrm{km}\).

Notes problem of regression leverage for some attenuation studies. Lots of data in fairly narrow distance band, e.g. records from San Fernando earthquake, can dominate regression and lead to biased coefficients.

Divides data into ten distance bands (A-J) which are \(10\,\mathrm{km}\) wide up to \(60\,\mathrm{km}\) and then \(60\)-\(99.9\,\mathrm{km}\), \(100\)â\(139.9\,\mathrm{km}\), \(140\)â\(199.9\,\mathrm{km}\) and \(\geq 200\,\mathrm{km}\). Fits \(\log_{10} a=b M-c\) to data in each band and fits Ground-motion model to selected point set in \(M\), \(R\) and \(a\).

Also fits equation using all data using normal least squares.

Adds 52 records (\(3.2\leq M \leq 6.5\), \(5\leq R \leq 15\,\mathrm{km}\)) and repeats; finds little change.

Ground-motion model is: \[\mathrm{PGA}=a_1 10^{a_2 M} (\Delta+10)^{a_3}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{gal}\), for type I sites \(a_1=46.0\), \(a_2=0.208\) and \(a_3=-0.686\) , for type II sites \(a_1=24.5\), \(a_2=0.333\) and \(a_3=-0.924\), for type III sites \(a_1=59.0\), \(a_2=0.261\) and \(a_3=-0.886\), for type IV sites \(a_1=12.8\), \(a_2=0.432\), \(a_3=-1.125\) and for all sites \(a_1=34.1\), \(a_2=0.308\) and \(a_3=-0.925\) (\(\sigma\) not given).

Use four site categories:

Tertiary or older rock (defined as bedrock) or diluvium with depth to bedrock, \(H<10\,\mathrm{m}\), 29 records.

Diluvium with \(H\geq 10\,\mathrm{m}\) or alluvium with \(H<10\,\mathrm{m}\), 74 records.

Alluvium with \(H<25\,\mathrm{m}\) including soft layer (sand layer vulnerable to liquefaction or extremely soft cohesive soil layer) with thickness \(<5\,\mathrm{m}\), 130 records.

Other than above, usually soft alluvium or reclaimed land, 68 records.

Select earthquakes with Richter magnitude \(\geq 5.0\), hypocentral depth \(\leq 60\,\mathrm{km}\) and which include at least one record with PGA \(\geq 50\,\mathrm{gals}\) (\(0.5\,\mathrm{m/s^2}\)). Exclude records with PGA \(<10\,\mathrm{gals}\) (\(0.1\,\mathrm{m/s^2}\)).

All records for \(M\geq 7.0\) are from distance \(>60\,\mathrm{km}\).

Do regression separately for each soil category and also for combined data.

Ground-motion model is: \[Y_c=b_1 \mathrm{e}^{b_2 M} (R+b_4)^{-b_3}\] Coefficients unknown.

Ground-motion model is: \[A=10^{a_1 M-a_2 \log x +a_3}\] where \(A\) is in \(\,\mathrm{cm/s^2}\), for horizontal PGA \(a_1=0.440\), \(a_2=1.381\) and \(a_3=1.04\) and for vertical PGA \(a_1=0.485\), \(a_2=1.85\) and \(a_3=1.38\) (\(\sigma\) not given).

All records from free-field bedrock sites.

Ground-motion model is: \[\mathrm{PGA}=a \exp(b M)[R+c_1 \exp(c_2 M)]^{-d}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), for constrained model \(a=0.0782\), \(b=1.10\), \(c_1=0.343\), \(c_2=0.629\), \(d=1.75\) and \(\sigma=0.457\) (in terms of natural logarithm).

Similar to Campbell (1981) (see SectionÂ 2.35) but different data.

Ground-motion model is: \[\mathrm{PGA}=a \exp(b M)[R+c_1 \exp(c_2 M)]^{-d}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), for unconstrained model \(a=0.0159\), \(b=0.868\), \(c_1=0.0606\), \(c_2=0.700\), \(d=1.09\) and \(\sigma=0.372\) (on natural logarithm) and for constrained model \(a=0.0185\), \(b=1.28\), \(c_1=0.147\), \(c_2=0.732\), \(d=1.75\) and \(\sigma=0.384\) (in terms of natural logarithm).

Uses this functional form because capable of modelling possible nonlinear distance scaling in near field and because distance at which transition from near field to far field occurs probably proportional to fault rupture zone size.

Considers six site classifications but does not model:

Recent alluvium: Holocene Age soil with rock \(\geq 10\,\mathrm{m}\) deep, 71 records.

Pleistocene deposits: Pleistocene Age soil with rock \(\geq 10\,\mathrm{m}\) deep, 22 records.

Soft rock: Sedimentary rock, soft volcanics, and soft metasedimentary rock, 14 records.

Hard rock: Crystalline rock, hard volcanics, and hard metasedimentary rock, 9 records.

Shallow soil deposits: Holocene or Pleistocene Age soil \(<10\,\mathrm{m}\) deep overlying soft or hard rock, 17 records. Not used in analysis.

Soft soil deposits: extremely soft or loose Holocene Age soils, e.g. beach sand or recent floodplain, lake, swamp, estuarine, and delta deposits, 1 record. Not used in analysis.

Notes that data from areas outside western USA may be substantially different than those from western USA due to tectonics and recording practices but far outweighed by important contribution these data can make to understanding of near-source ground motion.

Notes use of only near-source data has made differences in anelastic attenuation negligible to inherent scatter from other factors.

Selects data from shallow tectonic plate boundaries generally similar to western N. America, deep subduction events excluded because of differences in travel paths and stress conditions.

Selects data from instruments with similar dynamic characteristics as those used in USA to avoid bias, therefore excludes data from SMAC accelerographs in Japan.

Selects data which meet these criteria:

Epicentres known with an accuracy of \(5\,\mathrm{km}\) or less, or accurate estimate of closest distance to fault rupture surface known.

Magnitudes accurate to within \(0.3\) units.

Distances were within \(20\), \(30\), and \(50\,\mathrm{km}\) for magnitudes less than \(4.75\) between \(4.75\) and \(6.25\) and greater than \(6.25\) respectively. Only uses data from earthquakes with magnitude \(\geq 5.0\) because of greatest concern for most design applications.

Hypocentres or rupture zones within \(25\,\mathrm{km}\) of ground surface.

PGA\(\geq 0.2\,\mathrm{m/s^2}\) for one component, accelerographs triggered early enough to capture strong phase of shaking.

Accelerograms either free-field, on abutments of dams or bridges, in lowest basement of buildings, or on ground level of structures without basements. Excluded Pacoima Dam record, from San Fernando (9/2/1971) earthquake due to topographic, high-frequency resonance due to large gradation in wave propagation velocities and amplification due to E-W response of dam.

Well distributed data, correlation between magnitude and distance only \(6\%\).

Uses PGA from digitised, unprocessed accelerograms or from original accelerograms because fully processed PGAs are generally smaller due to the \(0.02\,\mathrm{s}\) decimation and frequency band-limited filtering of records.

Uses mean of two horizontal components because more stable peak acceleration parameter than either single components taken separately or both components taken together.

Magnitude scale chosen to be generally consistent with \(M_w\). Division point between using \(M_L\) and \(M_s\) varied between \(5.5\) and \(6.5\); finds magnitudes quite insensitive to choice.

Notes \(r_{rup}\) is a statistically superior distance measure than epicentral or hypocentral and is physically consistent and meaningful definition of distance for earthquakes having extensive rupture zones.

Does not use all data from San Fernando earthquake to minimize bias due to large number of records.

Uses seven different weighting schemes, to control influence of well-recorded earthquakes (e.g. San Fernando and Imperial Valley earthquakes). Giving each record or each earthquake equal weight not reasonable representation of data. Uses nine distance dependent bins and weights each record by a relative weighting factor \(1/n_{i,j}\), where \(n_{i,j}\) is total number of recordings from \(i\)th earthquake in \(j\)th interval.

Finds unconstrained coefficients and all coefficients statistically significant at \(99\%\).

Finds coefficients with \(d\) constrained to \(1.75\) (representative of far-field attenuation of PGA) and \(c_2=b/d\), which means PGA is independent of magnitude at the fault rupture surface. All coefficients statistically significant at \(99\%\). Notes similarity between two models.

Plots normalised weighted residuals against distance, magnitude

^{8}and predicted acceleration. Finds that residuals uncorrelated, at \(99\%\), with these variables.

Normal probability plots, observed distribution of normalised weighted residuals and Kolmogorov-Smirnov test, at \(90\%\), confirms that PGA can be accepted as being lognormally distributed.

Finds effects of site geology, building size, instrument location and mechanism to be extensively interrelated so selects only records from free-field or small structures.

Analyses all selected data, find sites of classes E and F significantly higher PGA , at \(90\%\) level, so removes records from E and F.

Finds differences in PGA from other site categories to be negligible but notes that it cannot be extended to PGV, PGD, spectral ordinates or smaller magnitudes or further distances.

Distribution with mechanism is: 69 from strike-slip, 40 from reverse, 5 from normal and 2 records from oblique. Finds that reverse fault PGAs are systematically higher, significant at \(90\%\), than those from other fault types although size of bias is due to presence of data from outside N. America.

Considers soil (A and B) records from small buildings (115 components) and in free-field and those obtained in lowest basement of large buildings (40 components). Finds PGA significantly lower, at \(90\%\) level, in large buildings.

Finds topographic effects for 13 components used in final analysis (and for 11 components from shallow soil stations) to be significantly higher, at \(90\%\), although states size of it may not be reliable due to small number of records.

Removes Imperial Valley records and repeats analysis. Finds that saturation of PGA with distance is not strongly dependent on this single set of records. Also repeats analysis constraining \(c_2=0\), i.e. magnitude independent saturation, and also constraining \(c_1=c_2=0\), i.e. no distance saturation, finds variance when no distance saturation is significantly higher, at \(95\%\), than when there is saturation modelled.

Finds that magnitude saturation effects in modelling near-source behaviour of PGA is important and \(c_2\) is significantly greater than zero at levels of confidence exceeding \(99\%\). Also variance is reduced when \(c_2\neq0\) although not at \(90\%\) or above.

Repeats analysis using distance to surface projection of fault, finds reduced magnitude saturation but similar magnitude scaling of PGA for larger events.

Ground-motion model is: \[\log a =b_0 + b_{AN} X_{AN} + b_{AB} X_{AB} + b_M M_L +b_d \log d\] where \(a\) is in \(\,\mathrm{g}/100\), \(b_0=0.04\), \(b_{AN}=0.24\), \(b_{AB}=0.23\), \(b_M=0.41\) and \(b_d=-0.99\) (\(\sigma\) not given).

Use three site categories for Friuli records, although note that information is rather superficial:

Alluvium with depth \(>20\,\mathrm{m}\), 36 records.

Rock-like: hard rock or stiff soil, 24

^{9}records.Alluvium-like with depth \(\leq 20\,\mathrm{m}\): includes sites for which thickness of deposit is reported to be very small which accounts for a few metres of weathering of underlying bedrock, 60 records.

Alpide belt records divided into two categories: rock-like (25 records) and alluvium-like (40 records).

Use data from free-field instruments or from instruments in basements of small structures and divide data into three regions: those from 1976 Friuli shocks (120 records) \(\Rightarrow X_{AN}=X_{AB}=0\), those from 1972 Ancona swarm (40 records) \(\Rightarrow X_{AN}=1\) & \(X_{AB}=0\) and those from Alpide Belt (Azores to Pakistan excluding those from Friuli and Ancona) (64 records) \(\Rightarrow X_{AN}=0\) & \(X_{AB}=1\). Exclude records with PGA \(<0.15\,\mathrm{m/s^2}\) to avoid possible bias at low acceleration values.

Assume average focal depth of \(6\,\mathrm{km}\).

Note some PGA values derived from velocity records which are retained because compatible with other data. No instrument corrections applied to Friuli records because correction does not substantially alter PGA.

Use \(M_L\) because determined at short distances and allows homogenous determination from lowest values up to saturation at \(M_L=7.0\) and it is determined at frequencies of nearly \(1\,\mathrm{Hz}\), close to accelerographic band.

Perform regression on PGAs from each of the three regions and each soil types considered within that region.

Group rock-like (R) and thick alluvium (ThA) records together for Friuli. Find \(b_d\) for Friuli equations derived for thin alluvium-like and rock and thick alluvium not significantly different but \(b_M\) is significantly different, at \(95\%\) level. Repeat analysis using only Tolmezzo records because of large scatter in residuals but decide it is in thA category.

For Alpide belt equations find \(b_M\) is almost the same for Rl and Al records and the difference in \(b_d\) is less than standard error, thus repeat analysis using a dummy variable \(X_{Al}\) which equals \(0\) for Rl and \(1\) for Al records.

Ground-motion model is: \[\log A=b_0+b_1M+b_2\log(\Delta+b_3)\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(b_0=2.305\), \(b_1=0.178\), \(b_2=-0.666\) and \(b_3=30\) (\(\sigma\) not reported

^{10}).Use \(N\)-value profiles from standard penetration tests (SPTs) to characterise sites. Use data from alluvial and diluvial sites. Exclude data from rock and very soft soils. Define \(S_n\) as a weighting function for SPT profile to characterise softness of surface layers. Plot residuals from model against \(S_n\) and find correlation. Derive site correction factors for model. Find coefficient of variation decreases after applying correction.

Use 346 uncorrected components (magnitudes from \(5\) to about \(7.8\) and \(r_{epi}\) from about \(7\) to \(500\,\mathrm{km}\)) to derive preliminary model without site term: \(\bar{A}=b_0 10^{b_1M}/(r_{epi}+30)^{b_2}\). Derive models using different data selections: all data, \(M<6.6\), \(M\geq 6.6\), \(r_{epi}\leq 119\,\mathrm{km}\), \(r_{epi}>119\,\mathrm{km}\), \(M\)-\(r_{epi}\) region where expected PGA (from model using all data) \(\geq 39\,\mathrm{cm/s^2}\) or expected PGA \(<39\,\mathrm{cm/s^2}\) (these selected divide data into two equal halves). Examine scaling of the various models in 3D plots. Based on this analysis, conclude that model depends on \(M\)-\(r_{epi}\) range used for data selection. Because of engineering interest in PGA\(>10\,\mathrm{gal}\) believe model should be derived using \(M\)-\(r_{epi}\) region defined by expected PGA.

18 records from 1978 Off Miyagi earthquakes and 6 records from 1978 Izu-oshima-kinkai earthquake.

Strong correlation between \(M\) and \(r_{epi}\) with almost all data from \(M>7\) being from \(r_{epi}>100\,\mathrm{km}\).

Try different \(b_3\) values but find influence on coefficient of variation minimal so fix to \(30\,\mathrm{km}\).

For final model use corrected accelerograms (PGAs generally 10% to 30% higher than uncorrected values). Most data from SMAC instruments.

Plot residuals w.r.t. \(r_{epi}\) and \(M\) and find no trends.

Ground-motion model is: \[\begin{aligned} \log y&=&\alpha +\beta \mathbf{M}-\log r +b r\\ \mbox{where }r&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(y\) is in \(\,\mathrm{g}\), \(\alpha=-1.02\), \(\beta=0.249\), \(b=-0.00255\), \(h=7.3\) and \(\sigma=0.26\).

Use two site categories (not all records have category):

Rock: sites described as granite, diorite, gneiss, chert, greywacke, limestone, sandstone or siltstone and sites with soil material less than \(4\) to \(5\,\mathrm{m}\) thick overlying rock, 29 records. Indicate caution in applying equations for \(\mathbf{M}>6.0\) due to limited records.

Soil: sites described as alluvium, sand, gravel, clay, silt, mud, fill or glacial outwash except where soil less than \(4\) to \(5\,\mathrm{m}\) thick, 96 records.

Restrict data to western North American shallow earthquakes, depth less than \(20\,\mathrm{km}\), with \(\mathbf{M}>5.0\). Most records from earthquakes with magnitudes less than \(6.6\).

Exclude records from base of buildings three or more storeys high and from abutments of dams.

Exclude records associated with distances which had an uncertainty greater than \(5\,\mathrm{km}\).

Exclude records from distances greater than or equal to the shortest distance to an instrument which did not trigger.

Six earthquakes recorded at only one station so not included in second stage regression.

Include quadratic dependence term, \(\gamma \mathbf{M}^2\), but not significant at \(90\%\) level so omitted.

Include site term, \(cS\), but not significant so omitted.

Examine residuals against distance for difference magnitude ranges, no obvious differences in trends are apparent among the different magnitude classes.

Consider a magnitude dependent \(h=h_1 \exp (h_2 [\mathbf{M}-6.0])\) but reduction in variance not significant. Also prefer magnitude independent \(h\) because requires fewer parameters.

Examine effect of removing records from different earthquakes from data.

Examine effect of different \(h\) on residuals and \(b\). Note coupling between \(h\) and \(b\).

Note coincidence of anelastic coefficient, \(b\), and measured \(Q\) values. Also note similarity between \(h\) and proportions of depth of seismogenic zone in California.

Ground-motion model is: \[y=a\{(x+d)^2+1\}^c \mathrm{e}^{-b (x+d)}\] where \(y\) is in \(\,\mathrm{g}\), for \(5\leq \mathbf{M}<6\) \(a=1.2\), \(b=0.066\), \(c=0.033\), \(d=23\) and standard error for one observation of \(0.06\,\mathrm{g}\), for \(6 \leq \mathbf{M}<7\) \(a=1.2\), \(b=0.044\), \(c=0.042\), \(d=25\) and standard error for one observation of \(0.10\,\mathrm{g}\), for \(7 \leq \mathbf{M}\leq 7.7\) \(a=0.24\) \(b=0.022\), \(c=0.10\), \(d=15\) and standard error for one observation of \(0.05\,\mathrm{g}\) and for \(6\leq \mathbf{M}\leq 7.7\) \(a=1.6\), \(b=0.026\), \(c=-0.19\), \(d=8.5\) and standard error for one observation of \(0.09\,\mathrm{g}\).

Use data of Joyner and Boore (1981).

Form of equation chosen to satisfy plausible physical assumptions but near-field behaviour is not determined from overwhelming contributions of far-field data.

Apply nonlinear regression on \(y\) not on \(\log y\) to give more weight to near-field values.

Split data into four magnitude dependent groups: \(5 \leq \mathbf{M} < 6\), \(6 \leq \mathbf{M} < 7\), \(7 \leq \mathbf{M} \leq 7.7\) and \(6 \leq \mathbf{M} \leq 7.7\).

Use form of equation and regression technique of Joyner and Boore (1981), after removing 25 points from closer than \(8\,\mathrm{km}\) and find very similar coefficients to Joyner and Boore (1981). Conclude from this experiment and their derived coefficients for the four magnitude groups that using their form of equation predicted near-field accelerations are not governed by far-field data.

Find no evidence of systematic increase in PGA near the source as a function of magnitude and that the large scatter prevents attaching significance to differences in near-field PGA which are predicted using their attenuation relations for different magnitude ranges.

Ground-motion model is: \[\begin{aligned} \log y&=&\alpha+\beta(M-6)+\gamma(M-6)^2-p \log r+b r+c S\\ r&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(y\) is in \(\,\mathrm{g}\), \(\beta=0.23\), \(\gamma=0\), \(p=1\), \(b=-0.0027\), \(c=0\), \(h=8.0\) and \(\sigma=0.28\) and for randomly oriented component \(\alpha=0.43\) and for larger component \(\alpha=0.49\).

Use same data and method as Joyner and Boore (1981), see SectionÂ 2.38, for PGA.

Use data from shallow earthquakes, defined as those for which fault rupture lies mainly above a depth of \(20\,\mathrm{km}\).

Ground-motion model is: \[\ln (a)=C_1+C_2 M+C_3 \ln [R+C_4 \exp(C_5 M)]\] where \(a\) is in \(\,\mathrm{g}\), \(C_1=-1.17\), \(C_2=0.587\), \(C_3=-1.26\), \(C_4=2.13\), \(C_5=0.25\) and \(\sigma=0.543\).

Use data from Italy (6 records, 6 earthquakes), USA (18 records, 8 earthquakes), Greece (13 records, 9 earthquakes), Iran (3 records, 3 earthquakes), Pakistan (3 records, 1 earthquake), Yugoslavia (3 records, 1 earthquake), USSR (1 record, 1 earthquake), Nicaragua (1 record, 1 earthquake), India (1 record, 1 earthquake) and Atlantic Ocean (1 record, 1 earthquake).

Develop for use in .

Ground-motion model is: \[\log A_{\mathrm{mean}}=a M -b\log R +c\] where \(A_{\mathrm{mean}}\) is in \(\,\mathrm{cm/s^2}\), \(a=1.1143\), \(b=1.576\) and \(c=2.371\) (\(\sigma\) not given).

Fits equation by eye because least squares method is often strictly dependent on marginal observations, particularly for little pronounced dependence.

Ground-motion model is: \[A^{1/3}=a_1+a_2 M+a_3 \ln (d^2+a_4^2)\] where \(A\) is in \(\,\mathrm{g}\), \(a_1=0.432 (0.072)\), \(a_2=0.110 (0.012)\), \(a_3=-0.0947 (0.0101)\), \(a_4=6.35 (3.24)\), \(\sigma_1=0.0351 (0.0096)\) (inter-event) and \(\sigma_2=0.0759 (0.0042)\) (intra-event), where numbers in brackets are the standard errors of the coefficients.

Use exploratory data analysis (EDA) and alternating conditional expectations (ACE) techniques.

Firstly sought to determine functions \(\theta(A)\), \(\phi (M)\) and \(\psi (d)\) so that \(\theta (A) \doteq \phi(M)+\psi(d)\), i.e. an approximately additive relationship. Prefer additivity because of linearity, ease of interpolation and interpretation and departures from fit are more easily detected.

Use ACE procedure to find model. For set of data, with response \(y_i\) and predictors \(w_i\) and \(x_i\) find functions to minimize: \(\sum_{i=1}^n [\theta(y_i)-\phi(w_i)-\psi(x_i)]^2\) subject to \(\sum \phi (w_i)=0\), \(\sum \psi (x_i)=0\), \(\sum \theta (y_i)=0\) and \(\sum \theta (y_i)^2=n\). Search amongst unrestricted curves or unrestricted monotonic curves. Use EDA to select specific functional forms from the estimates of \(\theta\), \(\phi\) and \(\psi\) at each data point.

Do not use weighting because does not seem reasonable from statistical or seismological points of view.

Do not want any individual earthquake, e.g. one with many records, overly influencing results.

Note that because each earthquake has its own source characteristics its records are intercorrelated. Therefore use ârandom effects modelâ which accounts for perculiarities of individual earthquakes and correlation between records from same event.

On physical grounds, restrict \(\theta\), \(\phi\) and \(\psi\) to be monotonic and find optimal transformation of magnitude is approximately linear, optimal transformation of distance is logarithmic and cube root is optimal for acceleration transformation.

Note that need correlations between coefficients, which are provided, to attach uncertainties to estimated PGAs.

Provide method of linearization to give \(95\%\) confidence interval for acceleration estimates.

Also provide a graphical procedure for estimating accelerations that does not rely on an assumed functional form.

Examine residual plots (not shown) and found a candidate for an outlying observation (the record from the Hollister 1974 earthquake of \(0.011\,\mathrm{g}\) at \(17.0\,\mathrm{km}\)).

Find that assumption of normality after transformation seems reasonable.

Ground-motion model is: \[\begin{aligned} \ln y&=&a+b M+d \ln [r+h_1 \exp (h_2 M)]+s\\ \mbox{where }s&=&e_1 K_1+e_2 K_2+e_3 K_3+e_4 K_4+e_5 K_5+e_6(K_4+K_5)\tanh(e_7 r)\end{aligned}\] where \(y\) is in \(\,\mathrm{g}\), \(a=-2.817\), \(b=0.702\), \(d=-1.20\), \(h_1=0.0921\), \(h_2=0.584\), \(e_1=0.32\), \(e_2=0.52\), \(e_3=0.41\), \(e_4=-0.85\), \(e_5=-1.14\), \(e_6=0.87\), \(e_7=0.068\) and \(\sigma=0.30\).

^{11}Uses two site categories:

Soils \(\leq 10\,\mathrm{m}\) deep.

Other.

Uses three embedment categories:

Basements of buildings \(3\)â\(9\) storeys.

Basements of buildings \(\geq 10\) storeys.

Other.

Selects data using these criteria:

Largest horizontal component of peak acceleration was \(\geq 0.02\,\mathrm{g}\) [\(\geq 0.2\,\mathrm{m/s^2}\)].

Accelerograph triggered early enough to record strongest phase of shaking.

Magnitude of earthquake was \(\geq 5.0\).

Closest distance to seismogenic rupture was \(<30\) or \(<50\,\mathrm{km}\), depending on whether magnitude of earthquake was \(<6.25\) or \(>6.25\).

Shallowest extent of seismogenic rupture was \(\leq 25\,\mathrm{km}\).

Recording site located on unconsolidated deposits.

Excludes records from abutments or toes of dams.

Derives two equations: unconstrained (coefficients given above) and constrained which includes a anelastic decay term \(kr\) which allows equation to be used for predictions outside near-source zone (assumes \(k=-0.0059\) for regression, a value appropriate for region of interest should be chosen).

Uses two source mechanism categories:

Strike-slip.

Reverse.

Uses two directivity categories:

Rupture toward site.

Other.

Ground-motion model is: \[\begin{aligned} \log y&=&c_0+c_1 (\mathbf{M}-6) +c_2(\mathbf{M}-6)^2 +c_3 \log r +c_4 r +S\\ \mbox{where }r&=&(d^2+h^2)^{1/2}\\ \mbox{and: }S&=&\left\{ \begin{array}{r@{\quad \mbox{for}\quad}l} 0&\mbox{rock site}\\ c_6 \log \frac{V}{V_0}&\mbox{soil site}\\ \end{array} \right.\end{aligned}\] where \(y\) is in \(\,\mathrm{g}\), coefficients \(c_0\) to \(c_4\), \(h\) and \(\sigma\) are from Joyner and Boore (1981) and \(c_6\) and \(V_0\) are not significant at \(90\%\) level so do not report them.

Use data of Joyner and Boore (1981).

Continuous site classification for soil sites in terms of shear-wave velocity, \(V\), to depth of one quarter wavelength of waves of period of concern. \(V\) measured down to depths of at least \(30\,\mathrm{m}\) and then extrapolated using geological data. \(V\) known for 33 stations.

Soil amplification factor based on energy conservation along ray tubes, which is a body wave argument and may not hold for long periods for which surface waves could be important. Does not predict resonance effects.

Regress residuals, \(R_{ij}\), w.r.t. motion predicted for rock sites on \(\log R_{ij}=P_i + c_6 V_j\), where \(j\) corresponds to \(j\)th station and \(i\) to \(i\)th earthquake. Decouples site effects variation from earthquake-to-earthquake variation. Find unique intercept by requiring average site effect term calculated using shear-wave velocity to be same as that calculated using rock/soil classification.

No significant, at \(90\%\), correlation between residuals and \(V\) for PGA.

Repeat regression on residuals using \(V\) and depth to underlying rock (defined as either shear-wave velocity \(>750\,\mathrm{m/s}\) or \(>1500\,\mathrm{m/s}\)). Find no correlation.

Ground-motion model is: \[X(M,\Delta,\mathrm{GC}_i)=a(\mathrm{GC}_i) 10^{b(\mathrm{GC}_i) M} (\Delta+30)^{c}\] where \(X(M,\Delta,\mathrm{GC}_i)\) is in \(\,\mathrm{gal}\), \(c=-1.218\), for group 1 sites \(a(\mathrm{GC}_1)=987.4\), \(b(\mathrm{GC}_1)=0.216\) and \(\sigma=0.216\), for group 2 sites \(a(\mathrm{GC}_2)=232.5\), \(b(\mathrm{GC}_2)=0.313\) and \(\sigma=0.224\) and for group 3 sites \(a(\mathrm{GC}_3)=403.8\), \(b(\mathrm{GC}_3)=0.265\) and \(\sigma=0.197\).

Use three site categories:

Tertiary or older rock (defined as bedrock) or diluvium with \(H<10\,\mathrm{m}\) or fundamental period \(T_G<0.2\,\mathrm{s}\).

Diluvium with \(H\geq 10\,\mathrm{m}\), alluvium with \(H<10\,\mathrm{m}\) or alluvium with \(H<25\,\mathrm{m}\) including soft layer with thickness \(<5\,\mathrm{m}\) or fundamental period \(0.2<T_G<0.6\,\mathrm{s}\).

Other than above, normally soft alluvium or reclaimed land.

Only includes free-field records with \(M_{\mathrm{JMA}}\geq 5.0\) and focal depths \(D_p<60\,\mathrm{km}\). Excludes records from structures with first floor or basement.

Records instrument corrected, because Japanese instruments substantially suppress high frequencies, considering accuracy of digitization for frequencies between \(\frac{1}{3}\) and \(12\,\mathrm{Hz}\).

Note that \(M_{\mathrm{JMA}}\) and \(\Delta\) not necessarily most suitable parameters to represent magnitude and distance but only ones for all records in set.

Note lack of near-field data for large magnitude earthquakes, approximately \(\frac{3}{4}\) of records from \(M_{\mathrm{JMA}}<7.0\).

Use \(30\,\mathrm{km}\) in distance dependence term because focal depth of earthquakes with magnitudes between \(7.5\) and \(8.0\) are between \(30\) and \(100\,\mathrm{km}\) so \(30\) is approximately half the fault length.

Try equation: \(\log X= f_1+f_2M+f_3 \log(\Delta+30)+f_4 D_p +f_5 M \log (\Delta+30)+f_6 M D_p+f_7 D_p \log (\Delta+30)+f_8 M^2+f_9\{\log(\Delta+30)\}^2+f_{10} D_p^2\) where \(f_i\) are coefficients to be found considering each soil category separately. Apply multiple regression analysis to 36 combinations of retained coefficients, \(f_i\), and compute multiple correlation coefficient, \(R\), and adjusted multiple correlation coefficient, \(R^*\). Find that inclusion of more than three coefficients does not give significant increase in \(R^*\), and can lead to unrealistic results. Conclude due to insufficient data.

Consider \(a\), \(b\) and \(c\) dependent and independent of soil type and examine correlation coefficient, \(R\), and adjusted correlation coefficient, \(R^*\). Find that \(c\) is not strongly dependent on soil type.

Find match between normal distribution and histograms of residuals.

Four Ground-motion models: \[\begin{aligned} \log_{10} Y&=&a+b M+d \log_{10} [(R^2+h^2)^{1/2}] \mbox{\qquad Model I}\\ \log_{10} Y&=&a+b M+d \log_{10} [R+c_1 \exp (c_2 M)] \hfill \mbox{\qquad Model II}\\ \log_{10} Y&=&a+b M+d \log_{10} \left[ \frac{c_1}{R^2} + \frac{c_2}{R} \right] +e R \mbox{\qquad Model III} \\ \log_{10} Y&=&a+b M+d \log_{10} [R+25] \mbox{\qquad Model IV}\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), for model I \(a=-1.320\), \(b=0.262\), \(d=-0.913\), \(h=3.852\) and \(\sigma=0.158\), for model II \(a=-1.115\), \(b=0.341\), \(c_1=1.000\), \(c_2=0.333\), \(d=-1.270\) and \(\sigma=0.154\), for model III \(a=-2.000\), \(b=0.270\), \(c_1=0.968\), \(c_2=0.312\), \(d=0.160\), \(e=-0.0105\) and \(\sigma=0.175\) and for model IV \(a=1.009\), \(b=0.222\), \(d=-1.915\) and \(\sigma=0.174\).

Note \(25\) in Model IV should not be assumed but should be found by regression.

Note tectonics and travel paths may be different between N. American and foreign records but consider additional information in near field more relevant.

Selection procedure composite of Campbell (1981) and Joyner and Boore (1981). Exclude data from buildings with more than two storeys.

Weighted least squares, based on distance, applied to control influence of well recorded events (such as San Fernando and Imperial Valley). Similar to Campbell (1981)

Test assumption that logarithm of residuals are normally distributed. Cannot disprove assumption.

Variability between models not more than \(\pm 20\%\) at distances \(>10\,\mathrm{km}\) but for distances \(<1\,\mathrm{km}\) up to \(\pm 50\%\).

Ground-motion model is: \[\log A_{\mathrm{mean}}=aM-b \log R+c\] where \(A_{\mathrm{mean}}\) is in \(\,\mathrm{cm/s^2}\), \(a=0.37\), \(b=1.58\) and \(c=2.35\) (\(\sigma\) not given).

Considers two site conditions but does not model:

Solid

Soft

Fits equation by eye.

States applicable approximately for: \(R_{\mathrm{lower}} \leq R \leq R_{\mathrm{upper}}\) where \(\log R_{\mathrm{lower}} \doteq 0.1 M+0.5\) and \(\log R_{\mathrm{upper}} \doteq 0.35 M +0.4\), due to distribution of data.

Notes great variability in recorded ground motions up to \(R=30\,\mathrm{km}\) due to great influence of different site conditions.

Notes for \(M\leq 4\) source can be assumed spherical but for \(M>4\) elongated (extended) shape of focus should be taken into account.

Ground-motion model is: \[\mathrm{PGA}=a_1 \exp (a_2 M) (R+a_3)^{-a_4}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), \(a_1=0.1548\), \(a_2=0.5442\), \(a_3=8\) and \(a_4=1.002\) (\(\sigma\) not given).

All records from aftershocks of 1975 Haicheng earthquake and from 1976 Tangshan earthquake and aftershocks.

Most records from earthquakes with magnitude less than \(5.8\) and from distances \(<30\,\mathrm{km}\).

Exclude records with PGA \(<0.5\,\mathrm{m/s^2}\) to avoid too much contribution from far field.

Due to small number of records simple regression technique justified.

States valid for \(4\leq M\leq6.5\) and \(R\leq 100\,\mathrm{km}\).

Also use 158 records from western N. America to see whether significantly different than N. Chinese data. Derive equations using both western N. American and N. Chinese data and just western N. American data and find that predicted PGAs are similar, within uncertainty.

Insufficient data to find physically realistic anelastic term.

Ground-motion model is: \[\begin{aligned} \log A&=&a_1+a_2 M-\log r+a_3 r\\ \mbox{where} \quad r^2&=&d^2+a_4^2\end{aligned}\] where \(A\) is in \(\,\mathrm{g}\), \(a_1=-1.229 (0.196)\), \(a_2=0.277 (0.034)\), \(a_3=-0.00231 (0.00062)\), \(a_4=6.650 (2.612)\), \(\sigma_1=0.1223 (0.0305)\) (inter-event) and \(\sigma=0.2284 (0.0127)\) (intra-event), where numbers in brackets are the standard errors of the coefficients.

Provide algorithm for random effects regression.

Note that the functional form adopted in Brillinger and Preisler (1984) is strictly empirical and hence repeat analysis using functional form of Joyner and Boore (1981), which is based on physical reasoning.

Note that need correlations between coefficients, which are provided, to attach uncertainties to estimated PGAs.

Use very similar data to Kawashima, Aizawa, and Takahashi (1984); do not use some records because missing due to recording and digitizing processes. Use equation and method (although do not check all 36 combinations of forms of equation) used by Kawashima, Aizawa, and Takahashi (1984), see sectionÂ 2.46.

\(X(M,\Delta,\mathrm{GC}_i)\) is in \(\,\mathrm{gal}\). Coefficients are: \(c=-1.190\) and for ground group 1 \(a=117.0\) and \(b=0.268\) and for ground group 2 \(a=88.19\) and \(b=0.297\) and for group ground 3 \(a=13.49\) and \(b=0.402\) with \(\sigma=0.253\).

Ground-motion model is: \[A=b_1 \exp(b_2 M) (R+h)^{-b_3}\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(b_1=2164\), \(b_2=0.7\pm 0.03\), \(h=20\) and \(b_3=-1.8\pm 0.02\) (\(\sigma\) is not reported).

Derived by averaging (at \(M=7.5\)) eight previous models: Donovan (1973), Orphal and Lahoud (1974), Esteva (1974), Katayama (1974) and Trifunac (1976a).

Check predictions against eight Greek accelerograms and find agreement.

Ground-motion model is: \[\log_{10} a =A+B M+C \log_{10} R +DR\] where \(a\) is in \(\,\mathrm{cm/s^2}\), for N.E. China \(A=-0.474\), \(B=0.613\), \(C=-0.873\) and \(D=-0.00206\) (\(\sigma\) not given) and for S.W. China \(A=0.437\), \(B=0.454\), \(C=-0.739\) and \(D=-0.00279\) (\(\sigma\) not given).

Consider two site conditions for NE records but do not model:

Rock: 28 records.

Soil: 45 records.

Consider all records to be free-field.

Note that Chinese surface-wave magnitude, \(M\), is different than \(M_s\) and may differ by \(0.5\) or more. Use \(m_b\) or \(M_s\) and find larger residuals.

Most records from \(M\leq 5.8\).

Note isoseismals are not elongated for these earthquakes so use of another distance measure will not change results by much.

Also derives equation for SW China (\(3.7 \leq M \leq 7.2\), \(6.0 \leq R \leq 428.0\,\mathrm{km}\) all but one record \(\leq 106.0\,\mathrm{km}\) , 36 records from 23 earthquakes) and note difference between results from NE China although use less data.

Note that some scatter may be due to radiation pattern.

Note that data is from limited distance range so need more data to confirm results.

Ground-motion model is: \[\begin{aligned} \log A_m&=&a_1+a_2M-\log R-a_3 R\\ R&=&\sqrt{d^2+h^2}\end{aligned}\] where \(A_m\) is \(\,\mathrm{g}\), \(a_1=-1.49\), \(a_2=0.31\), \(a_3=0.0248\), \(h=9.4\,\mathrm{km}\) and \(\sigma=0.32\) (for horizontal components) and \(a_1=-1.92\), \(a_2=0.29\), \(a_3=0.0146\), \(h=6.7\,\mathrm{km}\) and \(\sigma=0.36\) (for vertical components).

Data from experimental strong-motion array consisting of 12 Kinemetrics PDR-1 instruments deployed in the epicentral area of the \(M_s=7.8\) Tangshan earthquake of 28th July 1976. Provide details of site geology at each station; most stations are on soil.

Records from earthquakes recorded by only one station were excluded from analysis.

Note that equations are preliminary and more refined equations await further studies of magnitudes and distances used in analysis.

Note that high anelastic attenuation coefficient may be due to biases introduced by the distribution in magnitude-distance space and also because of errors in magnitude and distances used.

Ground-motion model is: \[\ln (a)=C_1+C_2 M+C_3 \ln [R+C_4 \exp(C_5 M)]+C_6 F\] where \(a\) is in \(\,\mathrm{g}\), \(C_1=-0.855\), \(C_2=0.46\), \(C_3=-1.27\), \(C_4=0.73\), \(C_5=0.35\), \(C_6=0.22\) and \(\sigma=0.49\).

Use data from Italy (47 records, 9 earthquakes), USA (128 records, 18 earthquakes), Greece (11 records, 8 earthquakes), Iran (2 records, 2 earthquakes), Yugoslavia (7 records, 2 earthquake), Nicaragua (1 record, 1 earthquake), New Zealand (3 records, 3 earthquakes), China (2 records, 2 earthquakes) and Canada (2 records, 1 earthquake).

Develop for use in .

Select earthquakes with \(M_s<7\) and \(R\leq 40\,\mathrm{km}\).

Focal depths \(<40\,\mathrm{km}\).

Use two source mechanism categories (40 records have no source mechanism given):

Strike-slip and normal, 85 records.

Thrust, 78 records.

Also derive equation not considering source mechanism, i.e. \(C_6=0\).

Ground-motion model is: \[A=a_1 (\mathrm{e}^{a_2 M_L})(d_h)^{a_3}\] where \(A\) is in \(\,\mathrm{g}\), \(a_1=0.00205\), \(a_2=1.72\) and \(a_3=-1.58\) (\(\sigma\) not given).

Ground-motion model is: \[\ln y=a+bM_s+cM_s^2+d \ln (r+1)+kr\] where \(y\) is in \(\,\mathrm{gal}\), \(a=2.48456\), \(b=0.73377\), \(c=-0.01509\), \(d=-0.50558\), \(k=-0.00935\) and \(\sigma=0.58082\).

Records from deep soil sites (generally greater than \(60\,\mathrm{m}\) in thickness).

Data from shallow crustal earthquakes.

Ground-motion model is (for shallow earthquakes): \[\log A=a_1+a_2M-\log r+a_3 r\] where A is in \(\,\mathrm{cm/s^2}\), \(a_1=1.23\) (for hard sites), \(a_1=1.41\) (for soft sites), \(a_2=0.385\) and \(a_3=-0.00255\) (\(\sigma\) is not given).

Ground-motion model is (for subduction zone earthquakes): \[\log A=b_1+b_2 M-\log \sqrt{r^2+100^2}+b_3 r\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(b_1=2.08\) (for hard sites), \(b_1=2.32\) (for soft sites), \(b_2=0.35\) and \(b_3=-0.0025\) (\(\sigma\) is not given).

Use four site categories:

Rock

Stiff soil

Deep cohesionless soil (\(\geq 16\,\mathrm{m}\))

Soft to medium stiff clay (\(\geq 16\,\mathrm{m}\))

Categories 1 and 2 are combined into a hard (H) class and 3 and 4 are combined into a soft (S) class. This boundary established using field evidence at a shear-wave velocity of \(400\,\mathrm{m/s}\) and at an SPT N count of 60.

Use data from ground floors and basements of small or low structures (under 3 stories) because believe that small structures have little effect on recorded ground motions.

Separate earthquakes into shallow (\(h\leq 19\,\mathrm{km}\)) and subduction (\(h\geq 20\,\mathrm{km}\)) because noted that ground motions have different characteristics.

Use epicentral distance for Japanese data because practical means of representing deep subduction earthquakes with distant and imprecise fault locations.

Do not use rupture distance or distance to surface projection of rupture because believe unlikely that stress drop and peak motions will occur with equal strength along the fault length and also because for most records fault locations are not reliably determinable.

Note that there is a paucity of data but believe that the few high peak values observed (e.g. Pacoima Dam and Morgan Hill) cannot be dismissed without the possibility that interpretations will be affected dangerously.

For subduction equations, use records from Japanese SMAC instruments that have not been instrument corrected, even though SMAC instruments show reduced sensitivity above \(10\,\mathrm{Hz}\), because ground motions \(>10\,\mathrm{Hz}\) are not significant in subduction earthquakes. Do not use records from SMAC instruments for shallow earthquakes because high frequency motions may be significant.

Examine differences between ground motions in extensional (strike-slip and normal faulting) and compressional (reverse) regimes for shallow earthquakes but do not model. Find that the extensional ground motions seem to be higher than compressional motions, which suggest is because rupture propagation comes closer to ground surface in extensional faults than in compressional faults.

Group records into \(1\,M\) unit intervals and plot ground motions against distance. When data is numerous enough the data points are encompassed in boxes (either one, two or three) that have a range equal to the distribution of data. The positions of the calculated values within the boxes were used as guides for shaping appropriate curves. Initially curves developed for \(M=6.5\) were there is most data and then these were extended to smaller and larger magnitudes.

Ground-motion model is: \[\log y=a+bM-\log (R^2+h^2)^{1/2}+eS\] where \(y\) is in \(\,\mathrm{g}\) and for distance to surface projection of fault \(a=-1.562\), \(b=0.306\), \(e=0.169\), \(h=5.8\) and \(\sigma=0.173\).

Use two site categories:

Stiff and deep soil: limestone, sandstone, siltstone, marl, shale and conglomerates (\(V_s>800\,\mathrm{m/s}\)) or depth of soil, \(H\), \(>20\,\mathrm{m}\), 74 records.

Shallow soil: depth of soil, \(H\), \(5\leq H\leq 20\,\mathrm{m}\), 21 records.

Select records which satisfy these criteria:

Reliable identification of the triggering earthquake.

Magnitude greater than \(4.5\) recorded by at least two stations.

Epicentres determined with accuracy of \(5\,\mathrm{km}\) or less.

Magnitudes accurate to within \(0.3\) units.

Accelerograms from free-field. Most are from small electric transformer cabins, 4 from one- or two-storey buildings with basements and 5 from near abutments of dams.

Depths between \(5.0\) and \(16.0\,\mathrm{km}\) with mean \(8.5\,\mathrm{km}\).

Focal mechanisms are: normal and oblique (7 earthquakes, 48 records), thrust (9 earthquakes, 43 records) and strike-slip (1 earthquake, 4 records).

Notes lack of records at short distances from large earthquakes.

Records baseline-, instrument-corrected and filtered with cutoff frequencies determined by visual inspection in order to maximise signal to noise ratio within band. Cutoff frequencies ranged from \(0.2\) to \(0.4\,\mathrm{Hz}\) and from \(25\) to \(35\,\mathrm{Hz}\). This correction routine thought to provide reliable estimates of PGA so uncorrected PGA do not need to be used.

For well separated multiple shocks, to which magnitude and focal parameters refer, use only first shock.

Magnitude scale assures a linear relationship between logarithm of PGA and magnitude and avoids saturation effects of \(M_L\).

Distance to surface projection of fault rupture thought to be a more physically consistent definition of distance for earthquakes having extensive rupture zones and is easier to predict for future earthquakes. Also reduces correlation between magnitude and distance.

Use Exploratory Data Analysis using the ACE procedure to find transformation functions of distance, magnitude and PGA.

Include anelastic attenuation term but it is positive and not significant.

Include magnitude dependent \(h\) equal to \(h_1\exp(h_2M)\) but find \(h_2\) not significantly different than zero. Note distribution of data makes test not definitive.

Find geometric attenuation coefficient, \(c\), is close to \(-1\) and highly correlated with \(h\) so constrain to \(-1\) so less coefficients to estimate.

Consider deep soil sites as separate category but find difference between them and stiff sites is not significant.

Also use two-stage method but coefficients and variance did not change significantly with respect to those obtained using one-stage method, due to uniform distribution of recordings among earthquakes.

Find no significant trends in residuals, at \(99\%\) level and also no support for magnitude dependent shape for attenuation curves.

Exclude records from different seismotectonic and geological regions and repeat analysis. Find that predicted PGA are similar.

Plot residuals from records at distances \(15\,\mathrm{km}\) or less against magnitude; find no support for magnitude dependence of residuals.

Note some records are affected by strong azimuthal effects, but do not model them because they require more coefficients to be estimated, direction of azimuthal effect different from region to region and azimuthal effects have not been used in other relationships.

Ground-motion model is: \[\ln y=a+b \mathbf{M}+c_1(8.5-\mathbf{M})^{c_2}+d\ln[r+h_1 \exp(h_2 \mathbf{M})]\] where \(y\) is in \(\,\mathrm{g}\). For strike-slip earthquakes: \(b=1.1\), \(c_1=0\), \(c_2=2.5\), for PGA at soil sites \(a=-2.611\) and \(d=-1.75\), for \(\mathbf{M}<6.5\) \(h_1=0.8217\), \(h_2=0.4814\) and for \(\mathbf{M}\geq 6.5\) \(h_1=0.3157\) and \(h_2=0.6286\), for PGA at rock sites \(a=-1.406\) and \(d=-2.05\), for \(\mathbf{M}<6.5\) \(h_1=1.353\) and \(h_2=0.406\) and for \(\mathbf{M}\geq 6.5\) \(h_1=0.579\) and \(h_2=0.537\). For reverse-slip increase predicted values by \(20\%\). For \(\mathbf{M}<6.5\) \(\sigma=1.26-0.14\mathbf{M}\) and for \(\mathbf{M} \geq 6.5\) \(\sigma=0.35\).

Uses two site categories:

Soil

Rock

Use two source mechanism categories:

Strike-slip

Reverse-slip

Supplement data with significant recordings of earthquakes with focal depths \(<20\,\mathrm{km}\) from other parts of world.

Different equations for \(\mathbf{M}<6.5\) and \(\mathbf{M}\geq 6.5\).

Ground-motion model is: \[\log y_{\mathrm{max}}=\alpha M_s -c \log R +\beta\] where \(y_{\mathrm{max}}\) is in \(\,\mathrm{cm/s^2}\), \(\alpha=0.429\), \(c=2.976\), \(\beta=5.396\) and \(\sigma=0.15\).

More complicated functional form unwarranted due to limited distance range.

Depths between \(15\) and \(20\,\mathrm{km}\).

Only use data from a single firm site (Ciudad Universitaria), on a surface layer of lava flow or volcanic tuff.

Only records from coastal earthquakes.

Residuals plotted against distance, no trends seen.

Give amplification factor for lake bed sites (\(25\) to \(80\,\mathrm{m}\) deposit of highly compressible, high water content clay underlain by resistant sands), but note based on only a few sites so not likely to be representative of entire lake bed.

Ground-motion model is: \[\ln (A)=a_1+a_2M_s+a_3 \ln(R)+a_4R\] where \(A\) is in \(\,\mathrm{g}\), \(a_1=-1.987\), \(a_2=0.604\), \(a_3=-0.9082\), \(a_4=-0.00385\) and \(\sigma=0.68\).

Ground-motion model is: \[\log A =C_m M+C_h H -C_d \log (R+A\exp BM )+C_o\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(A\) and \(B\) so PGA becomes independent of magnitude at fault rupture, \(H\) is depth of point on fault plane when \(R\) becomes closest distance to fault plane, \(C_m=0.627\), \(C_h=0.00671\), \(C_d=2.212\), \(C_o=1.711\) and \(\sigma=0.211\).

Focal depths \(<100\,\mathrm{km}\).

Convert records from sites with \(V_s<300\,\mathrm{m/s}\) into records from sites with \(V_s>300\,\mathrm{m/s}\) using 1-D wave propagation theory.

Introduce term \(C_h H\) because it raises multiple correlation coefficient for PGA.

Note equations apply for site where \(300 \leq V_s \leq 600\,\mathrm{m/s}\).

Ground-motion model is: \[\log A =a M -\log (R+c 10^{a M}) -b R +d\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(a=0.41\), \(b=0.0034\), \(c=0.032\), \(d=1.30\) and \(\sigma=0.21\).

Use four site categories for some Japanese stations (302 Japanese records not classified):

Rock: 41 records

Hard: ground above Tertiary period or thickness of diluvial deposit above bedrock \(<10\,\mathrm{m}\), 44 records.

Medium: thickness of diluvial deposit above bedrock \(>10\,\mathrm{m}\), or thickness of alluvial deposit above bedrock \(<10\,\mathrm{m}\), or thickness of alluvial deposit \(<25\,\mathrm{m}\) and thickness of soft deposit is \(<5\,\mathrm{m}\), 66 records.

Soft soil: other soft ground such as reclaimed land, 33 records.

Use 1100 mean PGA values from 43 Japanese earthquakes (\(6.0\leq M_{\mathrm{JMA}} \leq 7.9\), focal depths \(\leq 30\,\mathrm{km}\)) recorded at many stations to investigate one and two-stage methods. Fits \(\log A=c-b\log X\) (where \(X\) is hypocentral distance) for each earthquake and computes mean of \(b\), \(\bar{b}\). Also fits \(\log A =a M -b^{*} \log X +c\) using one-stage method. Find that \(\bar{b}>b^{*}\) and shows that this is because magnitude and distance are strongly correlated (\(0.53\)) in data set. Find two-stage method of Joyner and Boore (1981) very effective to overcome this correlation and use it to find similar distance coefficient to \(\bar{b}\). Find similar effect of correlation on distance coefficient for two other models: \(\log A=a M-b \log (\Delta +30)+c\) and \(\log A=aM-\log X -b X +c\), where \(\Delta\) is epicentral distance.

Japanese data selection criteria: focal depth \(<30\,\mathrm{km}\), \(M_{\mathrm{JMA}}>5.0\) and predicted PGA \(\geq 0.1\,\mathrm{m/s^2}\). US data selection criteria: \(d_r \leq 50\,\mathrm{km}\), use data from Campbell (1981).

Because \(a\) affects distance and magnitude dependence, which are calculated during first and second steps respectively use an iterative technique to find coefficients. Allow different magnitude scaling for US and Japanese data.

For Japanese data apply station corrections before last step in iteration to convert PGAs from different soil conditions to standard soil condition using residuals from analysis.

Two simple numerical experiments performed. Firstly a two sets of artificial acceleration data was generated using random numbers based on attenuation relations, one with high distance decay and which contains data for short distance and one with lower distance decay, higher constant and no short distance data. Find that the overall equation from regression analysis has a smaller distance decay coefficient than individual coefficients for each line. Secondly find the same result for the magnitude dependent coefficient based on similar artificial data.

Exclude Japanese data observed at long distances where average acceleration level was predicted (by using an attenuation relation derived for the Japanese data) to be less than the trigger level (assume to be about \(0.05\,\mathrm{m/s^2}\)) plus one standard deviation (assume to be \(0.3\)), i.e. \(0.1\,\mathrm{m/s^2}\), to avoid biasing results and giving a lower attenuation rate.

Use the Japanese data and same functional form and method of Joyner and Boore (1981) to find an attenuation relation; find the anelastic coefficient is similar so conclude attenuation rate for Japan is almost equal to W. USA.

Find difference in constant, \(d\), between Japanese and W. USA PGA values.

Plot residuals against distance and magnitude and find no bias or singularity.

Ground-motion model is: \[\log \mathrm{PGA} =[(a_1\log R+a_2)/a_3] (M_L-a_4)-a_5\log R -a_6 R +a_7\] where \(\mathrm{PGA}\) is in \(\,\mathrm{m/s^2}\), \(a_1=5\), \(a_2=3\), \(a_3=20\), \(a_4=6\), \(a_5=0.77\), \(a_6=0.0045\) and \(a_7=1.2\) (\(\sigma\) not given).

Considers three site categories but does not model:

Rock: 6 records

Alluvium: 5 records

Average site: 10 records

Most records from earthquakes with magnitudes about \(3\) and most from distances below about \(20\,\mathrm{km}\).

Band pass filter records to get PGA associated with waves with periods between \(0.1\) and \(0.5\,\mathrm{s}\) because high frequency PGA from uncorrected records not of engineering significance.

Adds 4 near source (\(5\leq R \leq 10\,\mathrm{km}\)) records from US, Indian and New Zealand earthquakes with magnitudes between \(6.3\) and \(6.7\) to supplement high magnitude range.

Add some PGA points estimated from intensities associated with 14/10/1968 \(M_L=6.9\) Meckering earthquake in Western Australia.

Plot 6 records from one well recorded event with \(M_L=4.5\) and fit an attenuation curve of form \(\log \mathrm{PGA}=b_1 -b_2 \log R-b_3 R\) by eye. Plot PGA of all records with \(2\leq R\leq 20\,\mathrm{km}\) against magnitude, fit an equation by eye. Use these two curves to normalise all PGA values to \(M_L=4.5\) and \(R=5\,\mathrm{km}\) from which estimates attenuation relation.

Ground-motion model is: \[A=a (\exp(bM)) \left( \frac{R}{R_0}+c \right)^{-d}\] where \(A\) is in \(\,\mathrm{g}\), \(\ln a=-5.75\), \(b=1.72\), \(c=0\), \(d=1.69\) and \(R_0=1\) (\(\sigma\) not given).

Few records from free-field, most are in dams or special structures.,

Because only 62 records, set \(R_0=1\) and \(c=0\).

Most records from earthquakes with \(M_L\) between \(1.5\) and \(2.0\).

Maximum PGA in set \(3.05\,\mathrm{m/s^2}\).

Nonuniform distribution of focal distances. One quarter of records from same hypocentral distance. Therefore plot PGA of these records against magnitude (\(1.2\lesssim M_L \lesssim 4.3\) most less than \(2.1\)) to find \(b\). Then plot \(bM-\ln A\) against \(\ln (R/R_0)\) for all records to find \(a\) and \(d\).

Notes limited data.

Ground-motion model is: \[\ln (a)=b_1'+b_2 M+b_3 \ln (R+c)\] where \(a\) is in \(\,\mathrm{cm/s^2}\), \(b_1'=6.4830\), \(b_2=0.5438\), \(b_3=-1.3330\), \(c=20\,\mathrm{km}\) and \(\sigma=0.6718\) (for horizontal PGA) and \(b_1=5.6440\), \(b_2=0.5889\), \(b_3=-1.3290\), \(c=20\,\mathrm{km}\) and \(\sigma=0.6690\) (for vertical PGA) (also give coefficients for other choices of \(c\)).

Data from âmoderateâ soil conditions.

Data mainly from SMA-1s but 17 from RFT-250s.

Data from northern Greece (5 records, 4 stations, 3 earthquakes), northern Italy (45 records, 18 stations, 20 earthquakes) and former Yugoslavia (70 records, 42 stations, 23 earthquakes).

Data from free-field or in basements of structures.

Select records from earthquakes with \(3\leq M \leq 7\). Most earthquakes with \(M\leq 5.5\). 4 earthquakes (4 records) with \(M\leq 3.5\), 20 (27 records) with \(3.5<M\leq 4.5\), 13 (25 records) with \(4.5<M\leq 5.5\), 8 (50 records) with \(5.5<M\leq 6.5\) and 1 (14 records) with \(M>6.5\).

Select records from earthquakes with \(h\leq 40\,\mathrm{km}\). Most earthquakes with \(h\leq 10\,\mathrm{km}\). 6 earthquakes with \(h\leq 5\,\mathrm{km}\), 30 with \(5<h\leq 10\,\mathrm{km}\), 5 with \(10<h\leq 20\,\mathrm{km}\), 4 with \(20<h \leq 30\,\mathrm{km}\) and 1 with \(h>30\).

Select records that satisfied predetermined processing criteria so that their amplitude would be such as to give negligible errors after processing.

Select records to avoid concentration of records w.r.t. certain sites, magnitudes, hypocentral distances or earthquakes. Most well-recorded earthquakes is 15/4/1979 Montenegro earthquake with 14 records.

Try values of \(c\) between \(0\) and \(40\,\mathrm{km}\). Find standard deviation does not vary much for different choices.

Test assumption of the log-normal probability distribution of data using graph in a coordinate system for log-normal distribution of probability, by \(\chi^2\) test and by the Kolmogorov-Smirnov test (not shown). Find assumption is acceptable.

Ground-motion model is: \[\ln y=C_0+C_1 M_s+C_2 \ln R\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(C_0=4.75\), \(C_1=0.52\), \(C_2=-1.00\) and \(\sigma=0.53\) for hard ground and horizontal; \(C_0=3.77\), \(C_1=0.70\), \(C_2=-1.24\) and \(\sigma=0.64\) for hard ground and vertical; \(C_0=4.37\), \(C_1=0.57\), \(C_2=-0.91\) and \(\sigma=0.52\) for medium ground and horizontal; \(C_0=3.20\), \(C_1=0.75\), \(C_2=-1.07\) and \(\sigma=0.55\) for medium ground and vertical; \(C_0=4.68\), \(C_1=0.55\), \(C_2=-0.99\) and \(\sigma=0.54\) for soft ground and horizontal; and \(C_0=3.53\), \(C_1=0.71\), \(C_2=-1.10\) and \(\sigma=0.66\) for soft ground and vertical.

Use 3 site classes (and develop independent models for each):

Equivalent to rock. 57 records.

Soil depth \(<20\,\mathrm{m}\) or âshallowâ. Includes sites described as âmediumâ.53 records.

Soil depth \(>20\,\mathrm{m}\) or âdeepâ. Includes sites described as âsoftâ or âalluviumâ. 53 records.

Notes that difficult to classify due to lack of information for some stations (particularly for soft and medium sites).

Extends models of PML (1982, 1985) to spectral ordinates.

Develop for use in the UK.

Use earthquakes with focal depths \(\leq 30\,\mathrm{km}\).

Prioritize data away from well-defined plate-boundaries, e.g. central and eastern North America (Miramichi, Nahanni, New Madrid), mainland China (e.g. Tangshan) and the Alpide Belt (Ancona, Friuli, Irpinia, Cephalonia, Koyna, Montenegro, Imotski, Gazli, Tabas), over data from California. When Californian data is required use near-field records so that model not affected by specific attenuation of California. California data was required for soft ground model because of lack of near-field data from other areas.

Vast majority of data from \(M_w 4\) to \(7\) and from \(\leq 100\,\mathrm{km}\).

Because of limited information on stress field and diversity of focal mechanisms in the UK do not consider style of faulting when selecting data.

Correct data for instrument response and apply elliptic filter for majority of data (some data already processed using similar schemes). Note that differences below \(0.04\,\mathrm{s}\) between records corrected in different ways are minor.

Use PGAs from processed data, which note may be slightly lower than those from uncorrected records.

Ground-motion model is: \[\log \bar{A}=\alpha M-\beta \log (\Delta+10)+\gamma T +\delta\] where \(\bar{A}\) is in \(\,\mathrm{gal}\), \(T\) is predominant period of site, \(\alpha=0.509\), \(\beta=2.32\), \(\gamma=0.039\) and \(\delta=2.33\) (\(\sigma\) not given).

Correlation coefficient between magnitude and distance is \(0.84\), so magnitude and distance cannot be considered independent, so attenuation rate, \(\beta\), is difficult to find.

First step fit \(\log \bar{A}=-\beta_i \log (\Delta+10)+\delta_i\) to each earthquake. Define reliability parameter, \(\psi_i=N_i R_i^2\), where \(N_i\) is degrees of freedom for \(i\) earthquake and \(R_i\) is correlation coefficient. Plot \(\psi_i\) against \(\beta_i\) and find attenuation rate scattered, between \(-6\) and \(9\), for \(\psi_i<1\) (Group B) and for \(\psi_1>1\) attenuation rate converges (Group U).

Group B includes earthquakes with focal depths \(>388\,\mathrm{km}\), earthquakes with small magnitudes and records from distances \(\approx 100\,\mathrm{km}\), earthquakes with records from great distances where spread of distances is small, earthquakes recorded by only 3 stations and earthquakes with abnormal records. Exclude these records.

Apply multiple regression on Group U to find \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\) simultaneously. Also fix \(\beta=\sum \psi_i \beta_i/\sum \psi_i\) and find \(\alpha\), \(\gamma\) and \(\delta\). Find different coefficients but similar correlation coefficient. Conclude due to strong correlation between \(M\) and \(\Delta\) so many regression planes exist with same correlation coefficient.

Perform Principal Component Analysis (PCA) on \(\log A\), \(M\), \(\log (\Delta+10)\), \(T\) and \(\log \bar{A}/A\) and find that equation found by fixing \(\beta\) is not affected by ill-effect of correlation between \(M\) and \(\Delta\).

Omit \(T\) from regression and find little effect in estimation.

Ground-motion model is: \[\begin{aligned} \log_{10} a&=&\beta-\nu \log_{10} x\\ \mbox{where }\beta&=&b_1 +b_2 M\\ \mbox{and: }\nu&=&c_1+c_2 M\end{aligned}\] where \(a\) is in \(\,\mathrm{gal}\), \(b_1=-3.64\), \(b_2=1.29\), \(c_1=-0.99\) and \(c_2=0.38\) (\(\sigma\) not given).

Focal depths between \(0\) and \(130\,\mathrm{km}\).

Regress recorded PGA of each earthquake, \(i\), on \(\log_{10} a=\beta_i -\nu_i \log_{10} x\), to find \(\beta_i\) and \(\nu_i\). Then find \(b_1\) and \(b_2\) from \(\beta=b_1+b_2 M\) and \(c_1\) and \(c_2\) from \(\nu=c_1+c_2 M\).

Also consider \(\nu=d_1 \beta\).

Find \(\beta\) and \(\nu\) from 6 earthquakes (magnitudes between \(5.4\) and \(6.1\)) from Tokyo-Yokohama area are much higher than for other earthquakes, so ignore them. Conclude that this is due to effect of buildings on ground motion.

Ground-motion model is: \[\ln (a_{\mathrm{max}})=C_1+C_2M_w-C_3 \ln[R+C_4 \exp (C_5 M_w)]+B Z_t\] where \(a_{\mathrm{max}}\) is in \(\,\mathrm{g}\), \(C_1=19.16\), \(C_2=1.045\), \(C_3=4.738\), \(C_4=205.5\), \(C_5=0.0968\), \(B=0.54\) and \(\sigma=1.55-0.125 M_w\).

Use only rock records to derive equation but use some (389 records) for other parts of study. Classification using published shear-wave velocities for some sites.

Exclude data from very soft lake deposits such as those in Mexico City because may represent site with special amplification characteristics.

Data from subduction zones of Alaska, Chile, Peru, Japan, Mexico and Solomon Islands.

Use two basic types of earthquake:

Interface earthquakes: low angle, thrust faulting shocks occurring on plate interfaces.

Intraslab earthquakes: high angle, predominately normal faulting shocks occurring within down going plate.

Classification by focal mechanisms or focal depths (consider earthquakes with depths \(>50\,\mathrm{km}\) to be intraslab). Note that possible misclassification of some intraslab shocks as interface events because intraslab earthquakes do occur at depths \(<50\,\mathrm{km}\).

Plots PGA from different magnitude earthquakes against distance; find near-field distance saturation.

Originally include anelastic decay term \(-C_6 R\) but \(C_6\) was negative (and hence nonphysical) so remove.

Plot residuals from original PGA equation (using rock and soil data) against \(M_w\) and \(R\); find no trend with distance but reduction in variance with increasing \(M_w\). Assume standard deviation is a linear function of \(M_w\) and find coefficients using combined rock and soil data (because differences in variance estimation from rock and soil are not significant).

Use derived equation connecting standard deviation and \(M_w\) for weighted (weights inversely proportional to variance defined by equation) nonlinear regression in all analyses.

Plot residuals from original PGA equation; find that hypothesis that coefficients of equations for interface and intraslab earthquakes are the same can be rejected (using likelihood ratio test for nonlinear regression models) at \(0.05\) percentile level for both soil and rock. Try including a term proportional to depth of rupture into equation (because intraslab deeper than interface events) but find no significant reduction in standard error. Introduce \(B Z_t\) term into equation; find \(B\) is significant at \(0.05\) percentile level. Try including rupture type dependence into other coefficients but produces no further decrease in variance so reject.

Use only data from sites with multiple recordings of both interface and intraslab earthquakes and include dummy variables, one for each site, to remove differences due to systematic site effects. Fix \(C_1\) to \(C_5\) to values from entire set and find individual site terms and \(B\); find \(B\) is very similar to that from unconstrained regression.

Examine residuals for evidence of systematic differences between ground motion from different subduction zones; find no statistically significant differences in PGA among different subduction zones.

Use geometric mean of two horizontal components to remove effect of component-to-component correlations that affect validity of statistical tests assuming individual components of motion represent independent measurements of ground motion. Results indicate no significant difference between estimates of variance about median relationships obtained using geometric mean and using both components as independent data points.

Extend to \(M_w>8\) using finite difference simulations of faulting and wave propagation modelled using ray theory. Method and results not reported here.

Ground-motion model is: \[\log_{10} a=\alpha +\beta M -\bar{c} \log_{10} [r+\exp(h_2 M)] +F\phi+E b r\] where \(F=1\) for reverse or reverse oblique events and \(0\) otherwise and \(E=1\) for interplate events and \(0\) otherwise, \(a\) is in \(\,\mathrm{g}\), for horizontal PGA \(\alpha=-0.62\), \(\beta=0.177\), \(\bar{c}=0.982\), \(h_2=0.284\), \(\phi=0.132\), \(b=-0.0008\) and \(\sigma=0.277\) and for vertical PGA \(\alpha=-1.15\), \(\beta=0.245\), \(\bar{c}=1.096\), \(h_2=0.256\), \(\phi=0.096\), \(b=-0.0011\) and \(\sigma=0.296\).

Consider three site classifications, based on Joyner and Boore (1981):

Rock: corresponds to C, D & E categories of Campbell (1981), 159 records.

Soil: corresponds to A,B & F categories of Campbell (1981), 324 records.

Unclassified: 102 records.

Use to examine possible dependence in residuals not in regression because of many unclassified stations.

Data based on Campbell (1981).

Fault mechanisms are: strike-slip (256 records from 28 earthquakes), normal (14 records from 7 earthquakes), normal oblique (42 records from 12 earthquakes), reverse (224 records from 21 earthquakes) and reverse oblique (49 records from 8 earthquakes). Grouped into normal-strike-slip and reverse events. Weakly correlated with magnitude (\(0.23\)), distance (\(0.18\)) and tectonic environment (\(0.03\)).

Tectonic environments are: interplate (555 records from 66 earthquakes) and intraplate (30 records from 10 earthquakes) measurements. Weakly correlated with magnitude (\(-0.26\)), distance (\(-0.17\)) and fault mechanism (\(0.03\)).

Depths less than \(25\,\mathrm{km}\).

Use array average (37 instruments are in array) from 10 earthquakes recorded at array in Taiwan.

Most records from distances less than \(100\,\mathrm{km}\) and magnitude distribution is reasonably uniform but correlation between magnitude and distance of \(0.52\).

Try two-stage technique and model (modified to include fault mechanism and tectonic environment parameters) of Joyner and Boore (1981), find inadmissable positive anelastic coefficient, so do not use it.

Use a hybrid regression technique based on Joyner and Boore (1981) and Campbell (1981). A method to cope with highly correlated magnitude and distance is required. First step: fit data to \(f_2(r)=\bar{c} \log_{10}(r+h)\) and have separate constants for each earthquake (like in two-stage method of Joyner and Boore (1981)). Next holding \(\bar{c}\) constant find \(\alpha\), \(\beta\), \(b\) and \(h_2\) from fitting \(h=\exp(h_2 M)\). Weighting based on Campbell (1981) is used.

Form of \(h\) chosen using nonparametric function, \(H(M)\), which partitions earthquakes into \(0.5\) unit bins. Plot \(H(M)\) against magnitude. Find that \(H(M)=h_1 \exp (h_2 M)\) is controlled by Mexico (19/9/1985) earthquake and \(h_1\) and \(h_2\) are highly correlated, \(0.99\), although does given lower total variance. Choose \(H(M)=\exp (h_2 M)\) because Mexico earthquake does not control fit and all parameters are well-determined, magnitude dependent \(h\) significant at \(90\%\).

Try removing records from single-recorded earthquakes and from shallow or soft soil but effect on predictions and variance small (\(<10\%\)).

Plot weighted residuals within \(10\,\mathrm{km}\) no significant, at \(90\%\), trends are present.

Find no significant effects on vertical PGA due to site classification.

Ground-motion model is: \[\ln \mathrm{PHA} = a+b M_L - 1.0 \ln [R+c_1]\] where \(\mathrm{PHA}\) is in \(\,\mathrm{g}\), \(a=-2.501\), \(b=0.623\), \(c_1=7.28\) and \(\sigma=0.506\).

Selects records from deep soil (\(>10\,\mathrm{m}\)). Excludes data from shallow soil (\(\leq 10\,\mathrm{m}\)) and rock sites and those in basements of buildings or associated with large structures, such as dams and buildings taller than two storeys. Selects records with epicentral distances \(\leq 20\,\mathrm{km}\) for \(M_L<4.75\) and distances \(\leq 30\,\mathrm{km}\) for \(M_L\geq 4.75\) to minimize regional differences in anelastic attenuation and potential biases associated with nontriggering instruments and unreported PGAs.

Focal depths, \(H\), between \(1.8\) and \(24.3\,\mathrm{km}\) with mean of \(8.5\,\mathrm{km}\).

PGAs scaled from either actual or uncorrected accelerograms in order to avoid potential bias due to correction.

Uses weighted nonlinear least squares technique of Campbell (1981).

Tries two other forms of equation: \(\ln \mathrm{PHA}=a+b M_L-1.0 \ln [R+c_1] +e_1 H\) and \(\ln \mathrm{PHA}=a+b M_L -1.0 \ln [R+c_1]+e_2 \ln H\) for epicentral and hypocentral distance. Allows saturation of PGA for short distances but finds nonsignificant coefficients, at \(90\%\). Also tries distance decay coefficient other than \(-1.0\) but finds instability in analysis.

Examines normalised weighted residuals against focal depth, \(M_L\) and distance. Finds that although residuals seem to be dependent on focal depth there are probably errors in focal depth estimation for deep earthquakes in the study so the dependence may not be real. Finds residuals not dependent on magnitude or distance.

Uses 171 records (\(0.9\leq R\leq 28.1\,\mathrm{km}\)) from 75 earthquakes (\(2.5 \leq M_L \leq 5.0\), \(0.7 \leq H \leq 24.3\,\mathrm{km}\)) excluded from original analysis because they were on shallow soil, rock and/or not free-field, to examine importance of site geology and building size. Considers difference between PGA from records grouped according to instrument location, building size, embedment, and site geology and the predicted PGA using the attenuation equation to find site factors, \(S\). Groups with nonsignificant, at \(90\%\), values of \(S\) are grouped together. Finds two categories: embedded alluvial sites from all building sizes (38 records) and shallow-soil (depth of soil \(\leq 10\,\mathrm{m}\)) sites (35 records) to have statistically significant site factors.

Performs regression analysis on all records (irrespective of site geology or building size) from Oroville (172 records from 32 earthquakes) and Imperial Valley (71 records from 42 earthquakes) to find individual sites that have significant influence on prediction of PGA (by using individual site coefficients for each station). Finds equations predict similar PGA to those predicted by original equation. Finds significant differences between PGA recorded at different stations in the two regions some related to surface geology but for some finds no reason.

Uses 27 records (\(0.2 \leq R \leq 25.0\,\mathrm{km}\)) from 19 earthquakes (\(2.5 \leq M_{bLG} \leq 4.8\), \(0.1 \leq H \leq 9\,\mathrm{km}\)) from E. N. America to examine whether they are significantly different than those from W. N. America. Finds residuals significantly, at \(99\%\) level, higher than zero and concludes that it is mainly due to site effects because most are on shallow soils or other site factors influence ground motion. Correcting the recorded PGAs using site factors the difference in PGA between E. N. America and W. N. America is no longer significant although notes may not hold for all of E. N. America.

Ground-motion model is: \[\log Y=c_1+c_2M+c_4\log[R+c_5\exp(c_6M)]\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(c_1=0.207\), \(c_2=0.808\), \(c_4=-2.026\), \(c_5=0.183\), \(c_6=0.703\) and \(\sigma=0.247\) (for rock sites) and \(c_1=0.716\), \(c_2=0.647\), \(c_4=-1.706\), \(c_5=0.187\), \(c_6=0.703\) and \(\sigma=0.251\) (for soil sites).

Use 2 site classes and derive separate models:

Ground-motion model is unknown.

Ground-motion model for near field is: \[\log(A)=a_1+a_2 M_s-\log (r^2+a_3^2)^{\frac{1}{2}}\] where \(A\) is in \(\,\mathrm{g}\), \(a_1=-1.116\), \(a_2=0.312\), \(a_3=7.9\) and \(\sigma=0.21\).

Ground-motion model for far field is: \[\log(A)=b_1+b_2M_s+b_3 \log (r^2+b_4^2)^{\frac{1}{2}}\] where \(A\) is in \(\,\mathrm{g}\), \(b_1=-1.638\), \(b_2=0.438\), \(b_3=-1.181\), \(b_4=70.0\) and \(\sigma=0.21\).

Separate crustal and subduction data because of differences in travel path and stress conditions:

Near field

Far field, 20 records from San Salvador, 20 earthquakes, \(4.2\leq M_s \leq 7.2\), depths between \(36\) and \(94\,\mathrm{km}\), \(31 \leq r \leq 298\,\mathrm{km}\).

Ground-motion model is: \[\begin{aligned} \log y&=&\alpha +\beta M_w-\log r +b r\\ \mbox{where }r&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(y\) is in \(\,\mathrm{g}\), \(\alpha=-1.101\), \(\beta=0.2615\), \(b=-0.00255\), \(h=7.2\) and \(\sigma=0.25\).

Uses data and method of Joyner and Boore (1981) but re-evaluates \(M_w\) for all earthquakes. Finds some large changes, e.g. Santa Barbara changes from \(M_w=5.1\) to \(M_w=5.85\). Uses \(M_L\) for 2 earthquakes (\(M_L=5.2\), \(6.2\)).

Find effect of uncertainty in \(M_w\) causes less than \(10\%\) change in \(\sigma\).

Also calculates equation using \(M_s\) instead of \(M_w\).

Finds assumption \(M_s=M_w\) introduces bias, particularly for small magnitude shocks, on unsafe side, and this can be significant in cases where there is a preponderance of small earthquakes in set.

Ground-motion model is: \[\begin{aligned} \ln(Y)&=&a+bM+d \ln [R+c_1 \exp (c_2 M)]+eF+f_1\tanh[f_2(M+f_3)]\\ &&{}+g_1 \tanh(g_2 D)+h_1K_1+h_2K_2+h_3 K_3\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(a=-2.245\), \(b=1.09\), \(c_1=0.361\), \(c_2=0.576\), \(d=-1.89\), \(e=0.218\), \(f_1=0\), \(f_2=0\), \(f_3=0\), \(g_1=0\), \(g_2=0\), \(h_1=-0.137\), \(h_2=-0.403\) and \(h_3=0\). \(\sigma=0.517\) for \(M\leq 6.1\) and \(\sigma=0.387\) for \(M\geq 6.2\). Also given is \(\sigma=0.450\) for \(M\geq 4.7\).

Records from firm soil and soft rock sites. Characterises site conditions by depth to basement rock (sediment depth) in \(\,\mathrm{km}\), \(D\).

Records from different size buildings. \(K_1=1\) for embedded buildings 3â11 storeys, \(K_2=1\) for embedded buildings with \(>\)11 storeys and \(K_3=1\) for non-embedded buildings \(>\)2 storeys in height. \(K_1=K_2=K_3=0\) otherwise.

Uses two fault mechanisms:

Strike-slip

Reverse

Ground-motion model is: \[\begin{aligned} \ln A&=&c_1+c_2 M+c_4 R +\ln G(R,R_0)\\ \mbox{where } G(R,R_0)&=&R^{-1} \quad \mbox{for} \quad R \leq R_0\\ \mbox{and: } G(R,R_0)&=&R_0^{-1}\left(\frac{R_0}{R}\right)^{5/6} \quad \mbox{for} \quad R>R_0\end{aligned}\] where \(A\) is in \(\,\mathrm{m/s^2}\), \(c_1=-1.471\), \(c_2=0.849\), \(c_4=-0.00418\) and \(\sigma=0.83\).

Use records from rock sites (presumably with hard rock or firm ground conditions).

Assume intraplate refers to area that are tectonically stable and geologically more uniform than plate boundary areas. Select records from several âreasonablyâ intraplate areas (eastern N. America, China, Australia, and some parts of Europe), due to lack of data.

Select records which are available unprocessed and with sufficient information on natural frequency and damping of instrument.

Use \(M_s\), when available, because reasonably unbiased with respect to source dimensions and there is globally consistent calculation method.

Most (\(72\%\)) records from earthquakes with \(M\leq5.5\). Tangshan and Friuli sequence comprise a large subset. Correlation coefficient between magnitude and distance is \(0.31\).

Instrument correct records and elliptical filter with pass band \(0.25\) to \(25.0\,\mathrm{Hz}\).

If depth unknown assume \(15\,\mathrm{km}\).

Choose \(R_0=100\,\mathrm{km}\) although depends on crustal structure and focal depth. It is distance at which spherical spreading for S waves overtaken by cylindrical spreading for Lg waves.

PGA attenuation relation is pseudo-acceleration equation for \(0.025\,\mathrm{s}\) period and \(5\%\) damping.

Plot residuals against magnitude and distance.

Note âfirst orderâ results, because data from several geological regions and use limited data base.

Ground-motion model is: \[A=10^{(a_1+a_2M+a_3 \log d+a_4 d)}\] where \(A\) is in \(\,\mathrm{g}\), \(a_1=-1.43\), \(a_2=0.31\), \(a_3=-0.62\) and \(a_4=-0.0026\) (\(\sigma\) not given).

Note equation only for hard rock sites.

Equation from a composite of two separate regressions: one using data from 6 earthquakes, \(4.7\leq M\leq 6.4\) and \(d\) primarily between \(40\) and \(820\,\mathrm{km}\) and one using the same data supplemented with data from 2 earthquakes with \(M=1.8\) and \(M=3.2\) and \(d\leq 20\,\mathrm{km}\) to extend results to smaller \(M\) and \(d\). Give no details of this composite regression.

Note regressions are preliminary and should be tested against more data.

Note careful assessment of uncertainties is required.

Ground-motion model is: \[\ln \mathrm{PGA}=a+bM+c \ln (r+h)+\phi F\] where \(\mathrm{PGA}\) is in \(\,\mathrm{cm/s^2}\), \(a=1.375\), \(b=1.672\), \(c=-1.928\) and \(\phi=0.213\) (\(h\) not given). Standard deviation is composed of two parts, inter-site \(\tau=0.261\) and intra-site \(\sigma=0.653\). \(F=1\) for thrust mechanism and \(0\) otherwise.

Computes theoretical radiation pattern and finds a linear trend between residuals and radiation pattern but does not model.

Ground-motion model is: \[a_{\mathrm{peak}}=\alpha_0 \exp(\alpha_1M)\exp(-\alpha_2R)R^{-\alpha}P\] where \(P=1\).

Notes that data are very limited and any definite conclusions should, therefore, be avoided.

Does not give coefficients, only predictions.

Ground-motion model is: \[\ln y=C_0+C_1M+C_2 (8.5-M)^{2.5}+C_3\ln [D+C_4 \exp(C_5M)]\] where \(y\) is in \(\,\mathrm{g}\), \(C_3=-2.1\), \(C_4=0.616\), \(C_5=0.524\) and for \(M\geq 6.5\) \(C_0=-1.092\), \(C_1=1.10\), \(C_2=0\) and \(\sigma=0.36\) and for \(M<6.5\) \(C_0=-0.442\), \(C_1=1.0\), \(C_2=0\) and \(\sigma=1.27-0.14M\).

All records from rock or rock-like sites.

Separate equation for \(M<6.5\) and \(M\geq 6.5\).

Use only shallow crustal thrust earthquakes.

Use another database of rock and soil site records and simulated acceleration time histories to find conversion factors to predict strike-slip and oblique ground motions from the thrust equation given above. For strike-slip conversion factor is \(0.83\) and for oblique conversion factor is \(0.91\).

Standard deviation, \(\sigma\), for \(M\geq 6.5\) from regression whereas \(\sigma\) for \(M<6.5\) from previous results. Confirm magnitude dependence of standard deviation using 803 recordings from 124 earthquakes, \(3.8\leq M_w \leq 7.4\), \(D<100\,\mathrm{km}\).

Ground-motion model is: \[\begin{aligned} \log a&=&\alpha +\beta M -\log r +b r\\ \mbox{where } r&=&(d^2+h_0^2)^{1/2}\\ \mbox{or: } r&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(a\) is in \(\,\mathrm{g}\), for horizontal PGA \(\alpha=-1.09\), \(\beta=0.238\), \(b=-0.00050\), \(h=6.0\) and \(\sigma=0.28\) and for vertical PGA \(\alpha=-1.34\), \(\beta=0.230\), \(b=0\), \(h=6.0\) and \(\sigma=0.27\). When use focal depth explicitly: for horizontal PGA \(\alpha=-0.87\), \(\beta=0.217\), \(b=-0.00117\) and \(\sigma=0.26\) and for vertical PGA \(\alpha=-1.10\), \(\beta=0.200\), \(b=-0.00015\) and \(\sigma=0.26\).

Consider two site classifications (without regard to depths of deposits) but do not model:

Rock

Alluvium

Select records which have: \(M_s\geq 4.0\) and standard deviation of \(M_s\) known and reliable estimates of source-site distance and focal depth, \(h\leq25\,\mathrm{km}\), regardless of local soil conditions from free-field and bases of small buildings. No reliable data or outliers excluded. Records from instruments at further distances from the source than the closest non-triggered instrument were non-excluded because of non-homogeneous and irregularly spaced networks and different and unknown trigger levels.

Most data, about \(70\%\), with distances less than \(40\,\mathrm{km}\). Note strong bias towards smaller values of magnitude and PGA.

PGA read from analogue and digitised data, with different levels of processing. Differences due to different processing usually below \(5\%\), but some may be larger.

Errors in distances for small shocks may be large.

Prefer one-stage technique because second step of two-stage method would ignore records from singly-recorded earthquakes which compose over half the events, also find more realistic, \(b\), and \(h_0\) using one-stage method. Do not use weighting because involves assumptions which are difficult to verify.

Find inadmissable and positive \(b\) for vertical PGA so remove and repeat.

Remove records from distances less than or equal to half their focal depth and also less than or equal to their focal depth, find that \(h_0\) is governed by near-field data.

Use focal depth explicitly, by replacing \(r=(d^2+h_0^2)^{1/2}\) by \(r=(d^2+h^2)^{1/2}\). Find lower standard deviation and that it is very significant.

Repeat analysis on subsets of records grouped by focal depth. Find no correlation between \(h_0\) and focal depth of subset. Use \(h_0\) equal to mean focal depth in each subset and find similar results to when focal depth used explicitly.

Repeat analysis with geometric attenuation coefficient equal to \(-0.83\), corresponding to the Airy phase, as opposed to \(-1.0\).

Find small dependence of horizontal PGA on site classification, note due to level of information available.

Ground-motion model is: \[\ln \mathrm{PGA}=p_1+p_2M+p_4 \ln [R+p_5 \exp(p_6M)]+p_7h\] where \(\mathrm{PGA}\) is in \(\,\mathrm{gal}\), using all PGA values \(p_1=6.36\), \(p_2=1.76\), \(p_4=-2.73\), \(p_5=1.58\), \(p_6=0.608\), \(p_7=0.00916\) and \(\sigma=0.773\).

Use data from stiff soil sites (depth of soil \(<25\,\mathrm{m}\)).

Include data from any zones with strong seismic coupling, such as the younger subduction zones (S.W. Japan, Alaska, C. America (Mexico), C. Chile, Peru and northern Honshu and Kuril subduction zones in Japan) unless compelling reasons to exclude data. Do this because lack of data from Cascadia. Most (\(>70\%\)) are from Japan.

Focal depths, \(h\), between \(0\) and \(238\,\mathrm{km}\).

Compare Japanese and Cascadia PGA values for earthquakes with similar magnitude and depths and find similar.

Do not exclude data from buildings or which triggered on S-wave. Note could mean some PGAs are underestimated.

Plot ground motion amplitude (PGA and also some maximum displacements from seismograms) against distance for a number of large magnitude shocks (including some data from rock sites which not included in set for regression). Find that rate of attenuation becomes smaller for shorter distances and process is magnitude dependent. Also plot Japanese PGA data, from earthquakes with \(h\leq50\,\mathrm{km}\), split into three distance groups (between \(50\) and \(75\,\mathrm{km}\), between \(100\) and \(150\,\mathrm{km}\) and between \(250\) and \(300\,\mathrm{km}\)) find as distance increases magnitude scaling becomes larger and possible saturation in PGA for large magnitudes. Fit \(\ln \mathrm{PGA}=p_1+p_2 \ln (R+C)\) to some PGA values from large magnitude shocks for \(C=0\) and \(C>0\), find lower standard deviation for \(C>0\).

Fit \(\ln \mathrm{PGA}=a+bM\) and \(\ln \mathrm{PGA}=a+bM+cM^2\) to Japanese data split into the three distance groups (mentioned above); find \(b\) increases with increasing distance range but both equations fit data equally well.

Constrain \(p_4\) to negative value and \(p_5\) and \(p_6\) to positive values.

Include quadratic magnitude term, \(p_3 M^2\), but find equal to zero.

Plot residuals against \(M\); find uniformly distributed and evidence for smaller residuals for larger \(M\).

Plot residuals against \(R\)

^{12}and find decreasing residuals for increasing \(R\).Give equation using only those records available in digital form (235 records).

Ground-motion model is: \[\ln \mathrm{PGA}=\ln C_0+C_1 M-0.5 \ln r -\gamma r\] where \(\mathrm{PGA}\) is in \(\,\mathrm{cm/s^2}\), for Iberian Peninsula \(\ln C_0=-5.13\), \(C_1=2.12\) and \(\gamma=0.0039\), for NE region \(\ln C_0=-4.74\), \(C_1=2.07\) and \(\gamma=0.0110\) and for SSE region \(\ln C_0=-5.30\), \(C_1=2.21\) and \(\gamma=0.0175\) (\(\sigma\) is not given).

Derive equations for two regions:

South south-east part of the Iberian peninsula, from the Guadalquivir basin to the Mediterranean Sea, including the Betic Cordillera, 140 records from 5 stations.

North-east part of the Iberian peninsula, including the Pyrenees, the Catalan Coastal Ranges, the Celtiberian chain and the Ebro basin, 107 records from 3 stations.

Use vertical-component short-period analogue records of Lg-waves (which are believed to have the largest amplitudes for the period range \(0.1\) to \(1s\)) from regional earthquakes in Iberian Peninsula.

Processing procedure is: digitise seismogram using irregular sampling rate to get better sampling at peaks and âkinksâ, select baseline, apply cubic spline interpolation and compare original and digitised seismograms. Next the Fourier amplitude spectrum is computed and the instrument amplitude response is removed.

Estimate PGA using the maximum value of pseudo-absolute acceleration obtained from Fourier amplitude spectra. Derived equations are for characteristic frequency of \(5\,\mathrm{Hz}\).

Compare estimated PGAs with observed PGAs from five earthquakes and find good agreement.

Use \(5\,\mathrm{Hz}\) \(\gamma\) values from Garcia-Fernandez and Canas (1992) and Vives and Canas (1992).

Ground-motion model is: \[\ln\mathrm{PGA}=C_1+C_2M+C_3\ln\left(r_{\mathrm{rup}}+C_4\mathrm{e}^{C_5M}\right)+C_6Z_T\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), for horizontal PGA, rock sites and strike-slip faulting \(C_3=0\) and \(C_4=-2.100\), for \(M\leq 6.5\) \(C_1=-0.624\), \(C_2=1.0\), \(C_5=1.29649\) and \(C_6=0.250\) and for \(M>6.5\), \(C_1=-1.274\), \(C_2=1.1\), \(C_5=-0.48451\) and \(C_6=0.524\). For reverse and thrust earthquakes multiply strike-slip prediction by \(1.2\). \(\sigma=1.39-0.14M\) for \(M<7.21\) and \(\sigma=0.38\) for \(M\geq 7.21\). For horizontal PGA and deep soil \(C_2=1.0\), \(C_3=1.70\) and \(C_6=0\), for strike-slip faulting \(C_1=-2.17\) and for reverse or thrust faulting \(C_1=-1.92\), for \(M\leq 6.5\) \(C_4=2.1863\) and \(C_5=0.32\) and for \(M>6.5\) \(C_4=0.3825\) and \(C_5=0.5882\). \(\sigma=1.52-0.16M\) for \(M\leq 7\) and \(\sigma=0.40\) for \(M=7\).

For vertical PGA, rock sites and strike-slip faulting \(C_3=0\) and \(C_4=-2.300\), for \(M\leq 6.5\) \(C_1=-0.430\), \(C_2=1.0\), \(C_5=1.2726\) and \(C_6=0.228\) and for \(M>6.5\), \(C_1=-1.080\), \(C_2=1.1\), \(C_5=-0.3524\) and \(C_6=0.478\). For reverse and thrust earthquakes multiply strike-slip prediction by \(1.1\) and for oblique faulting multiply by \(1.048\). \(\sigma=0.48\) for \(M\geq 6.5\), \(\sigma=3.08-0.40M\) for \(6<M<6.5\) and \(\sigma=0.68\) for \(M\leq 6\).

Use two site categories (for horizontal motion):

Rock: bedrock within about a metre of surface. Note that many such sites are soft rock with \(V_s \leq 750\,\mathrm{m/s}\) and a strong velocity gradient because of near-surface weathering and fracturing, 274 records.

Deep soil: greater than \(20\,\mathrm{m}\) of soil over bedrock. Exclude data from very soft soil sites such as those from San Francisco bay mud, 690 records.

Vertical equations only for rock sites.

Crustal earthquakes defined as those that occur on faults within upper \(20\) to \(25\,\mathrm{km}\) of continental crust.

Use source mechanism: RV=reverse (26+2) \(\Rightarrow Z_T=1\) and SS=strike-slip (and some normal) (89+0) \(\Rightarrow Z_T=0\). Classified as RV if rake\(>45^{\circ}\) and SS if rake\(<45^{\circ}\). Find peak motions from small number of normal faulting earthquakes not to be significantly different than peak motions from strike-slip events so were including in SS category.

Records from instruments in instrument shelters near ground surface or in ground floor of small, light structures.

4 foreign records (1 from Gazli and 3 from Tabas) supplement Californian records.

Separate equations for \(M_w <6.5\) and \(M_w \geq 6.5\) to account for near-field saturation effects and for rock and deep soil sites.

Ground-motion model is (case II): \[\log y =C_1+C_2 M-C_4 \log [R+C_5 \exp (C_6 M)]\] where \(y\) is in \(\,\mathrm{gal}\), \(C_5=0.231\) and \(C_6=0.626\), for rock \(C_1=0.894\), \(C_2=0.563\), \(C_4=1.523\) and \(\sigma=0.220\) and for soil \(C_1=1.135\), \(C_2=0.462\), \(C_4=1.322\) and \(\sigma=0.243\) (these coefficients are from regression assuming \(M\) and \(R\) are without error).

Use two site categories:

Rock

Soil

Supplement western USA data in large magnitude range with 25 records from 2 foreign earthquakes with magnitudes \(7.2\) and \(7.3\).

Note that there are uncertainties associated with magnitude and distance and these should be considered in derivation of attenuation relations.

Develop method, based on weighted consistent least-square regression, which minimizes residual error of all random variables not just residuals between predicted and measured ground motion. Method considers ground motion, magnitude and distance to be random variables and also enables inverse of attenuation equation to be used directly.

Note prediction for \(R>100\,\mathrm{km}\) may be incorrect due to lack of anelastic attenuation term.

Use both horizontal components to maintain their actual randomness.

Note most data from moderate magnitude earthquakes and from intermediate distances therefore result possibly unreliable outside this range.

Use weighted analysis so region of data space with many records are not overemphasized. Use \(M\)-\(R\) subdivisions of data space: for magnitude \(M<5.5\), \(5.5\leq M \leq 5.9\), \(6.0 \leq M \leq 6.4\), \(6.5 \leq M \leq 6.9\), \(7.0 \leq M \leq 7.5\) and \(M>7.5\) and for distance \(R<3\), \(3 \leq R \leq 9.9\), \(10 \leq R \leq 29.9\), \(30 \leq R \leq 59.9\), \(60 \leq R \leq 99.9\), \(100 \leq R \leq 300\) and \(R>300\,\mathrm{km}\). Assign equal weight to each subdivision, and any data point in subdivision \(i\) containing \(n_i\) data has weight \(1/n_i\) and then normalise.

To find \(C_5\) and \(C_6\) use 316 records from 7 earthquakes (\(5.6 \leq M \leq 7.2\)) to fit \(\log Y=\sum^m_{i=1} C_{2,i} E_i -C_4 \log [r+\sum^m_{i=1} R_{0,i} E_i]\), where \(E_i=1\) for \(i\)th earthquake and \(0\) otherwise. Then fit \(R_0=C_5 \exp (C_6 M)\) to results.

Also try equations: \(\log y=C_1+C_2 M -C_4 \log [R+C_5]\) (case I) and \(\log y=C_1 +C_2 M-C_3M^2-C_4 \log [R+C_5 \exp (C_6 M)]\) (case III) for \(M\leq M_c\), where impose condition \(C_3 =(C_2-C_4C_6/\ln 10)/(2M_c)\) so ground motion is completely saturated at \(M=M_c\) (assume \(M_c=8.0\)).

Find equations for rock and soil separately and for both combined.

Ground-motion model is: \[\ln(Y)=[\alpha_0+\exp(\alpha_1+\alpha_2 M)]+[\beta_0-\exp(\beta_1+\beta_2 M)]\ln (R+20)+a F\] where \(Y\) is in \(\,\mathrm{g}\), \(a=0.2\), for \(M\leq 6\) \(\alpha_0=-0.150\), \(\alpha_1=2.261\), \(\alpha_2=-0.083\), \(\beta_0=0\), \(\beta_1=1.602\), \(\beta_2=-0.142\) and \(\sigma=1.39-0.14M\) and for \(M>6\) \(\alpha_0=-0.050\), \(\alpha_1=3.477\), \(\alpha_2=-0.284\), \(\beta_0=0\), \(\beta_1=2.475\), \(\beta_2=-0.286\) and for \(M<7\frac{1}{4}\) \(\sigma=1.39-0.14M\) and for \(M\geq 7\frac{1}{4}\) \(\sigma=0.38\).

Records from rock sites.

Uses three fault mechanisms:

Strike slip

Oblique

Reverse

Separate equations for \(M\leq 6\) and \(M>6\).

Examines residuals for PGA. Finds average residual almost zero over entire distance range; trend reasonable up to about \(60\,\mathrm{km}\) but beyond \(60\,\mathrm{km}\) relationship would underestimate recorded PGA.

Finds standard deviation to be linear function of magnitude.

Ground-motion model is: \[a=b_1 \mathrm{e}^{b_2 M} (R+b_4)^{-b_3}\] where \(a\) is in \(\,\mathrm{g}\), \(b_1=1.128\), \(b_2=0.728\), \(b_3=1.743\), \(b_4=32\,\mathrm{km}\) and \(\sigma=0.563\) (in terms of \(\ln\)).

Use only data from rock sites.

Focal depths, \(h\), between \(0.2\) and \(97.4\,\mathrm{km}\). Most records from \(h<30\,\mathrm{km}\).

Also derive equations for PGA using \(\log_{10}(a)=b_1+b_2 M+b_3 \log \sqrt{R^2+b_5^2}\) and \(a=b_1 \mathrm{e}^{b_2 M} (R+b_4 \mathrm{e}^{b_5 M})^{-b_3}\) in order to have diversity in the characterisation of ground motion.

Use \(r_{hypo}\) because no clear fault ruptures identified for Taiwanese earthquakes.

All data from SMA-1s.

PGAs between \(7.3\) and \(360.2\,\mathrm{cm/s^2}\).

Exact functional form unknown but based on those of Campbell (1981), Fukushima and Tanaka (1990) and Abrahamson and Litehiser (1989).

Use three site classes.

Develop separate equations for each site class. Only possible for two classes. Therefore, modify equation derived for site class C to obtain coefficients for other two classes.

Digitization sampling rate of records used is \(50\,\mathrm{Hz}\). Most data low-pass filtered at \(24.5\,\mathrm{Hz}\).

Most data high-pass filtered with cut-offs above \(0.25\,\mathrm{Hz}\).

Due to limited data, advise caution when using model.

Ground-motion model is: \[\ln Y=a+b M+d \ln [R+c_1 \mathrm{e}^{c_2 M}]\] where \(Y\) is in \(\,\mathrm{g}\), for horizontal PGA \(a=-5.503\), \(b=0.936\), \(c_1=0.407\), \(c_2=0.455\), \(d=-0.816\) and \(\sigma=0.461\) and for vertical PGA \(a=-5.960\), \(b=0.989\), \(c_1=0.013\), \(c_2=0.741\), \(d=-1.005\) and \(\sigma=0.551\).

All records from array so essentially identical site conditions and travel paths.

All records from free-field instruments mounted on \(4\mathrm{inch}\) (\(10\,\mathrm{cm}\)) thick concrete base mats, approximately \(2\) by \(3\,\mathrm{feet}\) (\(60\) by \(90\,\mathrm{cm}\)) across.

Select earthquakes to cover a broad range of magnitude, distance and azimuth and ensuring thorough coverage of the array. Criteria for selection is: at least 25 stations recorded shock, focal depth \(<30\,\mathrm{km}\), hypocentral distance \(<50\,\mathrm{km}\) except for two large earthquakes from beyond \(50\,\mathrm{km}\) to constrain distance dependence.

Focal depths between \(0.2\) and \(27.2\,\mathrm{km}\) with all but one \(\leq 13.9\,\mathrm{km}\).

Azimuths between \(60^{\circ}\) and \(230^{\circ}\).

Most records (\(78\%\)) have magnitudes between \(5.9\) and \(6.5\). Note magnitude and distance are not independent (correlation coefficient is \(0.6\)).

Records have sampling interval of \(0.01\,\mathrm{s}\). Processed using trapezoidal band passed filter with corner frequencies \(0.07\), \(0.10\), \(25.0\) and \(30.6\,\mathrm{Hz}\).

Not enough information to use distance to rupture zone.

Source mechanisms of earthquakes are: 4 normal, 2 reverse, 1 reverse oblique and 1 normal oblique with 4 unknown. Do not model source mechanism dependence because of 4 unknown mechanisms.

Use weighted regression, give equal weight to recordings from each earthquake within each of 10 distance bins (\(<2.5\), \(2.5\)â\(5.0\), \(5.0\)â\(7.5\), \(7.5\)â\(10.0\), \(10.0\)â\(14.1\), \(14.1\)â\(20.0\), \(20\)â\(28.3\), \(28.3\)â\(40.0\), \(40.0\)â\(56.6\) and \(56.6\)â\(130\,\mathrm{km}\)). Do this so earthquakes with smaller number of recordings are not overwhelmed by those with a larger coverage and also to give additional weight to shocks recorded over multiple distance bins. Apply two-stage regression, because of high correlation between magnitude and distance, excluding 3 earthquakes (\(M=3.6\), \(5.0\), \(7.8\)) with 162 records from first stage to reduce correlation between \(M\) and \(R\) to \(0.1\). Also do one-stage regression although do not give coefficients.

Use mean horizontal component because reduces uncertainty in prediction.

Examine coefficient of variation for each earthquake using median and normalized standard deviation of recordings in inner ring of array. Find evidence for magnitude dependent uncertainty (large magnitude shocks show less uncertainty). Find that main contribution to scatter is inter-event variations again by examining coefficient of variation; although note may be because using dense array data.

Examine mean residuals of observations from each earthquake. Find evidence for higher than predicted vertical PGA from reverse faulting earthquakes and lower than predicted vertical PGA from normal faulting earthquakes, although due to lack of information for 4 earthquakes note that difficult to draw any conclusions.

Examine mean residuals of observations from each station in inner ring. Find mean residuals are relatively small compared with standard deviation of regression so variation between stations is less than variation between earthquakes. Find for some stations some large residuals.

Ground-motion model is: \[\log a_p=a_1+0.36M-0.002R+a_2 \log R+a_3 S_1+a_4 S_1 \log R+a_5 S_5+a_6 S_5 \log R+a_7 S_6 \log R\] where \(a_p\) is in \(\,\mathrm{g}\), \(a_1=-1.62\), \(a_2=-1.01\), \(a_3=0.246\), \(a_4=0.212\), \(a_5=0.59\), \(a_6=-0.29\), \(a_7=0.21\) and \(\sigma=0.29\).

Use six local site classifications:

Holocene

Pleistocene soil

Soft rock

Hard rock

Shallow (\(<10\,\mathrm{m}\) depth) soil

Soft soil (e.g. bay mud)

Data from about 800 different stations.

Note that inclusion of subduction-zone events in analysis may affect results with unmodelled behaviour, particularly with regard to distance scaling although believe use of \(r_{rup}\) partially mitigates this problem.

Firstly compute an equation does not include site coefficients. Conduct regression analysis on site-condition subsets of the residuals using \(M\) or \(\log R\) as dependent variable. Find several regressions are not statistically significant at the \(5\%\) level and/or the predicted effects are small at the independent variable extremes. Find strongest effects and most significant results are for shallow soil sites and soft soil sites although because of the high correlation between \(M\) and \(\log R\) in the set used it is difficult to construct unbiased models.

Use a stochastic random-vibration approach to find theoretical equations for estimating PGA that include the effect of local site conditions as distance-dependent terms. Using the results from this analysis construct equation based on the observed PGAs. Try including terms for \(S_1\), \(S_2\), \(S_5\), \(S_6\) and corresponding \(\log R\) terms for each site type but iterate to retain only the significant terms.

Fix magnitude scaling (\(0.36M\)) and anelastic attenuation (\(0.002R\)). Do not try to optimise the fit other than using fixed values similar to those given by the stochastic analysis.

Note that anelastic coefficient may be too low but it produces an acceptable geometric spreading term.

Note that because Moho critical reflections can increase amplitudes beyond about \(50\,\mathrm{km}\) the effects of anelastic or geometric attenuation may be masked.

Allowing all the coefficients in the equation to be free produces a smaller magnitude scaling coefficient, a smaller geometric spreading coefficient, and a non-significant anelastic attenuation term.

Note that data from \(S_5\) and \(S_6\) are sparse.

Compare estimated PGAs with data from within small magnitude ranges. Find that PGAs from Morgan Hill earthquake are overestimated, which believe is due to the unilateral rupture of this earthquake masking the effect of the local site conditions.

Ground-motion model is: \[\mathrm{Acc}=b_1 \exp(b_2M)(R_h+c)^{b_3}\] \(\mathrm{Acc}\) is in \(\,\mathrm{cm/s^2}\), \(b_1=534.355\), \(b_2=0.46087\), \(b_3=-1.14459\), \(c=25\) and \(\sigma_{\ln \mathrm{Acc}}=0.72936\).

Data from 141 different sites, which are considered to have average soil conditions.

Data from Yugoslavia (23 earthquakes), Italy (45 earthquakes), northern Greece (3 earthquakes), Romania (1 earthquake), Mexico (1 earthquake) and the USA (5 earthquakes). Select earthquakes to have range of magnitudes and focal depths.

Data processed using standard procedure.

Conduct Pearson \(\chi^2\) and Kolmogorov-Smirnov tests to test acceptability of log-normal assumption using a \(5\%\) significance level. Conclude that assumption is justified.

Note the strong influence of the data used on results and the need to improve it.

Ground-motion model is: \[\ln y=a+b M+d \ln (r+c)+e F\] where \(a=0.0586\), \(b=0.696\), \(c=12.0\), \(d=-1.858\), \(e=0.205\), \(\sigma=0.399\) (intra-event) and \(\tau=0.201\) (inter-event) (units of \(y\) are not given but probably \(\,\mathrm{g}\)).

\(F\) is fault type (details not given).

Develop new algorithm for one-stage maximum-likelihood regression, which is more robust than previous algorithms.

Ground-motion model is: \[\begin{aligned} \log (a)&=&c_1+c_2 M+c_3 r+c_4 \log r\\ r&=&(d^2+h_0^2)^{\frac{1}{2}}\end{aligned}\] where \(a\) is in \(\,\mathrm{g}\), \(c_1=-1.038\), \(c_2=0.220\), \(c_3=-0.00149\), \(c_4=-0.895\), \(h_0=5.7\) and \(\sigma=0.260\).

Investigate equations of PML (1982) and PML (1985) using criteria:

Is the chosen data set of earthquake strong-motion records suitable to represent the seismic environment?

Are the associated seismological and geophysical parameters used in these reports reliable and consistent?

Is the methodology used to derive attenuation laws and design spectra from the data set reliable?

Investigate effect of different Ground-motion model, one and two-stage regression technique, record selection technique and recalculation of associated parameters. Find these choice cause large differences in predictions.

Coefficients given above are for PML (1985) data with recalculated magnitudes and distances and addition of extra records from some earthquakes.

Ground-motion model is

^{13}: \[\ln Y=a_1+a_2M+a_3\ln[R+a_4\exp(a_5M)]\] where \(Y\) is in \(\,\mathrm{gal}\), \(a_1=0.1497\), \(a_2=1.9088\), \(a_3=-2.049\), \(a_4=0.1818\) and \(a_5=0.7072\) (\(\sigma\) is unknown).Use macroseismic intensities and strong-motion data to derive model. Details unknown.

There is another model by these authors Huo and Hu (1991) (see SectionÂ 2.88) to which this model is probably similar.

Ground-motion model is (note that there is a typographical error in Kamiyama, OâRourke, and Flores-Berrones (1992; Kamiyama 1995) because \(r_t\) has been replaced by \(r_c\) in equations): \[\begin{aligned} \log_{10} a_{\mathrm{max}}&=&-1.64 R_0+b_1 R_1+b_2 R_2 +c_a +\sum_{i=1}^{N-1} A_i S_i\\ R_0&=&\left\{ \begin{array}{r@{\quad \mbox{for}\quad}l} 0&r \leq r_t\\ \log_{10}r-\log_{10}r_c&r >r_t\\ \end{array} \right.\\ R_1&=&\left\{ \begin{array}{r@{\quad \mbox{for}\quad}l} 0&r \leq r_t\\ 1&r >r_t\\ \end{array} \right.\\ R_2&=&\left\{ \begin{array}{r@{\quad \mbox{for}\quad}l} 0&r \leq r_t\\ M&r >r_t\\ \end{array} \right.\end{aligned}\] where \(S_i=1\) for \(i\) station, \(S_0=0\) otherwise, \(a_{\mathrm{max}}\) is in \(\,\mathrm{cm/s^2}\), \(b_1=-1.164\), \(b_2=0.358\), \(c_a=2.91\), \(r_c=5.3\,\mathrm{km}\) and \(\sigma=0.247\) (\(A_i\) given in publications but not reported here due to lack of space).

Instrument correct records and filter with pass band between \(0.24\) and \(11\,\mathrm{Hz}\).

Model individual soil conditions at each site as amplification factors, \(\mathrm{AMP}_i\), as described by Kamiyama and Yanagisawa (1986).

Most records are from hypocentral distances between \(30\) and \(200\,\mathrm{km}\).

Focal depths between \(0\) and \(130\,\mathrm{km}\).

Models peak ground accelerations independent of magnitude and distance in a fault zone, \(r_t\), where \(r_t=r_c 10^{(b_1+b_2M)/1.64}\).

Constrain decay with distance in far field to \(-1.64\) using results from other studies to avoid problems due to correlation between \(M\) and \(\log_{10} r\).

Use trial and error method to find \(r_c\) so that resulting values of \(r_t\) are consistent with empirical estimates of fault length from past studies.

Also give expression using shortest distance to fault plane (rupture distance), \(R\), by replacing the expression for \(r\leq r_c\) and \(r>r_c\) by one expression given by replacing \(r\), hypocentral distance, by \(R+r_c\) in expression for \(r>r_c\). This gives PGA independent of magnitude at distance \(R=0\,\mathrm{km}\).

Note that use of \(r_{hypo}\) is not necessarily best choice but use it due to simplicity.

Check residual plots; find no trends so conclude adequate from statistical point of view.

Ground-motion model is: \[\begin{aligned} \log A&=&\alpha +\beta M -\log R +b R\\ \mbox{with: }R&=&\sqrt{d^2+h^2}\end{aligned}\] where \(A\) is in \(\,\mathrm{g}\), for average horizontal PGA and \(4<M<6\) \(\alpha=-1.98\), \(\beta=0.365\), \(b=-0.0039\) and \(\sigma=0.30\), for larger horizontal PGA and \(4<M<6\) \(\alpha=-1.72\), \(\beta=0.327\), \(b=-0.0043\) and \(\sigma=0.30\) and for both horizontal PGAs and \(2<M<6\) \(\alpha=-2.28\), \(\beta=0.386\), \(b=0\) and \(\sigma=0.29\).

Find that Icelandic data does not fit other published relations.

Find equation using only records with \(M\geq 4.0\), \(h\) equal to focal depth and both the horizontal components.

Find equation using only records with \(M\geq 4.0\), \(h\) equal to focal depth and larger horizontal component.

Also repeated with all data. Anelastic coefficient constrained to zero because otherwise positive.

Also done with \(h\) free.

Note that large earthquakes have \(h \approx 10\,\mathrm{km}\) while small events have \(h \approx 5\,\mathrm{km}\).

Ground-motion model is: \[\ln \mathrm{pga}=c_1+1.2 M+c_3\ln(r+20)+0.25 F\] where \(\mathrm{pga}\) is in \(\,\mathrm{g}\), \(c_1=-3.27\), \(c_3=-1.79\) and \(\sigma_{total}=0.46\) for deep soil and \(c_1=-3.56\), \(c_3=-1.67\) and \(\sigma_{total}=0.46\) for rock/shallow soil.

Originally use five site classes (chosen based on site response analyses using broad categories and generic site profiles):

Rock. 66 records

Shallow soil (\(<250\,\mathrm{ft}\). 6 records.)

Intermediate depth soil (\(250\)â\(1000\,\mathrm{ft}\)). 2 records.

Deep soil (\(>1000\,\mathrm{ft}\)). 51 records.

Alluvium of unknown depth. 10 records.

but insufficient records in shallow and intermediate classes to evaluate separately so combine rock and shallow classes and intermediate, deep and unknown depth categories to leave two classes: \(<250\,\mathrm{ft}\) and \(>250\,\mathrm{ft}\).

Use two faulting mechanisms:

Strike-slip

Reverse or oblique

Process data by: 1) interpolation of uncorrected unevenly sampled records to 400 samples per second; 2) frequency domain low-pass filtering using a causal five-pole Butterworth filter with corner frequencies selected based on visual examination of Fourier amplitude spectrum; 3) removal of instrument response; 4) decimation to 100 or 200 samples per second depending on low-pass filter corner frequencies; and 5) application of time-domain baseline correction, using polynomials of degrees zero to ten depending on integrated displacements, and final high-pass filter chosen based on integrated displacements that is flat at corner frequency and falls off proportional to frequency on either side, which is applied in the time domain twice (forward and backwards) to result in zero phase shift.

Note that due to limited magnitude range of data, magnitude dependence is not well constrained nor is dependency on mechanism. Hence these coefficients are fixed based on previous studies.

Plot residuals w.r.t. distance. Find slight increase at \(70\)â\(100\,\mathrm{km}\). To test if due to Moho bounce repeat regression assuming functional form that is flat between \(70\) and \(90\,\mathrm{km}\) but this produced a smaller likelihood. Conclude that data does not support significant flattening at \(<100\,\mathrm{km}\).

Note that model is preliminary.

Ground-motion model is: \[\ln (A)=a_1+a_2 M_s+a_3 \ln(R)+a_4 R\] where \(A\) is in \(\,\mathrm{m/s^2}\), \(a_1=0.339\), \(a_2=0.455\), \(a_3=-0.67\), \(a_4=-0.00207\) and \(\sigma=0.61\).

Ground-motion model is: \[\begin{aligned} \ln \mathrm{PGA}&=&b_1+b_2M+b_3R-\ln R\\ \mbox{where }R&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{gal}\), \(b_1=4.73\), \(b_2=0.52\), \(b_3=-0.00216\), \(h\) is mean focal depth of group into which each earthquake is classified and \(\sigma=0.67\).

Most records from distances between \(10\,\mathrm{km}\) and \(40\,\mathrm{km}\).

Correction technique based on uniform Caltech correction procedure. Most (125) were automatically digitised, rest were manually digitised. Roll-on and cutoff frequencies of Ormsby filter were selected by adopting a record dependent criteria. Cutoff frequencies range between \(0.13\,\mathrm{Hz}\) and \(1.18\,\mathrm{Hz}\) with a median of \(0.38\,\mathrm{Hz}\).

Records included from analysis were from free-field stations. Excluded those not complete (e.g. started during strong-motion phase). Excluded those with epicentral distances greater than that of first nontriggered station.

Note relatively small influence of form of equation adopted although two step method seems preferable.

Note correction procedure plays a relevant role in analysis.

Note using \(d\) instead of \(R\) causes greater scatter in data.

Note moderate underestimation for low magnitude in near field and for high magnitude in far field.

Ground-motion model is: \[\ln Y=C_1+C_2 M+C_3 \ln (R+R_0)+C_4 S\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(C_1=3.88\), \(C_2=1.12\), \(C_3=-1.65\), \(R_0=15\), \(C_4=0.41\) and \(\sigma=0.71\).

Use two site categories (mean opinion of seven specialists who classified sites into three categories: soft alluvium, crystalline rock and intermediate):

Rock: 34+4 records. Japanese sites have diluvium with depth to bedrock \(H<10\,\mathrm{m}\). Alaskan sites have \(\mathrm{PGV}/\mathrm{PGA} \approx 66\pm 7\,\mathrm{cms^{-1}g^{-1}}\).

Alluvium: 71+12 records. Japanese sites have diluvium \(H>10\,\mathrm{m}\) or alluvium \(H<10\,\mathrm{m}\), and alluvium with \(H<25\,\mathrm{m}\) as well as soft layers with thickness \(<5\,\mathrm{m}\). Alaskan sites have \(\mathrm{PGV}/\mathrm{PGA} > 66\pm 7\,\mathrm{cms^{-1}g^{-1}}\).

\(70\%\) of records from ground level or basement of buildings with two storeys or less. Rest from buildings with up to eight storeys.

Some (16) Greek records manually digitized and baseline corrected, some (22) Greek records manually digitized and filtered and rest of the Greek records automatically digitized and filtered.

Due to lack of data for \(7.0<M_s<7.5\) include shallow subduction data from other regions with similar seismotectonic environments (Japan and Alaska) using criteria i) depth \(<35\,\mathrm{km}\), ii) \(M_w\) or \(M_{\mathrm{JMA}}\) between \(7.0\) and \(7.5\), iii) instruments triggered before S-wave, iv) free-field recording, v) surface geology known at station. Note \(M_s\), \(M_w\) and \(M_{\mathrm{JMA}}\) are equivalent between \(6.0\) and \(8.0\).

Focal depths between \(0\,\mathrm{km}\) (\(13\,\mathrm{km}\)) and \(18\,\mathrm{km}\) (\(31\,\mathrm{km}\)).

Most data from \(M_s<5.5\) and from \(R<50\,\mathrm{km}\).

Use four step regression procedure. First step use only Greek data from \(M_s>6.0\) (\(9 \leq R \leq 128\,\mathrm{km}\), 14 records) for which distances are more reliable (use both hypocentral and epicentral distance find epicentral distance gives smaller standard deviation) to find geometrical coefficient \(C_{31}\) and \(R_0\) ignoring soil conditions. Next find constant (\(C_{12}\)), magnitude (\(C_{22}\)) and soil (\(C_{42}\)) coefficients using all data. Next recalculate geometrical (\(C_{33}\)) coefficient using only Greek data with \(M_s>6.0\). Finally find constant (\(C_{14}\)), magnitude (\(C_{24}\)) and soil (\(C_{44}\)) coefficients using all the data; final coefficients are \(C_{14}\), \(C_{24}\), \(C_{33}\) and \(C_{44}\).

Plot residuals against \(M_s\) and \(R\) and find no apparent trends. Find residuals (binned into \(0.2\) intervals) fit normal distribution.

Ground-motion model is: \[\begin{aligned} \ln \mbox{pga}_{rock}&=&\theta_1+\theta_2 M+\theta_3\ln [r+\exp(\theta_4+\theta_5M)]+\theta_{11}F_1\\ \ln \mbox{pga}_{soil}&=&\theta_6+\theta_7 M+\theta_8\ln [r+\exp(\theta_9+\theta_{10})]+\theta_{11}F_1\end{aligned}\] where \(\mbox{pga}\) is in \(\,\mathrm{g}\), \(\theta_1=-4.364\), \(\theta_2=1.016\), \(\theta_3=-1.285\), \(\theta_4=-3.34\), \(\theta_5=0.79\), \(\theta_6=-8.698\), \(\theta_7=1.654\), \(\theta_8=-1.166\), \(\theta_9=-6.80\), \(\theta_{10}=1.40\), \(\theta_{11}=0.17\), \(\sigma=0.44\), \(\tau=0.00\) (sic) and \(\sigma_{total}=0.44\).

Originally use five site classes (chosen based on site response analyses using broad categories and generic site profiles):

Rock. 78 records

Shallow soil (\(<250\,\mathrm{ft}\). 25 records.)

Intermediate depth soil (\(250\)â\(1000\,\mathrm{ft}\)). 5 records.

Deep soil (\(>1000\,\mathrm{ft}\)). 62 records.

Alluvium of unknown depth. 31 records.

but insufficient records in shallow and intermediate classes to evaluate separately so combine rock and shallow classes and intermediate, deep and unknown depth categories to leave two classes: \(<250\,\mathrm{ft}\) and \(>250\,\mathrm{ft}\).

Use two faulting mechanisms:

Strike-slip or normal

Reverse

Based on Silva and Abrahamson (1992) (see SectionÂ 2.100.

Only use Nahanni records for spectral ordinates and not PGA because more representative of eastern US rock than western US rock.

Ground-motion model is: \[\begin{aligned} \log Y&=&b_1+b_2 (\mathbf{M}-6)+b_3 (\mathbf{M}-6)^2+b_4 r+b_5\log r+b_6 G_B+b_7 G_C\\ \mbox{where }r&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), for randomly-oriented horizontal component (or geometrical mean) \(b_1=-0.105\), \(b_2=0.229\), \(b_3=0\), \(b_4=0\), \(b_5=-0.778\), \(b_6=0.162\), \(b_7=0.251\), \(h=5.57\) and \(\sigma=0.230\) (for geometrical mean \(\sigma=0.208\)) and for larger horizontal component \(b_1=-0.038\), \(b_2=0.216\), \(b_3=0\), \(b_4=0\), \(b_5=-0.777\), \(b_6=0.158\), \(b_7=0.254\), \(h=5.48\) and \(\sigma=0.205\).

Due to an error in Equation (3) of Boore, Joyner, and Fumal (1994a) and Equation (6) of Boore, Joyner, and Fumal (1997) \(\sigma_c\) reported in Boore, Joyner, and Fumal (1994a, 1997) are too large by a factor of \(\sqrt{2}\). Therefore correct values of standard deviations are: \(\sigma_f=0.431\), \(\sigma_c=0.160\), \(\sigma_r=0.460\), \(\sigma_s=0.184\) and \(\sigma_{\ln Y}=0.495\).

Use three site categories:

\(V_{s,30} >750\,\mathrm{m/s}\), some categorised using measured shear-wave velocity, most estimated \(\Rightarrow G_B=0, G_C=0\), 48 records

\(360<V_{s,30} \leq 750\,\mathrm{m/s}\), some categorised using measured shear-wave velocity, most estimated \(\Rightarrow G_B=1, G_C=0\), 118 records.

\(180<V_{s,30} \leq 360\,\mathrm{m/s}\),some categorised using measured shear-wave velocity, most estimated \(\Rightarrow G_B=0, G_C=1\), 105 records.

where \(V_{s,30}\) is average shear-wave velocity to \(30\,\mathrm{m}\).

Define shallow earthquakes as those for which fault rupture lies mainly above a depth of \(20\,\mathrm{km}\).

Peak acceleration scaled directly from accelerograms, in order to avoid bias from sparsely sampled older data.

Do not use data from structures three storeys or higher, from dam abutments or from base of bridge columns. Do not use data from more than one station with the same site condition within a circle of radius \(1\,\mathrm{km}\) (note that this is a somewhat arbitrary choice).

Exclude records triggered by S wave.

Do not use data beyond cutoff distance which is defined as equal to lesser of distance to the first record triggered by S wave and closest distance to an operational nontriggered instrument.

Note that little data beyond \(80\,\mathrm{km}\).

Due to positive values of \(b_4\) when \(b_5=-1\), set \(b_4\) to zero and let \(b_5\) vary.

Ground-motion model is: \[\begin{aligned} \ln (Y)&=&\beta_0+a_1 M+ \beta_1 \tanh[a_2(M-4.7)]-\ln (R^2+[a_3\exp(a_1M)]^2)^{1/2}\\ &&{}-(\beta_4+\beta_5M)R+a_4 F+[\beta_2+a_5\ln(R)]S+\beta_3\tanh(a_6D)\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(\beta_0=-3.15\), \(\beta_1=0\), \(\beta_2=0\), \(\beta_3=0\), \(\beta_4=0.0150\), \(\beta_5=-0.000995\), \(a_1=0.683\), \(a_2=0.647\), \(a_3=0.0586\), \(a_4=0.27\), \(a_5=-0.105\), \(a_6=0.620\) and \(\sigma=0.50\).

Uses two site categories:

Quaternary deposits (soil).

Tertiary or older sedimentary, metamorphic, and igneous deposits (rock).

Also includes depth to basement rock (\(\,\mathrm{km}\)), \(D\).

Uses two fault mechanisms:

Strike-slip.

Reverse, reverse-oblique, thrust, and thrust-oblique.

Recommends use \(F=0.5\) for normal or unknown mechanisms.

Gives estimates of average minimum depths to top of seismogenic rupture zone.

Uses stochastic simulation model to find anelastic coefficients \(\beta_4\) and \(\beta_5\) because uses only near-source records.

Uses weighted nonlinear regression method based on Campbell (1981) to control dominance of well-recorded earthquakes.

Ground-motion model is: \[\begin{aligned} \log y&=&\alpha +\beta \mathbf{M}-\log r +b r\\ \mbox{where }r&=&(d^2+h^2)^{1/2}\end{aligned}\] Coefficients are unknown.

Data from earthquakes occurring between 1987 and 1991.

Ground-motion model is: \[\log Y=a+bM-\log \sqrt{r^2+h^2}-c r\] where \(Y\) is in \(\,\mathrm{g}\), \(a=-5.026\), \(b=0.989\), \(h=2.7\) and \(c=0.00443\) (\(\sigma\) not reported).

Some data from velocity sensors have been used, after differentiation, to increase amount of data at moderate and long distances.

Ground-motion model is (Type A): \[\log_{10} \mathrm{PGA}=a+bM_w-cr-d\log_{10}r\] where \(PGA\) is in \(\,\mathrm{g}\), \(a=-1.434\pm0.339\), \(b=0.209\pm0.036\), \(c=0.00297\pm0.00093\), \(d=-0.449\pm0.186\) and \(\sigma=0.276\).

Find that ground motions in previous earthquakes were significantly higher than the motions predicted by equations derived from W. N. America data.

Only include records from earthquakes for which \(M_w\) is known because of poor correlation between \(M_L\) and \(M_w\) in New Zealand.

Focal depths, \(h_e\leq 122\,\mathrm{km}\).

140 records from reverse faulting earthquakes.

Divide records into crustal and deep earthquakes.

Only use records for which reliable event information is available, regardless of their distances with respect to untriggered instruments.

Only use records which triggered on the P-wave.

Also derive separate equations for shallow, upper crustal earthquakes (\(h_e\leq 20\,\mathrm{km}\), 102 records, \(5.1\leq M_w \leq 7.3\), \(13\leq r \leq 274\,\mathrm{km}\)) and crustal earthquakes (\(h_e\leq 50\,\mathrm{km}\), 169 records, \(5.1\leq M_w \leq 7.3\), \(13\leq r \leq 274\,\mathrm{km}\)).

Also try equations of form: \(\log_{10} \mathrm{PGA}=a+bM_w-d\log_{10}r\) (Type B) and \(\log_{10} \mathrm{PGA}=a+bM_w-cr-\log_{10}r\) (Type C) because of large standard errors and highly correlated estimates for some of the coefficients (particularly \(c\) and \(d\)). Find Type B usually gives much reduced standard errors for \(d\) than Type A model and have lowest correlation between coefficients, but are sceptical of extrapolating to distance ranges shorter and longer than the range of data. Type C usually has similar standard deviations to Type A. Find that usually all three models give similar predictions over distance range of most of the data, but sometimes considerably different values at other distances.

Derive separate equations for reverse faulting earthquakes only and usually find similar results to the combined equations.

Find deep earthquakes produce significantly higher PGAs than shallow earthquakes for similar \(r\).

Ground-motion model is: \[\log y=c_1M-\log(d+c_2 10^{c_1M})+c_3d+c_4\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(c_1=0.42\), \(c_2=0.025\), \(c_3=-0.0033\) and \(c_4=1.22\) (\(\sigma\) unknown).

Ground-motion model is unknown.

Used by Tanner and Shedlock (2004).

Ground-motion model is: \[\begin{aligned} \log (A)&=&a_1+a_2M+a_3\log [G(R_0)]+a_4R_0\\ \mbox{where }R_0^2&=&R^2+(\mathrm{e}^{a_5 M})^2\\ G(R_0)&=&R_0 \mbox{ for: } R_0 \leq 100\,\mathrm{km}\\ \mbox{and: }G(R_0)&=&\sqrt{(100 R_0)} \mbox{ for: } R_0>100\,\mathrm{km}\end{aligned}\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(a_1=2.74\), \(a_2=0.212\), \(a_3=-0.99\), \(a_4=-0.000943\), \(a_5=0.47\) and \(\sigma=0.26\).

Use same data as Taylor Castillo et al. (1992).

Employ several different regression techniques.

Select equation found by Bayesian method (given above) for hazard study.

Ground-motion model is: \[\log(A_{\max})=a_1 M+a_2 \log(D+a_3)+a_4\] where \(A_{\max}\) is in \(\,\mathrm{cm/s^2}\), \(a_1=0.54\), \(a_2=-1.5\), \(a_3=10\) and \(a_4=1.25\) (\(\sigma\) not reported).

Ground-motion model is: \[\ln A = a+b M -c \ln (R+h) +d T_s\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(a=7.7\), \(b=0.49\), \(c=1.45\), \(d=0.19\), \(h=25.0\) and \(\sigma=0.46\).

Model soil using its fundamental period of the overburden soil, \(T_s\). Thickness of deposit defined as depth to rock base, defined either as \(V_s > 800\,\mathrm{m/s}\) or when ratio of shear-wave velocity in \(i\)th layer to shear-wave velocity in \(i-1\)th layer is greater than 2 (only calculate period to \(100\,\mathrm{m}\) because only have important effect on structure). For outcropping rock, \(T_s=0.05\,\mathrm{s}\).

Eight distance intervals used for weighting, five \(10\,\mathrm{km}\) wide up to \(50\,\mathrm{km}\), \(50\)â\(69.9\,\mathrm{km}\), \(70\)â\(99.9\,\mathrm{km}\) and \(100\)â\(200\,\mathrm{km}\). Within each interval each earthquake received equal weight, inversely proportional to number of records from that earthquake in interval.

Use resolve accelerations in direction, \(\theta\), which gives largest value. Find scatter is lower than for larger horizontal component.

Many (27) earthquakes only have one record associated with them and 60 records are from San Fernando.

Ground-motion model is: \[\begin{aligned} \log a&=&b_1+b_2M_s+b_3r+b_4\log r\\ \mbox{where }r&=&(d^2+h_0^2)^{0.5}\end{aligned}\] where \(a\) is in \(\,\mathrm{g}\), \(b_1=-1.58\), \(b_2=0.260\), \(b_3=-0.00346\), \(b_4=-0.625\), \(h_0=4\) and \(\sigma=0.26\).

Do not consider effect of site geology but expect it to be statistically insignificant for PGA.

Focal depths, \(h<25\,\mathrm{km}\). Mean focal depth is \(10\pm 4\,\mathrm{km}\).

Mean magnitude of earthquakes considered is \(6.0\pm 0.7\).

Most records from \(d<100\,\mathrm{km}\).

Only use records with \(\mathrm{PGA}>0.01\,\mathrm{g}\).

Records mainly from SMA-1s located at ground floor or in basements of buildings and structures and free-field sites regardless of topography.

Records from thrust earthquakes (\(46\%\) of total), normal earthquakes (\(26\%\)) and strike-slip earthquakes (\(28\%\)).

Baseline correct and low-pass filter records. Select cut-offs from visual examination of Fourier amplitude spectrum of uncorrected time-histories and choose cut-off below which the Fourier amplitude spectrum showed an unrealistic energy increase due to digitization noise and instrument distortions.

Find (from reprocessing about 300 records) that with very few exceptions differences in PGAs arising from different methods of processing are not significant, remaining below \(3\%\).

Also derive equation which includes focal depth explicitly.

Based on Boore, Joyner, and Fumal (1993) see SectionÂ 2.105

Ground-motion model is: \[\begin{aligned} \log Y&=&b_1+b_2 (\mathbf{M}-6)+b_3 (\mathbf{M}-6)^2+b_4 r+b_5\log r+b_V (\log V_S -\log V_A)\\ \mbox{where }r&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(b_1\) to \(b_5\), \(h\) and \(\sigma\) are same as for Boore, Joyner, and Fumal (1993) (see SectionÂ 2.105) and for randomly oriented component \(b_V=-0.371\) and \(V_A=1400\) and for larger horizontal component \(b_V=-0.364\) and \(V_A=1390\).

Model site effect as a continuous function of average shear-wave velocity to \(30\,\mathrm{m}\) deep, \(V_S\).

Coefficients \(b_1\), \(b_2\), \(b_3\),\(b_4\) and \(b_5\) from Boore, Joyner, and Fumal (1993).

Find no basis for different magnitude scaling at different distances.

Find evidence for magnitude dependent uncertainty.

Find evidence for amplitude dependent uncertainty.

Find marginal statistical significance for a difference between strike-slip (defined as those with a rake angle within \(30^{\circ}\) of horizontal) and reverse-slip motions but do not model it. Modelled in Boore, Joyner, and Fumal (1994b) (by replacing \(b_1\) by \(b_{SS}G_{SS}+b_{RS}G_{RS}\) where \(G_{SS}=1\) for strike-slip shocks and \(0\) otherwise and \(G_{RS}=1\) for reverse-slip shocks and \(0\) otherwise) and reported in Boore, Joyner, and Fumal (1997). Coefficients for randomly oriented horizontal component are: \(b_{SS}=-0.136\) and \(b_{RS}=-0.051\)

^{14}.Analysis done using one and two-stage maximum likelihood methods; note that results are very similar.

Earthquakes with magnitudes below \(6.0\) are poorly represented.

Note that few Class A records.

Note that \(V_S\) does not model all the effects of site because it does not model effect of the thickness of attenuating material on motion.

Note that ideally would like to model site in terms of average shear-wave velocity to one-quarter wavelength.

Note lack measurements from distances greater than \(100\,\mathrm{km}\) so that weak-motion data from seismographic stations maybe should be used.

Note that use of cutoff distances independent of geology or azimuth may be over strict but it is simple and objective. Note that methods based on data from nontriggered stations or using seismogram data may be better.

Ground-motion model is: \[\log a=C_1+C_2M+C_3 \log(R+C_4)\] where \(a\) is in \(\,\mathrm{cm/s^2}\), \(C_1=8.65\), \(C_2=0.71\), \(C_3=-1.6\), \(C_4=40\) and \(\sigma=0.6\).

May not be an empirical GMPE but derived through a intensity-PGA relations.

Ground-motion model is: \[\ln A=a_1 M+a_2 \ln \Delta +a_3\] where \(A\) is in \(\,\mathrm{gal}\), \(a_1=1.812\), \(a_2=-0.796\), \(a_3=-3.616\) and \(\sigma=0.558\).

Use velocity data recorded at GRW. Differentiate to acceleration. Use frequency range \(1\) to \(6\,\mathrm{Hz}\).

Ground-motion model is: \[\log Y=aM+bX-\log X+\sum \delta_i c_i\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(\delta_i=1\) at \(i\)th receiver and \(0\) otherwise, for horizontal PGA \(a=0.918\) and \(b=-0.00846\) (\(\sigma\) not given) and for vertical PGA \(a=0.865\) and \(b=-0.00741\) (\(\sigma\) not given). \(c_i\) given in paper but are not reported here due to lack of space.

Data from three vertical arrays in Japan so predictions at surface and at different depths down to \(950\,\mathrm{m}\).

Different definition of \(M_{\mathrm{JMA}}\) for focal depths \(>60\,\mathrm{km}\) so exclude such data. Focal depths between \(2\) and \(60\,\mathrm{km}\).

Exclude data from earthquakes \(M<5.0\) because errors are larger for smaller events.

Exclude data for which predicted, using a previous attenuation relation, \(\mathrm{PGV}<0.1\,\mathrm{cm/s}\) in order to find precise attenuation rate.

Most data from earthquakes with \(M\leq 6.0\) and most from \(X\leq 100\,\mathrm{km}\).

Records low-pass filtered with cutoff frequency \(25\,\mathrm{Hz}\) for records from 2 sites and \(30\,\mathrm{Hz}\) for records from 1 site.

Use two-stage method because positive correlation between \(M\) and \(X\). Also apply one step; find it is biased and two-stage method is most effective method to correct bias.

Check residuals (not shown) against \(M\) and \(X\) find no remarkable bias.

Ground-motion model is: \[\log Y=a+b(M-6)+c(M-6)^2+d \sqrt{R^2+h^2}+e\log \sqrt{R^2+h^2}+fS_B+gS_C\]

Use three site categories:

Firm to hard rock: granite, igneous rocks, sandstones and shales with close to widely spaced fractures, \(750\leq V_{s,30}\leq 1400\,\mathrm{m/s}\) \(\Rightarrow S_B=0\), \(S_C=0\).

Gravelly soils and soft to firm rocks: soft igneous rocks, sandstones and shales, gravels and soils with \(>20\%\) gravel, \(360\leq V_{s,30}\leq 750\,\mathrm{m/s}\) \(\Rightarrow S_B=1\) , \(S_C=0\).

Stiff clays and sandy soils: loose to very dense sands, silt loams and sandy clays, and medium stiff to hard clay and silty clays (\(N>5\,\mathrm{blows/ft}\)), \(180\leq V_{s,30}\leq360\,\mathrm{m/s}\) \(\Rightarrow S_B=0\), \(S_C=1\).

For shallow (fault rupture within \(20\,\mathrm{km}\) of earth surface) crustal earthquakes.

Use free-field records. Records not significantly contaminated by structural feedback, excludes records from structures with \(>\)2 stories.

Chooses Ground-motion model because of simplicity. Note that other possible forms of equation may have significant effect on results, but including more terms complicates relationships without reducing variability.

Do not give coefficients only predictions.

Ground-motion model is: \[\ln \mathrm{PGA}=c_1+c_2M_w+c_3 \ln R+c_4 h\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), \(c_1=-2.122\), \(c_2=1.885\), \(c_3=-1.011\), \(c_4=-0.012\) and \(\sigma=0.502\).

Focal depth, \(h\), between \(79\) and \(131\,\mathrm{km}\).

Consider to separate areas of \(90^{\circ}\) to investigate variation with respect to azimuth; find azimuthal dependence.

Find individual attenuation equations for three earthquakes. Note faster attenuation for smaller magnitude and faster attenuation for deeper events.

Ground-motion model is (model 1): \[\ln A=a+b M-\ln (R)+d R\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(a=2.11\), \(b=1.23\) and \(d=-0.014\).

Ground-motion model is (model 2): \[\begin{aligned} \ln A&=&c_1+c_2 M+c_4 R +\ln G(R,R_0)\\ \mbox{where } G(R,R_0)&=&R^{-1} \quad \mbox{for} \quad R \leq R_0\\ \mbox{and: } G(R,R_0)&=&R_0^{-1}\frac{R_0}{R}^{5/6} \quad \mbox{for} \quad R>R_0\end{aligned}\] where \(A\) is in \(\,\mathrm{m/s^2}\), \(c_1\) and \(c_2\) are from Dahle, Bungum, and Kvamme (1990), \(c_4=-0.0148\) and \(\sigma\) is recommended as \(0.65\) (although this is from an earlier study and is not calculated in regression).

Use data from Canada (Saguenay earthquake and Nahanni sequence) and Belgium (Roermond earthquake).

Focal depths, \(h\), between \(1\) and \(30\,\mathrm{km}\) with average \(14.4\,\mathrm{km}\).

Assume peak ground acceleration equals pseudo-acceleration at \(30\,\mathrm{Hz}\) due to few unclipped horizontal records and because instrument response of instruments means records unreliable above \(30\,\mathrm{Hz}\). Use only digital records for \(30\,\mathrm{Hz}\) model.

Note poorness of data due to data and other data being widely separated thus preventing a comparison between the two sets. Also means straightforward regression methods would be inadequate as there would be little control on shape of curves derived.

Note earlier models over predict data.

Use two-stage least squares method to give model 1. First stage fit only /Belgian data to find \(b\), in second stage use this value of \(b\) and use all data to find \(a\) and \(d\).

Do not recommend model 1 for general use because too influenced by limitations of data to be considered reliable. Canadian data probably insufficient to anchor curves at small \(R\)/large \(M\) and extremely high Saguenay earthquake records carry undue weight.

Use model of Dahle, Bungum, and Kvamme (1990) to get model 2. Fix \(c_1\) and \(c_2\) to those of Dahle, Bungum, and Kvamme (1990) and find \(c_4\). Prefer this model.

Ground-motion model is: \[\ln \mathrm{PGA}=c_1+c_2 M+c_3 \ln R+c_4 h\] where \(\mathrm{PGA}\) is in \(\,\mathrm{cm/s^2}\), \(c_1=5.432\), \(c_2=1.035\), \(c_3=-1.358\), \(c_4=-0.0072\) and \(\sigma=0.397\).

Sites have different soil conditions, some medium and stiff sites and some very soft soil sites.

Use some records from Moldova and Bulgaria.

Focal depths, \(h\), between \(91\) and \(133\,\mathrm{km}\).

Records from free-field or from basements of buildings.

Originally include data from a shallower (focal depth \(79\,\mathrm{km}\)), smaller magnitude (\(M_L=6.1\), \(M_w=6.3\)) earthquake with shorter return period than other three earthquakes, but exclude in final analysis.

Originally do attenuation analysis for two orthogonal directions N45E (which is in direction of fault plane) and N35E (which is normal to fault plane). From this define 3 \(90^{\circ}\) circular sectors based roughly on tectonic regions, and calculate attenuation relations for each of these sectors as well as for all data. Find azimuthal dependence.

Remove 1 to 3 anomalous records per sector.

Remove the only record from the 4/3/1977 earthquake, because it has a strong influence on results, and repeat analysis using model \(\ln \mathrm{PGA}=b_1+b_2M+b_3\ln R\), find lower predicted PGA.

Find slower attenuation in direction of fault plane compared with normal to fault plane.

Find faster attenuation and larger standard deviation (by finding attenuation equations for two different earthquakes) for deeper focus and larger magnitude shocks.

Ground-motion model is: \[\begin{aligned} a_h&=&a_1 (a_2+d+H)^{a_5} \exp(a_6 M_s)\\ H&=&|d-a_3|^{a_4}\end{aligned}\] where for horizontal peak acceleration \(a_h\) is in \(\,\mathrm{cm/s^2}\), \(a_1=4000\), \(a_2=20\), \(a_3=16\) and \(a_4=0.63\) for soil sites \(a_5=-2.02\) and \(a_6=0.8\) and for rock sites \(a_5=-2.11\) and \(a_6=0.79\) (\(\sigma\) not given). For vertical peak acceleration on soil sites \(a_v\) is in \(\,\mathrm{cm/s^2}\) \(a_1\) to \(a_3\) are same as horizontal and \(a_4=0.48\), \(a_5=-1.75\) and \(a_6=0.53\) (\(\sigma\) not given).

Use two site categories (from original of four) for which derive two separate equations:

Rock: mainly category (2) a) loose igneous rocks (tuffs), friable sedimentary rocks, foliated metamorphic rock and rocks which have been loosened by weathering, b) conglomerate beds, compacted sand and gravel and stiff clay (argillite) beds where soil thickness \(>60\,\mathrm{m}\) from bed rock. 29 records.

Soil: mainly category (4) a) soft and wet deposits resulting from high level of water table, b) gravel and sand beds with weak cementation and/or uncementated unindurated clay (clay stone) where soil thickness \(>10\,\mathrm{m}\) from bed rock. 54 records.

Focal depths between \(10\) and \(69\,\mathrm{km}\).

Find equations using hypocentral distance but find that poor fit for Rudbar (Manjil) earthquake (\(M_s=7.7\)) which conclude due to use of hypocentral rather than rupture distance.

Find equations using rupture distance

^{15}for Rudbar (Manjil) earthquake and hypocentral distances for other earthquakes. Coefficients given above. They conclude that it is important that equations are derived using rupture distance rather than hypocentral distance because most destructive earthquakes rupture surface in Iran.Do not know physical meaning of \(H\) term but find that it causes curves to fit data better.

Ground-motion model is: \[A_p=a \mathrm{e}^{b M_s} (R+\Delta)^c\] where \(A_p\) is in \(\,\mathrm{cm/s^2}\) and for combined Yunnan and W. N. American data \(a=1291.07\), \(b=0.5275\), \(c=-1.5785\), \(\Delta=15\) and \(\sigma=0.5203\) (in terms of natural logarithm).

All records from basement rock.

Most Yunnan data from main and aftershocks of Luquan and Luncang-Gengma earthquakes.

Records from Lancang-Gengma sequence corrected.

Most Yunnan records with \(3 \leq M_s \leq 5\) and \(10 \leq R \leq 40\,\mathrm{km}\).

To overcome difficulty due to shortage of large magnitude records and sample heterogeneous distribution in near and far fields use W. N. America data, because intensity attenuation is similar.

Fit curves to Yunnan and Yunnan with W. N. American data. Find curve for combined data has lower variance and fit to observation data for large magnitudes is better (by plotting predicted and observed PGA).

Ground-motion model is: \[\log(a^{1/M})=b_1-b_3 \log(R)\] where \(a\) is in \(\,\mathrm{cm/s^2}\), \(b_1=0.433\), \(b_3=0.073\) and \(\sigma=0.037\).

Data from three earthquakes with \(M_B\) of \(5.7\), one of \(M_B\) of \(5.8\) and the other \(M_B\) of \(7.2\).

Compare predicted and observed ground motions for 20/10/1991 Uttarkashi earthquake (\(M 6.1\)) and find good fit.

Ground-motion model is: \[\begin{aligned} \log a&=&A+B M_s+C r +D \log r\\ \mbox{where }r^2&=&d^2+h_0^2\end{aligned}\] where \(a\) is in \(\,\mathrm{g}\), for \(4.0\leq M \leq 7.4\): for horizontal PGA not including focal depth \(A=-1.43\), \(B=0.245\), \(C=-0.0010\), \(D=-0.786\), \(h_0=2.7\) and \(\sigma=0.24\), for vertical PGA not including focal depth \(A=-1.72\), \(B=0.243\), \(C=-0.00174\), \(D=-0.750\), \(h_0=1.9\) and \(\sigma=0.24\), for horizontal PGA including focal depth \(A=-1.06\), \(B=0.245\), \(C=-0.00045\), \(D=-1.016\), \(h_0=h\) and \(\sigma=0.25\) and for vertical PGA including focal depth \(A=-1.33\), \(B=0.248\), \(C=-0.00110\), \(D=-1.000\), \(h_0=h\) and \(\sigma=0.25\).

Reviews and re-evaluates distances, focal depths, magnitudes and PGAs because data from variety of sources with different accuracy and reliability. For \(M_s>6.0\) distances have acceptable accuracy but for \(M_s<6.0\) distance, depths and magnitudes are poorly known. Errors in locations for \(M_s<6.0\) still large with no foreseeable means of improving them. Use of \(r_{epi}\) for \(M_s<6.0\) justified because difference between \(r_{jb}\) and \(r_{epi}\) for small earthquakes is not larger than uncertainty in epicentre. Check and redetermine station locations; find large differences in excess of \(15\,\mathrm{km}\) for some stations.

Focal depths poorly determined. Revises 180 depths using S-start times (time between P and S-wave arrival).

Focal depths \(h<26\,\mathrm{km}\); most (60%+) between \(4\) and \(14\,\mathrm{km}\).

Does not use \(M_L\) because no \(M_L\) values for Algeria, Iran, Pakistan, Turkey and former USSR and unreliable for other regions. Does not use magnitude calculated from strong-motion records because magnitude calculation requires point source approximation to be valid. Conversion from \(M_L\) to \(M_s\) should not be done because of uncertainty in conversion which should be retained.

Notes that \(M_s\) results in nonlinear scaling on PGA with \(M_w\) due to nonlinear relationship between \(\log M_0\) and \(M_s\).

Uses PGAs in four forms: maximum values from accelerograms read by others (34%), from corrected records (30%), scaled directly from accelerograms (13%) and from digitised plots (23%). Notes potential bias in using both corrected and uncorrected PGAs but neglects it because small difference (\(\lesssim 4\%\) for those checked). Excludes PGAs near trigger level because processing errors can be large. Some unfiltered digital records which require additional processing to simulate SMA-1 could be associated with larger differences (\(\lesssim 10\%\)).

Excludes records from basements and ground floors of structures with more than 3 levels. Retains the few records from dam abutments and tunnel portals.

Excludes records generated by close small magnitude earthquakes triggered by S-wave.

Does not exclude records obtained at distances greater than shortest distance to an operational but not triggered instrument because of non-constant or unknown trigger levels and possible malfunctions of instruments.

Uses weighted regression of Joyner and Boore (1988) for second stage.

Splits data into five magnitude dependent subsets: \(2.0 \leq M_s \leq 7.3\) (1260 records from 619 shocks), \(3.0 \leq M_s \leq 7.3\) (1189 records from 561 shocks), \(4.0 \leq M_s \leq 7.3\) (830 records from 334 shocks), , \(5.0 \leq M_s \leq 7.3\) (434 records from 107 shocks), and \(3.0 \leq M_s \leq 6.0\) (976 records from 524 shocks). Calculates coefficients for each subset. Finds only small differences \(\pm 15\%\) over distance range \(1\)â\(200\,\mathrm{km}\) between predictions and uncertainties. Concludes results stable. Prefers results from subset with \(4.0 \leq M_s \leq 7.3\).

Finds it difficult to obtain some vertical accelerations due to low ground motion so ignores data from \(>100\,\mathrm{km}\) with PGA \(<1\%\mathrm{g}\) (\(0.1\,\mathrm{m/s^2}\)).

Repeats regression using \(r^2=d^2+h^2\). Finds depth important.

Calculates using one-stage method; finds very similar results for \(10 < d <100\,\mathrm{km}\).

Considers magnitude dependent function: \(\log a = b_1+b_2 M_s +b_3 r +b_4 [r+b_5 \exp (b_6 M_s)]\). Finds \(b_5\) is zero so drops \(b_3\) and repeats. Finds \(b_5\) close to zero so magnitude dependent function not valid for this dataset.

Local shear-wave velocity, \(V_s\), profiles known for 44 stations (268 records from 132 earthquakes between \(2.5\) and \(7.2\)) although only 14 from \(>40\,\mathrm{km}\) so barely sufficient to derive equation. Use 145 records from 50 earthquakes with \(M_s>4.0\) to fit \(\log a=A+B M_s +Cr +D \log r +E \log V_{s30}\), where \(V_{s30}\) is average shear-wave velocity to reference depth of \(30\,\mathrm{m}\). Finds \(C\) positive so constrain to zero. Find no reduction in standard deviation.

Uses residuals from main equation to find \(E\). Notes that should not be used because of small number of records. Considers different choices of reference depth; finds using between \(5\) and \(10\,\mathrm{m}\) leads to higher predicted amplifications. Notes better to use \(V_{s30}\) because no need for subjective selection of categories.

Ground-motion model is: \[\begin{aligned} \ln A&=&c_1+c_2 M_w+c_3 \ln R+c_4 R+c_5 S\\ \mbox{with: }R&=&\sqrt{r^2+r_h^2}\end{aligned}\] where \(A\) is in \(\,\mathrm{m/s^2}\), \(c_1=-1.579\), \(c_2=0.554\), \(c_3=-0.560\), \(c_4=-0.0032\), \(c_5=0.326\), \(r_h=6\) and \(\sigma=0.3535\)

Use records from Costa Rica, Mexico, Nicaragua and El Salvador. Only Mexican earthquakes with \(M_w \geq 6.5\) were used.

Use two site categories:

Rock: 92 records

Soil: 88 records

Use a Bayesian one-stage regression method (Ordaz, Singh, and Arciniega 1994) to yield physically possible coefficients.

Consider tectonic type: subduction or shallow crustal but do not model.

Find no significant difference between Guerrero (Mexico) and other data.

Find no significant difference between subduction and shallow crustal data.

Ground-motion models are (if define site in terms of local geological site classification): \[\log a_{\max}=M+\mathrm{Att}(\Delta/L,M,T)+b_1M+b_2s+b_3v+b_4+b_5M^2+\sum_i b_6^i S_L^i+b_{70} r R+b_{71} (1-r) R\] or (if define site in terms of depth of sediment): \[\log a_{\max}=M+\mathrm{Att}(\Delta/L,M,T)+b_1M+b_2h+b_3v+b_4+b_5M^2+\sum_i b_6^i S_L^i+b_{70} r R+b_{71} (1-r) R\] where: \[\begin{aligned} \mathrm{Att}(\Delta,M,T)&=&\left\{ \begin{array}{r} b_0\log_{10} \Delta \quad \mbox{for}\quad R\leq R_{\max}\\ b_0 \log_{10} \Delta_{\max}-(R-R_{\max})/200 \quad \mbox{for}\quad R>R_{\max}\\ \end{array} \right.\\ \Delta&=&S\left(\ln\frac{R^2+H^2+S^2}{R^2+H^2+S_0^2}\right)^{-1/2}\\ \Delta_{\max}&=&\Delta(R_{\max},H,S)\\ R_{\max}&=&\frac{1}{2}(-\beta+\sqrt{\beta^2-4H^2})\end{aligned}\] \(S_0\) is correlation radius of source function and can be approximated by \(S_0\sim \beta T/2\) (for PGA assume \(T\approx 0.1\,\mathrm{s}\) so use \(S_0=0.1\,\mathrm{km}\)), \(\beta\) is shear-wave velocity in source region, \(T\) is period, \(S\) is âsource dimensionâ approximated by \(S=0.2\) for \(M<3\) and \(S=-25.34+8.51M\) for \(3 \leq M \leq 7.25\), \(L\) is rupture length of earthquake approximated by \(L=0.01 \times 10^{0.5M}\) \(\,\mathrm{km}\) and \(v\) is component direction (\(v=0\) for horizontal \(1\) for vertical). Different \(b_0\), \(b_{70}\) and \(b_{71}\) are calculated for five different path categories. Coefficients are not reported here due to lack of space.

Use four types of site parameter:

Local geological site classification (defined for all records):

Sites on sediments.

Intermediate sites.

Sites on basement rock.

Depth of sediments from surface to geological basement rock beneath site, \(h\) (defined for 1675 records out of 1926).

Local soil type parameter describes average soil stiffness in top \(100\)â\(200\,\mathrm{m}\) (defined for 1456 records out of 1926):

âRockâ soil sites \(\Rightarrow\) \(S_L^1=1\), \(S_L^2=0\) and \(S_L^3=0\). Characterises soil up to depth of less than \(10\,\mathrm{m}\).

Stiff soil sites \(\Rightarrow\) \(S_L^1=1\), \(S_L^2=0\) and \(S_L^3=0\) (shear-wave velocities \(<800\,\mathrm{m/s}\) up to depth of \(75\)â\(100\,\mathrm{m}\)).

Deep soil sites \(\Rightarrow\) \(S_L^2=1\), \(S_L^1=0\) and \(S_L^3=0\). (shear-wave velocities \(<800\,\mathrm{m/s}\) up to depth of \(150\)â\(200\,\mathrm{m}\)).

Deep cohesionless soil sites \(\Rightarrow\) \(S_L^3=1\), \(S_L^1=0\) and \(S_L^2=0\) (only use for one site with 10 records).

Average soil velocity in top \(30\,\mathrm{m}\), \(v_L\) (if unavailable then use soil velocity parameter, \(s_T\)) (defined for 1572 records out of 1926):

\(v_L>750\,\mathrm{m/s}\).

\(360\,\mathrm{m/s}<v_L\leq 750\,\mathrm{m/s}\).

\(180\,\mathrm{m/s}<v_L\leq 360\,\mathrm{m/s}\).

\(v_L\leq 180\,\mathrm{m/s}\).

Only include records for which significant subset of site parameters (\(s\), \(h\), \(s_L\), \(v_L\)) exist.

Almost all earthquakes have focal depths \(H \leq 15\,\mathrm{km}\); all focal depths \(H\leq 43\,\mathrm{km}\).

Use records from 138 aftershocks of Imperial Valley earthquake (15/10/1979), which contribute most of \(M\leq 3\) records.

Use records from 109 earthquakes with \(M\leq 3\).

Use free-field records.

Characterise path by two methods:

Fraction of wave path travelled through geological basement rock measured at surface, from epicentre to station, \(0\leq r \leq 1\).

Generalised path type classification:

Sediments to sediments.

Rock-to-sediments, vertically.

Rock-to-sediments, horizontally.

Rock-to-rock.

Rock-to-rock through sediments, vertically.

Rock-to-sediments through rock and sediments, vertically.

Rock-to-sediments though rock and sediments, horizontally.

Rock-to-rock through sediments, horizontally.

Due to lack of data combine path types 2 and 6 in new category 2â, combine path types 3 and 7 in new category 3â, combine path types 4, 5 and 8 in new category 4â (when \(r\neq 1\)) and combine 4, 5 and 8 in new category 5â (when \(r=1\)).

Plot PGA against magnitude and distance to get surface by interpolation. Plot without smoothing and with light and intense smoothing. Find for small magnitude (\(M\approx 3\)â\(4\)) earthquakes attenuation is faster than for large magnitude (\(M\approx 6\)â\(7\)) earthquakes.

Use a multi-step residue regression method. First fit \(\log a_{\max}=M+\mathrm{Att}(\Delta,M,T)+b_1M+b_2s+b_3v+b_4+b_5 M^2\) (or \(\log a_{\max}=M+\mathrm{Att}(\Delta,M,T)+b_1M+b_2h+b_3v+b_4+b_5 M^2\)) and calculate residuals \(\epsilon=\log a_{\max}-\log \hat{a}_{\max}\) where \(a_{\max}\) is estimated PGA and \(\hat{a}_{\max}\) is recorded PGA. Fit \(\epsilon=b_7^{(-1)}S_L^{(-1)}+b_7^{(0)}S_L^{(0)}+b_7^{(1)}S_L^{(1)}+b_7^{(2)}S_L^{(2)}+b_7^{(3)}S_L^{(3)}\) where \(S_L^{(i)}=1\) if \(s_L=i\) and \(S_L^{(i)}=0\) otherwise. Find significant dependence. Try including \(v_L\) both as a continuous and discrete parameter in model but not significant at \(5\%\) significance level. Next calculate residuals from last stage and fit \(\epsilon=b_0' \log_{10} (\Delta/L)+b_4'+b_{60}rR+b_{61}(1-r)R\) for each of the five path type groups (1â to 5â). Lastly combine all the individual results together into final equation.

Note that \(b_{70}\) and \(b_{71}\) can only be applied for \(R\lesssim 100\,\mathrm{km}\) where data is currently available. For \(R\gtrsim 100\,\mathrm{km}\) the predominant wave type changes to surface waves and so \(b_{70}\) and \(b_{71}\) do not apply.

Study almost identical to Radu et al. (1994), see SectionÂ 2.123, but different coefficients given: \(c_1=3.672\), \(c_2=1.318\), \(c_3=-1.349\), \(c_4=-0.0093\) and \(\sigma=0.395\).

Ground-motion model is: \[\log y=b_0+b_1M+b_2r+b_3\log r+b_4 h+c_i\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(b_0=0.206\), \(b_1=0.477\), \(b_2=-0.00144\), \(b_3=-1\), \(b_4=0.00311\), \(\sigma=0.276\) and \(c_i\) is site coefficient for site \(i\) (use 76 sites), given in paper but are not reported here due to lack of space.

Records from accelerometers on small foundations detached from structures; thus consider as free-field.

Exclude records with one horizontal component with \(\mathrm{PGA}<1\,\mathrm{cm/s^2}[0.01\,\mathrm{m/s^2}]\) because weaker records not reliable due to resolution (\(\pm 0.03\,\mathrm{cm/s^2}[0.0003\,\mathrm{m/s^2}]\)) of instruments.

Exclude earthquakes with focal depths equal to \(0\,\mathrm{km}\) or greater than \(200\,\mathrm{km}\), due to lack of such data. Depths (depth of point on fault plane closest to site), \(h\), between about \(1\,\mathrm{km}\) to \(200\,\mathrm{km}\).

Apply a low-cut filter with cosine-shaped transition from \(0.01\) to \(0.05\,\mathrm{Hz}\).

Positive correlation between magnitude and distance so use two-stage method.

Note different definition for \(M_{\mathrm{JMA}}\) for focal depths \(>60\,\mathrm{km}\).

Firstly do preliminary analysis with \(b_4=0\) and no site coefficients; find \(b_2\) is positive so constrain to \(0\) but find \(b_3<-1.0\) so constrain \(b_3\) to \(-1.0\) and unconstrain \(b_2\). Find linear dependence in residuals on \(h\) especially for \(h<100\,\mathrm{km}\). Find significant improvement in coefficient of determination, \(R^2\), using terms \(b_4h\) and \(c\).

Find singularity in matrices if apply two-stage method, due to number of coefficients, so propose a iterative partial regression method.

Also separate data into five depth ranges (A: \(h=0.1\) to \(30\,\mathrm{km}\), 553 records from 111 earthquakes; B: \(h=30\) to \(60\,\mathrm{km}\), 778 records from 136 earthquakes; C: \(h=60\) to \(90\,\mathrm{km}\), 526 records from 94 earthquakes; D: \(h=90\) to \(120\,\mathrm{km}\), 229 records from 31 earthquakes; E: \(h=120\) to \(200\,\mathrm{km}\), 112 records from 19 earthquakes) and find attenuation equations for each range. Note results from D & E may not be reliable due to small number of records. Find similar results from each group and all data together.

Find weak correlation in station coefficients with soil categories, as defined in Iwasaki, Kawashima, and Saeki (1980), but note large scatter.

Ground-motion model is: \[\begin{aligned} \log(a_h)&=&C_1+C_2M+C_3M^2+C_4\log(R)+C_5R+C_6S\\ \mbox{where }R&=&\sqrt{d^2+h_0^2}\end{aligned}\] where \(a_h\) is in \(\,\mathrm{g}\), \(C_1=-3.4360\), \(C_2=0.8532\), \(C_3=-0.0192\), \(C_4=-0.9011\), \(C_5=-0.0020\), \(C_6=-0.0316\), \(h_0=4.24\) and \(\sigma=0.424\).

Use two site categories:

Rock

Soil

Use one-stage method because of the predominance of earthquakes with single recordings in the set.

Note that it is very important to choose a functional form based as much as possible on physical grounds because the data is sparse or non-existent for important ranges of distance and magnitude.

Carefully verify all the distances in set.

Use focal depths from (in order of preference): special reports (such as aftershock monitoring), local agencies and ISC and NEIS determinations. Focal depths \(<30\,\mathrm{km}\).

Do not use \(M_L\) or \(m_b\) because of a variety of reasons. One of which is the saturation of \(M_L\) and \(m_b\) at higher magnitudes (\(M_L, m_b>6\)).

If more than one estimate of \(M_w\) made then use average of different estimates.

Use PGAs from: a) digital or digitised analogue records which have been baseline corrected and filtered, b) data listings of various agencies and c) other literature. Difference between PGA from different sources is found to be small.

Also derive equations assuming \(C_3=0\) (using rock and soil records and only soil records) and \(C_3=0\), \(C_4=-1\) and \(C_6=0\) (using only rock records).

Include records from Nahanni region and find similar results.

Also derive equations for Australia (115 records from 86 earthquakes, \(2.4\leq M_w \leq 6.1\), \(1\leq d_e \leq 188\,\mathrm{km}\)) and N. E. China (Tangshan) (193 records from 64 earthquakes, \(3.5\leq M_w \leq 7.5\), \(2\leq d_e \leq 199\,\mathrm{km}\)) . Find considerable difference in estimated PGAs using the equations for the three different regions.

Ground-motion model is: \[\begin{aligned} \log y&=&C'_1+C_2M+C_4\log r+C_A S_A+C_S S_S\\ \mbox{where }r&=&\sqrt{d^2+h_0^2}\end{aligned}\] where \(y\) is in \(\,\mathrm{g}\), \(C'_1=-1.48\), \(C_2=0.266\), \(C_4=-0.922\), \(C_A=0.117\), \(C_S=0.124\), \(h_0=3.5\) and \(\sigma=0.25\).

Use four site conditions but retain three (because only three records from very soft (L) soil which combine with soft (S) soil category):

Rock: \(V_s>750\,\mathrm{m/s}\), \(\Rightarrow S_A=0, S_S=0\), 106 records.

Stiff soil: \(360<V_s\leq750\,\mathrm{m/s}\), \(\Rightarrow S_A=1, S_S=0\), 226 records.

Soft soil: \(180<V_s\leq360\,\mathrm{m/s}\), \(\Rightarrow S_A=0, S_S=1\), 81 records.

Very soft soil: \(V_s\leq180\,\mathrm{m/s}\), \(\Rightarrow S_A=0, S_S=1\), 3 records.

Lower limit of \(M_s=4.0\) because smaller earthquakes are generally not of engineering significance.

Focal depths less than \(30\,\mathrm{km}\), \(81\%\) between \(5\) and \(15\,\mathrm{km}\).

Note for some records distances have uncertainty of about \(10\,\mathrm{km}\).

Most records from distances less than about \(40\,\mathrm{km}\).

For some small events need to estimate \(M_s\) from other magnitude scales.

Most records from free-field stations although some from basements or ground floors of relatively small structures, and tunnel portals. Do not exclude records from instruments beyond cutoff distance because of limited knowledge about triggered level.

All uncorrected records plotted, checked and corrected for spurious points and baseline shifts.

Uniform correction procedure was applied for all records. For short records (\(<5\,\mathrm{s}\)) a parabolic adjustment was made, for long records (\(>10\,\mathrm{s}\)) filtering was performed with pass band \(0.20\) to \(25\,\mathrm{Hz}\) and for intermediate records both parabolic and filtering performed and the most realistic record was chosen. Instrument correction not applied due to limited knowledge of instrument characteristics.

Also analyze using one-stage method, note results comparable.

Based on N. N. Ambraseys, Simpson, and Bommer (1996), see SectionÂ 2.133.

Coefficients are: \(C'_1=-1.74\), \(C_2=0.273\), \(C_4=-0.954\), \(C_A=0.076\), \(C_S=0.058\), \(h_0=4.7\) and \(\sigma=0.26\).

Ground-motion model is: \[a_{\max}=a_1 [\exp(a_2 M_s) \exp (a_3 R)-a_4]\] where \(a_{\max}\) is in \(\,\mathrm{gal}\), \(a_1=2.8\), \(a_2=0.9\), \(a_3=-0.025\) and \(a_4=1\) (\(\sigma\) is not given).

Most records from \(r_{hypo}>20\,\mathrm{km}\).

Note that data from Turkey is limited and hence equation may be refined as amount of data increases.

Also give equation to estimate ratio of vertical PGA (\(a_v\)) to horizontal PGA (\(a_h\)): \(a_v/a_h=0.217+0.046M_s\) (\(\sigma\) is not given).

Ground-motion model is: \[\ln (A)=a+bM+d\ln(R)+qh\] where \(h\) is focal depth, \(A\) is in \(\,\mathrm{g}\), \(a=-1.47\), \(b=0.608\), \(d=-1.181\), \(q=0.0089\) and \(\sigma=0.54\).

Only use subduction earthquakes.

Do not recommend equation used for hazard analysis, since derive it only for investigating equations of Climent et al. (1994).

Ground-motion model is: \[\ln Y=a+b M+d \ln (R+c_1 \exp\{c_2 M\})+eF\] where \(Y\) is in \(\,\mathrm{g}\), for site category B: \(a=-2.342699\), \(b=1.091713\), \(c_1=0.413033\), \(c_2=0.623255\), \(d=-1.751631\), \(e=0.087940\) and \(\sigma=0.427787\) and for site category C: \(a=-2.353903\), \(b=0.838847\), \(c_1=0.305134\), \(c_2=0.640249\), \(d=-1.310188\), \(e=-0.051707\) and \(\sigma=0.416739\).

Use four site categories, \(\bar{V}_s\) is shear-wave velocity in upper \(100\,\mathrm{ft}\) (\(30\,\mathrm{m}\)):

Rock: \(\bar{V}_s \geq 2500\,\mathrm{fps}\) (\(\bar{V}_s \geq 750\,\mathrm{m/s}\)), 33 records

Soft rock or stiff soil: \(1200 \leq \bar{V}_s \leq 2500\,\mathrm{fps}\) (\(360 \leq \bar{V}_s < 750\,\mathrm{m/s}\)), 88 records

Medium stiff soil: \(600 \leq \bar{V}_s < 1200\,\mathrm{fps}\) (\(180 \leq \bar{V}_s < 360\,\mathrm{m/s}\)), 101 records

Soft clay: \(\bar{V}_s <600\,\mathrm{fps}\) (\(\bar{V}_s <180\,\mathrm{m/s}\)), 16 records

Use two source mechanisms: reverse (R): \(\Rightarrow F=1\), 81 records and strike-slip (S) \(\Rightarrow F=0\), 157 records. Most (77) reverse records from \(M_s\leq 6.7\).

Most (231) records from small building (up to 3 storeys in height) or from instrument shelters to reduce effect of soil-structure interaction. 6 records from 6 storey buildings and 1 record from a 4 storey building, included because lack of data in site or distance range of these records. Structures thought not to appreciably affect intermediate or long period and at large distances short period ground motion more greatly diminished than long period so less effect on predictions.

Exclude records from Eureka-Ferndale area in N. California because may be associated with subduction source, which is a different tectonic regime than rest of data. Also excluded Mammoth Lake records because active volcanic region, atypical of rest of California.

Include one record from Tarzana Cedar Hills although exclude a different record from this station due to possible topographic effects.

Most records between \(6\leq Ms \leq 7.25\) and \(10\leq R \leq 80\,\mathrm{km}\).

Apply weighted regression separately for site category B and C. Data space split into 4 magnitude (\(6.0\)â\(6.25\), \(6.25\)â\(6.75\), \(6.75\)â\(7.25\), \(7.25+\)) and 5 distance intervals (\(\leq 10\,\mathrm{km}\), \(10\)â\(20\,\mathrm{km}\), \(20\)â\(40\,\mathrm{km}\), \(40\)â\(80\,\mathrm{km}\), \(80\,\mathrm{km}+\)). Each recording within bin given same total weight.

So that \(Y\) is increasing function of \(M\) and decreasing function of \(R\) for all positive \(M\) and \(R\) apply constraints. Define \(g=b/d\) and \(h=-(g+c_2)\), then rewrite equation \(\ln Y=a+d \{gM+\ln [R+c_1 \exp(c_2 M)]\}+eF\) and apply constraints \(g\leq 0\), \(d \leq 0\), \(c \geq 0\), \(c_2 \geq 0\) and \(h\geq 0\).

Check plots of residuals (not shown in paper), find uniform distribution.

Find \(e\) not significantly different than \(0\) and inconsistency in results between different soil classes make it difficult to attach any significance to fault type.

Lack of records for A and D site categories. Find scale factors \(k_1=0.998638\) and \(k_2=1.200678\) so that \(Y_A=k_1 Y_B\) and \(Y_D=k_2 Y_C\), where \(Y_S\) is predicted ground motion for site class \(S\). Find no obvious dependence of \(k_1\) or \(k_2\) on acceleration from examining residuals. Find \(k_1\) and \(k_2\) not significantly different than \(1\).

Note limited data for \(R<10\,\mathrm{km}\), advise caution for this range.

Note equation developed to estimate site-amplification factors not for seismic hazard analysis.

Ground-motion model is: \[\begin{aligned} \log (Y)&=&C_1+C_2 \mathbf{M}+C_3 \mathbf{M}^2+C_4 \log(R)+C_5(R) +C_6(S)\\ R&=&\sqrt{d^2+h_0^2}\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), for \(\mathbf{M}>1.5\) using acceleration and velocity records, for horizontal PGA \(C_1=-4.2318\), \(C_2=1.1962\), \(C_3=-0.0651\), \(C_4=-1\), \(C_5=-0.0019\), \(C_6=0.261\), \(h_0=2.9\) and \(\sigma=0.432\) and for vertical PGA \(C_1=-4.1800\), \(C_2=1.0189\), \(C_3=-0.0404\), \(C_4=-1\), \(C_5=-0.0019\), \(C_6=0.163\), \(h_0=2.7\) and \(\sigma=0.415\).

Use two site categories:

Rock, H: 470 records, V: 395 records.

Soil, H: 88 records, V: 83 records.

Note that not most accurate approach but due to lack of site information consider this technique makes most consistent use of available information.

Select data using these criteria:

Epicentre and recording station must be within the stable continental region boundaries defined by Johnston and others (1994) because a) such regions form end of spectrum of regions described by âintraplateâ and hence allows differences with interplate regions to be seen, b) they are clearly delineated regions and c) intraplate oceanic crust is excluded.

Minimum magnitude level \(\mathbf{M}=1.5\).

Use records from dam abutments and downstream free-field sites but excludes records from crests, slopes, toes, galleries, or basements.

Use records from acceleration and velocity instruments.

Specify no minimum PGA.

Specify no maximum source distance. Do not exclude records from distances greater than shortest distance to a non-triggered station.

Data from Australia, N.W. Europe, Peninsular India and E. N. America.

Focal depths, \(2\leq h \leq 28\,\mathrm{km}\).

Most records from \(\mathbf{M}<4.0\).

Visually inspect all records including integrated velocities and displacements, identify and remove traces dominated by noise, identify and correct transient errors (spikes, ramps, linear sections, back time steps and clipped peaks), identify scaling errors, identify and remove multiple event records. Linear baseline correct and elliptically filter with cut-off \(0.25\) to \(0.5\,\mathrm{Hz}\) (determine frequency by visual inspection of adjusted record) and \(33\) to \(100\,\mathrm{Hz}\) (generally pre-determined by Nyquist frequency).

Large proportion of records from velocity time histories which differentiate to acceleration. Test time domain method (central difference technique) and frequency domain method; find very similar results. Use time domain method.

Distribution with respect to magnitude did not allow two-stage regression technique.

In many analyses distribution of data with respect to distance did not allow simultaneous determination of coefficients \(C_4\) and \(C_5\), for these cases constrain \(C_4\) to \(-1\).

Test effect of minimum magnitude cut-off for two cut-offs \(\mathbf{M}=1.5\) and \(\mathbf{M}=3.5\). Find if include data from \(\mathbf{M}<3.5\) then there is substantial over prediction of amplitudes for \(d<10\,\mathrm{km}\) for large magnitudes unless include \(C_3\) term. \(C_3\) effectively accounts for large number of records from small magnitudes and so predictions using the different magnitude cut-offs are very similar over broad range of \(\mathbf{M}\) and \(d\).

Try including focal depth, \(h\), explicitly by replacing \(h_0\) with \(h\) because \(h_0\) determined for whole set (which is dominated by small shocks at shallow depths) may not be appropriate for large earthquakes. Find improved fit at small distances but it does not result in overall improvement in fit (\(\sigma\) increases); this increase thought due to large errors in focal depth determination.

Find larger standard deviations than those found in previous studies which note may be due to intrinsic differences between regional subsets within whole set. Repeat analysis separately for Australia (for horizontal and vertical), N. America (for horizontal and vertical) and N.W. Europe (horizontal); find reduced standard deviations (although still large), \(C_5\) varies significantly between 3 regions.

Repeat analysis excluding velocity records.

Also repeat analysis using only rock records.

Ground-motion model is: \[\log \mathrm{PGA}=aM+b\log R+c\] where \(\mathrm{PGA}\) is in an unknown unit but it is probably in \(\,\mathrm{gal}\), \(a=0.65\), \(b=-0.9\) and \(c=-0.44\) (\(\sigma\) not reported).

Ground-motion model is: \[\log S(T)=a(T) M-\log X_{eq} -b(T) X_{eq} +c(T)+q \Delta{s(T)}\] where \(S(0.02)\) is in \(\,\mathrm{gal}\), \(a(0.02)=0.318\), \(b(0.02)=0.00164\) and \(c(0.02)=1.597\) (\(\Delta{s(0.02)}\) and \(\sigma\) only given in graphs).

Use two site conditions:

Pre-Quaternary: Rock (sandstone, siltstone, shale, granite, mudstone, etc.); thickness of surface soil overlying rock is less than \(10\,\mathrm{m}\); shallow soil or thin alluvium, 160 records. S-wave velocities \(>600\,\mathrm{m/s}\).

Quaternary: Soil (alluvium, clay, sand, silt, loam, gravel, etc.), 336 records. S-wave velocities \(\leq 600\,\mathrm{m/s}\).

Exclude records from very soft soil such as bay mud or artificial fill because few such records and ground motions may be strongly affected by soil nonlinearity.

Use equivalent hypocentral distance, \(X_{eq}\), because strong motion in near-source region affected from points other than nearest point on fault plane.

Use portion of record after initial S-wave arrival.

Approximates PGA by spectral acceleration for period of \(0.02\,\mathrm{s}\) and \(5\%\) damping.

Plot the amplitude factors from first stage against \(M_w\); find well represented by linear function.

Ground-motion model is: \[\log \mathrm{PHA} = a_1+a_2M_w-\log (d^2+h^2)^{1/2} + a_3 S\] where \(\mathrm{PHA}\) is in \(\,\mathrm{g}\), \(a_1=-1.870 \pm 0.182\), \(a_2=0.366 \pm 0.032\), \(a_3=0.168 \pm 0.045\), \(h=6\,\mathrm{km}\) and \(\sigma=0.173\) for \(r_{jb}\) and \(a_1=-2.238 \pm 0.200\), \(a_2=0.438 \pm 0.035\), \(a_3=0.195 \pm 0.049\), \(h=5\,\mathrm{km}\) and \(\sigma=0.190\) for \(r_{epi}\).

Use two site categories:

Rock or stiff soils and deep alluvium.

All other sites.

Use data and functional form of Sabetta and Pugliese (1987) but use \(M_w\) instead of magnitudes used by Sabetta and Pugliese (1987).

Ground-motion model is: \[\begin{aligned} \log(A_p/\mathrm{g})&=&b_1+b_2M_s+b_3 \log r+b_4 r\\ \mbox{where } r&=&(d^2+h_0^2)^{0.5}\end{aligned}\] where \(A_p\) is in \(\,\mathrm{g}\), using both horizontal components \(b_1=-1.617\), \(b_2=0.248\), \(b_3=-0.5402\), \(b_4=-0.00392\), \(h_0=3.2\) and \(\sigma=0.26\) and for larger horizontal component \(b_1=-1.507\), \(b_2=0.240\), \(b_3=-0.542\), \(b_4=-0.00397\), \(h_0=3.0\) and \(\sigma=0.26\).

Consider two soil categories but do not model:

Rock

Soil

Classify sites without regard to depth and shear-wave velocity of deposits.

Most records from W. USA but many from Europe and Middle East.

Focal depths between \(2\) and \(29\,\mathrm{km}\).

Records from instruments on ground floor or in basements of buildings and structures up to 3 storeys and at free-field sites, regardless of topography.

Records baseline corrected and low-pass filtered using elliptic filter.

Ground-motion model is: \[\log_{10} \mathrm{AGM}=b_1+0.31M-b_3 \log R\] where \(\mathrm{AGM}\) is in \(\,\mathrm{cm/s^2}\), \(b_1=1.14\) and \(b_3=0.615\) (\(\sigma\) is not given). Note there are typographical errors in the abstract.

Data from three earthquakes with \(m_b=5.7\), one with \(m_b=5.8\) and one with \(m_b=7.2\).

Adopt magnitude scaling coefficient (\(0.31\)) from Boore (1983).

Ground-motion model is: \[\begin{aligned} \log_{10}Y&=&b_1+b_2(M-6)+b_3(M-6)^2+b_4R+b_5\log_{10}R+b_6\Gamma\\ \mbox{where }R&=&\sqrt{r_{jb}^2+h^2}\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(b_1=0.156\), \(b_2=0.229\), \(b_3=0\), \(b_4=0\), \(b_5=-0.945\), \(b_6=0.077\), \(h=5.57\), \(\sigma=\sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2}\) where \(\sigma_1=0.216\), \(\sigma_2=0\), for randomly orientated component \(\sigma_3=0.094\) and for geometric mean \(\sigma_3=0\).

Use two site categories (following classification of Joyner and Boore (1981)):

Rock: 35 records

Soil: 93 records

Applicable for extensional regimes, i.e. those regions where lithosphere is expanding areally.

Reject records from structures of more than two storeys or from deeply embedded basements or those which triggered on S wave.

Include records from those instruments beyond cutoff distance, i.e. beyond first instrument which did not trigger.

Correction technique based on uniform correction and processing. Determine passband for filtering based on visual inspection of Fourier amplitude spectra and doubly-integrated displacements. Apply instrument correction.

Not enough data to be able to find all coefficients so use \(b_2\) and \(b_3\) from Boore, Joyner, and Fumal (1994a)

Note that should only be used in distance range \(0\) to \(70\,\mathrm{km}\) because further away ground motions tend to be over predicted.

Ground-motion model is: \[\begin{aligned} \mathrm{Acc}&=&\exp(b)\exp(b_M) (R_h+C)^{b_R}\\ \mbox{where } R_h^2&=&(R_e/\rho)^2+h^2\\ \mbox{and } \rho&=&\sqrt{\frac{1+t g^2 \alpha}{a^{-2}+t g^2 \alpha}}\end{aligned}\] where \(\mathrm{Acc}\) is in \(\,\mathrm{cm/s^2}\), \(\alpha\) is the azimuth of the site with respect to energy propagation pattern, \(b=3.49556\), \(b_M=1.35431\), \(C=30\), \(b_R=-1.58527\), \(a=1.2\) and \(\sigma=0.48884\) (definitions of \(t\) and \(g\) are not given).

Correct PGAs for local site effects so that PGAs used correspond to a site with a shear-wave velocity of \(700\,\mathrm{m/s}\). Do not state how this is performed.

Most records from SMA-1s.

Not all records from free-field.

Records from strong intermediate depth earthquakes in Vrancea region.

Focal depths, \(89.1 \leq h \leq 131\,\mathrm{km}\).

For each of the four earthquakes, calculate coefficents in equation \(\ln \mathrm{Acc}=b_0+b_1 \ln (R_e/\rho)\), the main direction of energy propagation and the relation between the semi-axes of the ellipse in two orthogonal directions (\(a:b\)).

Also calculate coefficents in equation \(\ln \mathrm{Acc}=b+b_M M+ b_R \ln (R_h+C)\) for different azimuth by normalising the values of \(R_e/\rho\) by the azimuth. Give coefficients for Bucharest, Valeni and Cerna Voda.

Note that uncertainty is high and suggest this is because of distribution of data with respect to \(M\), \(R_e\) and \(h\), the use of data processed in different ways, soil-structure interaction and the use of an approximate correction method for local site effects.

Ground-motion model is: \[\log A_p=a_1 M+a_2 R+a_3 \log R+a_4\] where \(A_p\) is in \(\,\mathrm{gal}\), \(a_1=0.329\), \(a_2=-0.00327\), \(a_3=-0.792\) and \(a_4=1.177\) (\(\sigma\) is not known).

Ground-motion model (horizontal component) is: \[\begin{aligned} \ln A_H&=&a_1+a_2 M+a_3 \ln \sqrt{R_{\mathrm{SEIS}}^2+[a_4 \exp(a_5 M)]^2}\\ &&{}+[a_6+a_7\ln R_{\mathrm{SEIS}}+a_8M]F+[a_9+a_{10}\ln R_{\mathrm{SEIS}}]S_{\mathrm{SR}}\\ &&{}+[a_{11}+a_{12}\ln R_{\mathrm{SEIS}}]S_{\mathrm{HR}}+f_A(D)\\ f_A(D)&=&\left\{ \begin{array}{r@{\quad \mbox{for}\quad}l} 0&D \geq 1\,\mathrm{km}\\ \{[a_{11}+a_{12}\ln (R_{\mathrm{SEIS}})]-[a_9+a_{10} \ln (R_{\mathrm{SEIS}})]S_{\mathrm{SR}}\}(1-D)(1-S_{\mathrm{HR}})&D<1\,\mathrm{km}\\ \end{array} \right.\end{aligned}\] where \(A_H\) is in \(\,\mathrm{g}\), \(a_1=-3.512\), \(a_2=0.904\), \(a_3=-1.328\), \(a_4=0.149\), \(a_5=0.647\), \(a_6=1.125\), \(a_7=-0.112\), \(a_8=-0.0957\), \(a_9=0.440\), \(a_{10}=-0.171\), \(a_{11}=0.405\), \(a_{12}=-0.222\), \(\sigma=0.55\) for \(A_H<0.068\,\mathrm{g}\), \(\sigma=0.173-0.140\ln (A_H)\) for \(0.068\,\mathrm{g}\leq A_H \leq 0.21\,\mathrm{g}\) and \(\sigma=0.39\) for \(A_H>0.21\,\mathrm{g}\) (when expressed in terms of acceleration) and \(\sigma=0.889-0.0691M\) for \(M<7.4\) and \(\sigma=0.38\) for \(M \geq 7.4\) (when expressed in terms of magnitude).

Ground-motion model (vertical component) is: \[\begin{aligned} \ln A_V&=&\ln A_H + b_1+b_2 M + b_3 \ln [R_{\mathrm{SEIS}}+b_4 \exp (b_5 M)]\\ &&{}+b_6 \ln[R_{\mathrm{SEIS}}+b_7 \exp(b_8 M)]+b_9 F\end{aligned}\] where \(A_V\) is in \(\,\mathrm{g}\), \(b_1=-1.58\), \(b_2=-0.10\), \(b_3=-1.5\), \(b_4=0.079\), \(b_5=0.661\), \(b_6=1.89\), \(b_7=0.361\), \(b_8=0.576\), \(b_9=-0.11\) and \(\sigma_V=\sqrt{\sigma^2+0.36^2}\) (where \(\sigma\) is standard deviation for horizontal PGA prediction).

Uses three site categories:

A

Hard rock: primarily Cretaceous and older sedimentary deposits, metamorphic rock, crystalline rock and hard volcanic deposits (e.g. basalt).

Soft rock: primarily Tertiary sedimentary deposits and soft volcanic deposits (e.g. ash deposits).

Alluvium or firm soil: firm or stiff Quaternary deposits with depths greater than \(10\,\mathrm{m}\).

Also includes sediment depth (\(D\)) as a variable.

Restricts to near-source distances to minimize influence of regional differences in crustal attenuation and to avoid complex propagation effects that have been observed at longer distances.

Excludes recordings from basement of buildings greater than two storeys on soil and soft rock, greater than five storeys on hard rock, toe and base of dams and base of bridge columns. Excludes recordings from shallow and soft soil because previous analyses showed such sites have accelerations significantly higher than those on deep, firm alluvium. Include records from dam abutments because comprise a significant number of rock recordings and due to stiff foundations are expected to be only minimally affected by dam. Some of these could be strongly affected by local topography.

Includes earthquakes only if they had seismogenic rupture within shallow crust (depths less than about \(25\,\mathrm{km}\)). Includes several large, shallow subduction interface earthquakes because previous studies found similar near-source ground motions to shallow crustal earthquakes.

Includes only earthquakes with \(M\) about \(5\) or larger to emphasize those ground motions of greatest engineering interest and limit analysis to more reliable, well-studied earthquakes.

Notes that distance to seismogenic rupture is a better measure than distance to rupture or distance to surface projection because top layer of crust is non-seismogenic and will not contribute to ground motion. Give estimates for average depth to top of seismogenic rupture for hypothetical earthquakes.

Considers different focal mechanisms: reverse (H:6, V:5), thrust (H:9, V:6), reverse-oblique (H:4, V:2) and thrust-oblique (0), total (H:19, V:13) \(\Rightarrow F=1\) (H:278 records, V:116 records) (reverse have a dip angle greater than or equal to \(45^{\circ}\)), strike-slip (H:27, V:13) \(\Rightarrow F=0\) (H:367 records, V:109 records) (strike-slip have an absolute value of rake less than or equal to \(22.5^{\circ}\) from the horizontal as measured along fault plane). There is only one normal faulting earthquakes in set of records (contributing four horizontal records) so difference is not modelled although \(F=0.5\) given as first approximation (later revised to \(F=0\)).

Mostly W. USA with 20 records from Nicaragua(1) Mexico (5), Iran (8), Uzbekistan (1), Chile (3), Armenia (1) and Turkey (1).

Does regression firstly with all data. Selects distance threshold for each value of magnitude, style of faulting and local site condition such that the 16th percentile estimate of \(A_H\) was equal to \(0.02\,\mathrm{g}\) (which corresponds to a vertical trigger of about \(0.01\,\mathrm{g}\)). Repeats regression repeated only with those records within these distance thresholds. Avoids bias due to non-triggering instruments.

Finds dispersion (uncertainty) to be dependent on magnitude and PGA, models as linear functions. Finds better fit for PGA dependency.

Ground-motion model is: \[\begin{aligned} \log_{10} \mathrm{PGA}&=&b_0+b_1(M-6)+b_2r-\log_{10} r+b_4S\\ \mbox{where }r&=&\sqrt{d^2+h^2}\end{aligned}\] \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), \(b_0=0.518\), \(b_1=0.387\), \(b_2=-0.00256\), \(b_4=0.335\), \(h=11.29\) and \(\sigma=0.237\).

Use two site categories:

Lava: 38 records

Ash: \(60 \lesssim V_s \lesssim 200\,\mathrm{m/s}\), 13 records

Depths between \(4\) and \(14\,\mathrm{km}\) with average \(9.6\,\mathrm{km}\) (standard deviation \(2.3\,\mathrm{km}\)). Limit of \(15\,\mathrm{km}\) chosen to differentiate between large tectonic earthquakes and deeper mantle events.

Attenuation greater than for western USA due to highly fractured volcanic pile.

Peak acceleration measured directly from accelerograms. Check against one from corrected records, small difference.

Excludes records triggered on S-wave and those beyond cutoff distance (the distance to first nontriggered instrument).

Does weighted and unweighted least squares analysis; find some differences.

Ground-motion model is: \[\begin{aligned} \log y&=&\alpha +\beta \mathbf{M}-\log r +b r\\ \mbox{where }r&=&(d^2+h^2)^{1/2}\end{aligned}\] Coefficients are unknown.

Also develop model using functional form of Molas and Yamazaki (1995).

All data from rock sites.

Data from seismographs of New Zealand National Seismograph Network and temporary deployments on East Cape of the North Island, the Marlborough region of the South Island and the central volcanic zone of the North Island.

Most data from more than \(100\,\mathrm{km}\) from the source.

Ground-motion model is: \[\begin{aligned} \log_{10} a&=&\alpha+\beta M-\log_{10} r+\gamma r\\ \mbox{where } r&=&(d^2+h^2)^{1/2}\end{aligned}\] where \(a\) is in \(\,\mathrm{g}\), \(\alpha=-1.237 \pm 0.254\), \(\beta=0.278 \pm 0.043\), \(\gamma=-0.00220 \pm 0.00042\), \(h=6.565 \pm 0.547\), \(\tau^2=0.00645 \pm 0.00382\) and \(\sigma^2=0.0527 \pm 0.00525\) (where \(\tau^2\) is the inter-earthquake variance and \(\sigma^2\) is the intra-earthquake variance and \(\pm\) signifies the standard error of the estimate.

Notes that errors in magnitude determination are one element that contributes to the between-earthquake component of variance and could thus cause apparent differences between earthquakes, even if none existed.

Develops a method to explicitly include consideration of magnitude uncertainties in a random earthquake effects model so that the between-earthquake component of variance can be split into the part that is due only to magnitude uncertainty (and is therefore of no physical consequence) and the part for which a physical explanation may be sought.

Applies method to data of Joyner and Boore (1981). Assume two classes of magnitude estimates: those with estimates of \(M_w\), which assumes to be associated with a standard error of \(0.1\), and those for which \(M_L\) was used as a surrogate for \(M_w\), which assumes to be associated with a standard error of \(0.3\). Find that the inter-earthquake variance is much lower than that computed assuming that the magnitudes are exact but that other coefficients are similar. Believes that the high inter-earthquake variance derived using the exact magnitudes model is largely explained by the large uncertainties in the magnitude estimates using \(M_L\).

Ground-motion model is: \[\begin{aligned} \ln A&=&c_1+c_2 M+c_3 \ln r+c_4 r+c_5 S_1 +c_6 S_2\\ \mbox{where }r&=&\sqrt{R^2+6^2}\end{aligned}\] where \(A\) is in \(\,\mathrm{m/s^2}\), \(c_1=-1.589\), \(c_2=0.561\), \(c_3=-0.569\), \(c_4=-0.003\), \(c_5=0.173\), \(c_6=0.279\) and \(\sigma=0.80\) (for all earthquakes), \(c_1=-1.725\), \(c_2=0.687\), \(c_3=-0.742\), \(c_4=-0.003\), \(c_5=0.173\), \(c_6=0.279\) and \(\sigma=0.83\) (for shallow crustal earthquakes) and \(c_1=-0.915\), \(c_2=0.543\), \(c_3=-0.692\), \(c_4=-0.003\), \(c_5=0.173\), \(c_6=0.279\) and \(\sigma=0.74\) (for subduction zone earthquakes).

Use three site categories:

Rock, 54 records.

Hard soil, 63 records.

Soft soil, 83 records.

Most records from SMA-1s with 6 records from SSA-2.

Use PSA at \(40\,\mathrm{Hz}\) (\(0.025\,\mathrm{s}\)) as peak ground acceleration.

Records instrument corrected and bandpass filtered with cut-offs of \(0.2\) and \(20\,\mathrm{Hz}\).

Use data from shallow crustal earthquakes (133 records) and subduction zone earthquakes (67 records).

Perform regression on combined shallow crustal and subduction zone records, on just the shallow crustal records using \(r_{hypo}\) and using \(r_{epi}\) and on just subduction zone records.

Note that distribution w.r.t. distance improves in the near field when epicentral distance is used but only possible to use \(r_{epi}\) for shallow crustal earthquakes because for subduction zone earthquakes hypocentral distance is much greater than epicentral distance so should use \(r_{hypo}\) instead.

For \(4 \leq M \leq 6\) distribution w.r.t. epicentral distance is quite good but for \(M>6\) no records from \(d_e<40\,\mathrm{km}\).

Use a two step procedure. Firstly use entire set and both horizontal components and compute two soil terms (one for hard and one for soft soil). In second step use soil terms to correct motions for rock conditions and then repeat regression.

Use Bayesian analysis (Ordaz, Singh, and Arciniega 1994) so that derived coefficients comply with physics of wave propagation because include

*a priori*information on the coefficients to avoid physically unrealistic values. Choose initial values of coefficients based on theory and previous resultsCannot find coefficient in \(r\) by regression so adopt \(6\,\mathrm{km}\) from previous study.

Examine residuals w.r.t. distance and magnitude and find no trends.

Ground-motion model for soil is: \[\begin{aligned} \ln \mathrm{PGA}&=&C_1^*+C_2 \mathbf{M} +C_3^* \ln \left[ r_{\mathrm{rup}}+\mathrm{e}^{C_4^*-\frac{C_2}{C_3^*} \mathbf{M}}\right] +C_5 Z_t+C_9H+C_{10}Z_{ss}\\ \mbox{with: }C_1^*&=&C_1+C_6Z_r\\ C_3^*&=&C_3+C_7Z_r\\ C_4^*&=&C_4+C_8Z_r\end{aligned}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), \(C_1=-0.6687\), \(C_2=1.438\), \(C_3=-2.329\), \(C_4=\ln(1.097)\), \(C_5=0.3643\), \(C_9=0.00648\) and \(\sigma=1.45-0.1\mathbf{M}\) (other coefficients in equation not needed for prediction on deep soil and are not given in paper).

Ground-motion model for rock is: \[\begin{aligned} \ln \mathrm{PGA}&=&C_1^*+C_2 \mathbf{M} +C_3^* \ln \left[ r_{\mathrm{rup}}+\mathrm{e}^{C_4^*-\frac{C_2}{C_3^*}\mathbf{M}}\right] +C_5 Z_{ss}+C_8Z_t+C_9H\\ \mbox{with: }C_1^*&=&C_1+C_3C_4-C_3^*C_4^*\\ C_3^*&=&C_3+C_6Z_{ss}\\ C_4^*&=&C_4+C_7Z_{ss}\end{aligned}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), \(C_1=0.2418\), \(C_2=1.414\), \(C_3=-2.552\), \(C_4=\ln(1.7818)\), \(C_8=0.3846\), \(C_9=0.00607\) and \(\sigma=1.45-0.1\mathbf{M}\) (other coefficients in equation not needed for prediction on rock and are not given in paper).

Use different models to force rock and soil accelerations to same level in near field.

Use three site categories to do regression but only report results for rock and deep soil:

A

Rock: Consists of at most about a metre of soil over weathered rock, 96 records.

Deep soil: Depth to bedrock is greater than \(20\,\mathrm{m}\), 284 records.

Shallow soil: Depth to bedrock is less than \(20\,\mathrm{m}\) and a significant velocity contrast may exist within \(30\,\mathrm{m}\) of surface, 96 records.

Use free-field recordings, i.e. instruments in basement or ground-floor of buildings less than four storeys in height. Data excluded if quality of time history poor or if portion of main shaking not recorded.

Consider tectonic type: interface (assumed to be thrust) (98 records) \(\Rightarrow Z_t=0\), intraslab (assumed to be normal) (66 records) \(\Rightarrow Z_t=1\)

Focal depths, \(H\), between \(10\) and \(229\,\mathrm{km}\)

Not enough data to perform individual regression on each subset so do joint regression analysis.

Both effect of depth and tectonic type significant.

Large differences between rock and deep soil.

Note differences between shallow crustal and interface earthquake primarily for very large earthquakes.

Assume uncertainty to be linear function of magnitude.

Ground-motion model (Model 1) is: \[\log_{10} \mathrm{PGA}= A_1 M_w + A_2 \log_{10} \sqrt{r^2+d^2} + A_3 h_c+A_4+A_5 \delta_R+A_6 \delta_A+A_7 \delta_I\] where \(\mathrm{PGA}\) is in \(\,\mathrm{m/s^2}\), \(\delta_R=1\) for crustal reverse \(0\) otherwise, \(\delta_A=1\) for rock \(0\) otherwise, \(\delta_I=1\) for interface \(0\) otherwise, \(A_1=0.298\), \(A_2=-1.56\), \(A_3=0.00619\), \(A_4=-0.365\), \(A_5=0.107\), \(A_6=-0.186\), \(A_7=-0.124\), \(d=19\) and \(\sigma=0.230\).

Models also given for soil sites only (Model 2), unspecified site (Model 3), focal mechanism and tectonic type unknown (Model 4) and only magnitude, depth and distance known (Model 5)

Records from ground or base of buildings. 33 from buildings with more than 3 storeys; find no significant differences.

Retain two site categories:

Rock: Topographic effects expected, very thin soil layer (\(\leq 3\,\mathrm{m}\)) overlying rock or rock outcrop.

Soil: everything else

Use depth to centroid of rupture, \(h_c\), \(4\leq h_c \leq 149\). Only nine are deeper than \(50\,\mathrm{km}\). Exclude records from deep events which travelled through mantle.

Consider tectonic type: C=crustal (24+17 records), I=interface (7+0 records) and S=slab (20+0 records)

Consider source mechanism: N=normal (15+1 records), R=reverse (22+5 records) and S=strike-slip (12+11 records). Classify mixed mechanisms by ratio of components \(\geq 1.0\).

For only five records difference between the distance to rupture surface and the distance to centroid could be more than 10%.

66 foreign near-source records (\(d_r \leq 10\,\mathrm{km}\)) from 17 crustal earthquakes supplement NZ data. Mainly from western North America including 17 from Imperial Valley and 12 from Northridge.

Exclude one stationâs records (Atene A) due to possible topographical effects.

Exclude records which could have been affected by different attenuation properties in the volcanic region.

Note regional difference between Fiordland and volcanic region and rest of country but do model.

Retain coefficients if significant at \(\alpha=0.05\).

Anelastic term not significant.

Ground-motion model is: \[\begin{aligned} \ln \mathrm{PGA}&=&a_1+a_2M+a_3\ln R+a_4R\\ \mbox{where } \quad R&=&\sqrt{R_{\mathrm{epi}}^2+a_5^2}\end{aligned}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{cm/s^2}\), \(a_1=0.4\), \(a_2=1.2\), \(a_3=-0.76\), \(a_4=-0.0094\) and \(a_5=10\) (\(\sigma\) not given).

This article has not been seen. The model presented may not be a fully empirical model.

Ground-motion model is: \[A=c \exp(\alpha M)[R^k+a]^{-\beta-\gamma R}\]

Coefficients not given, only predictions.

Ground-motion model is: \[\log(A)=a+bM+c\log(r)\] where \(A\) is in \(\,\mathrm{g}\), \(a=-1.879\), \(b=0.431\) and \(c=-1.908\) (for vertical components) and \(a=-2.114\), \(b=0.480\) and \(c=-1.693\) (for horizontal components).

All records from digital instruments.

Try including a term \(d\log(M)\) but tests show that \(d\) is negligible with respect to \(a\), \(b\) and \(c\).

Ground-motion model is: \[\begin{aligned} \log (A)&=&c_1+c_2 M+c_3 \log(D)+c_4 D+c_5 S\\ D&=&(R^2+d_0^2)^{1/2}\end{aligned}\] where \(A\) is in \(\,\mathrm{g}\), \(c_1=-1.664\), \(c_2=0.333\), \(c_3=-1.093\), \(c_4=0\), \(c_5=0.236\), \(d_0=6.6\) and \(\sigma=0.254\).

Uses four site categories (following N. N. Ambraseys, Simpson, and Bommer (1996)) but only two have data within them:

Rock (R): \(v_s>750\,\mathrm{m/s}\), 92 records.

Stiff soil (A): \(360< v_s \leq 750\,\mathrm{m/s}\), 184 records.

where \(v_s\) is average shear-wave velocity in upper \(30\,\mathrm{m}\).

Uses both horizontal components to get a more reliable set of data.

Tries using \(M_L\) rather than \(M_s\), epicentral distance rather than hypocentral distance and constraining anelastic decay coefficient, \(c_4\), to zero. Chooses combination which gives minimum \(\sigma\).

Ground-motion model is: \[\ln \mathrm{Sa}=\alpha_1+\alpha_2(M-6)+\alpha_3(M-6)^2+\alpha_4\ln R+\alpha_5R\] where \(\mathrm{Sa}\) is in \(\,\mathrm{cm/s^2}\), \(\alpha_1=5.8929\), \(\alpha_2=1.2457\), \(\alpha_3=-9.7565\times 10^{-2}\), \(\alpha_4=-0.50\), \(\alpha_5=-6.3159\times 10^{-3}\) and \(\sigma=0.420\).

Use data from one station, University City (CU) in Mexico City, a relatively firm site.

Ground-motion model is: \[\ln Y=C_{14}+C_{22}M+C_{31}\ln(R+15)+C_{43}S+C_{54}F\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(C_{14}=5.57\), \(C_{22}=0.82\), \(C_{31}=-1.59\), \(C_{43}=-0.14\), \(C_{54}=-0.18\) and \(\sigma=0.68\). Assume \(15\,\mathrm{km}\) inside \(\ln (R+\ldots)\) from Theodulidis and Papazachos (1992).

Use two site categories:

Rock: includes stiff sites.

Alluvium: includes both shallow and deep soil sites.

Use two source mechanism categories:

Thrust and strike-slip earthquakes.

Normal earthquakes.

Use epicentral distance because in Italy and Greece the surface geology does not show any evident faulting, consequently it is impossible to use a fault distance definition.

Good distribution and coverage of data with respect to site category and source mechanism.

Consider six strong-motion records (three Italian and three Greek) with different associated distances, magnitudes and record length and apply the different processing techniques of ENEA-ENEL and ITSAK to check if data from two databanks can be merged. Digitise six records using same equipment. ITSAK technique: subtract the reference trace (either fixed trace or trace from clock) from uncorrected accelerogram and select band-pass filter based on either Fourier amplitude spectra of acceleration components or selected using a different technique. ENEA-ENEL technique: subtract the reference trace from uncorrected accelerogram and select band-pass filter by comparing Fourier amplitude spectra of acceleration components with that of fixed trace. Find small differences in PGA, PGV, PGD so can merge Italian and Greek data into one databank.

Use four step regression procedure, similar to that Theodulidis and Papazachos (1992) use. First step use only data with \(M\geq 6.0\) (\(7 \leq R \leq 138\,\mathrm{km}\)) for which distances are more accurate to find geometrical coefficient \(C_{31}\). Next find constant (\(C_{12}\)) and magnitude (\(C_{22}\)) coefficients using all data. Next find constant (\(C_{13}\)) and soil (\(C_{43}\)) coefficients using all data. Finally find constant (\(C_{14}\)) and source mechanism (\(C_{54}\)) coefficients using data with \(M \geq 6.0\) for which focal mechanism is better constrained; final coefficients are \(C_{14}\), \(C_{22}\), \(C_{31}\), \(C_{43}\) and \(C_{54}\). Investigate influence of distance on \(C_{54}\) by subdividing data in final step into three categories with respect to distance (\(7\leq R \leq 140\,\mathrm{km}\), \(7 \leq R \leq100\,\mathrm{km}\) and \(7\leq R \leq 70\,\mathrm{km}\)).

Equation intended as first attempt to obtain attenuation relations from combined databanks and site characteristics and fault rupture properties could and should be taken into account.

Based on Sadigh et al. (1997), see SectionÂ 2.87.

Ground-motion model is: \[\ln \mathrm{PGA}= C_1+C_2 M+C_3 \ln [r_{\mathrm{rup}} + \exp(C_4+C_5M)]\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), for \(M<6.5\) \(C_4=1.29649\) and \(C_5=0.25\) and for \(M\geq 6.5\) \(C_4=-0.48451\) and \(C_5=0.524\). For rock sites: \(C_3=-2.100\), for strike-slip mechanism and \(M<6.5\) \(C_1=-0.949\) and \(C_2=1.05\), for strike-slip mechanism and \(M\geq 6.5\) \(C_1=-1.274\) and \(C_2=1.10\), for reverse-slip and \(M<6.5\) \(C_1=0.276\) and \(C_2=0.90\) and for reverse-slip and \(M\geq 6.5\) \(C_1=-1.024\) and \(C_2=1.10\). For soil sites: \(C_3=-1.75\), for strike-slip mechanism and \(M<6.5\) \(C_1=-1.1100\) and \(C_2=0.875\), for strike-slip mechanism and \(M\geq 6.5\) \(C_1=-1.3830\) and \(C_2=0.917\), for reverse-slip mechanism and \(M<6.5\) \(C_1=-0.0895\) and \(C_2=0.750\) and for reverse-slip mechanism and \(M\geq 6.5\) \(C_1=-1.175\) and \(C_2=0.917\) (\(\sigma\) not given).

Use two site categories:

Rock: bedrock within about a metre of surface. Note that many such sites are soft rock with \(V_s \leq 750\,\mathrm{m/s}\) and a strong velocity gradient because of near-surface weathering and fracturing, 274 records.

Deep soil: greater than \(20\,\mathrm{m}\) of soil over bedrock. Exclude data from very soft soil sites such as those from San Francisco bay mud, 690 records.

Define crustal earthquakes as those that occur on faults within upper \(20\) to \(25\,\mathrm{km}\) of continental crust.

Consider source mechanism: RV=reverse (26+2) and SS=strike-slip (and some normal) (89+0). Classified as RV if rake\(>45^{\circ}\) and SS if rake\(<45^{\circ}\). Find peak motions from small number of normal faulting earthquakes not to be significantly different than peak motions from strike-slip events so include in SS category.

Separate equations for \(M_w <6.5\) and \(M_w \geq 6.5\) to account for near-field saturation effects, for rock and deep soil sites and reverse and strike-slip earthquakes.

Records from instruments in instrument shelters near ground surface or in ground floor of small, light structures.

4 foreign records (1 from Gazli and 3 from Tabas) supplement Californian records.

Ground-motion model is: \[\log (a_p/g)=C_1+C_2M_s+C_3d+C_4 \log d\] where \(a_p\) is in \(\,\mathrm{g}\), for soil sites \(C_1=-1.86\), \(C_2=0.23\), \(C_3=-0.0062\), \(C_4=-0.230\) and \(\sigma=0.28\) and for rock sites \(C_1=-1.874\), \(C_2=0.299\), \(C_3=-0.0029\), \(C_4=-0.648\) and \(\sigma=0.33\).

Use two site categories because of limited available information (based on nature of top layer of site regardless of thickness) for which derive separate equations:

Soil

Rock

Use record from free-field or in basements of buildings \(\leq 3\) storeys high.

Use \(M_s\) because better represents size of shallow earthquakes and is determined from teleseismic readings with much smaller standard errors than other magnitude scales and also saturates at higher magnitudes than all other magnitude scales except \(M_w\) which is only available for relatively small portion of earthquakes. For some small earthquakes convert to \(M_s\) from other magnitude scales.

For very short records, \(\leq 5\,\mathrm{s}\) long, correct using parabolic baseline, for records \(> 10\,\mathrm{s}\) long correct using elliptical filter and for records between \(5\) and \(10\,\mathrm{s}\) long both parabolic correction and filtering applied and select best one from appearance of adjusted time histories.

Equations not any more precise than other attenuation relations but are simply included for completeness and for a comparison of effects of dataset used with other dataset. Data did not allow distinction between different source mechanisms.

Ground-motion model is: \[\log A=c_1+c_2 M-b\log (X+\mathrm{e}^{c_3 M})\] where \(A\) is in \(\,\mathrm{g}\), \(c_1=-1.072\), \(c_2=0.3903\), \(b=1.21\), \(c_3=0.5873\) and \(\sigma=0.14\).

Considers two site categories but does not model:

Rock: generally granite/quartzite/sandstone, 41 records.

Soil: exposed soil covers on basement, 25 records.

Focal depths between \(7.0\) and \(50.0\,\mathrm{km}\).

Most records from distances \(>50\,\mathrm{km}\). Correlation coefficient between \(M\) and \(X\) is \(0.63\).

Does not include source mechanism as parameter because not well defined and including many terms may lead to errors. Also neglects tectonic type because set is small and small differences are expected.

Fit \(\log A=-b \log X+c\) to data from each earthquake separately and find average \(b\) equal to \(1.292\). Then fit \(\log A=aM-b\log X+c\) to data from all earthquakes and find \(b=0.6884\). Fit \(\log A=-b \log X+\sum d_i l_i\) to all data, where \(l_i=1\) for \(i\)th earthquake and \(0\) otherwise and find \(b=1.21\), use this for rest of analysis.

Use weighted regression, due to nonuniform sampling over all \(M\) and \(X\). Divide data into distance bins \(2.5\,\mathrm{km}\) wide up to \(10\,\mathrm{km}\) and logarithmically dependent for larger distances. Within each bin each earthquake is given equal weight by assigning a relative weight of \(1/n_{j,l}\), where \(n_{j,l}\) is the number of recordings for \(j\)th earthquake in \(l\)th distance bin, then normalise so that sum to total number of recordings.

Original data included two earthquakes with focal depths \(91.0\,\mathrm{km}\) and \(119.0\,\mathrm{km}\) and \(M=6.8\) and \(6.1\) which caused large errors in regression parameters due to large depths so excluded them.

Check capability of data to compute coefficients by deleting, in turn, \(c_1\), \(c_2\) and \(c_3\), find higher standard deviation.

Makes one coefficient at a time equal to values given in Abrahamson and Litehiser (1989), finds sum of squares increases.

Notes lack of data could make relationship unreliable.

Ground-motion model is: \[\log Y =a+b M-\log R+d R\] where \(Y\) is in \(\mathrm{nm/s^2}\), \(b=0.868\), \(d=-0.001059\), \(\sigma=0.35\), for horizontal PGA \(a=5.230\) and for vertical PGA \(a=5.054\).

Most records from rock sites.

Focal depths between \(0\) and about \(27\,\mathrm{km}\) (most less than \(10\,\mathrm{km}\)).

Most records from \(M_L<3.5\).

Most earthquakes have strike-slip mechanism.

Uses records from high gain short period seismographs and from strong-motion instruments.

Records are instrument corrected.

Eliminates some far-field data from small magnitude earthquakes using signal to noise ratio criterion.

Records cover entire azimuthal range.

Notes that need more data in near field.

Notes that care must be taken when using equations for prediction of ground motion in strong earthquakes (\(M\approx 6\)) because of lack of data.

Ground-motion models are (using \(r_{hypo}\)): \[\ln \mathrm{PGA}=C_1+C_2 M+C_3 \ln R\] where \(\mathrm{PGA}\) is in \(\,\mathrm{cm/s^2}\), \(C_1=0.47\), \(C_2=1.15\), \(C_3=-1.22\) and \(\sigma=0.64\); and (using \(r_{epi}\)): \[\ln \mathrm{PGA}=C_1+C_2 M+C_3 \ln (r+R_0)\] where \(C_1=2.18\), \(C_2=1.19\), \(C_3=-1.64\), \(R_0=10\) and \(\sigma=0.63\).

Data from 7 free-field surface stations (STE, STC, FRM, TST, GRA, GRB and PRO) of the EuroSeisTest 3D array, which is located in an alluvial valley, recorded from April 1994 to January 1997.

Believes model corresponds to intermediate soil.

Data from ETNA and SSA-16 instruments.

Focal depths from \(0\) to \(15\,\mathrm{km}\).

Most data from \(<40\,\mathrm{km}\).

Examines site effects by re-calculating \(C_1\) for each station, called \(C_{sta}\), but keeping \(C_2\) and \(C_3\) fixed. Report \(C_{sta}\) for each station in graph. Find effect of soil is negigible (\(-0.22\leq C_{sta} \leq 0.33\)).

Prefers model with \(r_{hypo}\) as focal depths highly accurate.

Compares observations normalized to \(20\,\mathrm{km}\) against predictions and total residuals w.r.t. \(r_{hypo}\) and \(M_w\). Finds no trends.

Ground-motion model is: \[\ln Y=C_1+C_2M+C_3 \ln (\Delta+15)+0.31 S\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(C_1=4.85\), \(C_2=1.02\), \(C_3=-1.90\) and \(\sigma=0.50\) (the coefficients \(15\) and \(0.31\) were taken from Theodulidis and Papazachos (1992) since they cannot be determined by the data).

Use 2 site classes:

Bedrock

Alluvium

Use data from mainshock and aftershocks of 13 May 1995 Kozani-Grevena earthquake.

Use a four-step approach: derive relation between PGA and intensity (using only data from largest events), assess decay of PGA with distance, adjust PGA to a fixed distance using this equation and finally regression to find remaining coefficients

^{16}.Adjust all data to \(M_w 6.6\) and compare observed and predicted PGAs. Find reasonable fit.

Ground-motion model is: \[\ln A=C_1 + C_2 M+C_3(R+R_0)+C_4 \ln (R+R_0)+C_5 S\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(C_1=0\), \(C_2=0.664\), \(C_3=0.009\), \(C_4=-2.206\), \(R_0=20\), \(C_5=8.365\) (for S1), \(C_5=8.644\) (for S2), \(C_5=8.470\) (for S3) and \(C_5=8.565\) (for S4) for horizontal PGA using \(r_{epi}\) and \(M_s\) and all Mediterranean data, \(C_1=0\), \(C_2=0.658\), \(C_3=0.008\), \(C_4=-2.174\), \(R_0=20\), \(C_5=7.693\) (for S1), \(C_5=7.915\) (for S2) and \(C_5=7.813\) (for S4) (\(C_5\) not derived for S3) for vertical PGA using \(r_{epi}\) and \(M_s\) and all Mediterranean data. \(\sigma\) is not given (\(R^2\) is reported).

Use four site categories:

Hard basement rock.

Sedimentary rock and conglomerates.

Glacial deposits.

Alluvium and consolidated sediments.

Derive separate equations using data from Mediterranean region and also just using data from Spain.

Equations for Spain derived using \(m_{bLg}\).

Spanish data all from earthquakes with \(2.5 \leq m_{bLg} \leq 6.0\) and \(0 \leq r_{hypo} \leq 300\,\mathrm{km}\).

Ground-motion model is: \[\log_{10}Y=a+b(M-6)+c(M-6)^2+d \log (r^2+h^2)^{1/2}+e G_1 +f G_2\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(a=3.098\), \(b=0.3065\), \(c=-0.07570\), \(d=-0.8795\), \(h=6.910\), \(e=0.1452\), \(f=0.1893\) and \(\sigma=0.2124\).

Use three site categories:

\(V_{s,30}>760\,\mathrm{m/s}\), 24 records \(\Rightarrow G_1=0, G_2=0\).

\(360 < V_{s,30} \leq 760 \,\mathrm{m/s}\), 116 records \(\Rightarrow G_1=1, G_2=0\).

\(180 < V_{s,30} \leq 360 \,\mathrm{m/s}\), 164 records \(\Rightarrow G_1=0, G_2=1\).

Uses records from ground level or in basements of structures of two stories or less, and excludes records from dam or bridge abutments.

Selects records which include major motion portion of strong-motion episode, represented by S wavetrain. Excludes records triggered late on S wave or those of short duration terminating early in coda.

Most records already corrected. Some records instrument corrected and 4-pole causal Butterworth filtered (corner frequencies \(0.1\) and \(25\,\mathrm{Hz}\)). Other records instrument corrected and 4-pole or 6-pole causal Butterworth bandpass filtered (corner frequencies \(0.2\) and \(25\,\mathrm{Hz}\)). All data filtered using 6-pole causal high-pass Butterworth filter with corner frequency \(0.2\,\mathrm{Hz}\) and velocity and displacement curves examined.

Uses method of Campbell (1997) to reduce bias due to non-triggered instruments, for some recent shocks. Firstly uses all data to determine minimum distances (which are functions of magnitude and site condition) at which \(16\)th percentile values of PGA are \(<0.02\,\mathrm{g}[0.2\,\mathrm{m/s}]\) (corresponding to \(0.01\,\mathrm{g}[0.1\,\mathrm{m/s}]\) vertical component trigger threshold). Next delete records from larger distances and repeat regression.

Check residuals against distance and magnitude for each site class; find no obvious non-normal magnitude or distance dependent trends.

Based on Zhao, Dowrick, and McVerry (1997) see SectionÂ 2.153

Ground-motion model is: \[\begin{aligned} \log_{10} \mathrm{PGA}&=&A_1 M_w + A_2 \log_{10} R + A_3 h_c+A_4+A_5+A_6+A_7 R+A_8 M_w +A_9\\ &&{}+A_{10} R_v\end{aligned}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{m/s^2}\), \(R=\sqrt{r^2+d^2}\) and \(R_v\) is distance travelled by direct seismic wave through volcanic region. \(A_5\) only for crustal reverse, \(A_6\) only for interface, \(A_7\) only for strong and weak rock, \(A_8\) only for strong rock, \(A_9\) only for strong rock, \(A_1=0.2955\), \(A_2=-1.603\), \(A_3=0.00737\), \(A_4=-0.3004\), \(A_5=0.1074\), \(A_6=-0.1468\), \(A_7=-0.00150\), \(A_8=0.3815\), \(A_9=-2.660\), \(A_{10}=-0.0135\), \(d=19.0\) and \(\sigma=0.24\).

Originally considers five site categories but retain three:

Strong rock: \(V_s > 700\,\mathrm{m/s}\)

Weak rock: \(375 \leq V_s \leq 700\,\mathrm{m/s}\) and category AV those sites with a very thin layer (\(\leq 3\,\mathrm{m}\)) overlying rock

Soil: everything else

Depth to centroid of rupture, \(h_c\), used, \(4\leq h_c \leq 94\,\mathrm{km}\).

\(60\%\) on soil, \(40\%\) on rock

Consider tectonic type: C=Crustal (12+17), I=Interface (5+0) and S=Slab(8+0)

Consider source mechanism: N=normal (6+1), R=reverse (12+5) and S=strike-slip (7+11). Mixed classified by ratio of components \(\geq 1.0\).

Mixture of analogue and digital accelerograms (72%) and seismograms (28%)

Accelerograms sampled at \(100\)â\(250\) samples/sec. Bandpass frequencies chosen by analysis of Fourier amplitude spectrum compared with noise spectrum. \(f_{\mathrm{min}}\) between \(0.15\) and \(0.5\,\mathrm{Hz}\) and \(f_{\mathrm{max}}\) equal to \(25\,\mathrm{Hz}\). Instrument correction applied to analogue records.

Seismograms sampled at \(50\)â\(100\) samples/sec. Differentiated once. Instrument corrected and high pass filtered with \(f_{\mathrm{min}}=0.5\,\mathrm{Hz}\). No low pass filter needed.

Clipped seismograms usually retained.

Directional effect noticed but not modelled.

Most records from more than \(100\,\mathrm{km}\) away. Note lack of near-source data.

Records from accelerograms further away than first operational non-triggering digital accelerograph, which had a similar triggering level, were excluded.

Models difference between high attenuating volcanic and normal regions.

No details known.

Ground-motion model is: \[\log (a_{\max})=\phi_1+\phi_2 \log M_0 -\phi_3 \log (R)\] where \(a_{\max}\) is in \(\,\mathrm{cm/s^2}\), \(M_0\) is in \(\mathrm{dyn}\,\mathrm{cm}\) and \(R\) is in \(\,\mathrm{cm}\), \(\phi_1=0.0451\), \(\phi_2=0.3089\), \(\phi_3=0.9642\) and \(\sigma=0.3148\).

Instruments in basement of buildings located on rock or very stiff ground.

Records from 21 different stations.

Focal depths between \(1\) and \(11\,\mathrm{km}\).

Most records from digital instruments with \(200\,\mathrm{Hz}\) sampling frequency and high dynamic range.

Seismic moments calculated using the strong-motion data.

Most data from \(M_0\leq 5 \times 10^{23} \mathrm{dyn}\,\mathrm{cm}\) and from \(d_e\leq 40\,\mathrm{km}\).

Ground-motion model for rupture distance is: \[\log A=a M_w +h D +\sum d_i S_i +e -\log (X+c_1 10^{c_2 M_w}) -k X\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(a=0.50\), \(h=0.0036\), \(d_1=0\), \(d_2=0.09\), \(d_3=0.28\), \(e=0.60\), \(k=0.003\) and \(\sigma=0.27\) (\(c_1\) and \(c_2\) are not given).

Ground-motion model for equivalent hypocentral distance (EHD) is: \[\log A=a M_w + h D +\sum d_i S_i +e -\log X_{eq} -k X_{eq}\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(a=0.50\), \(h=0.0043\), \(d_1=0\), \(d_2=0.01\), \(d_3=0.22\), \(e=0.61\), \(k=0.003\) and \(\sigma=0.28\).

Use two site categories for most records following Joyner and Boore (1981):

Rock

Soil

Records from free-field or small buildings where soil-structure interaction effects are negligible.

Records from three different type of instrument so instrument correct. Filter with corner frequencies, chosen according to noise level, a) \(0.08\) & \(0.15\,\mathrm{Hz}\), b) \(0.10\) & \(0.20\,\mathrm{Hz}\) or c) \(0.15\) to \(0.33\,\mathrm{Hz}\).

Exclude records obviously affected by soil liquefaction.

Focal depth (defined as average depth of fault plane), \(D\), between \(6\) and \(120\,\mathrm{km}\); most less than \(40\,\mathrm{km}\).

Select records satisfying: distances \(<300\,\mathrm{km}\) for \(M_w>7\), distances \(<200\,\mathrm{km}\) for \(6.6 \leq M_w \leq 7\), distances \(<150\,\mathrm{km}\) for \(6.3 \leq M_w \leq 6.5\) and distances \(<100\,\mathrm{km}\) for \(M_w<6.3\).

Fix \(k=0.003\).

Multiply rock PGAs by \(1.4\) to get soil PGA based on previous studies.

Use three fault types: crustal (\(<\)719 records from 9 earthquakes) \(\Rightarrow S_1=1, S_2=0, S_3=0\), inter-plate (\(<\)291 records from 7 earthquakes) \(\Rightarrow S_2=1, S_1=0, S_3=0\) and intra-plate (\(<\)127 records from 5 earthquakes) \(\Rightarrow S_3=1, S_1=0, S_2=0\).

Use weighted regression giving more weight to near-source records (weight factor of \(8\) for records \(<25\,\mathrm{km}\), \(4\) for records between \(20\) and \(50\,\mathrm{km}\), \(2\) for records between \(50\) and \(100\,\mathrm{km}\) and \(1\) for records \(>100\,\mathrm{km}\)). Use only three earthquakes with sufficient near-source data to find \(c_1\) and \(c_2\) then use all earthquakes to find \(a\), \(h\), \(d_i\), \(e\) in second stage using weighted regression dependent on number of recordings for each earthquake (weight factor of \(3\) for \(>\)83 records, \(2\) for between 19 and 83 records, \(1\) for \(<\)19 records.

Note that \(M_w\) and \(D\) are positively correlated so \(a\) and \(h\) may not be correctly determined when using rupture distance. Constrain \(a\) for rupture distance model to that obtained for EHD and constrain PGA to be independent of magnitude at \(0\,\mathrm{km}\) and repeat regression. Coefficients given above.

Update of Spudich et al. (1997) see SectionÂ 2.144.

Ground-motion model is: \[\begin{aligned} \log_{10} Z&=&b_1+b_2 (M-6)+b_3(M-6)^2+b_5 \log_{10} D +b_6 \Gamma\\ \mbox{with: } D&=&\sqrt{r_{jb}^2+h^2}\end{aligned}\] where \(Z\) is in \(\,\mathrm{g}\), \(b_1=0.299\), \(b_2=0.229\), \(b_3=0\), \(b_5=-1.052\), \(b_6=0.112\), \(h=7.27\) and \(\sigma=\sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2}\) where \(\sigma_1=0.172\), \(\sigma_2=0.108\) and for randomly oriented horizontal component \(\sigma_3=0.094\) and for larger horizontal component \(\sigma_3=0\).

Values of \(\sigma_3\) (used to compute standard deviation for a randomly orientated component) reported in Spudich et al. (1999) are too large by a factor of \(\sqrt{2}\).

Use two site categories (could not use more or \(V_{s,30}\) because not enough data):

Rock: includes hard rock (12 records) (plutonic igneous rocks, lava flows, welded tuffs and metamorphic rocks unless severely weathered when they are soft rock), soft rock (16 records) (all sedimentary rocks unless there was some special characteristic noted in description, such as crystalline limestone or massive cliff-forming sandstone when they are hard rock) and unknown rock (8 records). 36 records in total.

Soil (alluvium, sand, gravel, clay, silt, mud, fill or glacial outwash of more than \(5\,\mathrm{m}\) deep): included shallow soil (8 records) (\(5\) to \(20\,\mathrm{m}\) deep), deep soil (77 records) (\(>20\,\mathrm{m}\) deep) and unknown soil (21 records). 106 records in total.

Applicable for extensional regimes, i.e. those regions where lithosphere is expanding areally. Significantly different ground motion than non-extensional areas.

Criteria for selection of records is: \(M_w\geq 5.0\), \(d_f\leq 105\,\mathrm{km}\). Reject records from structures of more than two storeys or from deeply embedded basements or those which triggered on S wave. Also reject those close to dams which may be affected by dam. Also only use records already digitised.

Include records from those instrument beyond cutoff distance, i.e. beyond first instrument which did not trigger, because of limited records and lack of data on non-triggering.

Not enough data to be able to find all coefficients so use \(b_2\) and \(b_3\) from Boore, Joyner, and Fumal (1993) and \(b_6\) from Boore, Joyner, and Fumal (1994a).

One-stage maximum likelihood method used because many events used which only have one record associated with them and the two-stage method underestimates the earthquake-to-earthquake component of variation in that case.

Correction technique based on uniform correction and processing using upper, \(f_h\), and lower, \(f_l\), frequencies for passband based on a visual inspection of Fourier amplitude spectrum and baseline fitting with a polynomial of degree 5.

Check to see whether normal and strike-slip earthquakes give significantly different ground motions. No significant difference.

Ground-motion model is: \[\log A=a+b M_s+c\log R+d R\] where \(A\) is in \(\,\mathrm{cm/s^2}\), using just soil records \(a=0.430\), \(b=0.428\), \(c=-0.764\), \(d=-0.00480\) and \(\sigma=0.271\).

Use records from aftershocks of Tangshan earthquake.

Focal depths between \(5.7\) and \(12.9\,\mathrm{km}\).

Note \(M_s\) values used may have some systematic deviation from other regions and errors, which decrease with increasing magnitude, can reach \(\pm 0.5\).

Errors in epicentral locations not less than \(2\,\mathrm{km}\). Reject 3 records because have \(R<2\,\mathrm{km}\), if include then find standard deviation increases and \(c\) obtained is unreasonable.

Fit equation to all data (both rock and soil) but note that only for reference. Also fit equation to soil data only (\(2.1 \leq R \leq 41.3\,\mathrm{km}\), \(3.7 \leq M_s \leq 4.9\), 33 records from 6 earthquakes).

Remove all four earthquakes with \(M_s<4.0\), for which error in magnitude determination is large, and fit equation to soil data only (\(2.8 \leq R \leq 41.1\,\mathrm{km}\), \(4.5 \leq M_s \leq 4.9\), 13 records from 2 earthquakes). Find smaller uncertainties.

Also fit data to \(\log A=a+b M_s-c \log (R+R_0)\); find similar results.

Also use resultant of both horizontal components; find similar results to using larger component.

Also fit eastern North America data (\(3.9 \leq R \leq 61.6\,\mathrm{km}\), \(2.3 \leq M_s \leq 3.8\), 7 records from 3 earthquakes); find similar attenuation characteristics.

All equations pass F-tests.

Ground-motion model is: \[\log A=aM-bX-d\log X+c_i S_i\] where units of \(A\) not given (but probably \(\,\mathrm{m/s^2}\)), for vertical PGA \(a=0.362\), \(b=0.0002\), \(c_1=-1.124\), \(c_2=-1.150\), \(c_3=-1.139\), \(c_4=-1.064\), \(d=1\) and \(\sigma=0.336\) and for horizontal PGA \(a=0.360\), \(b=0.0003\), \(c_1=-0.916\), \(c_2=-0.862\), \(c_3=-0.900\), \(c_4=-0.859\), \(d=1\) and \(\sigma=0.333\).

Use four site categories, which were based on \(H/V\) receiver function (RF) measurements (use geotechnical measurements at 50 sites and strong-motion accelerograms at other sites):

RF does not exhibit any significant amplification below \(15\,\mathrm{Hz}\). Corresponds to rock and stiff sediment sites with average S-wave velocity in top \(30\,\mathrm{m}\) (\(V_{s,30}\)) \(>700\,\mathrm{m/s}\). Use \(c_1\).

RF exhibits a fundamental peak exceeding \(3\) at a frequency between \(5\) and \(15\,\mathrm{Hz}\). Corresponds to stiff sediments and/or soft rocks with \(500<V_{s,30}\leq 700\,\mathrm{m/s}\). Use \(c_2\).

RF exhibits peaks between \(2\) and \(5\,\mathrm{Hz}\). Corresponds to alluvial sites with \(300<V_{s,30}\leq 500\,\mathrm{m/s}\). Use \(c_3\).

RF exhibits peaks for frequencies \(<2\,\mathrm{Hz}\). Corresponds to thick soft alluvium. Use \(c_4\).

Only 100 records are associated with earthquakes with known focal mechanisms, 40 correspond to strike-slip/reverse, 31 to pure strike-slip, 24 to pure reverse and 4 to a pure vertical plane. Note that use of equations should be limited to sources with such mechanisms.

Use only records for which the signal to noise ratio was acceptable.

Source parameters from teleseismic studies available for 279 records.

Calculate source parameters directly from the strong-motion records for the remaining 189 digital records using a source model. Hypocentral distance from S-P time and seismic moment from level of acceleration spectra plateau and corner frequency.

Focal depths from \(9\) to \(133\,\mathrm{km}\) but focal depth determination is very imprecise and majority of earthquakes are shallow.

Suggest that whenever estimation of depth of earthquake is impossible use distance to surface projection of fault rather than hypocentral distance because differences between hypocentral and epicentral distances are not significant for shallow earthquakes.

Also derive equations based only on data from the Zagros thrust fault zone (higher seismic activity rate with many earthquakes with \(4\leq M \leq 6\)) and based only on data from the Alborz-Central Iran zone (lower seismic activity rate but higher magnitude earthquakes). Find some differences between regions.

Investigate fixing \(d\) to \(1\) (corresponding to body waves) and to \(0.5\) (corresponding to surface waves).

Note that there are very few (only two) near-field (from less than \(10\,\mathrm{km}\) from surface fault rupture) records from earthquakes with \(M_w>6.0\) and so results are less certain for such combinations of magnitude and distance.

Ground-motion model is: \[\log y=b_1+b_2 M_s+b_3 d+b_A S_A+b_S S_S\] where \(y\) is in \(\,\mathrm{m/s^2}\), for horizontal PGA \(b_1=-0.659\), \(b_2=0.202\), \(b_3=-0.0238\), \(b_A=0.020\), \(b_S=0.029\) and \(\sigma=0.214\) and for vertical PGA \(b_1=-0.959\), \(b_2=0.226\), \(b_3=-0.0312\), \(b_A=0.024\), \(b_S=0.075\) and \(\sigma=0.270\).

Assume decay associated with anelastic effects due to large strains and cannot use both \(\log d\) and \(d\) because highly correlated in near field.

Use four site categories (often use shear-wave velocity profiles):

Very soft soil: approximately \(V_{s,30}<180\,\mathrm{m/s}\), (combine with category S) \(\Rightarrow S_A=0, S_S=1\), 4 records.

Soft soil: approximately \(180\leq V_{s,30}<360\,\mathrm{m/s}\) \(\Rightarrow S_A=0, S_S=1\), 87 records.

Stiff soil: approximately \(360\leq V_{s,30}<750\,\mathrm{m/s}\) \(\Rightarrow S_A=1, S_S=0\), 68 records.

Rock: approximately \(V_{s,30}>750\,\mathrm{m/s}\) \(\Rightarrow S_A=0, S_S=0\), 23 records.

where \(V_{s,30}\) is average shear-wave velocity to \(30\,\mathrm{m}\). Know no site category for 14 records.

Use only records from ânear fieldâ where importance of vertical acceleration is greatest. Select records with \(M_s\geq 5.8\), \(d\leq 15\,\mathrm{km}\) and focal depth \(h\leq 20\,\mathrm{km}\). Do not use magnitude dependent definition to avoid correlation between magnitude and distance for the records.

Focal depths, \(1\leq h \leq 19\,\mathrm{km}\).

Majority (133 records, 72%) of records from W. N. America, 40 records (22%) from Europe and rest from Canada, Nicaragua, Japan and Taiwan.

Consider three source mechanisms but do not model:

Normal, 8 earthquakes, 16 records.

Strike-slip, 18 earthquakes, 72 records.

Thrust, 16 earthquakes, 98 records.

Use only free-field records using definition of Joyner and Boore (1981), include a few records from structures which violate this criterion but feel that structure did not affect record in period range of interest.

Records well distributed in magnitude and distance so equations are well constrained and representative of entire dataspace. Note lack of records from normal earthquakes. Correlation coefficient between magnitude and distance is \(-0.10\).

Use same correction procedure (elliptical filter with pass band \(0.2\) to \(25\,\mathrm{Hz}\), roll-off frequency \(1.001\,\mathrm{Hz}\), sampling interval \(0.02\,\mathrm{s}\), ripple in pass-band \(0.005\) and ripple in stop-band \(0.015\) with instrument correction) for almost all records. Use 19 records available only in corrected form as well because in large magnitude range. Think different correction procedures will not affect results.

Try both one-stage and two-stage regression method for horizontal PGA; find large differences in \(b_2\) but very similar \(b_3\). Find that (by examining cumulative frequency distribution graphs for magnitude scaling of one-stage and two-stage methods) that two-stage better represents large magnitude range than one-stage method. Examine plot of amplitude factors from first stage of two-stage method against \(M_s\); find that amplitude factor of the two Kocaeli (\(M_s=7.8\)) records is far below least squares line through the amplitude factors. Remove the two Kocaeli records and repeat analysis; find \(b_2\) from two-stage method is changed by a lot but \(b_2\) from one-stage method is not. Conclude two-stage method is too greatly influenced by the two records from Kocaeli and hence use one-stage method.

Find \(b_2\) and \(b_3\) significantly different than \(0\) at \(5\%\) level but \(b_A\) and \(b_S\) not significant.

Ground-motion model is: \[\begin{aligned} \ln Y&=&c_1+c_2 M_w+c_3 (8.5-M_w)^2\\ &&{}+c_4 \ln(\{R_s^2+[(c_5 S_{HS}+c_6 \{S_{PS}+S_{SR}\}+c_7 S_{HR})\\ &&\exp(c_8 M_w+c_9 \{8.5-M_w\}^2)]^2\}^{1/2})+c_{10} F_{SS} +c_{11} F_{RV}+c_{12}F_{TH}\\ &&{}+c_{13}S_{HS}+c_{14}S_{PS}+c_{15}S_{SR}+c_{16}S_{HR}\end{aligned}\]

Use four site categories:

Holocene soil: recent alluvium \(\Rightarrow S_{HS}=1, S_{PS}=0, S_{SR}=0, S_{HR}=0\).

Pleistocene soil: older alluvium \(\Rightarrow S_{PS}=1, S_{HS}=0, S_{SR}=0, S_{HR}=0\).

Soft rock \(\Rightarrow S_{SR}=1, S_{HS}=0, S_{PS}=0, S_{HR}=0\).

Hard rock \(\Rightarrow S_{HR}=1, S_{HS}=0, S_{PS}=0, S_{SR}=0\).

Consider all records to be free-field.

All earthquakes occurred in shallow crustal tectonic environment.

Consider three source mechanisms: strike-slip (\(F_{SS}=1, F_{RV}=0, F_{TH}=0\)) 20+ earthquakes (including 1+ normal faulting shock), reverse (\(F_{RV}=1, F_{SS}=0, F_{TH}=0\)) 7+ earthquakes and thrust (\(F_{TH}=1, F_{SS}=0, F_{RV}=0\)) 6+ earthquakes.

Coefficients not given, only predictions.

Ground-motion model is: \[\begin{aligned} \ln Y&=&c_1+c_2 M_w +c_3 (8.5-M_w)^2+c_4 \ln(\{R_s^2+[(c_5+c_6\{S_{PS}+S_{SR}\}+c_7 S_{HR})\\ &&\exp(c_8 M_w+c_9\{8.5-M_w\}^2)]^2\}^{1/2})+c_{10}F_{SS}+c_{11}F_{RV}+c_{12}F_{TH}\\ &&{}+c_{13}S_{HS}+c_{14}S_{PS}+c_{15}S_{SR}+c_{16}S_{HR}\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), for horizontal uncorrected PGA \(c_1=-2.896\), \(c_2=0.812\), \(c_3=0\), \(c_4=-1.318\), \(c_5=0.187\), \(c_6=-0.029\), \(c_7=-0.064\), \(c_8=0.616\), \(c_9=0\), \(c_{10}=0\), \(c_{11}=0.179\), \(c_{12}=0.307\), \(c_{13}=0\), \(c_{14}=-0.062\), \(c_{15}=-0.195\), \(c_{16}=-0.320\) and \(\sigma=0.509\), for horizontal corrected PGA \(c_1=-4.033\), \(c_2=0.812\), \(c_3=0.036\), \(c_4=-1.061\), \(c_5=0.041\), \(c_6=-0.005\), \(c_7=-0.018\), \(c_8=0.766\), \(c_9=0.034\), \(c_{10}=0\), \(c_{11}=0.343\), \(c_{12}=0.351\), \(c_{13}=0\), \(c_{14}=-0.123\), \(c_{15}=-0.138\), \(c_{16}=-0.289\) and \(\sigma=0.465\), for vertical uncorrected PGA \(c_1=-2.807\), \(c_2=0.756\), \(c_3=0\), \(c_4=-1.391\), \(c_5=0.191\), \(c_6=0.044\), \(c_7=-0.014\), \(c_8=0.544\), \(c_9=0\), \(c_{10}=0\), \(c_{11}=0.091\), \(c_{12}=0.223\), \(c_{13}=0\), \(c_{14}=-0.096\), \(c_{15}=-0.212\), \(c_{16}=-0.199\) and \(\sigma=0.548\) and for vertical corrected PGA \(c_1=-3.108\), \(c_2=0.756\), \(c_3=0\), \(c_4=-1.287\), \(c_5=0.142\), \(c_6=0.046\), \(c_7=-0.040\), \(c_8=0.587\), \(c_9=0\), \(c_{10}=0\), \(c_{11}=0.253\), \(c_{12}=0.173\), \(c_{13}=0\), \(c_{14}=-0.135\), \(c_{15}=-0.138\), \(c_{16}=-0.256\) and \(\sigma=0.520\).

Use four site categories:

Holocene soil: soil deposits of Holocene age (11,000 years or less), generally described on geological maps as recent alluvium, approximate average shear-wave velocity in top \(30\,\mathrm{m}\) is \(290\,\mathrm{m/s}\) \(\Rightarrow S_{HS}=1, S_{PS}=0, S_{SR}=0, S_{HR}=0\).

Pleistocene soil: soil deposits of Pleistocene age (11,000 to 1.5 million years) , generally described on geological maps as older alluvium or terrace deposits, approximate average shear-wave velocity in top \(30\,\mathrm{m}\) is \(370\,\mathrm{m/s}\) \(\Rightarrow S_{PS}=1, S_{HS}=0, S_{SR}=0, S_{HR}=0\).

Soft rock: primarily includes sedimentary rock deposits of Tertiary age (1.5 to 100 million years), approximate average shear-wave velocity in top \(30\,\mathrm{m}\) is \(420\,\mathrm{m/s}\) \(\Rightarrow S_{SR}=1, S_{HS}=0, S_{PS}=0, S_{HR}=0\).

Hard rock: primarily includes older sedimentary rock deposits, metamorphic rock and crystalline rock, approximate average shear-wave velocity in top \(30\,\mathrm{m}\) is \(800\,\mathrm{m/s}\) \(\Rightarrow S_{HR}=1, S_{HS}=0, S_{PS}=0, S_{SR}=0\).

Earthquakes from shallow crustal active tectonic regions.

Most earthquakes with \(6 \leq M_w \leq 7\).

Use three source mechanism categories:

Strike-slip: primarily vertical or near-vertical faults with predominantly lateral slip (includes only normal faulting earthquake in set), \(\Rightarrow F_{SS}=1, F_{RV}=0, F_{TH}=0\).

Reverse: steeply dipping faults with either reverse or reverse-oblique slip, \(\Rightarrow F_{RV}=1, F_{SS}=0, F_{TH}=0\).

Thrust: shallow dipping faults with predominantly thrust slip including blind-thrust shocks, \(\Rightarrow F_{TH}=1, F_{SS}=0, F_{RV}=0\).

Consider all records to be free-field. Records from ground level in instrument shelter or a building \(<\)3 storeys high (\(<\)7 if located on hard rock). Include records from dam abutments to increase number of rock records. Exclude data from basements of buildings of any size or at toe or base of dams.

Exclude data from \(R_s>60\,\mathrm{km}\) to avoid complicating problems related to arrival of multiple reflections from lower crust. Distance range is believed to include most ground shaking amplitudes of engineering interest, except for possibly long period spectral accelerations on extremely poor soil.

Equations for uncorrected (Phase 1 standard level of processing) and corrected (Phase 2 standard level of processing).

Find sediment depth (depth to basement rock) has significant effect on amplitude of ground motion and should be taken into account; it will be included once its mathematical form is better understood.

Ground-motion model is: \[\mu (M,r_{\mathrm{jb}},V_s)=b_1+b_2 (M-6)+b_3 (M-6)^2+b_5 \ln [(r_{\mathrm{jb}}^2+h^2)^{0.5}]+b_v \ln (V_s/V_a)\] \(\mu (M,r_{\mathrm{jb}},V_s)\) is natural logarithm of ground-motion parameter (e.g. \(\ln (\mathrm{PGA})\) where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\)), \(b_{1,ss}=0.853 \pm 0.28\), \(b_{1,rv}=0.872 \pm 0.27\), \(b_2=0.442 \pm 0.15\), \(b_3=-0.067 \pm 0.16\), \(b_5=-0.960 \pm 0.07\), \(b_v=-0.154 \pm 0.14\), \(h=8.90\,\mathrm{km}\), \(V_a=760\,\mathrm{m/s}\), \(\sigma=0.47 \pm 0.02\) (intra-event) and \(\tau=0.23\) (inter-event). Also gives overall \(\sigma=(0.93-0.10 M_w)^{0.5}\) for \(M_w\leq 7.0\) and overall \(\sigma=0.48\) for \(M_w>7.0\).

Uses six site classes (from Wills et al. (2000)):

\(760\leq V_s \leq 1500\,\mathrm{m/s}\). Uses \(V_s=1000\,\mathrm{m/s}\) in regression. 12 records.

Boundary between B and C. Uses \(V_s=760\,\mathrm{m/s}\) in regression. 36 records.

\(360\leq V_s \leq 760\,\mathrm{m/s}\). Uses \(V_s=560\,\mathrm{m/s}\) in regression. 16 records.

Boundary between C and D. Uses \(V_s=360\,\mathrm{m/s}\) in regression. 166 records.

\(180 \leq V_s \leq 360\,\mathrm{m/s}\). Uses \(V_s=270\,\mathrm{m/s}\) in regression. 215 records.

Boundary between D and E. Uses \(V_s=180\,\mathrm{m/s}\) in regression. 2 records.

Uses data from the SCEC Phase III strong-motion database.

Uses three faulting mechanism classes:

Use \(b_{1,ss}\). 14 earthquakes, 103 records.

Use \(b_{1,rv}\). 6 earthquakes, 300 records.

Use \(0.5 (b_{1,ss}+b_{1,rv})\). 8 earthquakes, 46 records.

Notes that data is unbalanced in that each earthquake has a different number of records for each site type hence it is important to correct observations for the inter-event terms before examining residuals for site effects.

Plots average site class residuals w.r.t. BC category and the residuals predicted by equation and finds good match.

Uses 197 records with basin-depth estimates (depth defined to the \(2.5\,\mathrm{km/s}\) shear-wave velocity isosurface) to examine dependence of inter-event corrected residuals w.r.t. basin depth. Plots residuals against basin depth and fits linear function. Finds that all slopes are significantly different than zero by more than two sigmas. Finds a significant trend in subset of residuals where basin-depths are known w.r.t. magnitude hence needs to test whether basin-depth effect found is an artifact of something else. Hence derives Ground-motion models (coefficients not reported) using only subset of data for which basin-depth estimates are known and examines residuals w.r.t. basin-depth for this subset. Finds similar trends as before hence concludes found basin effect is truly an effect of the basin. Notes that basin-depth coefficients should be derived simultaneously with other coefficients but because only a subset of sites have a value this could not be done.

Tests for nonlinearity by plotting residuals for site class D w.r.t. predicted ground motion for BC boundary. Fits linear equation. Finds slope for PGA is significantly different than zero.

Notes that due to large number of class D sites site nonlinearity could have affected other coefficients in equation leading to less of a trend in residuals. Tests for this by plotting residuals for site classes B and BC combined w.r.t. predicted ground motion for BC boundary. Fits linear equation. Finds non-significant slopes. Notes that nonlinearity may lead to rock ground motions being underestimated by model but not enough data to conclude.

Investigates inter-event variability estimate through Monte Carlo simulations using 250 synthetic databases because uncertainty estimate of \(\tau\) was considered unreliable possibly due to limited number of events. Find that there could be a problem with the regression methodology adopted w.r.t. the estimation of \(\tau\).

Plots squared residuals w.r.t. magnitude and fits linear equations. Finds significant trends. Notes that method could be not statistically correct because squared residuals are not Gaussian distributed.

Plots squared residuals w.r.t. \(V_s\) and does not find a significant trend.

Provides magnitude-dependent estimates of overall \(\sigma\) up to \(M_w 7.0\) and constant overall \(\sigma\) for larger magnitudes.

Tests normality of residuals using Kolmogorov-Smirnov test and finds that the null hypothesis cannot be rejected. Also examines theoretical quantile-quantile plots and finds nothing notable.

Ground-motion model is: \[\ln(\mathrm{PGA})=b_1+b_2M+b_3R+b_4 \ln(R)\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), for central Himalayan earthquakes \(b_1=-4.135\), \(b_2=0.647\), \(b_3=-0.00142\), \(b_4=-0.753\) and \(\sigma=0.59\) and for non-subduction earthquakes in N.E. India \(b_1=-3.443\), \(b_2=0.706\), \(b_3=0\), \(b_4=-0.828\) and \(\sigma=0.44\) (coefficients of other equations not given here because they are for a particular earthquake).

Data from strong-motion accelerographs (SMA) and converted from structural response recorders (SRR), which consist of six seismoscopes with natural periods \(0.40\), \(0.75\) and \(1.25\,\mathrm{s}\) and damping levels \(5\) and \(10\%\). Conversion achieved by deriving spectral amplification factors (ratio of response ordinate and PGA) using SMA recordings close to SRR, checking that these factors were independent of distance. The mean of the six estimates of PGA (from the six spectral ordinates) from each SRR are then used as PGA values. Check quality of such PGA values through statistical comparisons and discard those few which appear inconsistent.

Data split into four categories for which derive separate equations:

Central Himalayan earthquakes (thrust): (32 SMA records, 117 SRR records), 3 earthquakes with \(5.5 \leq M \leq 7.0\), focal depths \(10\leq h\leq 33\,\mathrm{km}\) and \(2 \leq R \leq 322\,\mathrm{km}\).

Non-subduction earthquakes in NE India (thrust): (43 SMA records, 0 SRR records), 3 earthquakes with \(5.2 \leq M \leq 5.9\), focal depths \(33\leq h \leq 49\,\mathrm{km}\) and \(6 \leq R \leq 243\,\mathrm{km}\).

Subduction earthquakes in NE India: (33 SMA records, 104 SRR records), 1 earthquake with \(M=7.3\), focal depth \(h=90\,\mathrm{km}\) and \(39 \leq R \leq 772\,\mathrm{km}\).

Bihar-Nepal earthquake in Indo-Gangetic plains (strike-slip): (0 SMA records, 38 SRR records), 1 earthquake with \(M=6.8\), focal depth \(h=57\,\mathrm{km}\) and \(42 \leq R \leq 337\,\mathrm{km}\).

Limited details of fault ruptures so use epicentral distance.

Use epicentral locations which give best correlation between distance and PGA.

Find PGA not well predicted by earlier equations.

Simple model and regression method because of limited data.

Remove one PGA value from category b equation because significantly affecting equation and because epicentral location only approximate.

Constrain \(b_3\) for category b equation to zero because otherwise positive.

Category c originally contained another earthquake (14 SMA records, \(M=6.1\), \(200 \leq d \leq 320\,\mathrm{km}\)) but gave very small \(b_2\) so exclude it.

Equations for category c and category d have \(b_2\) equal to zero because only one earthquake.

Find considerable differences between predicted PGA in different regions.

Note lack of data hence use equations only as first approximation.

Ground-motion model is: \[\log_{10} y = a M-b x -\log (x+c 10^{d M}) +e h +S_k\] where \(h\) is focal depth, \(y\) is in \(\,\mathrm{cm/s^2}\), \(a=0.578\), \(b=0.00355\), \(e=0.00661\), \(S=-0.069\), \(S_R=-0.210\), \(S_H=-0.114\), \(S_M=0.023\), \(S_S=0.237\) and \(\sigma_T=\sqrt{\sigma^2+\tau^2}\) where \(\sigma=0.213\) and \(\tau=0.162\).

Use four site categories (most data from medium and hard soils):

Rock

Hard soil

Medium soil

Soft soil

\(S\) is the mean site coefficient, i.e. when do not consider site category.

Records interpolated in frequency domain from \(0.02\) to \(0.005\,\mathrm{s}\) interval and displacement time history calculated using a fast Fourier transform (FFT) method having perpended to beginning and appended to end at least \(5\,\mathrm{s}\) of zeros to record. Number of samples in FFT is large enough that duration used in FFT is at least twice that of selected duration for processing window so that numerical errors are small. Bandpass Ormsby filter used, with limits \(0.2\) and \(24.5\,\mathrm{Hz}\), and displacement time history plotted. If displacement in pre- and appended portions is large then increase lower frequency limit in filter until displacements are small, using smoothed Fourier spectral amplitudes from \(0.05\) to \(25\,\mathrm{Hz}\) to make choice.

Most earthquakes are intra-slab.

Note lack of near-field data for all magnitudes, most data from \(>100\,\mathrm{km}\), therefore use coefficients, \(c\) and \(d\), from an early study.

Excludes data from distances greater than the distance at which an earlier study predicts \(\mathrm{PGA}<0.02\,\mathrm{m/s^2}\).

Consider residuals of earthquakes in western Japan (a small subset of data) and find small difference in anelastic coefficient and focal depth coefficient but note may be due to small number of records or because type of source not modelled.

Note model predicts intraslab motions well but significantly over predicts interface motions.

Plots site correction factors (difference between individual site factor and mean factor for that category) and find rock sites have largest variation, which suggest due to hard and soft rock included.

Examine residual plots. Find no significant bias.

Ground-motion model is: \[\log a_m=C_0'+C_1M+C_2 \Delta +C_3 \log \Delta +C_4'S_r\] where \(a_m\) is in unknown unit, \(\Delta=\sqrt{\mathrm{DE}^2+H^2+S^2}\), \(\mathrm{DE}\) is epicentral distance, \(H\) is focal depth, \(S\) is fault area and \(C_0'=-1.23\), \(C_1=0.068\), \(C_2=-0.001\) and \(C_3=-0.043\) (\(\sigma\) is not given). Note that there are typographical inconsistencies in the text, namely \(S_r\) maybe should be replaced by \(S_{al}\).

Use two site categories (based on Argentinean seismic code):

Stiff soil (II\(_{\mathrm{A}}\)).

Intermediate stiff soil (II\(_{\mathrm{B}}\)).

Since there is no geotechnical data available, classify sites, assuming a uniform surface layer, using the predominant period of ground motions estimated using Fourier spectra to get an equivalent shear-wave velocity (mainly these are between \(100\) and \(400\,\mathrm{m/s}\)).

Records from instruments located in basements or ground floors of relatively small buildings.

Records from SMAC and SMA-1 instruments.

Uniform digitisation and correction procedure applied to all records to reduce noise in high and low frequency range.

Calculate fault area using \(\log S=M_s+8.13-0.6667\log (\sigma {\Delta \sigma}/\mu)\) where \({\Delta \sigma}\) is stress drop, \(\sigma\) is average stress and \(\mu\) is rigidity.

Most magnitudes between \(5.5\) and \(6.0\).

Most records from \(\mathrm{DE}<100\,\mathrm{km}\).

Most focal depths, \(H\leq 40\,\mathrm{km}\). One earthquake with \(H=120\,\mathrm{km}\).

Use weighted regression because of a correlation between magnitude and distance of \(0.35\). Weight each record by \(\omega_i=(\omega_M+\omega_{\mathrm{DH}})/2\) where (note there are typographical errors in formulae in paper): \[\begin{aligned} \omega_M&=&\frac{n_s(i_s) {\Delta M(n_i)} n_e (n_i,i_s) {\Delta M_T}}{n_{\mathrm{cat}}}\\ \omega_{\mathrm{DH}}&=&\frac{n_s (i_s) {\Delta \log \mathrm{DH}(n_i)} n_e(n_i,i_s) {\Delta \log \mathrm{DH}_T}}{n_{\mathrm{cat}}}\\ {\Delta M_T}&=&\frac{\sum{\Delta M(n_i)}}{n_{\mathrm{cat}}}\\ {\Delta \log \mathrm{DH}_T}&=&\frac{\sum{\Delta \log \mathrm{DH}(n_i)}}{n_{\mathrm{cat}}}\end{aligned}\] where \({\Delta M(n_i)}\) is the width of the \(n_i\)th magnitude interval and \({\Delta \log \mathrm{DH}(n_i)}\) is the width of the \(n_i\)th distance interval, \(n_{\mathrm{cat}}\) is total number of intervals, \(n_i\) the index of the interval, \(n_e(n_i,i_s)\) is the number of records in interval \(n_i\) from site classification \(i_s\) and \(n_s\) is the number of records from site classification \(i_s\). Use two site classifications, three magnitude intervals and four epicentral distance intervals so \(n_{\mathrm{cat}}=2 \times 3 \times 4=24\).

First do regression on \(\log a_i=C_0+C_1 M+C_2 \Delta +C_3 \log \Delta\) and then regress residuals, \(\epsilon_i\), against \(C_4 S_r+C_5 S_{al}\) where \(S_{al}=1\) if site is intermediate stiff soil and \(S_{al}=0\) otherwise. Then \(C_0'=C_0+C_5\) and \(C_4'=C_4+C_5\). Similar method to that used by N. N. Ambraseys, Simpson, and Bommer (1996).

Ground-motion model is: \[\ln Y=a+b M+c\ln \sqrt{R^2+h^2}+dS_B+eS_C+g\mathrm{FM}\] where \(Y\) is in \(\,\mathrm{cm/s^2}\) and when using \(M_w\): \(a=0.920\), \(b=1.128\), \(c=-0.997\), \(h=8.839\), \(d=0.643\), \(e=0.088\), \(g=-0.196\) and \(\sigma=0.647\).

Use 3 site classes:

Rock and stiff soil, \(V_s>800\,\mathrm{m/s}\). \(S_B=S_C=0\).

Shallow loose deposits, \(V_s<400\,\mathrm{m/s}\). \(S_B=1\), \(S_C=0\).

Deep medium-dense deposits, \(400\leq V_s \leq 800\,\mathrm{m/s}\). \(S_C=1\), \(S_B=0\).

Consider 2 mechanisms:

Thrust and strike-slip faulting.

Normal faulting.

Only select earthquakes recorded by \(\geq 3\) stations.

Exclude non-free-field stations and those with unknown site conditions.

Use \(r_{epi}\) because surface geology in Italy and Greece rarely shows evident seismogenic faults.

Note differences in the coefficients depending on whether \(M_w\) or \(M_s\) is used.

Based on Sharma (1998), see 2.162.

\(A\) is in \(\,\mathrm{g}\) and coefficients are: \(c_1=-2.87\), \(c_2=0.634\), \(c_3=0.62\), \(b=1.16\) and \(\sigma=0.142\).

Fit \(\log A=-b \log X+c\) to data from each earthquake separately and find average \(b\) equal to \(1.18\). Then fit \(\log A=aM-b\log X+c\) to data from all earthquakes and find \(b=0.405\). Fit \(\log A=-b \log X+\sum d_i l_i\) to all data, where \(l_i=1\) for \(i\)th earthquake and \(0\) otherwise and find \(b=1.16\), use this for rest of analysis.

Ground-motion model is: \[\begin{aligned} \log Y&=&a+b M-\log R+d R\\ \mbox{where }R&=&\sqrt{D^2+h^2}\end{aligned}\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(a=0.72\), \(b=0.44\), \(d=-0.00231\), \(h=4.5\) and \(\sigma=0.28\).

Records from soil or alluvium sites.

All records corrected.

Note that scatter can be reduced by increasing number of records used (especially in near field), improving all seismological and local site parameters and increasing number of variables (especially in near field and those modelling local site behaviour) but that this requires much more information than is available.

Ground-motion model is: \[\log_{10}[y]=aM-bx-\log_{10}(x+c10^{dM})+e(h-h_c)\delta_h+S_k\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(a=0.446\), \(b=0.00350\), \(c=0.012\), \(d=0.446\), \(e=0.00665\), \(S=0.941\), \(S_R=0.751\), \(S_H=0.901\), \(S_M=1.003\), \(S_S=0.995\), \(\sigma_T=\sqrt{\sigma^2+\tau^2}\) where \(\sigma=0.135\) (intra-event) and \(\tau=0.203\) (inter-event), \(h_c\) is chosen as \(20\,\mathrm{km}\) because gave positive depth term.

Use four site categories:

Rock

Hard soil

Medium soil

Soft soil

Note site conditions for many stations are uncertain. \(S\) is the mean site term for all data.

Note ISC focal depths, \(h\), significant reduce prediction errors compared with JMA depths. \(\delta_h=1\) for \(h \geq h_c\) and \(\delta_h=0\) otherwise.

Most Japanese data from \(x>50\,\mathrm{km}\).

Use 166 Californian and Chilean (from 2 earthquakes) records to control model in near source.

Due to lack of multiple records from many sites and because \(c\) and \(d\) require near-source records use a maximum likelihood regression method of two steps. Firstly, find all coefficients using all data except those from sites with only one record associated with them and unknown site class. Next, use individual site terms for all sites so as to reduce influence of uncertainty because of approximate site classifications and find \(a\), \(b\), \(e\) and site terms using \(c\) and \(d\) from first step.

Intra-event and inter-event residuals decrease with increasing magnitude.

Conclude variation in residuals against distance is due to small number of records at short and large distances.

Individual site factors means prediction error propagates into site terms when number of records per station is very small.

Note model may not be suitable for seismic hazard studies because model prediction errors are partitioned into \(\sigma_T\) and mean site terms for a given site class. Suitable model can be derived when accurate site classifications are available.

Ground-motion model is: \[\log Y=C+(\alpha+\beta M)\log (R+R_0)\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(C=4.053\), \(\alpha=-2.797\), \(\beta=0.251\), \(R_0=8.84\) and \(\sigma=0.257\).

Use same data as Joyner and Boore (1981), see SectionÂ 2.38.

Use a two-stage method based on Joyner and Boore (1981). Firstly fit data to \(\log Y=C+\sum_{i=1}^n (a_i E_i)\log (R_i+R_0)\), where \(E_i=1\) for records from \(i\)th earthquake and \(E_i=0\) otherwise, to find \(C\) and \(a_i\) for each earthquake. Next fit \(a=\alpha+\beta M\) to find \(\alpha\) and \(\beta\) using \(a_i\) from first stage.

Ground-motion model is: \[\log A=a_1+a_2M+a_3\log[R+a_4\exp(a_5M)]\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(a_1=2.304\), \(a_2=0.747\), \(a_3=-2.590\), \(a_4=2.789\) and \(a_5=0.451\) (\(\sigma\) is unknown).

Ground-motion model for shallow crustal earthquakes is: \[\ln A=c_1+c_2M-c_3\ln D_p -(c_4-c_5D_p) \ln D_e\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(c_1=2.8096\), \(c_2=0.8993\), \(c_3=0.4381\), \(c_4=1.0954\), \(c_5=0.0079\) and \(\sigma=0.60\).

Ground-motion model for subduction earthquakes is: \[\ln A=c'_1+c'_2M-c'_3 \ln D_p-c'_4\ln D_h\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(c'_1=4.7141\), \(c'_2=0.8468\), \(c'_3=0.17451\), \(c'_4=1.2972\) and \(\sigma=0.56\).

Note that there is limited site information available for strong-motion stations in Taiwan so do not consider local site effects.

Use strong-motion data from Central Weather Bureau from 1994 to 1998 because it is more numerous and of better quality than older data.

Separate earthquakes into shallow crustal and subduction earthquakes because of different seismic attenuation and seismogenic situation for the two types of earthquake.

Shallow crustal earthquakes are mostly due to continental deformation, shallow collision or back-arc opening or are the uppermost interface earthquakes. Focal depths depth between \(1.1\) and \(43.7\,\mathrm{km}\) with most shallower than \(20\,\mathrm{km}\). Most records from earthquakes with \(4.5\leq M_w \leq 6.0\).

Subduction earthquakes are located in the Wadati-Benioff zone or the deep lateral collision zone and are principally intraslab. Focal depth between \(39.9\) and \(146.4\,\mathrm{km}\).

Do not use records from earthquakes associated with coseismic rupture because they have complex near-field source effects.

To avoid irregularly large amplitudes at great distances reject distant data predicted to be less than trigger level plus \(1\) standard deviation using this threshold formula: \(a M_w-b \ln D +c \geq \ln V\), where \(V\) is geometric mean of PGA equal to threshold plus \(1\) standard deviation. For shallow crustal earthquakes: \(a=0.64\), \(b=0.83\), \(c=2.56\) and \(V=6.93\) and for subduction earthquakes: \(a=0.76\), \(b=1.07\), \(c=3.13\) and \(V=6.79\).

For shallow crustal earthquakes examine effect of focal depth on seismic attenuation by finding geometric attenuation rate using epicentral distance, \(D_e\), for earthquakes with \(5\,\mathrm{km}\) depth intervals. Find that deeper earthquakes have slower attenuation than shallow earthquakes. Therefore assume ground motion, \(A\), is product of \(f_{\mathrm{source}}\) (source effects) and \(f_{\mathrm{geometrical-spreading}}\) (geometrical spreading effects) where

\(f_{\mathrm{source}}=C_1 \exp(c_2 M)/D_p^{-c_3}\) and \(f_{\mathrm{geometrical-spreading}}=D_e^{-(c_4-c_5D_p)}\) where \(D_p\) is focal depth.For subduction earthquakes examine effect of focal depth in the same way as done for shallow crustal earthquakes but find no effect of focal depth on attenuation rate. Therefore use \(f_{\mathrm{geometrical-spreading}}=D_h^{-c_4}\).

Plot residuals of both equations against distance and find no trend.

Note that it is important to separate subduction and shallow crustal earthquakes because of the different role of focal depth and attenuation characteristics.

Plot residual maps of ground motion for Taiwan and find significant features showing the important effect of local structures on ground motion.

Cite various published models for Taiwan

^{17}.

Ground-motion model is: \[\log a_{\mathrm{max}}=c_1+c_2 M_L+c_3 \log \sqrt{c_4^2+D^2}\] where \(a_{\mathrm{max}}\) is in \(\,\mathrm{g}\), for horizontal PGA \(c_1=-1.300 \pm 0.192\), \(c_2=0.331 \pm 0.040\), \(c_3=-1.152 \pm 0.099\), \(c_4=11.8 \pm 4.8 \,\mathrm{km}\) and \(\sigma=0.311\) and for vertical PGA \(c_1=-1.518 \pm 0.293\), \(c_2=0.302 \pm 0.035\), \(c_3=-1.061 \pm 0.096\), \(c_4=11.0 \pm 5.5\) and \(\sigma=0.313\).

Records from 39 sites. Records from instruments on ground floor or in basements of relatively small structures.

Site information only available for a small portion of the recording sites and therefore is not considered. Believe that most sites are ârockâ or âstiff soilâ.

All records from Kinemetrics SMA-1s.

Select records with \(M_L\geq 4.5\) and \(D\leq 200\,\mathrm{km}\) because of poor reliability of SMA-1 records for small earthquakes and to avoid problems related to a possible change of geometrical spreading when surface waves start to dominate over body waves at large distances.

Bandpass filter with passbands selected for which signal-to-noise ratio is \(>1\). Widest passband is \(0.07\)â\(25\,\mathrm{Hz}\).

Do not use \(r_{jb}\) because do not accurately know causative fault geometry for majority of events.

Do not include an anelastic decay term because data is inadequate to independently determine geometric and anelastic coefficients.

Note correlation between magnitude and distance in data distribution therefore use two-stage regression. Because many earthquakes have only a few records data is divided into classes based on magnitude (details not given).

Most data from \(M_L<5.5\), particularly data from \(D<20\,\mathrm{km}\).

Find all coefficients significantly different than \(0\) at levels exceeding \(0.999\).

Also regress using one-stage method and find practically equal coefficients and larger standard errors.

Find residuals are approximately lognormally distributed with slight asymmetry showing longer tail on positive side. Relate this to site amplification at some stations balanced by larger than expected number of slightly negative residuals.

Find no distance or magnitude dependence of residuals.

Compute ratio between larger and average horizontal component as \(1.15\).

Believe that higher than normal \(\sigma\) is due to lack of consideration of site effects and due to the use of \(r_{epi}\) rather than \(r_{jb}\).

Ground-motion model is: \[\log \mathrm{PSA}(f)=a(f)M+b(f)R-\log R+c(i,f)\] where \(\mathrm{PSA}(f)\) is in \(\,\mathrm{cm/s^2}\), \(a(f)=3.71\times 10^{-1}\), \(b(f)=-2.54\times 10^{-3}\), \(c(A,f)=0.617\), \(c(B,f)=0.721\), \(c(C,f)=0.845\), \(c(D,f)=0.891\) and \(\sigma=3.13\times 10^{-1}\).

Use four site categories, based on \(V_{s,30}\) (average shear-wave velocity in top \(30\,\mathrm{m}\)) as proposed in Eurocode 8:

\(V_{s,30}>800\,\mathrm{m/s}\). Use \(c(A,f)\). 14 records.

\(400<V_{s,30}\leq 800\,\mathrm{m/s}\). Use \(c(B,f)\). 856 records.

\(200<V_{s,30} \leq 400\,\mathrm{m/s}\). Use \(c(C,f)\). 1720 records.

\(100<V_{s,30} \leq 200\,\mathrm{m/s}\). Use \(c(D,f)\). 421 records.

Good determination of site conditions between shear-wave velocities have been measured down to \(10\) to \(20\,\mathrm{m}\) at every site. Extrapolate shear-wave velocity data to \(30\,\mathrm{m}\) to find \(V_{s,30}\). \(V_{s,30}\) at stations is between about \(50\,\mathrm{m/s}\) and about \(1150\,\mathrm{m/s}\).

Use data from Kyoshin network from 1996, 1997 and 1998.

All data from free-field sites.

No instrument correction needed or applied.

Use data from earthquakes with \(M_{\mathrm{JMA}}>3.5\) and focal depth \(<20\,\mathrm{km}\) because want to compare results with N. N. Ambraseys, Simpson, and Bommer (1996) and Boore, Joyner, and Fumal (1997). Also this criteria excludes data from deep subduction earthquakes and data that is not significant for seismic hazard studies.

Homogeneous determination of JMA magnitude and hypocentral distance.

Roughly uniform distribution of records with magnitude and distance.

Assume pseudo-spectral acceleration for \(5\%\) damping at \(0.02\,\mathrm{s}\) equals PGA.

Note equation valid for \(3.5 \leq M_{\mathrm{JMA}} \leq 6.3\) and \(10\leq r_{hypo} \leq 200\,\mathrm{km}\).

Find inclusion of site classification has reduced standard deviation.

Ground-motion model is: \[\log(A_{\mathrm{max}})=a M_s+b \log R+c\] where the units of \(A_{\mathrm{max}}\) are not given

^{18}, \(a=0.444\), \(b=-2.254\) and \(c=4.059\) (\(\sigma\) is not given).Use one site category: firm ground.

Ground-motion model is: \[\log_{10}(Y)=C_1+C_2M_w-\log_{10}(r_{\mathrm{rup}}+h)+C_3r_{\mathrm{rup}}\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(C_1=0.00215\), \(C_2=0.581\), \(C_3=-0.00414\), \(h=0.00871 \times 10^{0.5M_w}\) from the square root of the expected rupture area and \(\sigma=0.79\) (in terms of natural logarithms not common logarithms).

Select data from events with \(M_L>5\) and focal depths \(<35\,\mathrm{km}\) to restrict interest to large shallow earthquakes, which cause most damage.

Focal depths between \(1.40\) and \(34.22\,\mathrm{km}\).

Relocate events using available data.

Develop empirical relationship to convert \(M_L\) to \(M_w\).

Develop relation for use in near real-time (within \(2\,\mathrm{min}\)) mapping of PGA following an earthquake.

Select records from the Taiwan Rapid Earthquake Information Release System (TREIRS) and records from the TSMIP if \(r_{\mathrm{rup}}<30\,\mathrm{km}\) so as not to bias the results at larger distances by untriggered instruments.

Most data from \(50\leq d_r \leq 200\,\mathrm{km}\) and \(5 \leq M_w \leq 6\).

Compute site correction factors for TSMIP stations (since these sites have not been well classified), \(S\), by averaging residuals between observed and predicted values. After applying these site amplifications in regression analysis obtain reduced \(\sigma\) of \(0.66\).

Display inter-event residuals w.r.t. \(M_w\) before and after site correction.

Ground-motion model is: \[\log_{10} \mathrm{PGA}=\theta_0+\theta_1 M+\theta_2 M^2+\theta_3 R+\theta_4 \log_{10} (R+\theta_5 10^{\theta_6 M})\] where \(\mathrm{PGA}\) is in \(\,\mathrm{cm/s^2}\), \(\theta_0=-4.366 \pm 2.020\), \(\theta_1=2.540 \pm 0.714\), \(\theta_2=-0.172 \pm 0.0611\), \(\theta_3=0.00173 \pm 0.000822\), \(\theta_4=-1.845 \pm 0.224\), \(\theta_5=0.0746 \pm 0.411\), \(\theta_6=0.221 \pm 0.405\), \(\sigma_e^2=0.0453 \pm 0.0113\) (earthquake-specific variance), \(\sigma_s^2=0.0259 \pm 0.00699\) (site-specific variance) and \(\sigma_r^2=0.0297 \pm 0.00235\) (record-specific variance). \(\pm\) signifies the estimated standard errors.

Records from 45 stations on rock and firm soil. All sites have more than two records.

Use a new estimation procedure where the residual variance is decomposed into components due to various source of deviations. Separate variance into earthquake-to-earthquake variance, site-to-site variance and the remainder.

Proposed method does not require additional regression or searching procedures.

Perform a simulation study and find proposed procedure yields estimates with smaller biases and take less computation time than do other similar regression techniques.

Visually examine the equation for various magnitude values before regressing.

Ground-motion model is (their model D): \[\begin{aligned} \ln \mathrm{GM}&=&\theta_1+\theta_2 M+(\theta_3+\theta_4M)\ln [D+\exp(\theta_5)]+\theta_6 (1-S)+\theta_7 (M-6)^2+\theta_8 F\\ &&{}+\theta_9/\tanh(D+\theta_{10})\end{aligned}\] where \(\mathrm{GM}\) is in \(\,\mathrm{g}\), \(\theta_1=4.31964\), \(\theta_2=-0.00175\), \(\theta_3=-2.40199\), \(\theta_4=0.19029\), \(\theta_5=2.14088\), \(\theta_6=0.09754\), \(\theta_7=-0.21015\), \(\theta_8=0.38884\), \(\theta_9=-2.29732\), \(\theta_{10}=448.88360\), \(\sigma=0.5099\) (intra-event) and \(\tau=0.4083\) (inter-event) for horizontal PGA using the static dataset without the Chi-Chi data and \(\theta_1=1.50813\), \(\theta_2=0.15024\), \(\theta_3=-2.52562\), \(\theta_4=0.17143\), \(\theta_5=2.12429\), \(\theta_6=0.10517\), \(\theta_7=-0.16655\), \(\theta_8=0.22243\), \(\theta_9=-0.11214\), \(\theta_{10}=19.85830\), \(\sigma=0.5141\) (intra-event) and \(\tau=0.4546\) (inter-event) for vertical PGA using the static dataset without the Chi-Chi data. Coefficients are also given for the three other models and for both the dynamic and the static datasets but are not reported here due to lack of space.

Use two site categories:

Soil: includes sites located on deep broad and deep narrow soil deposits.

Rock: includes sites that are located on shallow stiff soil deposits;

Use three rupture mechanism categories:

Strike-slip, 39 earthquakes, 387 records;

Reverse/oblique, 13 earthquakes, 194 records;

Thrust, 16 earthquakes, 412 records.

Process records using two procedures as described below.

Use the standard PEER procedure with individually chosen filter cut-offs.

Fit the original integrated velocity time-history with three different functional forms (linear in velocity; bilinear, piecewise continuous function; and quadratic in velocity). Choose the âbest-fitâ result and view it for reasonableness. Differentiate the velocity time-history and then low-pass filter with a causal Butterworth filter with cut-offs about \(50\,\mathrm{Hz}\).

PGA values from the two processing techniques are very similar.

Investigate using a nonlinear model for site response term but the resulting models did not improve the fit.

Also try three other functional forms: \[\begin{aligned} \ln (\mathrm{GM})&=&\theta_1+\theta_2M+\theta_3 \ln [D+\theta_4 \exp(\theta_5 M)]+\theta_6 (1-S)+\theta_7 F\\ \ln (\mathrm{GM})&=&\theta_1+\theta_2M+(\theta_3+\theta_4M)\ln [D+\exp(\theta_5)]+\theta_6 (1-S)+\theta_7 (M-6)^2+\theta_8 F\\ \ln (\mathrm{GM})&=&\theta_1+\theta_2 M+\theta_3 \ln [D+\exp(\theta_5 M)]+\theta_6 (1-S)+\theta_7 F + \theta_8/\tanh(D+\theta_9)\end{aligned}\] which all give similar standard deviations and predictions but prefer model D.

Models oversaturate slightly for large magnitudes at close distances. Therefore recommend that the PGA equations are not used because this oversaturation is based on very little data.

Because the Chi-Chi short period ground motions may be anomalous relative to California they develop equations including and excluding the Chi-Chi data, which only affects predictions for large magnitudes (\(M>7.5\)).

Ground-motion model is: \[\begin{aligned} \ln Y&=&b_1+b_2(M-6)+b_3(M-6)^2+b_5\ln r+b_V \ln(V_S/V_A)\\ \mbox{where }r&=&(r_{\mathrm{cl}}^2+h^2)^{1/2}\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(b_1=-0.682\), \(b_2=0.253\), \(b_3=0.036\), \(b_5=-0.562\), \(b_V=-0.297\), \(V_A=1381\), \(h=4.48\) and \(\sigma=0.562\).

Use three site categories:

Average shear-wave velocity, \(V_S\), is \(200\,\mathrm{m/s}\). 40 records.

Average shear-wave velocity, \(V_S\), is \(400\,\mathrm{m/s}\). 24 records.

Average shear-wave velocity, \(V_S\), is \(700\,\mathrm{m/s}\). 29 records.

Actual shear-wave velocities and detailed site descriptions are not available for most stations in Turkey. Therefore estimate site classification by analogy with information in similar geologic materials. Obtain type of geologic material in number of ways: consultation with geologists at Earthquake Research Division of Ministry of Public Works and Settlement, various geological maps, past earthquake reports and geological references prepared for Turkey.

Only used records from small earthquakes recorded at closer distances than large earthquakes to minimize the influence of regional differences in attenuation and to avoid the complex propagation effects coming from longer distances.

Only use records from earthquakes with \(M_w\gtrsim 5.0\) to emphasize ground motions of engineering significance and to limit analysis to more reliably recorded earthquakes.

During regression lock magnitudes within \(\pm 0.25\) magnitude unit bands centred at halves or integer magnitudes to eliminate errors coming from magnitude determination.

Note that use of epicentral distance for small earthquakes does not introduce significant bias because dimensions of rupture area of small earthquakes are usually much smaller than distance to recording stations.

Examine peak ground motions from the small number of normal- (14 records) and reverse-faulting (6 records) earthquakes in set and find that they were not significantly different from ground motions from strike-slip earthquakes (73 records). Therefore combine all data.

Records mainly from small buildings built as meteorological stations up to three stories tall. Note that this modifies the recorded accelerations and hence increases the uncertainty.

Exclude data from aftershocks (mainly of the Kocaeli and Duzce earthquakes) because it was from free-field stations and did not want to mix it with the data from the non-free-field records.

Exclude a few records for which PGA of mainshock is \(\lesssim 0.04\,\mathrm{g}\).

Note that there is limited data and the data is poorly distributed. Also note that there is near-total lack of knowledge of local geology and that some of the records could be affected by the building in which the instrument was housed.

More than half the records (49 records, 53% of total) are from two \(M_w>7\) earthquakes (Kocaeli and Duzce) so the results are heavily based on the ground motions recorded in these two earthquakes.

Ground-motion model is: \[\log A_{max}=a_1+a_2 M_w-\log R+a_3 R\] where \(A_{max}\) is in \(\,\mathrm{gal}\), \(a_1=-0.148\), \(a_2=0.623\), \(a_3=-0.0032\) and \(\sigma=0.273\).

Focal depths between \(40\) and \(65\,\mathrm{km}\).

Compare predictions with data from Copalillo (\(M_w 5.9\)) earthquake and find good match.

Compare predictions and observations at Ciudad Universitaria station and find good match (bias of \(-0.013\) and standard deviation of \(0.25\) for \(\log A_{max}\).

Ground-motion model is: \[Y=C_1 \exp(C_2M)((R+C_3 \exp(C_4 M))^{C_5})+C_6 S\] where \(Y\) is in \(\,\mathrm{g}\), \(C_1=0.040311\), \(C_2=0.417342\), \(C_3=0.001\), \(C_4=0.65\), \(C_5=-0.351119\) and \(C_6=-0.035852\) for horizontal PGA and \(C_1=0.0015\), \(C_2=0.8548\), \(C_3=0.001\), \(C_4=0.4\), \(C_5=-0.463\) and \(C_6=0.0006\) for vertical PGA.

Uses two site categories:

Rock, site categories I and II of Iranian building code.

Soil, site categories III and IV of Iranian building code.

Selection criteria are: i) causative earthquake, earthquake fault (if known) and respective parameters are determined with reasonable accuracy, ii) PGA of at least one component \(>50\,\mathrm{gal}\), iii) records from free-field conditions or ground level of low-rise buildings (< three stories), iv) some aftershocks have been eliminated to control effect of a few large earthquakes and v) records have been processed with acceptable filter parameters.

Regresses directly on \(Y\) not on logarithm of \(Y\). Therefore does not calculate standard deviation in normal way. Considers the deviation of individual records from predictive equations as being PGA dependent. Finds that a sigmoidal model fits the data well. Therefore \(Y=(ab+cx^d)/(b+x^d)\) where \(Y\) is the error term and \(x\) is the predicted ground motion, \(a=0.038723\), \(b=0.00207\), \(c=0.29094\) and \(d=4.97132\) for horizontal PGA and \(a=0.00561\), \(b=0.0164\), \(c=0.1648\) and \(d=1.9524\) for vertical PGA.

Ground-motion model is: \[\ln Y=c_0+c_1 M_w+c_2 \ln (R+R_0)+c_3 S\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(c_0=4.16\), \(c_1=0.69\), \(c_2=-1.24\), \(R_0=6\), \(c_3=0.12\) and \(\sigma=0.70\).

Use three site categories:

NEHRP and UBC category B. 145 records.

NEHRP and UBC category C. 378 records.

NEHRP and UBC category D. 221 records.

Selection criteria are: a) earthquake has \(M_w \geq 4.5\), b) \(\mathrm{PGA} \geq 0.05\,\mathrm{g}\) and c) \(\mathrm{PGA}<0.05\,\mathrm{g}\) but another record from same earthquake has \(\mathrm{PGA} \geq 0.05\,\mathrm{g}\).

Records mainly from normal faulting earthquakes.

Exclude data recorded in buildings with four stories or higher.

Automatically digitize records and process records homogenously, paying special attention to the filters used.

Correlation between \(M_w\) and \(R\) in set of records used. For \(4.5 \leq M_w \leq 5.0\) records exist at \(R\leq 40\,\mathrm{km}\) and for larger magnitudes records exist at intermediate and long distances. For \(M_w>6.0\) there is a lack of records for \(R<20\,\mathrm{km}\).

Use a two step regression method. In first step use all records to find \(c_1\). In second step use records from earthquakes with \(M_w\geq 5.0\) to find \(c_0\), \(c_2\) and \(c_3\).

Adopt \(R_0=6\,\mathrm{km}\) because difficult to find \(R_0\) via regression due to its strong correlation with \(c_2\). This corresponds to average focal depth of earthquakes used.

Also try Ground-motion model: \(\ln Y=c'_0+c'_1M_w+c'_2 \ln (R^2+h_0^2)^{1/2}+c'_3 S\). Coefficients are: \(c'_0=3.52\), \(c'_1=0.70\), \(c'_2=-1.14\), \(h_0=7\,\mathrm{km}\) (adopted), \(c'_3=0.12\) and \(\sigma=0.70\).

Find no apparent trends in residuals w.r.t. distance.

Due to distribution of data, equations valid for \(5\leq R \leq 120\,\mathrm{km}\) and \(4.5\leq M_w\leq 7.0\).

Ground-motion model is unknown.

Ground-motion model is: \[\begin{aligned} \log_{10} a_{H(V)}&=&c_1+c_2M_L+c_4 \log_{10} (r)+c_R S_R+c_A S_A+c_S S_S\\ \mbox{where } r&=&\sqrt{R_e^2+h_0^2}\end{aligned}\] where \(a_{H(V)}\) is in \(\,\mathrm{g}\), \(c_1=-3.0815\), \(c_2=0.5161\), \(c_4=-0.9501\), \(c_R=-0.1620\), \(c_A=-0.1078\), \(c_S=0.0355\), \(h_0=2.0\) and \(\sigma=0.3193\) for horizontal PGA and \(c_1=-2.8053\), \(c_2=0.4858\), \(c_4=-1.1842\), \(c_R=-0.1932\), \(c_A=-0.0210\), \(c_S=0.0253\), \(h_0=2.5\) and \(\sigma=0.3247\) for vertical PGA.

Use three site categories:

Rock, subsoil classes A1, (A2) \(V_s>800\,\mathrm{m/s}\) (according to E DIN 4149) or subsoil class B (rock) \(760<V_s\leq 1500\,\mathrm{m/s}\) (according to UBC 97). \(S_R=1\), \(S_A=0\), \(S_S=0\). 59 records.

Stiff soil, subsoil classes (A2), B2, C2 \(350\leq V_s\leq800\,\mathrm{m/s}\) (according to E DIN 4149) or subsoil class C (very dense soil and soft rock) \(360<V_s\leq 760\,\mathrm{m/s}\) (according to UBC 97). \(S_A=1\), \(S_R=0\), \(S_S=0\). 88 records.

Soft soil, subsoil classes A3, B3, C3 \(V_s<350\,\mathrm{m/s}\) (according to E DIN 4149) or subsoil class D (stiff clays and sandy soils) \(180<V_s \leq 360\,\mathrm{m/s}\) (according to UBC 97). \(S_S=1\), \(S_R=0\), \(S_A=0\). 536 records.

KOERI stations classified using UBC 97 and temporary stations of German TaskForce classified using new German code E DIN 4149. Classify temporary stations of German TaskForce using microtremor H/V spectral ratio measurements by comparing shapes of H/V spectral ratios from microtremors to theoretical H/V spectral ratios as well as with theoretical transfer functions determined for idealized subsoil profiles.

Use Kocaeli aftershock records from temporary German TaskForce stations (records from earthquakes with \(1 \lesssim M_L<4.9\) and distances \(R_e<70\,\mathrm{km}\), 538 records) and from mainshock and aftershocks records from Kandilli Observatory (KOERI) stations (\(4.8 \leq M_L \leq 7.2\) and distances \(10\leq R_e \leq 250\,\mathrm{km}\), 145 records).

Visually inspect all time-histories and only use those thought to be of sufficiently good quality.

Baseline correct all records.

Use technique of N. N. Ambraseys, Simpson, and Bommer (1996) to find the site coefficients \(c_R\), \(c_A\) and \(c_S\), i.e. use residuals from regression without considering site classification.

Note that equations may not be reliable for rock and stiff soil sites due to the lack of data and that equations probably only apply for \(2\leq M_L \leq 5\) due to lack of data from large magnitude earthquakes.

Ground-motion model is: \[\ln \mathrm{PGA}=b'+b_M M+b_R \ln \left\{ \left[ \left( \frac{R_e}{\rho} \right)^2+h^2 \right]^{1/2} +C \right\}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{cm/s^2}\). For Bucharest azimuth \(b'=-0.21056\), \(b_M=1.29099\), \(b_R=-0.80404\), \(C=40\) and \(\sigma=0.52385\), for Valeni azimuth \(b'=-1.52412\), \(b_M=1.42459\), \(b_R=-0.70275\), \(C=40\) and \(\sigma=0.51389\) and for Cherna Voda \(b'=4.16765\), \(b_M=1.11724\), \(b_R=-1.44067\), \(C=40\) and \(\sigma=0.47607\).

Focal depths, \(h\), between \(89\) and \(131\,\mathrm{km}\).

Incomplete data on local site conditions so not included in study.

Some strong-motion records are not from free-field locations.

Uses \(\rho\) to characterise the non-homogeneity of region. Includes effect of instrument location w.r.t. the main direction of propagation of seismic energy, as well as the non-homogeneous attenuation in two orthogonal directions. \(\rho=\sqrt{(1+t g^2 \alpha)/(a^{-2}+tg^2 \alpha)}\) where \(\alpha\) is angle between instrument and main direction of seismic energy or direction of fault projection on surface and \(a\) is parameter defining the non-homogeneous attenuation in two orthogonal directions, or relation between the semi-axes of the ellipse of seismic field.

Uses a two step method. In first step derive equations for each earthquake using \(\ln \mathrm{PGA}=b'_0 + b_1 \ln (R_e/\rho)\). In the second step the complete Ground-motion model is found by normalizing separately for each earthquake with a value of \(\rho\) defined for that earthquake according to the location for which the equation was defined.

Notes that there is limited data so coefficients could be unreliable.

Strong-motion records processed by different institutions.

Ground-motion model is: \[\begin{aligned} \log y&=&C_1+C_2 M_s +C_4 \log r +C_A S_A+C_S S_S\\ \mbox{where } r&=&\sqrt{d^2+h_0^2}\end{aligned}\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(C_1=2.080\), \(C_2=0.214\), \(h_0=7.27\), \(C_4=-1.049\), \(C_A=0.058\), \(C_S=0.085\) and \(\sigma=0.27\).

Use three site categories:

Soft soil, \(V_{s,30}\leq 360\,\mathrm{m/s}\). \(S_S=1\), \(S_A=0\). 25% of records.

Stiff soil, \(360<V_{s,30}<750\,\mathrm{m/s}\). \(S_A=1\), \(S_S=0\). 50% of records.

Rock, \(V_{s,30} \geq 750\,\mathrm{m/s}\). \(S_S=0\), \(S_A=0\). 25% of records.

If no \(V_{s,30}\) measurements at station then use agency classifications.

Supplement dataset of Bommer et al. (1998) with 66 new records using same selection criteria as Bommer et al. (1998) with a lower magnitude limit of \(M_s=5.5\). Remove 3 records from Bommer et al. (1998) with no site classifications.

Roughly uniform distribution of records w.r.t. magnitude and distance. New data contributes significantly to large magnitude and near-field ranges.

Correct records using an elliptical filter selecting an appropriate low-frequency cut-off, \(f_L\), individually for each record using the criterion of Bommer et al. (1998).

Plot PGA against \(f_L\) for two pairs of horizontal components of ground motion from the BOL and DZC stations from the Duzce earthquake (12/11/1999). Record from BOL was recorded on a GSR-16 digital accelerograph and that from DZC was recorded on a SMA-1 analogue accelerograph. Find PGA is stable for low-frequency cut-offs up to at least \(0.4\,\mathrm{Hz}\) for the selected records.

Ground-motion model is: \[\log_{10}(Y)=a+bM+c\log_{10} (R^2+h^2)^{1/2}+e \Gamma\] where \(Y\) is in \(\,\mathrm{g}\), \(a=-1.632\), \(b=0.304\), \(c=-1\), \(h=2.7\), \(e=0\) and \(\sigma=0.275\).

Use two site categories:

\(V_{s,30}\leq 750\,\mathrm{m/s}\), \(\Gamma=0\).

\(V_{s,30}>750\,\mathrm{m/s}\), \(\Gamma=1\).

Note that amount of data available for the Umbria-Marche area in central Italy is sufficiently large to perform statistical analysis at regional scale.

Focal depths between \(2\) and \(8.7\,\mathrm{km}\). Exclude data from an earthquake that occurred at \(47\,\mathrm{km}\).

Select only records from earthquakes with \(M_L\geq 4.5\) recorded at less than \(100\,\mathrm{km}\).

Exclude data from Nocera Umbra station because it shows a strong amplification effect due to the presence of a sub-vertical fault and to highly fractured rocks.

Uniformly process records using BAP (Basic strong-motion Accelerogram Processing software). Instrument correct records and band-pass filter records using a high-cut filter between \(23\) and \(28\,\mathrm{Hz}\) and a bi-directional Butterworth low-cut filter with corner frequency of \(0.4\,\mathrm{Hz}\) and rolloff parameter of \(2\).

Note that can use \(M_L\) because it does not saturate until about \(6.5\) and largest earthquake in set is \(M_L=5.9\).

More than half of records are from earthquakes with \(M_L\leq 5.5\).

State that equations should not be used for \(M_L>6\) because of lack of data.

Use similar regression method as N. N. Ambraseys, Simpson, and Bommer (1996) to find site coefficient, \(e\).

Ground-motion model is (his model 2): \[\log(a)=A+BM+Cr+D\log(r)\] where \(a\) is in \(\,\mathrm{gal}\), \(A=5.5766\), \(B=0.06052\), \(C=0.0039232\), \(D=-2.524849\) and \(\sigma=0.2597\).

Due to lack of information classify stations as soil or rock (stations with \(\leq10\,\mathrm{m}\) of soil). Only derives equation for rock.

Uses data from National Accelerometer Network managed by INGEOMINAS from 1993 to 1999.

Exclude data from subduction zone, focal depths \(h>60\,\mathrm{km}\).

Focal depths, \(11.4 \leq h \leq 59.8\,\mathrm{km}\).

Exclude data from earthquakes with \(M_L<4.0\).

Exclude data with \(\mathrm{PGA}<5\,\mathrm{gal}\). \(5 \leq \mathrm{PGA} \leq 100.1\,\mathrm{gal}\).

Derive equations using four different models: \[\begin{aligned} a&=&C_1 \mathrm{e}^{C_2 M} (R+C_3)^{-C_4}\\ \log(a)&=&A+BM+Cr+D\log(r)\\ \log(y)&=&C_0+C_1 (M-6)+C_2(M-6)+C_3\log(r)+C_4 r\\ \ln(a)&=&a+bM+d \ln(R)+qh\end{aligned}\]

Ground-motion model is: \[\log A=b_0+b_1 M_c+b_r \log r\] where \(A\) is in \(\,\mathrm{g}\), \(b_0=-1.939\), \(b_1=0.278\), \(b_2=-0.858\) and \(\sigma=0.259\) for horizontal PGA and \(b_0=-2.367\), \(b_1=0.244\), \(b_2=-0.752\) and \(\sigma=0.264\) for vertical PGA.

Use two site categories: bedrock (\(S=0\)) and sediments (\(S=1\)) but found the coefficient \(b_3\) in the term \(+b_3 S\) is close to zero so repeat analysis constraining \(b_3\) to \(0\).

Records from SSA-1 instruments.

Carefully inspect and select records.

Do not use record from the Aqaba (\(M=7.2\)) earthquake because it is very far and was only recorded at one station.

Do not use records from buildings or dams because they are affected by response of structure.

Instrument correct records. Apply bandpass filter (\(0.1\) to \(25\,\mathrm{Hz}\)) to some low-quality records.

Do regression using only records from earthquakes with \(4.8 \leq M \leq 5.8\) and also using only records from earthquakes with \(3.5 \leq M \leq 4.5\).

Most data from \(M\leq 5\) and \(r \leq 100\,\mathrm{km}\).

Note that use a small set of records and so difficult to judge reliability of derived equation.

Ground-motion model is: \[\begin{aligned} \log Y&=&c_1+c_2 \mathbf{M}+c_3 h+c_4 R-g \log R+c_5 \mathrm{sl} S_C+c_6 \mathrm{sl} S_D+c_7 \mathrm{sl} S_E\\ \mbox{where } R&=&\sqrt{D_{\mathrm{fault}}^2+\Delta^2}\\ \Delta&=&0.00724\,10^{0.507 \mathbf{M}}\\ \mathrm{sl}&=&\left\{ \begin{array}{r@{\quad \mbox{for}\quad}l} 1 &\mathrm{PGA}_{rx} \leq 100\,\mathrm{cm/s}\mathrm{or} f\leq 1\,\mathrm{Hz}\\ 1-\frac{(f-1) (\mathrm{PGA}_{rx}-100)}{400} & 100<\mathrm{PGArx} < 500\,\mathrm{cm/s}\mathrm{\&} 1 \,\mathrm{Hz}<f <2\,\mathrm{Hz}\\ 1-(f-1) & \mathrm{PGA}_{rx}\geq 500\,\mathrm{cm/s}\mathrm{\&} 1 \,\mathrm{Hz}<f <2\,\mathrm{Hz}\\ 1-\frac{\mathrm{PGA}_{rx}-100}{400} & 100<\mathrm{PGArx} < 500\,\mathrm{cm/s}\mathrm{\&} f\geq 2\,\mathrm{Hz}\\ 0 & \mathrm{PGA}_{rx}\geq 500\,\mathrm{cm/s}\mathrm{\&} f\geq 2\,\mathrm{Hz})\\ \end{array} \right.\end{aligned}\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(f\) is frequency of interest, \(\mathrm{PGA}_{rx}\) is predicted PGA on NEHRP B sites, \(c_1=2.991\), \(c_2=0.03525\), \(c_3=0.00759\), \(c_4=-0.00206\), \(\sigma_1=0.20\) (intra-event) and \(\sigma_2=0.11\) (inter-event) for interface events and \(c_1=-0.04713\), \(c_2=0.6909\), \(c_3=0.01130\), \(c_4=-0.00202\), \(\sigma_1=0.23\) and \(\sigma_2=0.14\) for in-slab events and \(c_5=0.19\), \(c_6=0.24\), \(c_7=0.29\) for all events. \(g=10^{1.2-0.18 \mathbf{M}}\) for interface events and \(g=10^{0.301-0.01 \mathbf{M}}\) for in-slab events. Recommended revised \(c_1\) for interface events in Cascadia is \(2.79\) and in Japan \(3.14\), recommended revised \(c_1\) for in-slab events in Cascadia is \(-0.25\) and in Japan \(0.10\).

Use four site categories:

NEHRP site class B, \(V_{s,30}>760\,\mathrm{m/s}\). \(S_C=0\), \(S_D=0\) and \(S_E=0\).

NEHRP site class C, \(360<V_{s,30} \leq 760\,\mathrm{m/s}\). \(S_C=1\), \(S_D=0\) and \(S_E=0\).

NEHRP site class D, \(180\leq V_{s,30} \leq 360\,\mathrm{m/s}\). \(S_D=1\), \(S_C=0\) and \(S_E=0\).

NEHRP site class E, \(V_{s,30}<180\,\mathrm{m/s}\). \(S_E=1\), \(S_C=0\) and \(S_D=0\).

Stations in KNET were classified using shear-wave velocity profiles using an statistical method to extrapolate measured shear-wave velocities to depths up to \(10\)â\(20\,\mathrm{m}\) to \(30\,\mathrm{m}\). Stations in Guerrero array assumed to be on rock, i.e. site class B. Broadband stations in Washington and British Columbia sited on rock (\(V_{s,30} \approx 1100\,\mathrm{m/s}\)), i.e. site class B. Strong-motion stations in Washington classified using map of site classes based on correlations between geology and \(V_{s,30}\) in Washington, and verified at 8 stations using actual borehole measurements. Converted Youngs et al. (1997) Geomatrix classifications by assuming Geomatrix A=NEHRP B, Geomatrix B=NEHRP C, Geomatrix C/D=NEHRP D and Geomatrix E=NEHRP E using shear-wave velocity and descriptions of Geomatrix classification.

Note that cannot develop equations using only Cascadia data because not enough data. Combine data of Crouse (1991) and Youngs et al. (1997) with additional data from Cascadia (strong-motion and broadband seismographic records), Japan (KNET data), Mexico (Guerrero array data) and El Salvador data.

Classify event by type using focal depth and mechanism as:

All earthquakes with normal mechanism. Earthquakes with thrust mechanism at depths \(>50\,\mathrm{km}\) or if occur on steeply dipping planes.

Earthquakes with thrust mechanism at depths \(<50\,\mathrm{km}\) on shallow dipping planes.

Exclude events of unknown type.

Exclude events with focal depth \(h>100\,\mathrm{km}\).

Exclude events that occurred within crust above subduction zones.

Use many thousands of extra records to explore various aspects of ground motion scaling with \(M\) and \(D_{\mathrm{fault}}\).

Data relatively plentiful in most important \(M\)-\(D_{\mathrm{fault}}\) ranges, defined according to deaggregations of typical hazard results. These are in-slab earthquakes of \(6.5 \leq \mathbf{M} \leq 7.5\) for \(40 \leq D_{\mathrm{fault}} \leq 100\,\mathrm{km}\) and interface earthquakes of \(\mathbf{M} \geq 7.5\) for \(20 \leq D_{\mathrm{fault}} \leq 200\,\mathrm{km}\).

Data from KNET from moderate events at large distances are not reliable at higher frequencies due to instrumentation limitations so exclude KNET data from \(\mathbf{M}<6\) at \(D_{\mathrm{fault}}>100\,\mathrm{km}\) and for \(\mathrm{M} \geq 6\) at \(D_{\mathrm{fault}}>200\,\mathrm{km}\). Excluded data may be reliable at low frequencies.

Estimate \(D_{\mathrm{fault}}\) for data from Crouse (1991) and for recent data using fault length versus \(\mathbf{M}\) relations of Wells and Coppersmith (1994) to estimate size of fault plane and assuming epicentre lies above geometric centre of dipping fault plane. Verified estimates for several large events for which fault geometry is known.

Perform separate regressions for interface and in-slab events because analyses indicated extensive differences in amplitudes, scaling and attenuation between two types.

Experiment with a variety of functional forms. Selected functional form allows for magnitude dependence of geometrical spreading coefficient, \(g\); the observed scaling with magnitude and amplitude-dependent soil nonlinearity.

For \(h>100\,\mathrm{km}\) use \(h=100\,\mathrm{km}\) to prevent prediction of unrealistically large amplitudes for deeper earthquakes.

\(R\) is approximately equal to average distance to fault surface. \(\Delta\) is defined from basic fault-to-site geometry. For a fault with length and width given by equations of Wells and Coppersmith (1994), the average distance to the fault for a specified \(D_{\mathrm{fault}}\) is calculated (arithmetically averaged from a number of points distributed around the fault), then used to determine \(\Delta\). Magnitude dependence of \(R\) arises because large events have a large spatial extent, so that even near-fault observation points are far from most of the fault. Coefficients in \(\Delta\) were defined analytically, so as to represent average fault distance, not be regression. Although coefficients in \(\Delta\) were varied over a wide range but did not improve accuracy of model predictions.

Determine magnitude dependence of \(g\) by preliminary regressions of data for both interface and in-slab events. Split data into \(1\) magnitude unit increments to determine slope of attenuation as a function of magnitude using only \(1\) and \(2\,\mathrm{s}\) data and records with \(50 \leq D_{\mathrm{fault}} \leq 300\,\mathrm{km}\) (\(50\,\mathrm{km}\) limit chosen to avoid near-source distance saturation effects). Within each bin regression was made to a simple functional form: \(\log Y'=a_1+a_2 \mathbf{M}-g \log R+a_3S\) where \(Y'=Y \exp(0.001R)\), i.e. \(Y\) corrected for curvature due to anelasticity, and \(S=0\) for NEHRP A or B and \(1\) otherwise. \(g\) is far-field slope determined for each magnitude bin.

Nonlinear soil effects not strongly apparent in database on upon examination of residuals from preliminary regressions, as most records have \(\mathrm{PGA}<200\,\mathrm{cm/s^2}\), but may be important for large \(M\) and small \(D_{\mathrm{fault}}\). To determine linear soil effects perform separate preliminary regressions for each type of event to determine \(c_5\), \(c_6\) and \(c_7\) assuming linear response. Smooth these results (weighted by number of observations in each subset) to fix \(c_5\), \(c_6\) and \(c_7\) (independent of earthquake type) for subsequent regressions. \(\mathrm{sl}\) was assigned by looking at residual plots and from consideration of NEHRP guidelines. Conclude that there is weak evidence for records with \(\mathrm{PGA}_{rx}>100\,\mathrm{cm/s^2}\), for NEHRP E sites at periods \(<1\,\mathrm{s}\). Use these observations to fix \(\mathrm{sl}\) for final regression.

Final regression needs to be iterated until convergence because of use of \(\mathrm{PGA}_{rx}\) in definition of dependent variable.

To optimize fit for \(M\)-\(D_{\mathrm{fault}}\) range of engineering interest limit final regression to data within: \(5.5 \leq \mathbf{M}<6.5\) and \(D_{\mathrm{fault}}\leq 80\,\mathrm{km}\), \(6.5 \leq \mathbf{M}<7.5\) and \(D_{\mathrm{fault}} \leq 150\,\mathrm{km}\) and \(\mathbf{M}\geq 7.5\) and \(D \leq 300\,\mathrm{km}\) for interface events and \(6.0 \leq \mathbf{M}<6.5\) and \(D_{\mathrm{fault}} \leq 100\,\mathrm{km}\) and \(\mathbf{M} \geq 6.5\) and \(D_{\mathrm{fault}} \leq 200\,\mathrm{km}\) for in-slab events. These criteria refined by experimentation until achieved an optimal fit for events that are important for seismic hazard analysis. Need to restrict \(M\)-\(D_{\mathrm{fault}}\) for regression because set dominated by records from moderate events and from intermediate distances whereas hazard is from large events and close distances.

Lightly smooth coefficients (using a weighted 3-point scheme) over frequency to get smooth spectral shape and allows for reliable linear interpolation of coefficients for frequencies not explicitly used in regression.

In initial regressions, use a \(\mathbf{M}^2\) term as well as a \(\mathbf{M}\) term leading to a better fit over a linear magnitude scaling but lead to a positive sign of the \(\mathbf{M}^2\) rather than negative as expected. Therefore to ensure the best fit in the magnitude range that is important for hazard and constrained by data quadratic source terms refit to linear form. Linear model constrained to provide same results in range \(7.0 \leq \mathbf{M} \leq 8.0\) for interface events and \(6.5 \leq \mathbf{M} \leq 7.5\) for in-slab events. To ensure that non-decreasing ground motion amplitudes for large magnitudes: for \(\mathbf{M}>8.5\) use \(\mathrm{M}=8.5\) for interface events and for \(\mathbf{M}>8.0\) use \(\mathrm{M}=8.0\) for in-slab events.

Calculate \(\sigma\) based on records with \(\mathbf{M} \geq 7.2\) and \(D_{\mathrm{fault}} \leq 100\,\mathrm{km}\) for interface events and \(\mathbf{M} \geq 6.5\) and \(D_{\mathrm{fault}} \leq 100\,\mathrm{km}\) for in-slab events. These magnitude ranges selected to obtain the variability applicable for hazard calculations. Do not use KNET data when computing \(\sigma\) because data appear to have greater high-frequency site response than data from same soil class from other regions, due to prevalence of sites in Japan with shallow soil over rock.

Determine \(\sigma_1\) using data for several well-recorded large events and determining average value. Then calculate \(\sigma_2\) assuming \(\sigma=\sqrt{\sigma_1^2+\sigma_2^2}\).

Examine residuals w.r.t. \(D_{\mathrm{fault}}\) using all data from \(\mathbf{M} \geq 5.5\) and \(D_{\mathrm{fault}} \leq 200\,\mathrm{km}\) and \(\mathbf{M} \geq 6.5\) and \(D_{\mathrm{fault}} \leq 300\,\mathrm{km}\). Find large variability but average residuals near \(0\) for \(D_{\mathrm{fault}} \leq 100\,\mathrm{km}\).

Find significantly lower variability for \(\mathbf{M} \geq 7.2\) events (\(\sigma=0.2\)â\(0.35\) for larger events and \(\sigma=0.25\)â\(0.4\) for smaller events).

Examine graphs and statistics of subsets of data broken down by magnitude, soil type and region. Find significant positive residuals for \(\mathbf{M}<6.6\) due to use of linear scaling with magnitude. Accept positive residuals because small magnitudes do not contribute strongly to hazard.

Find large positive residuals for class C sites for interface events (most records are from Japan) whereas residuals for class C sites for in-slab events (which are from both Japan and Cascadia) do not show trend. No other overwhelming trends. Differences in residuals for Japan and Cascadia class C sites likely due to differences in typical soil profiles in the two regions within the same NEHRP class. Sites in Japan are typically shallow soil over rock, which tend to amplify high frequencies, whereas in Cascadia most soil sites represent relatively deep layers over rock or till. Provide revised \(c_1\) coefficients for Japan and Cascadia to model these differences.

Note that debate over whether 1992 Cape Mendocino earthquake is a subduction zone or crustal earthquake. Excluding it from regressions has a minor effect on results, reducing predictions for interface events for \(\mathbf{M}<7.5\).

Ground-motion model is: \[\begin{aligned} \log \mathrm{PGA}&=&\psi(\mathbf{M})-\log g(r) - \eta'(\mathbf{M}) r\\ \mbox{where}&&\\ \psi(\mathbf{M})&=&\psi_1+\psi_2 (\mathbf{M}-5.5) \quad \mbox{for} \quad \mathbf{M} \leq 5.5\\ &=&\psi_1+\psi_3 (\mathbf{M}-5.5) \quad \mbox{for} \quad \mathbf{M} > 5.5\\ \eta'(\mathbf{M})&=&\eta_1 \quad \mbox{for} \quad \mathbf{M}\leq 5.5\\ &=&\eta_1 \times 10^{\rho(\mathbf{M}-5.5)} \quad \mathbf{M}>5.5\\ g(r)&=&r \quad \mbox{for} \quad r \leq r_0=27.5\,\mathrm{km}\\ &=&r_0(r/r_0)^{0.7} \quad \mbox{for} \quad r>r_0=27.5\,\mathrm{km}\end{aligned}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{m/s^2}\), \(\psi_1=1.45 \pm 0.24\), \(\psi_2=1.00\pm 0.01\), \(\psi_3=0.31 \pm 0.09\), \(\eta_1=0.0073 \pm 0.0003\), \(\rho=-0.30 \pm 0.06\), \(\sigma_e=0.170\) (inter-earthquake) and \(\sigma_r=0.361\) (intra-earthquake).

Classify station into four classes using the NEHRP categories using geological maps:

Rock. Amplification from category C \(0.79\).

Soft rock or stiff soil. Amplification from category C \(1.00\).

Soft soil. Amplification from category C \(1.35\).

Bay mud. Amplification from category D \(1.64\).

The amplifications (from Boore, Joyner, and Fumal (1997)) are used to correct for site effects.

For some stations in the broadband Berkeley Digital Seismic Network, which are in seismic vaults and mine adits and therefore have low site amplifications, use one-half the above site amplifications.

Use data from August 1999 and December 2002 from the northern California ShakeMap set of data. Extend set to larger earthquakes by adding data from nine previous large northern California earthquakes.

Focal depths, \(0.1 \leq h \leq 28..8\,\mathrm{km}\).

Use hypocentral distance because this distance is available to ShakeMap immediately after an earthquake. Note that this is a poor predictor of near-field ground motion from extended faults.

Plot decay of PGA with distance for two moderate earthquakes (\(\mathbf{M}=4.9\), \(\mathbf{M}=3.9\)) and find decay is poorly fit by a power-law function of distance and that fitting such an equation who require \(\mathrm{PGA} \propto r^{-2}\), which they believe is physically unrealistic for body-wave propagation.

Find that PGAs flatten or even increase at large distances, which is believed to be due to noise. Hence use a magnitude-dependent limit of \(r_{\mathrm{max}}=100 (\mathbf{M}-2) \leq 400\,\mathrm{km}\), determined by inspecting PGA and PGV data for all events, to exclude problem data.

Fit data from each event separately using \(\log \mathrm{PGA}=\psi -\eta r -\log g(r)+\log s_{\mathrm{BJF}}\). Find \(\eta\) varies between four groups: events near Eureka triple junction, events within the Bay Area, events near San Juan Bautista and those in the Sierras and the western Mojave desert.

Use a numerical search to find the segmentation magnitude \(\mathbf{M}'\). Choose \(\mathbf{M}'=5.5\) as the segmentation magnitude because it is the lowest segmentation magnitude within a broad minimum in the \(\chi^2\) error for the regression.

Fit magnitude-dependent part of the equation to the PGA values scaled to \(10\,\mathrm{km}\) and site class C.

Note that the PGAs predicted are significantly higher than those given by equations derived by Joyner and Boore (1981) and Boore, Joyner, and Fumal (1997) because of use of hypocentral rather than fault distance.

Recompute site amplifications relative to category C as: for B \(0.84\pm 0.03\), for D \(1.35 \pm 0.05\) and for E \(2.17 \pm 0.15\).

Ground-motion model is: \[\log y=C_1+C_2M+C_4 \log (\sqrt{r^2+h^2})+C_A S_A+C_S S_S +C_N F_N+C_R F_R\] where \(y\) is in \(\,\mathrm{g}\), \(C_1=-1.482\), \(C_2=0.264\), \(C_4=-0.883\), \(h=2.473\), \(C_A=0.117\), \(C_S=0.101\), \(C_N=-0.088\), \(C_R=-0.021\), \(\sigma_1=0.243\) (intra-event) and \(\sigma_2=0.060\) (inter-event).

Use four site conditions but retain three (because only three records from very soft (L) soil which combine with soft (S) soil category):

Rock: \(V_s>750\,\mathrm{m/s}\), \(\Rightarrow S_A=0, S_S=0\), 106 records.

Stiff soil: \(360<V_s\leq750\,\mathrm{m/s}\), \(\Rightarrow S_A=1, S_S=0\), 226 records.

Soft soil: \(180<V_s\leq360\,\mathrm{m/s}\), \(\Rightarrow S_A=0, S_S=1\), 81 records.

Very soft soil: \(V_s\leq180\,\mathrm{m/s}\), \(\Rightarrow S_A=0, S_S=1\), 3 records.

Use same data as N. N. Ambraseys, Simpson, and Bommer (1996).

Use three faulting mechanism categories:

Strike-slip: earthquakes with rake angles (\(\lambda\)) \(-30\leq \lambda \leq 30^{\circ}\) or \(\lambda \geq 150^{\circ}\) or \(\lambda \leq -150^{\circ}\), \(\Rightarrow F_N=0, F_R=0\), 47 records.

Normal: earthquakes with \(-150 < \lambda < -30^{\circ}\), \(\Rightarrow F_N=1, F_R=0\), 146 records.

Reverse: earthquakes with \(30 < \lambda <150^{\circ}\), \(\Rightarrow F_R=1, F_N=0\), 229 records.

Earthquakes classified as either strike-slip or reverse or strike-slip or normal depending on which plane is the main plane were included in the corresponding dip-slip category. Some records (137 records, 51 normal, 10 strike-slip and 76 reverse) from earthquakes with no published focal mechanism (80 earthquakes) were classified using the mechanism of the mainshock or regional stress characteristics.

Try using criteria of Campbell (1997) and Sadigh et al. (1997) to classify earthquakes w.r.t. faulting mechanism. Also try classifying ambiguously classified earthquakes as strike-slip. Find large differences in the faulting mechanism coefficients with more stricter criteria for the rake angle of strike-slip earthquakes leading to higher \(C_R\) coefficients.

Note that distribution of records is reasonably uniform w.r.t. to mechanism although significantly fewer records from strike-slip earthquakes.

Try to use two-stage maximum-likelihood method as employed by N. N. Ambraseys, Simpson, and Bommer (1996) but find numerical instabilities in regression.

Also rederive mechanism-independent equation of N. N. Ambraseys, Simpson, and Bommer (1996) using one-stage maximum-likelihood method.

Ground-motion model is: \[\begin{aligned} \ln Y&=&c_1+f_1(M_w)+c_4 \ln \sqrt{f_2 (M_w,r_{\mathrm{seis}},S)}+f_3(F)+f_4(S)\\ &&+f_5(\mathrm{HW},F,M_w,r_{\mathrm{seis}})\\ \mbox{where} \quad f_1(M_w)&=&c_2M_w+c_3(8.5-M_w)^2\\ f_2(M_w,r_{\mathrm{seis}},S)&=&r_{\mathrm{seis}}^2+g(S)^2(\exp[c_8M_w+c_9(8.5-M_w)^2])^2\\ g(S)&=&c_5+c_6(S_{VFS}+S_{SR})+c_7S_{FR}\\ f_3(F)&=&c_{10}F_{RV}+c_{11}F_{TH}\\ f_4(S)&=&c_{12}S_{VFS}+c_{13}S_{SR}+c_{14}S_{FR}\\ f_5(\mathrm{HW},F,M_w,r_{\mathrm{seis}})&=&\mathrm{HW} f_{\mathrm{HW}}(M_w) f_{\mathrm{HW}}(r_{\mathrm{seis}})(F_{\mathrm{RV}}+F_{\mathrm{TH}})\\ \mathrm{HW}&=&\left\{ \begin{array}{r@{\quad \mbox{for}\quad}l} 0 &r_{\mathrm{jb}} \geq 5\,\mathrm{km}\mbox{ or } \delta>70^{\circ}\\ (S_{VFS}+S_{SR}+S_{FR})(5-r_{\mathrm{jb}})/5& r_{\mathrm{jb}}<5\,\mathrm{km}\mbox{ \& } \delta \leq 70^{\circ}\\ \end{array} \right.\\ f_{\mathrm{HW}}(M_w)&=&\left\{ \begin{array}{r@{\quad \mbox{for}\quad}l} 0 &M_w<5.5\\ M_w-5.5& 5.5 \leq M_w \leq 6.5\quad\\ 1&M_w>6.5\\ \end{array} \right.\\ f_{\mathrm{HW}}(r_{\mathrm{seis}})&=&\left\{ \begin{array}{r@{\quad \mbox{for}\quad}l} c_{15}(r_{\mathrm{seis}}/8)&r_{\mathrm{seis}}<8\,\mathrm{km}\\ c_{15}&r_{\mathrm{seis}}\geq 8\,\mathrm{km}\\ \end{array} \right.\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(r_{\mathrm{jb}}\) is the distance to the surface projection of rupture and \(\delta\) is the dip of the fault; for uncorrected horizontal PGA: \(c_1=-2.896\), \(c_2=0.812\), \(c_3=0.0\), \(c_4=-1.318\), \(c_5=0.187\), \(c_6=-0.029\), \(c_7=-0.064\), \(c_8=0.616\), \(c_9=0\), \(c_{10}=0.179\), \(c_{11}=0.307\), \(c_{12}=-0.062\), \(c_{13}=-0.195\), \(c_{14}=-0.320\), \(c_{15}=0.370\) and \(\sigma=c_{16}-0.07M_w\) for \(M_w<7.4\) and \(\sigma=c_{16}-0.518\) for \(M_w \geq 7.4\) where \(c_{16}=0.964\) or \(\sigma=c_{17}+0.351\) for \(\mathrm{PGA} \leq 0.07\,\mathrm{g}\), \(\sigma=c_{17}-0.132 \ln (\mathrm{PGA})\) for \(0.07\,\mathrm{g}< \mathrm{PGA} < 0.25\,\mathrm{g}\) and \(\sigma=c_{17}+0.183\) for \(\mathrm{PGA}\geq 0.25\,\mathrm{g}\) where \(c_{17}=0.263\); for corrected horizontal PGA: \(c_1=-4.033\), \(c_2=0.812\), \(c_3=0.036\), \(c_4=-1.061\), \(c_5=0.041\), \(c_6=-0.005\), \(c_7=-0.018\), \(c_8=0.766\), \(c_9=0.034\), \(c_{10}=0.343\), \(c_{11}=0.351\), \(c_{12}=-0.123\), \(c_{13}=-0.138\), \(c_{14}=-0.289\), \(c_{15}=0.370\) and \(\sigma=c_{16}-0.07M_w\) for \(M_w<7.4\) and \(\sigma=c_{16}-0.518\) for \(M_w \geq 7.4\) where \(c_{16}=0.920\) or \(\sigma=c_{17}+0.351\) for \(\mathrm{PGA} \leq 0.07\,\mathrm{g}\), \(\sigma=c_{17}-0.132 \ln (\mathrm{PGA})\) for \(0.07\,\mathrm{g}< \mathrm{PGA} < 0.25\,\mathrm{g}\) and \(\sigma=c_{17}+0.183\) for \(\mathrm{PGA}\geq 0.25\,\mathrm{g}\) where \(c_{17}=0.219\); for uncorrected vertical PGA: \(c_1=-2.807\), \(c_2=0.756\), \(c_3=0\), \(c_4=-1.391\), \(c_5=0.191\), \(c_6=0.044\), \(c_7=-0.014\), \(c_8=0.544\), \(c_9=0\), \(c_{10}=0.091\), \(c_{11}=0.223\), \(c_{12}=-0.096\), \(c_{13}=-0.212\), \(c_{14}=-0.199\), \(c_{15}=0.630\) and \(\sigma=c_{16}-0.07M_w\) for \(M_w<7.4\) and \(\sigma=c_{16}-0.518\) for \(M_w \geq 7.4\) where \(c_{16}=1.003\) or \(\sigma=c_{17}+0.351\) for \(\mathrm{PGA} \leq 0.07\,\mathrm{g}\), \(\sigma=c_{17}-0.132 \ln (\mathrm{PGA})\) for \(0.07\,\mathrm{g}< \mathrm{PGA} < 0.25\,\mathrm{g}\) and \(\sigma=c_{17}+0.183\) for \(\mathrm{PGA}\geq 0.25\,\mathrm{g}\) where \(c_{17}=0.302\); and for corrected vertical PGA: \(c_1=-3.108\), \(c_2=0.756\), \(c_3=0\), \(c_4=-1.287\), \(c_5=0.142\), \(c_6=0.046\), \(c_7=-0.040\), \(c_8=0.587\), \(c_9=0\), \(c_{10}=0.253\), \(c_{11}=0.173\), \(c_{12}=-0.135\), \(c_{13}=-0.138\), \(c_{14}=-0.256\), \(c_{15}=0.630\) and \(\sigma=c_{16}-0.07M_w\) for \(M_w<7.4\) and \(\sigma=c_{16}-0.518\) for \(M_w \geq 7.4\) where \(c_{16}=0.975\) or \(\sigma=c_{17}+0.351\) for \(\mathrm{PGA} \leq 0.07\,\mathrm{g}\), \(\sigma=c_{17}-0.132 \ln (\mathrm{PGA})\) for \(0.07\,\mathrm{g}< \mathrm{PGA} < 0.25\,\mathrm{g}\) and \(\sigma=c_{17}+0.183\) for \(\mathrm{PGA}\geq 0.25\,\mathrm{g}\) where \(c_{17}=0.274\).

Use four site categories:

Generally includes soil deposits of Holocene age (less than 11,000 years old) described on geological maps as recent alluvium, alluvial fans, or undifferentiated Quaternary deposits. Approximately corresponds to \(V_{s,30}=298 \pm 92\,\mathrm{m/s}\) and NEHRP soil class D. Uncorrected PGA: 534 horizontal records and 525 vertical records and corrected PGA: 241 horizontal records and 240 vertical records. \(S_{VFS}=0\), \(S_{SR}=0\) and \(S_{FR}=0\).

Generally includes soil deposits of Pleistocene age (11,000 to 1.5 million years old) described on geological maps as older alluvium or terrace deposits. Approximately corresponds to \(V_{s,30}=368 \pm 80\,\mathrm{m/s}\) and NEHRP soil class CD. Uncorrected PGA: 168 horizontal records and 166 vertical records and corrected PGA: 84 horizontal records and 83 vertical records. \(S_{VFS}=1\), \(S_{SR}=0\) and \(S_{FR}=0\).

Generally includes sedimentary rock and soft volcanic deposits of Tertiary age (1.5 to 100 million years old) as well as âsofterâ units of the Franciscan Complex and other low-grade metamorphic rocks generally described as melange, serpentine and schist. Approximately corresponds to \(V_{s,30}=421 \pm 109\,\mathrm{m/s}\) and NEHRP soil class CD. Uncorrected PGA: 126 horizontal records and 124 vertical records and corrected PGA: 63 horizontal records and 62 vertical records. \(S_{SR}=1\), \(S_{VFS}=0\) and \(S_{FR}=0\).

Generally include older sedimentary rocks and hard volcanic deposits, high-grade metamorphic rock, crystalline rock and the âharderâ units of the Franciscan Complex generally described as sandstone, greywacke, shale, chert and greenstone. Approximately corresponds to \(V_{s,30}=830 \pm 339\,\mathrm{m/s}\) and NEHRP soil class BC. Uncorrected PGA: 132 horizontal records and 126 vertical records and corrected PGA: 55 horizontal records and 54 vertical records. \(S_{FR}=1\), \(S_{VFS}=0\) and \(S_{SR}=0\).

Note that for generic soil (approximately corresponding to \(V_{s,30}=310\,\mathrm{m/s}\) and NEHRP site class D) use \(S_{VFS}=0.25\), \(S_{SR}=0\), \(S_{FR}=0\) and for generic rock (approximately corresponding to \(V_{s,30}=620\,\mathrm{m/s}\) and NEHRP site class C) use \(S_{SR}=0.50\), \(S_{FR}=0.50\) and \(S_{VFS}=0\).

Use four fault types but only model differences between strike-slip, reverse and thrust:

Earthquakes with rake angles between \(202.5^{\circ}\) and \(337.5^{\circ}\). 4 records from 1 earthquake.

Includes earthquakes on vertical or near-vertical faults with rake angles within \(22.5^{\circ}\) of the strike of the fault. Also include 4 records from 1975 Oroville normal faulting earthquake. Uncorrected PGA: 404 horizontal records and 395 vertical records and corrected PGA: 127 horizontal and vertical records. \(F_{RV}=0\) and \(F_{TH}=0\)

Steeply dipping earthquakes with rake angles between \(22.5^{\circ}\) and \(157.5^{\circ}\). Uncorrected PGA: 186 horizontal records and 183 vertical records and corrected PGA: 58 horizontal records and 57 vertical records. \(F_{RV}=1\) and \(F_{TH}=0\).

Shallow dipping earthquakes with rake angles between \(22.5^{\circ}\) and \(157.5^{\circ}\). Includes some blind thrust earthquakes. Uncorrected PGA: 370 horizontal records and 363 vertical records and corrected PGA: 258 horizontal records and 255 vertical records. \(F_{TH}=1\) and \(F_{RV}=0\).

Note that for generic (unknown) fault type use \(F_{RV}=0.25\) and \(F_{TH}=0.25\).

Most records from \(5.5\leq M_w \leq 7.0\).

Note that equations are an update to equations in Campbell (1997) because they used a somewhat awkward and complicated set of Ground-motion models because there used a mixture of functional forms. Consider that the new equations supersede their previous studies.

Uncorrected PGA refers to the standard level of accelerogram processing known as Phase 1. Uncorrected PGAs are either scaled directly from the recorded accelerogram or if the accelerogram was processed, from the baseline and instrument-corrected Phase 1 acceleration time-history.

Corrected PGA measured from the Phase 1 acceleration time-history after it had been band-pass filtered and decimated to a uniform time interval.

Restrict data to within \(60\,\mathrm{km}\) of seismogenic rupture zone (\(r_{\mathrm{seis}}\leq 60\,\mathrm{km}\)) of shallow crustal earthquakes in active tectonic regions which have source and near-source attenuation similar to California. Most data from California with some from Alaska, Armenia, Canada, Hawaii, India, Iran, Japan, Mexico, Nicaragua, Turkey and Uzbekistan. Note some controversy whether this is true for all earthquakes (e.g. Gazli and Nahanni). Exclude subduction-interface earthquakes.

Restrict earthquakes to those with focal depths \(<25\,\mathrm{km}\).

Exclude data from subduction-interface earthquakes, since such events occur in an entirely different tectonic environment that the other shallow crustal earthquakes, and it has not been clearly shown that their near-source ground motions are similar to those from shallow crustal earthquakes.

Restrict to \(r_{\mathrm{seis}}\leq 60\,\mathrm{km}\) to avoid complications related to the arrival of multiple reflections from the lower crust. Think that this distance range includes most ground-motion amplitudes of engineering interest.

All records from free-field, which define as instrument shelters or non-embedded buildings \(<3\) storeys high and \(<7\) storeys high if located on firm rock. Include records from dam abutments to enhance the rock records even though there could be some interaction between dam and recording site. Exclude records from toe or base of dam because of soil-structure interaction.

Do preliminary analysis, find coefficients in \(f_3\) need to be constrained in order to make \(Y\) independent on \(M_w\) at \(r_{\mathrm{seis}}=0\), otherwise \(Y\) exhibits âoversaturationâ and decreases with magnitude at close distances. Therefore set \(c_8=-c_2/c_4\) and \(c_9=-c_3/c_4\).

Functional form permits nonlinear soil behaviour.

Do not include sediment depth (depth to basement rock) as a parameter even though analysis of residuals indicated that it is an important parameter especially at long periods. Do not think its exclusion is a serious practical limitation because sediment depth is generally not used in engineering analyses and not included in any other widely used attenuation relation.

Do not apply weights during regression analysis because of the relatively uniform distribution of records w.r.t. magnitude and distance.

To make regression analysis of corrected PGA more stable set \(c_2\) equal to value from better-constrained regression of uncorrected PGAs.

Examine normalised residuals \(\delta_i=(\ln Y_i-\ln \bar{Y})/\sigma_{\ln(\mathrm{Unc. PGA}}\) where \(\ln Y_i\) is the measured acceleration, \(\bar{Y}\) is the predicted acceleration and \(\sigma_{\ln(\mathrm{Unc. PGA}}\) is the standard deviation of the uncorrected PGA equation. Plot \(\delta_i\) against magnitude and distance and find models are unbiased.

Consider equations valid for \(M_w\geq 5.0\) and \(r_{\mathrm{seis}}\leq 60\,\mathrm{km}\). Probably can be extrapolated to a distance of \(100\,\mathrm{km}\) without serious compromise.

Note that should use equations for uncorrected PGA if only an estimate of PGA is required because of its statistical robustness. If want response spectra and PGA then should use corrected PGA equation because the estimates are then consistent.

Note that should include ground motions from Kocaeli (17/8/1999, \(M_w=7.4\)), Chi-Chi (21/9/1999, \(M_w=7.6\)), Hector Mine (16/10/1999, \(M_w=7.1\)) and Duzce (12/11/1999, \(M_w=7.1\)) earthquakes but because short-period motions from these earthquakes was significantly lower than expected their inclusion could lead to unconservative estimated ground motions for high magnitudes.

Prefer the relationship for \(\sigma\) in terms of \(\mathrm{PGA}\) because statistically more robust. Note that very few records to constrain value of \(\sigma\) for large earthquakes but many records to constrain \(\sigma\) for \(\mathrm{PGA} \geq 0.25\,\mathrm{g}\).

Find that Monte Carlo simulation indicates that all regression coefficients statistically significant at \(10\%\) level.

Ground-motion models are: \[\log A=a M-b \log R+c\] where \(A\) is in \(\,\mathrm{g}\), \(a=0.484\), \(b=1.4989\), \(c=-2.1640\) and \(\sigma=0.3091\), and: \[\log A=a M-\log R-b R+c\] \(a=0.4805\), \(b=0.0049\), \(c=-2.6860\) and \(\sigma=0.3415\).

Vast majority of data from south Iceland (18 earthquakes in SW Iceland and 4 in N Iceland).

Most data from less than \(50\,\mathrm{km}\) and \(M<5.5\). 76% of data is from \(5\) to \(50\,\mathrm{km}\).

Examine residual plots against distance and find no trends.

Recommend first equation.

Most data from five earthquakes (04/06/1998, 13/11/1998, 27/09/1999, 17/06/2000 and 21/06/2000).

Ground-motion model is: \[A_{max}=a 10^{b M} (\Delta+r)^c\] where \(A_{max}\) is in \(\,\mathrm{cm/s^2}\), \(a=459.0\), \(b=0.198\) and \(c=-1.175\) (\(\sigma\) is not reported). \(r\) can be \(5\), \(10\) or \(15\,\mathrm{km}\).

Data from Lancang-Gengma 1989 (\(M 7.6\)) earthquake.

Ground-motion model is: \[\log \mathrm{PGA}=a_1+a_2M+a_3D-\log (R+a_4 10^{a_5 M_{JMA}})+a_6 R D^{a_6}+S_j\] where \(\mathrm{PGA}\) is in \(\,\mathrm{m/s^2}\), \(a_1=-1.579\), \(a_2=0.739\), \(a_3=0.022\), \(a_4=0.0006\), \(a_5=0.69\), \(a_6=-0.0025\) and \(a_7=0.263\) (\(\sigma\) is unknown).

Use unknown number of site classes, \(S_j\)

\(D\) is focal depth.

Use data from K-Net.

Ground-motion model is: \[\log \mathrm{PGA}=a_1+a_2 M_s+a_3 \log[R+a_4 \exp(a_5 M_s)]\] where \(\mathrm{PGA}\) is in \(\,\mathrm{cm/s^2}\), \(a_1=1.3012\), \(a_2=0.6057\), \(a_3=-1.7216\), \(a_4=1.126\) and \(a_5=0.482\) (\(\sigma\) not reported).

Ground-motion model is: \[\begin{aligned} \log_{10}(\mathrm{PGA})&=&b_0+b_1 M-\log_{10} (R)+b_2 R\\ R&=&\sqrt{D^2+h^2}\end{aligned}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), \(b_0=-1.2780 \pm 0.1909\), \(b_1=0.2853 \pm 0.0316\), \(b_2=-1.730 \times 10^{-3} \pm 2.132 \times 10^{-4}\) and \(\sigma=0.3368\) (\(\pm\) indicates the standard deviation of the coefficients). \(h\) was fixed arbitrarily to \(8\,\mathrm{km}\).

Use data from ISESD (Ambraseys et al. 2004). Select using \(d_e<1000\,\mathrm{km}\), \(5\leq M \leq 7\) (where \(M\) is either \(M_w\) or \(M_s\)).

Focal depths \(<20\,\mathrm{km}\).

Only use data from strike-slip earthquakes.

Note that coefficient of variation for \(b\) coefficients is in range \(11\) to \(15\%\).

Note that \(b_0\) and \(b_1\) are very strongly negatively correlated (correlation coefficient of \(-0.9938\)), believed to be because PGA is governed by \(b_0+b_1M\) as \(D\) approaches zero, but they are almost uncorrelated with \(b_2\) (correlation coefficients of \(-0.0679\) and \(-0.0076\) for \(b_0\) and \(b_1\) respectively), believed to be because of zero correlation between \(M\) and \(D\) in the data used.

Also derive equation using \(\log_{10}(\mathrm{PGA})=b_0+b_1 M+b_2 R+b_3 \log_{10}(R)\) (do not report coefficients) and find slightly smaller residuals but similar behaviour of the \(b\) parameters.

Plot distribution of residuals (binned into intervals of 0.25 units) and the normal probability density function.

Ground-motion model is: \[\log Y=c_0+c_1 M+c_2 \log (R^2+h^2)^{1/2}+c_3 F+c_5 S\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(c_0=0.86\), \(c_1=0.45\), \(c_2=-1.27\), \(c_3=0.10\), \(c_5=0.06\) and \(\sigma=0.286\).

Use three site classes (from NEHRP):

B: 19 stations plus 6 stations between A and B

C: 68 stations

D: 25 stations

No stations in NEHRP class A or E. Use geotechnical information where available and geological maps for the other stations.

Focal depths, \(h\), between \(0.0\) and \(30.1\,\mathrm{km}\).

Classify earthquakes into three faulting mechanism classes:

Normal, 101 earthquakes

Strike-slip, 89 earthquakes

Thrust, 35 earthquakes

but only retain two categories: normal and strike-slip/thrust. Classify using plunges of P and T axes and also knowledge of the geotectonic environment. Have fault-plane solutions for 67 earthquakes.

Choose data that satisfies at least one of these criteria:

from earthquake with \(M_w\geq 4.5\);

record has PGA \(\geq 0.05\,\mathrm{g}\), independent of magnitude;

record has PGA \(<0.05\,\mathrm{g}\) but at least one record from earthquake has PGA \(\geq 0.05\,\mathrm{g}\).

Relocate all earthquakes.

Redigitise all records using a standard procedure and bandpass filter using cut-offs chosen by a comparison of the Fourier amplitude spectrum (FAS) of the record to the FAS of the digitised fixed trace. Find that PGAs from uncorrected and filtered accelerograms are almost identical.

Convert \(M_L\) to \(M_w\), for earthquakes with no \(M_w\), using a locally derived linear equation

Most data from earthquakes with \(M_w<6\) and \(r_{hypo}<60\,\mathrm{km}\).

Note correlation in data between \(M_w\) and \(r_{hypo}\).

Note lack of near-field data (\(R<20\,\mathrm{km}\)) for \(M_w>6.0\).

Plot estimated distance at which instruments would not be expected to trigger and find that all data lie within the acceptable distance range for mean trigger level and only 14 records fall outside the distance range for trigger level plus one \(\sigma\). Try excluding these records and find no effect. Hence conclude that record truncation would not affect results.

Use an optimization procedure based on the least-squares technique using singular value decomposition because two-step methods always give less precise results than one-step techniques. Adopted method allows the controlling of stability of optimization and accurate determination and analysis of errors in solution. Also method expected to overcome and quantify problems arising from correlation between magnitude and distance.

Test assumption that site coefficient for site class D is twice that for C by deriving equations with two site terms: one for C and one for D. Find that the site coefficient for D is roughly twice that of site coefficient for C.

Test effect of focal mechanism by including two coefficients to model difference between normal, strike-slip and thrust motions. Find that the coefficients for difference between strike-slip and normal and between thrust and normal are almost equal. Hence combine strike-slip and thrust categories.

Try including quadratic \(M\) term but find inadmissible (positive) value due to lack of data from large magnitude events.

Also derive equations using this functional form: \(\log Y=c_0+c_1 M+c_2 \log (R+c_4)+c_3 F+c_5 S\) where \(c_4\) was constrained to \(6\,\mathrm{km}\) from an earlier study due to problems in deriving reliable values of \(c_2\) and \(c_4\) directly by regression.

Plot observed data scaled to \(M_w 6.5\) against predictions and find good fit.

Find no systematic variations in residuals w.r.t. remaining variables.

Find reduction in \(\sigma\) w.r.t. earlier studies. Relate this to better locations and site classifications.

Ground-motion model is: \[\log A=a_1+a_2 M-\log (R+a_3 10^{0.5 M})+a_4 R\] where \(A\) is in \(\,\mathrm{gal}\), \(a_1=0.505171\), \(a_2=0.537579\), \(a_3=0.008347\) and \(a_4=-0.00242\) (\(\sigma\) is not known).

Ground-motion model is: \[A_{max}=a 10^{b M} (\Delta+10)^c\] where \(A_{max}\) is in \(\,\mathrm{cm/s^2}\), \(a=195.0\), \(b=0.38\) and \(c=-1.97\) (\(\sigma\) is not reported).

Data from Lancang-Gengma 1989 (\(M 7.6\)) earthquake.

Ground-motion model is: \[\log (\mathrm{PGA})=aM+bR-\log(R)+c\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), \(a=0.611377\), \(b=-0.00584334\), \(c=-3.216674\) and \(\sigma=0.5\).

Do not include terms for site effects due to uncertainty of site classifications (rock/soil). Suggest multiplying predictions by \(3\) to estimate PGA at soil sites.

Derive model to better estimate macroseismic intensities rapidly after an earthquake.

Select data from 21/11/2004 to 28/12/2004, which mainly come from earthquakes in the Les Saintes sequence but include some subduction events and crustal earthquakes in other locations.

Data from 13 stations on Guadeloupe.

Vast majority of data from \(M<4\) and \(20<d<100\,\mathrm{km}\).

Remove constant offset from accelerations but do not filter.

Use resolved maximum because other definitions (e.g. larger) can underestimate PGA by up to \(30\%\).

Plot residuals against \(M\) and find no trends. Observe some residuals of \(\pm 1.5\).

Apply model to other earthquakes from the region and find good match to observations.

Ground-motion model is: \[\log \mathrm{PGA}=a_1+a_2M_w^2+a_3 \log(r+a_4)\] where \(\mathrm{PGA}\) is in unknown unit (probably \(\,\mathrm{cm/s^2}\)), \(a_1=2.581\), \(a_2=0.029\), \(a_3=-1.305\), \(a_4=7\) and \(\sigma=0.712\)

^{19}.Data from rock sites.

Ground-motion model is: \[\log_{10}(y)=a+(b+cm)m+(d+em)\log_{10}(\sqrt{r^2+h^2})\] where \(y\) is in \(\,\mathrm{g}\), \(a=0.46\), \(b=0.35\), \(c=0.07\), \(d=-4.79\), \(e=0.60\), \(h=8.9\,\mathrm{km}\) and \(\sigma=0.33\).

Investigates effect of nontriggering stations on derivation of empirical Ground-motion model based on the assumption that the triggering level is known (or can be estimated from data) but do not know which stations triggered (called left truncated data).

Develops mathematical theory and computational method (after trying various alternative methods) for truncated regression analysis (TRA) and randomly truncated regression analysis (RTRA) (where triggering level changes with time).

Tests developed methods on 1000 lognormally-distributed synthetic data points simulated using the equation of N. N. Ambraseys, Simpson, and Bommer (1996) for \(4\leq M_s \leq 7\) and \(1\leq d_f \leq 100\,\mathrm{km}\). A fixed triggering threshold of \(0.02\,\mathrm{g}\) is imposed. Regresses remaining 908 samples using TRA and RTRA. Finds a very similar equation using TRA but large differences for \(d_f>20\,\mathrm{km}\) by using standard regression analysis (SRA) due to slower attenuation. Also apply TRA to randomly truncated synthetic data and find a close match to original curve, which is not found using SRA.

Applies method to 189 records from rock sites downloaded from ISESD with \(M>4.5\) (scale not specified) and \(d<80\,\mathrm{km}\) (scale not specified) using functional form: \(\log_{10}(y)=a+bm+c\log_{10}(\sqrt{r^2+h^2})\). Uses these selection criteria to allow use of simple functional form and to avoid complications due to crustal reflections that reduce attenuation. Discards the five points with PGA \(<0.01\,\mathrm{g}\) (assumed threshold of SMA-1s). Applies TRA and SRA. Finds both \(M\)-scaling and distance attenuation are larger with TRA than with SRA because TRA accounts for larger spread in original (not truncated) data. Differences are relevant for \(M<6\) and \(d>20\,\mathrm{km}\).

Applies method to dataset including, in addition, non-rock records (456 in total). Finds no differences between TRA and SRA results. Believes that this is due to lack of data in range possibly affected by truncation (small \(M\) and large \(d\)). Finds similar results to N. N. Ambraseys, Simpson, and Bommer (1996).

Applies method to NE Italian data from seven seismometric and ten accelerometric digital stations assuming: \(\log_{10}(y)=a+bm+c\log_{10}(\sqrt{r^2+h^2})\). Accelerometric stations used usually trigger at \(0.001\,\mathrm{g}\). Seismometric stations used trigger based on ratio of short-term and long-term averages (STA/LTA), which varies from station to station and acts like a random threshold. Firstly neglects randomness and assumes trigger level of each station equals lowest recorded PGA and applies TRA and SRA. Finds small differences for \(d<8\,\mathrm{km}\) and \(d>30\,\mathrm{km}\).

Applies method using functional form above, which believes is more physically justified. SRA does not converge. Studies reason for this by regressing on data from \(M\) intervals of \(0.3\) units wide. Finds behaviour of PGAs inverts for \(M<3\). Finds increasing \(\sigma\) with decreasing \(M\) for \(M>3\). TRA does converge and shows stronger magnitude saturation than SRA.

Notes that application of RTRA to model effect of STA/LTA for used data is not realistic since probably not enough data to constrain all 23 parameters and to computational expensive using adopted maximization technique for RTRA.

Estimates the random truncation parameters for one station (Zoufplan) and finds that the fixed threshold assumption made is acceptable since estimated random truncation parameters predict that only \(14\%\) of observations are lost at the earlier assumed fixed threshold level (the lowest PGA recorded).

Ground-motion model is: \[\ln y=a+b M+c \ln \sqrt{r^2+h^2}\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(a=-2.25\), \(b=1.95\), \(c=-1.65\) and \(h=6\) (\(\sigma\) is not known).

Ground-motion model is: \[\ln \mathrm{PGA}=C_1+C_2M+C_3\ln R_h+C_4 R_h+C_5 v\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), \(C_1=-7.515\), \(C_2=1.049\), \(C_3=-0.105\), \(C_4=-0.0211\), \(C_5=-0.287\) and \(\sigma=0.511\). \(v=0\) for horizontal PGA and \(1\) for vertical PGA.

Data from basalt sites (7 stations), thick hard lateritic soil underlain by basalt (1 station) and dam galleries (4 stations).

Data from 13-station strong-motion network (AR-240 and RFT-250 instrument types) close to Koyna Dam. Exclude data from dam top. Use data from foundation gallery because believe they can be considered as ground acceleration data. Select set of 31 significant records after scrutinizing all data.

Correct for instrument response and filter using cut-off frequencies based on a signal-to-noise ratio \(>1\).

Use a 2-stage regression method. Firstly, find \(C_1\), \(C_2\) and \(C_5\) (magnitude and component dependencies) and then find updated \(C_1\), \(C_3\) and \(C_4\) (distance dependence) using residuals from first stage.

Find that equation matches the observed data quite well.

Ground-motion model is: \[\log_{10} y=C_1+C_2 M-B \log_{10}(r+\mathrm{e}^{C_3 M})\] where \(y\) is in \(\,\mathrm{g}\), \(C_1=-1.5232\), \(C_2=0.3677\), \(B=1.0047\), \(C_3=0.41\) and \(\sigma=0.2632\).

Data from rock sites, which assume to have \(760<V_{s,30}<1500\,\mathrm{m/s}\).

38 records from Sharma (1998) and 23 are new data.

Ground-motion model is: \[\begin{aligned} \ln Y_V&=&C_1+C_2 (M-6)+C_3 (M-6)^2+C_4 (M-6)^3+C_5 \ln r+C_6 \Gamma_1+C_7 \Gamma_2\\ r&=&(r_{cl}^2+h^2)^{1/2}\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(C_1=0.055\), \(C_2=0.387\), \(C_3=-0.006\), \(C_4=0.041\), \(C_5=-0.944\), \(C_6=0.277\), \(C_7=0.030\), \(h=7.72\,\mathrm{km}\), \(\sigma_{\mathrm{rock}}=0.629\), \(\sigma_{\mathrm{soil}}=0.607\) and \(\sigma_{\mathrm{soft soil}}=0.575\).

Use three site classes:

Rock: average \(V_s=700\,\mathrm{m/s}\), 27 records

Soil: average \(V_s=400\,\mathrm{m/s}\), 26 records

Soft soil: average \(V_s=200\,\mathrm{m/s}\), 47 records

Classify using approximate methods due to lack of available information. Note that correspondence between average \(V_s\) values for each site class and more widely accepted soil categories is tenuous.

Focal depths from \(0\) to \(111.0\,\mathrm{km}\). State that all earthquakes were shallow crustal events. Only 4 records come from earthquakes with reported focal depths \(>33\,\mathrm{km}\).

Expand with data from after 1999 and update database of GÃ¼lkan and Kalkan (2002).

Faulting mechanism distribution is: normal (12 earthquakes, 14 records), strike-slip (33 earthquakes, 81 records) and reverse (2 earthquakes, 5 records). Note that poor distribution w.r.t. mechanism does not allow its effect to be modelled.

Use only records from earthquakes with \(M_w\geq 4.5\) to emphasize motions having greatest engineering interest and to include only more reliably recorded events. Include data from one \(M_w 4.2\) earthquake because of high vertical acceleration (\(31\,\mathrm{mg}\)) recorded.

Data reasonably well distribution w.r.t. \(M\) and \(d\) for \(d<100\,\mathrm{km}\).

Data mainly recorded in small and medium-sized buildings \(\leq 3\) storeys. Note that these buildings modify recorded motions and this is an unavoidable uncertainty of the study.

Data from main shocks. Exclude data from aftershocks, in particular that from the 1999 Kocaeli and DÃ¼zce aftershocks because these records are from free-field stations, which do not want to commingle with non-free-field data.

Exclude a few records for which PGA caused by main shock is \(<10\,\mathrm{mg}\). Exclude data from aftershocks from the same stations.

Note that data used is of varying quality and could be affected by errors.

Include cubic term for \(M\) dependence to compensate for the controversial effects of sparsity of Turkish data. Find that it gives a better fit.

Use two-step method of N. N. Ambraseys, Simpson, and Bommer (1996) to find site coefficients \(C_6\) and \(C_7\) after exploratory analysis to find regression method that gives the best estimates and the lowest \(\sigma\).

State equations can be used for \(4.5 \leq M_w\leq 7.4\) and \(d_f\leq 200\,\mathrm{km}\).

Find no significant trends in residuals w.r.t. \(M\) or \(d\) for all data and for each site category except for a few high residuals for soil and soft soil records at \(d_f>100\,\mathrm{km}\).

Compute individual \(\sigma\)s for each site class.

Find that observed ground motions for the Kocaeli earthquake are well predicted.

Ground-motion model is: \[\begin{aligned} \ln Y&=&b_1+b_2 (M-6)+b_3 (M-6)^2+b_5 \ln r+b_V \ln (V_S/V_A)\\ r&=&(r_{cl}^2+h^2)^{1/2}\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(b_1=0.393\), \(b_2=0.576\), \(b_3=-0.107\), \(b_5=-0.899\), \(b_V=-0.200\), \(V_A=1112\,\mathrm{m/s}\), \(h=6.91\,\mathrm{km}\) and \(\sigma=0.612\).

Use three site classes:

Average \(V_s=700\,\mathrm{m/s}\), 23 records

Average \(V_s=400\,\mathrm{m/s}\), 41 records

Average \(V_s=200\,\mathrm{m/s}\), 48 records

Use \(V_s\) measurements where available (10 stations, 22 records) but mainly classify using approximate methods. Note that correspondence between average \(V_s\) values for each site class and more widely accepted soil categories is tenuous.

Focal depths from \(0\) to \(111.0\,\mathrm{km}\). State that all earthquakes were shallow crustal events. Only 4 records come from earthquakes with reported focal depths \(>33\,\mathrm{km}\).

Expand with data from after 1999 and update database of GÃ¼lkan and Kalkan (2002).

Faulting mechanism distribution is: normal (12 earthquakes, 14 records), strike-slip (34 earthquakes, 82 records), reverse (2 earthquakes, 5 records), unknown (9 earthquakes, 11 records). Note that poor distribution w.r.t. mechanism does not allow its effect to be modelled.

Use only records from earthquakes with \(M_w\geq 4.0\) to include only more reliably recorded events.

Data reasonably well distribution w.r.t. \(M\) and \(d\) for \(d<100\,\mathrm{km}\).

Data from main shocks. Exclude data from aftershocks, in particular that from the 1999 Kocaeli and DÃ¼zce aftershocks because of high nonlinear soil behaviour observed during the mainshocks near the recording stations.

Data mainly recorded in small and medium-sized buildings \(\leq 3\) storeys. Note that these buildings modify recorded motions and this is an unavoidable uncertainty of the study.

State equations can be used for \(4.0 \leq M_w\leq 7.5\) and \(d_f\leq 250\,\mathrm{km}\).

Find no significant trends in residuals w.r.t. \(M\) or \(d\) for all data and for each site category.

Find that observed ground motions for the Kocaeli earthquake are well predicted.

Ground-motion model is not reported. Use six functional forms.

Use four site categories:

\(V_{s,30}<180\,\mathrm{m/s}\). 0 records.

\(180\leq V_{s,30}<360\,\mathrm{m/s}\). 1 record.

\(360 \leq V_{s,30}<750\,\mathrm{m/s}\). 34 records.

\(V_{s,30}\geq 750\,\mathrm{m/s}\). 93 records.

Site conditions are unknown for 35 records. Classify mainly using description of local site conditions owing to unavailability of \(V_s\) measurements.

Exclude data from \(M_w<3.0\) to exclude data from earthquakes that are likely to be associated with large uncertainties in their size and location and because ground motions from smaller earthquakes are likely to be of no engineering significance.

Exclude data from multi-storey buildings, on or in dams or on bridges.

Most data from \(M_w<5.5\) so believe use of \(r_{epi}\) is justified.

Records from: eastern N America (78 records), NW Europe (61 including 6 from UK) and Australia (24).

Locations from special studies, ISC/NEIC or local network determinations.

Note distinct lack of data from \(<10\,\mathrm{km}\) for \(M_w>5\).

Only retain good quality strong-motion data. No instrument correction applied because of the lack of instrument characteristics for some records. Individually bandpass filter each record with a Butterworth filter with cut-offs at \(25\,\mathrm{Hz}\) and cut-off frequencies chosen by examination of signal-to-noise ratio and integrated velocity and displacement traces.

Find use of different functional forms has significant influence on predicted PGA.

Regression on only rock data generally reduced PGA.

Predictions using the functional forms with quadratic \(M\)-dependence were unreliable for \(M_w>5.5\) because they predict decrease PGA with increasing \(M\) since there was insufficient data from large magnitude earthquakes to constrain the predictions.

Find different regression methods predict similar PGAs with differences of \(<5\%\) for a \(M_w 5\) event at \(5\,\mathrm{km}\) when all records were used but differences up to \(63\%\) when using only rock data. Prefer the one-stage maximum-likelihood method since allows for correlation between \(M\) and \(d\) in dataset and does not ignore earthquakes recorded by only a single station (25% of data).

Find, from analysis of residuals, that equation generally underpredicts PGA of data from eastern N America and Australia but overpredicts motions from Europe and UK.

Find no trends in residuals w.r.t. amplitude, distance, magnitude or fault mechanism.

Believe that large \(\sigma\)s found are due to: lack of data from close to large magnitude earthquakes, use of data from different regions with varying source and path characteristics and use of much data from small earthquakes that are probably associated with higher uncertainty w.r.t. magnitude and location since such earthquakes have not been as well studied as large earthquakes and there is a lack of data with high signal-to-noise ratio from which relocations can be made.

Do not recommend equations for practical use due to large uncertainties.

Ground-motion model is: \[\log_{10} \mathrm{PGA}=a_1+a_2M_L+a_3\log_{10} R\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), \(a_1=-3.93\), \(a_2=0.78\), \(a_3=-1.5\) and \(\sigma=0.55\).

All records from stiff bedrock. Shear-wave velocities estimated from geology gives: \(1200\)â\(2000\,\mathrm{m/s}\) for carbonated formations and \(>2500\,\mathrm{m/s}\) for eruptive formations (majority of data).

Derive equation since find previous equations are not consistent with recent data recorded in France and because of differences between \(M_L\) of LDG and other \(M_L\) scales.

Use data from the Alps, the Pyrenees and Armorican Massif recorded by LDG network of vertical seismometers between 1995 and 1996. Convert vertical PGAs to horizontal PGAs using empirical relation of Smit (1998).

Focal depths between \(2\) and \(12\,\mathrm{km}\).

11 records from \(3\leq d_e \leq 50\,\mathrm{km}\), 34 from \(50 < d_e \leq 200\,\mathrm{km}\) and 18 from \(d_e>200\,\mathrm{km}\) (all from two largest earthquakes with \(M_L 5.3\) and \(M_L 5.6\)).

Plot predictions and data from rock sites of all French earthquakes with \(M_L\geq 4\) recorded by RAP network (largest three earthquakes have \(M_L 5.5\), \(M_L 5.7\) and \(M_L 5.9\)) and find good agreement. State that this agreement shows that equation can be extrapolated to strongest earthquakes considered for France.

Note that it will be possible to establish a more robust equation using increasing number of data from RAP, especially from near field and large magnitudes.

Ground-motion models are: \[\begin{aligned} \log A&=&b-\log (X+c)-k X \quad \mbox{for $D \leq 30\,\mathrm{km}$}\\ \log A&=&b+0.6 \log (1.7 D+c)-1.6 \log (X+c)-kX \quad \mbox{for $D>30\,\mathrm{km}$}\\ \mbox{where} \quad b&=&a M_w+h D+d_i S_i +e\end{aligned}\] where \(A\) is in \(\,\mathrm{gal}\), \(a=0.59\), \(c=0.0060 \times 10^{0.5 M_w}\) (adopted from Si and Midorikawa (2000)), \(d_1=0.00\) (for crustal earthquakes), \(d_2=0.08\) (for inter-plate earthquakes), \(d_3=0.30\) (for intra-plate earthquakes), \(e=0.02\), \(h=0.0023\), \(k=0.003\) [adopted from Si and Midorikawa (2000)], \(\sigma_{\mathrm{intra-event}}=0.27\) and \(\sigma_{\mathrm{inter-event}}=0.16\).

Use two site categories [definitions of Joyner and Boore (1981)]:

Use \(V_{s,30}\) where available. Multiply PGA values from rock sites by \(1.4\) to normalise them w.r.t. PGA at soil sites.

All records from the free-field or small buildings where soil-structure interaction is negligible.

Data from different types of instruments hence instrument correct and bandpass filter.

Classify earthquakes into these three types:

\(S_1=1\), \(S_2=S_3=0\). 12 earthquakes, 1255 records. Focal depths, \(D\), between \(3\) and \(30\,\mathrm{km}\).

\(S_2=1\), \(S_1=S_3=0\). 10 earthquakes, 640 records. \(6\leq D \leq 49\,\mathrm{km}\).

\(S_3=1\), \(S_1=S_2=0\). 11 earthquakes, 1440 records. \(30 \leq D \leq 120\,\mathrm{km}\).

Most data from \(M_w<7\). No data between \(6.9\) and \(7.6\).

Use separate functional forms for \(D\leq 30\,\mathrm{km}\) and \(D>30\,\mathrm{km}\) because of significantly faster decay for deeper earthquakes.

Plot histograms of residuals and conclude that they are lognormally distributed.

Compute \(\sigma\) for 4 \(M\) ranges: \(5.5\)â\(5.9\), \(6.0\)â\(6.5\), \(6.6\)â\(6.9\) and \(7.6\)â\(8.3\). Find slight decrease in \(\sigma\) w.r.t. \(M\).

Compute \(\sigma\) for ranges of \(20\,\mathrm{km}\). Find significantly smaller \(\sigma\)s for distances \(<50\,\mathrm{km}\) and almost constant \(\sigma\)s for longer distances.

Compute \(\sigma\) for ranges of PGA of roughly \(50\,\mathrm{km}\). Find much larger \(\sigma\)s for small PGA than for large PGA.

Believe that main cause of \(M\)-dependent \(\sigma\) is that stress-drop is \(M\)-dependent and that radiation pattern and directivity are not likely to be significant causes.

Believe that distance-dependent \(\sigma\) is likely to be due to randomness of propagation path (velocity and \(Q\)-structure).

Believe site effects do not contribute greatly to the variance.

Plot PGA versus distance and observe a saturation at several hundred \(\,\mathrm{cm/s^2}\), which suggest may be due to nonlinear soil behaviour.

Plot \(\sigma\) w.r.t. PGA for three site categories: \(100\leq V_{s,30} \leq 300\,\mathrm{m/s}\), \(300 \leq V_{s,30} \leq 600\,\mathrm{m/s}\) and \(600 \leq V_{s,30} \leq 2600\,\mathrm{m/s}\). Find \(\sigma\) lower for soft soils than for stiff soils, which believe may demonstrate that nonlinear soil response is a cause of PGA-dependent \(\sigma\).

Note that because inter-event \(\sigma\) is significantly smaller than intra-event \(\sigma\), source effects are unlikely to be the main cause for observed \(\sigma\) dependencies.

Ground-motion model is: \[\log(Y)=a+b (M-6)+c (M-6)^2+d \log \sqrt{R^2+h^2}+eG_1+f G_2\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(a=3.287\), \(b=0.503\), \(c=-0.079\), \(d=-1.1177\), \(e=0.141\), \(f=0.331\), \(h=14.82\,\mathrm{km}\) and \(\sigma=0.260\).

Use three site classes:

A: shear-wave velocity \(>750\,\mathrm{m/s}\), 4 records, and B: shear-wave velocity \(360\)â\(750\,\mathrm{m/s}\), 20 records.

C: shear-wave velocity \(180\)â\(360\,\mathrm{m/s}\), 35 records.

D: shear-wave velocity \(<180\,\mathrm{m/s}\), 136 records.

Originally A and B were separate but combine due to lack of data for site class A.

Focal depths between \(5.4\) and \(25.0\,\mathrm{km}\).

Use \(M_w\) for \(M>6\) to avoid saturation effects.

Assume \(M_L=M_w\) for \(M\leq 6\).

Select records from earthquakes with \(M\geq 5.0\).

Most (15 earthquakes, 146 records) data from earthquakes with \(M\leq 5.8\).

Only use data from the Earthquake Research Department of General Directorate of Disaster Affairs from \(d_f \leq 100\,\mathrm{km}\).

Exclude record from Bolu because of possible instrument error.

Use mixed effects model to account for both inter-event and intra-event variability.

Find that the mixed effects model yields \(\sigma\)s lower than fixed effects model.

Compare predictions with observed data from the Kocaeli and DÃ¼zce earthquakes and find reasonable fit.

Plot coefficients and \(\sigma\)s against frequency and find dependence on frequency.

Plot inter-event and intra-event residuals against distance and magnitude and find not systematic trends.

Find intra-event residuals are significantly larger than inter-event residuals. Suggest that this is because any individual eventâs recordings used to develop model follow similar trends with associated parameters.

Recommend that equations are only used for ground-motion estimation in NW Turkey.

Ground-motion model is: \[\begin{aligned} \log_{10}(Z)&=&b_1+b_2 (M-6)+b_3(M-6)^2+b_5 \log_{10} D+b_6 \Gamma\\ D&=&(r_{jb}^2+h^2)^{1/2}\end{aligned}\] where \(Z\) is in \(\,\mathrm{g}\), \(b_1=0.237\), \(b_2=0.229\), \(b_3=0\), \(b_5=-1.052\), \(b_6=0.174\), \(h=7.27\,\mathrm{km}\) and \(\sigma_{\log Z}=0.203\) (see Spudich and Boore (2005) for correct value of \(\sigma_3\) for use in calculating \(\sigma\) for randomly-orientated component).

Use two site classes:

Rock: sites with soil depths of \(<5\,\mathrm{m}\).

Soil

Use data of Spudich et al. (1999).

Correct equations of Spudich et al. (1999) for \(20\%\) overprediction of motions for rock sites, which was due either to underestimation of shear-wave velocities for rock sites for extensional regimes (believed to be more likely) or an overestimation of shear-wave velocities at soil sites. Correction based on adjusting \(b_1\) and \(b_6\) to eliminate bias in rock estimates but leave soil estimates unchanged.

Verify that adjustment reduces bias in rock estimates.

Do not change \(\sigma_{\log Z}\) because changes to \(b_1\) and \(b_6\) have a negligible influence on \(\sigma_{\log Z}\) w.r.t. errors in determining \(\sigma_{\log Z}\).

Ground-motion model is: \[\log Y=c_0+c_1 M+c_2 \log (R^2+h^2)^{1/2}\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(c_0=1.03\), \(c_1=0.32\), \(c_2=-1.11\), \(h=7\,\mathrm{km}\) and \(\sigma=0.34\).

Classify stations into four NEHRP categories: A, B, C and D (through a site coefficient, \(c_4\)) but find practically no effect so neglect.

Aim to investigate scaling of ground motions for small magnitude earthquakes.

Most earthquakes have normal mechanisms from aftershock sequences.

Records from permanent and temporary stations of ITSAK network. Many from EuroSeisTest array.

Records from ETNA, K2, SSA-1 and SSA-2 plus very few SMA-1 instruments.

Filter records based on a consideration of signal-to-noise ratio. For digital records use these roll-off and cut-off frequencies based on magnitude (after studying frequency content of records and applying different bandpass filters): for \(2\leq M_w <3\) \(f_r=0.95\,\mathrm{Hz}\) and \(f_c=1.0\,\mathrm{Hz}\), for \(3\leq M_w <4\) \(f_r=0.65\,\mathrm{Hz}\) and \(f_c=0.7\,\mathrm{Hz}\) and for \(4 \leq M_w <5\) \(f_r=0.35\) and \(f_c=0.4\,\mathrm{Hz}\). Find that this method adequately removes the noise from the accelerograms used.

Use source parameters computed from high-quality data from local networks. Note that because focal parameters are from different institutes who use different location techniques may mean data set is inhomogeneous.

Note that errors in phase picking in routine location procedures may lead to less accurate locations (especially focal depths) for small earthquakes as opposed to large earthquakes due to indistinct first arrivals.

To minimize effects of focal parameter uncertainties, fix \(h\) as \(7\,\mathrm{km}\), which corresponds to average focal depth in Greece and also within dataset used.

Exclude data from \(d_e>40\,\mathrm{km}\) because only a few (3% of total) records exist for these distances and also to exclude far-field records that are not of interest.

Most records from \(d_e<20\,\mathrm{km}\) and \(2.5\leq M_w \leq 4.5\).

Also derive equations using this functional form: \(\log Y=c_0+c_1 M+c_2 \log (R+c_3)\) where \(c_3\) was constrained to \(6\,\mathrm{km}\) from an earlier study due to problems in deriving reliable values of \(c_2\) and \(c_3\) directly by regression.

Use singular value decomposition for regression following Skarlatoudis et al. (2003).

Combined dataset with dataset of Skarlatoudis et al. (2003) and regress. Find significant number of data outside the \(\pm 1\sigma\) curves. Also plot average residual at each \(M\) w.r.t. \(M\) and find systematically underestimation of PGA for \(M_w \geq 5\). Conclude that this shows the insufficiency of a common relation to describe both datasets.

Find no trends in the residuals w.r.t. magnitude or distance.

Find that the predominant frequencies of PGAs are \(<15\,\mathrm{Hz}\) so believe results not affected by low-pass filtering at \(25\)â\(27\,\mathrm{Hz}\).

Ground-motion model is: \[\log Y(T)=b_1(T)+b_2(T) M_{\mathrm{J}}-b_3(T)D-b_4(T) \log(R)\] where \(Y(T)\) is in \(\,\mathrm{cm/s^2}\), \(b_1(0)=1.1064\), \(b_2(0)=0.2830\), \(b_3(0)=0.0076\), \(b_4(0)=0.6322\) and \(\sigma=0.303\) for horizontal PGA and \(b_1(0)=0.7134\), \(b_2(0)=0.3091\), \(b_3(0)=0.0069\), \(b_4(0)=0.7421\) and \(\sigma=0.301\) for vertical PGA.

Records from 225 stations of K-Net network with \(39.29 \leq V_{s,30} \leq 760.25\,\mathrm{m/s}\) (mean \(V_{s,30}=330.80\,\mathrm{m/s}\).

Select earthquakes that occurred within the region of the boundary of the Okhotsk-Amur plates (NE Japan bordering Sea of Japan) defined by its horizontal location and vertically, to exclude earthquakes occurring in other plates or along other boundaries.

Focal depths, \(D\), between \(8\) and \(43\,\mathrm{km}\) with mean depth of \(20.8\,\mathrm{km}\).

Mean value of \(M\) is \(4.72\).

Mean \(r_{epi}\) is \(84.67\,\mathrm{km}\).

State that exclude records with PGA \(<5\,\mathrm{cm/s^2}\) (although ranges of PGAs given include records with PGA \(<5\,\mathrm{cm/s^2}\)).

Horizontal PGA range: \(4.15\)â\(411.56\,\mathrm{cm/s^2}\). Vertical PGA range: \(0.50\)â\(163.11\,\mathrm{cm/s^2}\).

Originally use this form: \(\log Y(T)=b_1(T)+b_2(T) M-b_3(T)D-\log(R)+b_5(T)R\) but find \(b_5(T)>0\). Regress using the 379 records from sites with \(V_{s,30}>300\,\mathrm{m/s}\) and still find \(b_5(T)>0\) but report results for investigating site effects.

Plot residuals w.r.t. \(r_{hypo}\) and find mean of residuals is zero but find some high residuals.

Note that need to refine model to consider site effects.

Ground-motion model is: \[\mathrm{PGA}=a_1 \mathrm{e}^{a_2(a_3M_w-R_e+a_4S_A+a_5S_B)}\] where \(\mathrm{PGA}\) is in \(\,\mathrm{gal}\), \(a_1=2.18\), \(a_2=0.0218\), \(a_3=33.3\), \(a_4=7.8427\), \(a_5=18.9282\) and \(\sigma=86.4\).

Use three site categories:

Rock, 55 records.

Soil, 94 records.

Soft soil, 72 records.

Classify by adopting those given by other authors, selecting the class reported by more than one source.

Most data from instruments in small buildings.

Use records with PGA \(>20\,\mathrm{gal}\) to avoid bias due to triggering.

PGAs of records between \(20\) and \(806\,\mathrm{gal}\).

Use records from earthquakes with \(M_w\geq 4\) because smaller earthquakes are generally not of engineering significance.

Derive linear conversion formulae (correlation coefficients \(>0.9\)) to transform \(M_s\) (39), \(m_b\) (18), \(M_d\) (10) and \(M_L\) (6) to \(M_w\) (73 events in total).

Note that rupture surfaces have not been accurately defined for most events therefore use \(r_{epi}\).

Note that accurate focal depths are often difficult to obtain and different data sources provide different estimates therefore do not use \(r_{hypo}\).

Use records from \(\geq 5\,\mathrm{km}\) because of assumed average error in epicentral locations.

Use records from \(\leq 100\,\mathrm{km}\) because this is the distance range where engineering significant ground motions occur.

Most data from \(M_w\leq 6\) and \(d_e\leq 50\,\mathrm{km}\).

Do not consider faulting mechanism because focal mechanism solutions for most earthquakes not available.

Plot observed versus predicted PGA and find that a few points fall above and below the lines with slopes \(1:0.5\) and \(1:2\) but most are between these lines.

Note that to improve precision of equation site characterisation based on \(V_s\) measurements should be included. Also note that directivity, fault type and hanging wall effects should be considered when sufficient data is available.

Ground-motion model is: \[\log S_a=C_1+C_2 M+ C_3 M^2+C_4 \log[R+C_5 \exp(C_6 M)]\] where \(S_a\) is in \(\,\mathrm{cm/s^2}\), \(C_1=-1.276\), \(C_2=1.442\), \(C_3=-0.067\), \(C_4=-1.884\), \(C_5=1.046\), \(C_6=0.451\) and \(\sigma=0.232\).

Almost all data from \(r_{epi}<100\,\mathrm{km}\).

Assume saturation at \(M=8\) and find \(C_5\) and \(C_6\). Once these coefficients are fixed find other coefficients by regression.

Ground-motion model is unknown.

Data from generic rock sites, equivalent to NEHRP class B.

Ground-motion model is: \[\log y=a_1+a_2 M_w+(a_3+a_4 M_w) \log \sqrt{d^2+a_5^2}+a_6 S_S+a_7 S_A +a_8 F_N+a_9 F_T+a_{10} F_O\] where \(y\) is in \(\,\mathrm{m/s^2}\), \(a_1=2.522\), \(a_2=-0.142\), \(a_3=-3.184\), \(a_4=0.314\), \(a_5=7.6\), \(a_6=0.137\), \(a_7=0.050\), \(a_8=-0.084\), \(a_9=0.062\), \(a_{10}=-0.044\), \(\sigma_1=0.665-0.065 M_w\) (intra-event) and \(\sigma_2=0.222-0.022 M_w\) (inter-event).

Use three site categories:

Soft soil (S), \(180<V_{s,30}\leq 360\,\mathrm{m/s}\). 143 records.

Stiff soil (A), \(360<V_{s,30}\leq 750\,\mathrm{m/s}\). 238 records.

Rock (R), \(V_{s,30}>750\,\mathrm{m/s}\). 203 records.

Originally include a fourth category, very soft soil (\(V_{s,30}\leq 180\,\mathrm{m/s}\)), but only included 11 records so combined with soft soil records. Note that measured \(V_{s,30}\) only exist for 89 of 338 stations contributing 161 records so use descriptions of local site conditions to classify stations. Exclude records from stations with unknown site conditions because could not be handled by chosen regression method.

Use only data from Europe and Middle East because believe their databank is reasonably complete for moderate and large earthquakes that occurred in region. Also these data have been carefully reviewed in previous studies. Finally based on a previous study believe motions in California could be significantly higher than those in Europe. Note that including these data would increase the quantity of high-quality near-source data available.

Combine data from all seismically active parts of Europe and the Middle East into a common dataset because a previous study shows little evidence for regional differences between ground motions in different regions of Europe.

Only use earthquakes with a \(M_0\) estimate for which to calculate \(M_w\). Do not convert magnitudes from other scales because this increases the uncertainty in the magnitude estimates. Exclude records from earthquakes with \(M_w<5\) in order to have a good distribution of records at all magnitudes. Note that this also excludes records from small earthquakes that are unlikely to be of engineering significance.

Use \(r_{jb}\) because does not require a depth estimate, which can be associated with a large error.

Exclude records from \(>100\,\mathrm{km}\) because: excludes records likely to be of low engineering significance, reduces possible bias due to non-triggering instruments, reduces effect of differences in anelastic decay in different regions and it gives a reasonably uniform distribution w.r.t. magnitude and distance, which reduces likelihood of problems in regression analysis.

Use only earthquakes with published focal mechanism in terms of trends and plunges of T, B and P axes because estimating faulting type based on regional tectonics or to be the same as the associated mainshock can lead to incorrect classification. Classify earthquakes using method of Frohlich and Apperson (1992):

Plunge of T axis \(>50^\circ\). 26 earthquakes, 91 records, \(F_T=1\), \(F_N=0\), \(F_O=0\).

Plunge of P axis \(>60^\circ\). 38 earthquakes, 191 records, \(F_T=0\), \(F_N=1\), \(F_O=0\).

Plunge of B axis \(>60^\circ\). 37 earthquakes, 160 records, \(F_T=0\), \(F_N=0\), \(F_O=0\).

All other earthquakes. 34 earthquakes, 153 records, \(F_T=0\), \(F_N=0\), \(F_O=1\).

Use this method because does not require knowledge of which plane is the main plane and which the auxiliary.

Do not exclude records from ground floors or basements of large buildings because of limited data.

Exclude records from instruments that triggered late and those that were poorly digitised.

Instrument correct records and then apply a low-pass filter with roll-off and cut-off frequencies of \(23\) and \(25\,\mathrm{Hz}\) for records from analogue instruments and \(50\) and \(100\,\mathrm{Hz}\) for records from digital instruments. Select cut-off frequencies for high-pass bidirectional Butterworth filtering based on estimated signal-to-noise ratio and also by examining displacement trace. For records from digital instruments use pre-event portion of records as noise estimate. For those records from analogue instruments with an associated digitised fixed trace these were used to estimate the cut-offs. For records from analogue instruments without a fixed trace examine Fourier amplitude spectrum and choose the cut-offs based on where the spectral amplitudes do not tend to zero at low frequencies. Note that there is still some subjective in the process. Next choose a common cut-off frequency for all three components. Use a few records from former Yugoslavia that were only available in corrected form.

Only use records with three usable components in order that ground-motion estimates are unbiased and that mutually consistent horizontal and vertical equations could be derived.

Note lack of data from large (\(M_w>6.5\)) earthquakes particularly from normal and strike-slip earthquakes.

Data from: Italy (174 records), Turkey (128), Greece (112), Iceland (69), Albania (1), Algeria (3), Armenia (7), Bosnia & Herzegovina (4), Croatia (1), Cyprus (4), Georgia (14), Iran (17), Israel (5), Macedonia (1), Portugal (4), Serbia & Montenegro (24), Slovenia (15), Spain (6), Syria (5) and Uzbekistan (1).

Note that much strong-motion data could not be used due to lack of local site information.

Select one-stage maximum-likelihood regression method because accounts for correlation between ground motion from same earthquake whereas ordinary one-stage method does not. Note that because there is little correlation between \(M_w\) and distance in the data used (correlation coefficient of \(0.23\)) ordinary one-stage and one-stage maximum-likelihood methods give similar coefficients. Do not use two-stage maximum-likelihood method because underestimates \(\sigma\) for sets with many singly-recorded earthquakes (35 earthquakes were only recorded by one station). Do not use method that accounts for correlation between records from same site because records are used from too many different stations and consequently method is unlikely to lead to an accurate estimate of the site-to-site variability (196 stations contribute a single record). Do not use methods that account for uncertainty in magnitude determination because assume all magnitude estimates are associated with the same uncertainty since all \(M_w\) are derived from published \(M_0\) values.

Apply pure error analysis of Douglas and Smit (2001). Divide dataspace into \(0.2 M_w\) units by \(2\,\mathrm{km}\) intervals and compute mean and unbiased standard deviation of untransformed ground motion in each bin. Fit a linear equation to graphs of coefficient of variation against ground motion and test if slope of line is significantly different (at \(5\%\) significance level) than zero. If it is not then the logarithmic transformation is justified. Find that slope of line is not significantly different than zero so adopt logarithmic transformation of ground motion.

Use pure error analysis to compute mean and unbiased standard deviation of logarithmically transformed ground motion in each \(0.2 M_w \times 2\,\mathrm{km}\) bin. Plot the standard deviations against \(M_w\) and fit linear equation. Test significance (\(5\%\) level) of slope. Find that it is significantly different than zero and hence magnitude-independent standard deviation is not justified. Use the reciprocals of fitted linear equations as weighting functions for regression analysis.

Using the standard deviations computed by pure error analysis for each bin estimate lowest possible \(\sigma\) for derived equations.

Investigate possible magnitude-dependence of decay rate of ground motions using ten best-recorded earthquakes (total number of records between 13 and 26). Fit PGAs for each earthquake with equation of form: \(\log y=a_1+a_2 \log \sqrt{d^2+a_3^2}\). Plot decay rates (\(a_2\)) against \(M_w\) and fit a linear equation. Find that the fitted line has a significant slope and hence conclude that data supports a magnitude-dependent decay rate. Assume a linear dependence between decay rate and \(M_w\) due to limited data.

Try including a quadratic magnitude term in order to model possible differences in scaling of ground motions for earthquakes that rupture entire seismogenic zone. Find that term is not significant at \(5\%\) level so drop.

Could not simultaneously find negative geometric and anelastic decay coefficients so assume decay attributable to anelastic decay is incorporated into geometric decay coefficient.

Test significance of all coefficients at \(5\%\) level. Retain coefficients even if not significant.

Note that there is not enough data to model possible distance dependence in effect of faulting mechanism or nonlinear soil effects.

Compute median amplification factor (anti-logarithm of mean residual) for the 16 stations that have recorded more than five earthquakes. Find that some stations show large amplifications or large deamplifications due to strong site effects.

Compute median amplification factor for the ten best recorded earthquakes. Find that most earthquakes do not show significant overall differences but that a few earthquakes do display consistently lower or higher ground motions.

Plot residual plots w.r.t. weighted \(M_w\) and weighted distance and find no obvious dependence of scatter on magnitude or distance.

Plot histograms of binned residuals.

Compare predicted and observed PGAs from the 2004 Parkfield earthquake and find a close match. Note that this may mean that the exclusion of data from California based on possible differences in ground motions was not justified.

Ground-motion model is: \[\log y=a_1+a_2 M_w+(a_3+a_4 M_w) \log \sqrt{d^2+a_5^2}+a_6 S_S+a_7 S_A +a_8 F_N+a_9 F_T+a_{10} F_O\] where \(y\) is in \(\,\mathrm{m/s^2}\), \(a_1=0.835\), \(a_2=0.083\), \(a_3=-2.489\), \(a_4=0.206\), \(a_5=5.6\), \(a_6=0.078\), \(a_7=0.046\), \(a_8=-0.126\), \(a_9=0.005\), \(a_{10}=-0.082\), \(\sigma_1=0.262\) (intra-event) and \(\sigma_2=0.100\) (inter-event).

Based on N. N. Ambraseys et al. (2005a). See SectionÂ 2.236.

Ground-motion model is: \[\log_{10} (\mathrm{PGA})=c_1+c_2 M_s +c_3 r\] where \(\mathrm{PGA}\) is in \(\,\mathrm{m/s^2}\), \(c_1=-2.09\), \(c_2=0.47\), \(c_3=-0.039\) and \(\sigma=0.3\) (note that the method given in the article must be followed in order to predict the correct accelerations using this equation).

Uses data (186 records) of N. Ambraseys and Douglas (2000; Ambraseys and Douglas 2003) for \(M_s\geq 5.8\). Add 57 records from ISESD (Ambraseys et al. 2004) for \(5.0 \leq M_s \leq 5.7\).

Investigates whether âmagnitude-dependent attenuationâ, i.e. PGA saturation in response to increasing magnitude, can be explained by PGA approaching an upper physical limit through an accumulation of data points under an upper limit.

Proposes model with: a magnitude-independent attenuation model and a physical mechanism that prevents PGA from exceeding a given threshold. Considers a fixed threshold and a threshold with random characteristics.

Develops the mathematical models and regression techniques for the truncated and the randomly clipped normal distribution.

Reduces number of parameters by not considering site conditions or rupture mechanism. Believes following results of N. Ambraseys and Douglas (2000; Ambraseys and Douglas 2003) that neglecting site effects is justified in the near-field because they have little effect. Believes that the distribution of data w.r.t. mechanism is too poor to consider mechanism.

Performs a standard one-stage, unweighted regression with adopted functional form and also with form: \(\log_{10} (\mathrm{PGA})=c_1+c_2 M+c_3 r+c_4 M r+c_5 M^2+c_6 r^2\) and finds magnitude saturation and also decreasing standard deviation with magnitude.

Performs regression with the truncation model for a fixed threshold with adopted functional form. Finds almost identical result to that from standard one-stage, unweighted regression.

Performs regression with the random clipping model. Finds that it predicts magnitude-dependent attenuation and decreasing standard deviation for increasing magnitude.

Investigates the effect of the removal of high-amplitude (\(\mathrm{PGA}=17.45\,\mathrm{m/s^2}\)) record from Tarzana of the 1994 Northridge earthquake. Finds that it has little effect.

Ground-motion model is: \[\begin{aligned} \log_{10}(Y)&=&a+(b+cM)M+(d+eM^3) \log_{10}(r)\\ r&=&\sqrt{d^2+h^2}\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(a=-3.27\), \(b=1.95\), \(c=-0.202\), \(d=-3.11\), \(e=0.00751\), \(h=8.9\,\mathrm{km}\) and \(\sigma=0.399\) for horizontal PGA and \(r_{epi}\), \(a=-3.37\), \(b=1.93\), \(c=-0.203\), \(d=-3.02\), \(e=0.00744\), \(h=7.3\,\mathrm{km}\) and \(\sigma=0.358\) for horizontal PGA and \(r_{jb}\), \(a=-2.96\), \(b=1.79\), \(c=-0.184\), \(d=-3.26\), \(e=0.00708\), \(h=11.3\,\mathrm{km}\) and \(\sigma=0.354\) for vertical PGA and \(r_{epi}\) and \(a=-3.18\), \(b=1.80\), \(c=-0.188\), \(d=-3.13\), \(e=0.00706\), \(h=9.1\,\mathrm{km}\) and \(\sigma=0.313\) for vertical PGA and \(r_{jb}\).

Believe relation valid for rather rigid soil.

Use data from the Seismometric Network of Friuli-Venezia Giulia (SENF) (converted to acceleration), the Friuli Accelerometric Network (RAF), data from the 1976 Friuli sequence and data from temporary seismometric (converted to acceleration) and accelerometric stations of Uprava RS za Geofiziko (URSG) of the 1998 Bovec sequence.

Data from 1976 Friuli sequence is taken from ISESD. Records have been bandpass filtered with cut-offs of \(0.25\) and \(25\,\mathrm{Hz}\). No instrument correction has been applied. Data from other networks has been instrument corrected and high-pass filtered at \(0.4\,\mathrm{Hz}\).

Hypocentral locations and \(M_L\) values adopted from local bulletins and studies.

Use running vectorial composition of horizontal time series because horizontal vector is the actual motion that intersects seismic hazard. Find that on average running vectorial composition is \(8\%\) larger than the larger horizontal peak and \(27\%\) larger than the geometric mean. Find that using other methods to combine horizontal components simply changes \(a\) by about \(0.1\) downwards and does not change the other coefficients.

Use data from 19 earthquakes with \(M_L\geq 4.5\) (161 vertical records, 130 horizontal records).

Note that distribution w.r.t. magnitude of earthquakes used roughly follows log-linear Gutenberg-Richter distribution up to about \(M_L \geq 4.5\).

Few records available for \(d<10\,\mathrm{km}\) and \(M_L>3\).

Focal depths between \(1.0\) and \(21.6\,\mathrm{km}\). Average depth is \(11.4 \pm 3.6\,\mathrm{km}\).

Apply multi-linear multi-threshold truncated regression analysis (TRA) of Bragato (2004) to handle the effect of nontriggering stations using the simplification that for SENF and URSG data the random truncation level can be approximated by the lowest value available in the data set for that station. For data from the 1976 Friuli sequence use a unique truncation level equal to the minimum ground motion for that entire network in the dataset. Use same technique for RAF data.

Develop separate equations for \(r_{epi}\) and \(r_{jb}\) (available for 48 records in total including all from \(M_L>5.8\)). Note that physically \(r_{jb}\) is a better choice but that \(r_{epi}\) is more similar to geometric distance used for seismic hazard assessment.

Use \(M_L\) because available for regional earthquakes eastern Alps since 1972.

Conduct preliminary tests and find that weak-motion data shows higher attenuation than strong-motion data. Investigate horizontal PGA using entire data set and data for \(0.5\)-wide magnitude classes. Find that attenuation is dependent on magnitude and it is not useful to include a coefficient to model anelastic attenuation.

Since data is not uniformly distributed with magnitude, inversely weight data by number of records within intervals of \(0.1\) magnitude units wide.

Because correlation between magnitude and distance is very low (\(0.03\) and \(0.02\) for vertical and horizontal components, respectively) apply one-stage method.

Note that large differences between results for \(r_{epi}\) and \(r_{jb}\) are due to magnitude-dependent weighting scheme used.

Plot predicted and observed ground motions binned into \(0.3\) magnitude intervals and find close match.

Plot residuals w.r.t. focal depth, \(r_{jb}\) and \(M_L\). Find that it appears equation over-estimates horizontal PGA for \(d_f>80\,\mathrm{km}\), \(M_L<3\) and focal depths \(>15\,\mathrm{km}\) but note that this is due to the truncation of low amplitude data. Check apparent trend using TRA and find no significant trend.

Note that difficult to investigate importance of focal depth on attenuation due to unreliability of depths particularly for small earthquakes. Find that focal depths seem to be correlated to magnitude but believe that this is an artifact due to poor location of small earthquakes. Try regression using \(r_{hypo}\) and find larger \(\sigma\) hence conclude that depth estimates are not accurate enough to investigate effect of depth on ground motions.

Investigate methods for incorporation of site effect information using their ability to reduce \(\sigma\) as a criteria.

Note that largest possible reduction is obtained using individual average station residuals for each site but that this is not practical because this method cannot be used to predict ground motions at arbitrary site and that it requires sufficient number of observations for each station. Using just those stations that recorded at least five earthquakes obtain estimate of lowest possible \(\sigma\) by adopting this method.

Try using a classification of stations into three site categories: rock (16 stations, 1020 records), stiff soil (9 stations, 117 records) and soft soil (4 stations, 27 records) and find no reduction in \(\sigma\), which believe is due to the uneven distribution w.r.t. site class. Find that the strong site effects at Tolmezzo has a significant effect on the obtained site coefficients.

Use Nakamura (H/V) ratios from ambient noise for a selection of stations by including a term \(g(S)=c_{\mathrm{HV}} N(S)\), where \(N(S)\) is the Nakamura ratio at the period of interest (\(0.125\)â\(1\,\mathrm{s}\) for PGA), in the equation. Find large reductions in \(\sigma\) and high correlations between Nakamura ratios and station residuals.

Use receiver functions from earthquake recordings in a similar way to Nakamura ratios. Find that it is reduces \(\sigma\) more than site classification technique but less than using the Nakamura ratios, which note could be because the geometry of the source affects the computed receiver functions so that they are not representative of the average site effects.

Believe equation is more appropriate than previous equations for \(M_L<5.8\) and equivalent to the others up to \(M_L 6.3\). Discourage extrapolation for \(M_L>6.3\) because it overestimates PGA in the far-field from about \(M_L 6.5\).

Ground-motion model is: \[\log(Y)=a+bM+cM^2+d\log(R)+eS\] where \(Y\) is in \(\,\mathrm{g}\), \(a=-3.19\pm 0.02\), \(b=0.87 \pm 0.01\), \(c=-0.042 \pm 0.002\), \(d=-1.92 \pm 0.01\), \(e=0.249 \pm 0.005\) and \(\sigma=0.316\).

Use two site classes, because lack local geological information (e.g. average \(V_s\)):

Rock, eight stations, 3790 records.

Soil, seven stations, 3109 records.

Classify station using geological reports, \(M_L\) station corrections and H/V spectral ratios computed over a \(30\,\mathrm{s}\) wide time window of S waves for entire waveform data set.

Data from Regional Seismic Network of Northwestern Italy and Regional Seismic Network of Lunigiana-Garfagnana (ten Lennartz LE3D-5s and five Guralp CMG-40 sensors with Lennartz Mars88/MC recording systems). Sampling rate either \(62.5\) or \(125\,\mathrm{samples/s}\). Records from broadband and enlarged band seismometers converted to acceleration by: correcting for instrument response, bandpass filtering between \(1\) and \(20\,\mathrm{Hz}\) and then differentiating. Accuracy of conversion verified by comparing observed and derived PGA values at one station (STV2), which was equipped with both a Kinemetrics K2 accelerometer and a Guralp CMG-40 broadband sensor.

Find strong attenuation for short distances (\(<50\,\mathrm{km}\)) and small magnitudes (\(M_L <3.0\)).

\(M_L\) calculated using a calibration formula derived for northwestern Italy using a similar dataset.

Compute signal-to-noise (S/N) ratio for the S phase using windows of \(3\,\mathrm{s}\) wide and find that data is good quality (85% of windows have S/N ratio greater than \(10\,\mathrm{dB}\). Only use records with S/N ratio \(>20\,\mathrm{dB}\).

Most earthquakes are from SW Alps and NW Apennines.

Most records from earthquakes with \(1\leq M_L \leq 3\), small number from larger earthquakes particularly those with \(M_L>4\). \(M_L<1\): 1285 records, \(1\leq M_L <2\): 2902 records, \(2\leq M_L<3\): 1737 records, \(3\leq M_L <4\): 693 records and \(M_L \geq 4\): 282 records.

Data shows strong magnitude-distance correlation, e.g. records from earthquakes with \(M_L<1\) are from \(0\leq R \leq 100\,\mathrm{km}\) and those from earthquakes with \(M_L>4\) are mainly from \(R>50\,\mathrm{km}\). Distribution is uniform for \(2\leq M_L \leq 4\) and \(0\leq R \leq 200\,\mathrm{km}\).

Originally include an anelastic decay term (\(d_1 R\)) in addition but the value of \(d_1\) was positive and not statistically significantly different than zero so it was removed.

Regression in two-steps: firstly without site effect coefficient (\(e\)) and then with \(e\) added.

Compare data to estimated decay within one magnitude unit intervals and find predictions are good up to \(M_L=4.0\).

Find no systematic trends in the residuals.

Ground-motion model is: \[\begin{aligned} \log Y&=&c_1+c_2 M_w +c_3 R -c_4 \log R+c_5 H\\ R&=&\sqrt{R_{\mathrm{cld}}^2+\Delta^2}\\ \Delta&=&0.00750 \times 10^{0.507 M_w}\end{aligned}\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), for horizontal PGA: \(c_1=-0.2\), \(c_2=0.59\), \(c_3=-0.0039\), \(c_4=1\), \(c_5=0.008\), \(\sigma_r=0.27\), \(\sigma_e=0.10\) and for vertical PGA: \(c_1=-0.4\), \(c_2=0.60\), \(c_3=-0.0036\), \(c_4=1\), \(c_5=0.006\), \(\sigma_r=0.25\) and \(\sigma_e=0.11\) where \(\sigma_r\) is the intra-event standard deviation and \(\sigma_e\) is the inter-event standard deviation.

All data from 51 hard (NEHRP B) sites.

All stations in the Valley of Mexico omitted.

All data from free-field stations: small shelters, isolated from any building, dam abutment, bridge, or structure with more than one storey.

Focal depths: \(35\leq H \leq 138\,\mathrm{km}\), most records (13 earthquakes, 249 records) from \(35\leq H\leq 75\,\mathrm{km}\).

Exclude data from \(M_w<5.0\) and \(R>400\,\mathrm{km}\).

Exclude data from deep earthquakes where wave paths cross the mantle edge.

All data from normal-faulting earthquakes.

Use about 27 records from velocity records from broadband seismograph network that were differentiated to acceleration.

Adopt \(\Delta\) from Atkinson and Boore (2003).

Investigate a number of functional forms. Inclusion of \(\Delta\) substantially improves fit, leading to a decrease in random variability at close distances, and an increase in \(c_2\) and \(c_3\) coefficients. Find worse correlation when add a quadratic magnitude term. A magnitude-dependent \(c_4\) leads to higher \(\sigma\)s. Find unrealistically high ground motions at close distances using the form of \(c_4\) used by Atkinson and Boore (2003).

If exclude three deep earthquakes then little dependence on \(H\).

Do not find any noticeable bias in residuals w.r.t. distance, magnitude or depth (not shown).

Note that decrease in variability w.r.t. magnitude is only apparent for frequencies \(<1\,\mathrm{Hz}\).

Discuss observed dependence of, particularly high-frequency, ground motions on focal depth.

Ground-motion model is: \[\ln Y =a \ln (X+h)+b X+c M_w+d\] where \(Y\) is in \(\,\mathrm{cm/s^2}\) for horizontal PGA (for whole Taiwan) \(a=-0.852\), \(b=-0.0071\), \(c=1.027\), \(d=1.062\), \(h=1.24\,\mathrm{km}\) and \(\sigma=0.719\) and for vertical PGA (for whole Taiwan) \(a=-1.340\), \(b=-0.0036\), \(c=1.101\), \(d=1.697\), \(h=1.62\,\mathrm{km}\) and \(\sigma=0.687\). Also report coefficients for equations derived for three different sub-regions.

Do not differentiate site conditions.

Focal depths, \(h\), between \(2.72\) and \(29.98\,\mathrm{km}\).

Data from high-quality digital strong-motion networks of Taiwan Strong Motion Instrumentation Program (TSMIP) and Central Mountain Strong Motion Array (CMSMA).

Select data from earthquakes with \(h\leq 30\,\mathrm{km}\) and with records from \(\geq 6\) stations at \(d_e\leq 20\,\mathrm{km}\).

Select events following the 1999 Chi-Chi earthquake (\(M_w 7.7\)) with \(M_L>6\).

Do not use data from the Chi-Chi earthquake because: a) earlier analysis of Chi-Chi data showed short-period ground motion was significantly lower than expected and b) the Chi-Chi rupture triggered two \(M 6\) events on other faults thereby contaminating the ground motions recorded at some stations.

Data uniformly distributed for \(M_w\leq 6.5\) and \(20\leq r_{hypo}\leq 100\,\mathrm{km}\). Significant number of records for \(r_{hypo}>100\,\mathrm{km}\).

Use data from the Chi-Chi earthquake and the 2003 Cheng-Kung earthquake (\(M_w 6.8\)) for testing applicability of developed equations.

For 32 earthquakes (mainly with \(M_w < 5.3\)) convert \(M_L\) to \(M_w\) using empirical equation developed for Taiwan.

Develop regional equations for three regions: CHY in SW Taiwan (16 earthquakes, 1382 records), IWA in NE Taiwan (14 earthquakes, 2105 records) and NTO in central Taiwan (13 earthquakes, 3671 records) and for whole Taiwan to compare regional differences of source clustering in ground-motion characteristics.

Use \(M_w\) since corresponds to well-defined physical properties of the source, also it can be related directly to slip rate on faults and avoids saturation problems of other \(M\)-scales.

Use relocated focal depths and epicentral locations.

Do not use \(r_{jb}\) or \(r_{rup}\) because insufficient information on rupture geometries, particularly those of small earthquakes, even though believe such distance metrics are justified. However, for small earthquakes do not think using \(r_{hypo}\) rather than \(r_{rup}\) will introduce significant bias into the equations. Also use \(r_{hypo}\) because it is quickly determined after an earthquake hence early ground-motion maps can be produced.

From equations derived for different sub-regions and from site residual contour maps that ground motions in CHY are about four times higher than elsewhere due to thick, recent alluvial deposits.

Find predictions for Chi-Chi and Cheng-Kung PGAs are close to observations.

Plot contour maps of residuals for different sites and relate the results to local geology (alluvial plains and valleys and high-density schist).

Divide site residuals into three classes: \(>0.2 \sigma\), \(-0.2\)â\(0.2\sigma\) and \(<-0.2\sigma\) for four NEHRP-like site classes. Find the distribution of residuals is related to the site class particularly for the softest class. Find residuals for C (very dense soil and soft rock) and D (stiff soil) are similar so suggest combining them. Believe geomorphology may also play an important role in site classification because a geomorphologic unit is often closely related to a geologic unit.

Ground-motion model is: \[\log(y)=a+b M+d \log(R)+kR+s_1+s_2\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(a=-0.9892\), \(b=0.8824\), \(d=-1.355\), \(k=-0.1363\), \(s_1=0.337\) (for stations on surface), \(s_2=0\) (for station at depth) and \(\sigma=0.483\).

Use data from seven stations, one of which (TU1) is located underground within the mine. Determine site factors (constrained to be between \(0\) and \(1\)) from PGV data. Originally group into three site categories: one for stations with close to horizontal straight-line ray paths, one for stations with steeper ray paths and one for underground station. Find site factors for first two categories similar so combine, partly because there is no precedent for topographic site factors in empirical ground-motion estimation equations. Believe that low site factors found are because stations are on solid rock \(V_s>1.5\,\mathrm{km/s}\).

Most data from Trail Mountain coal mine from between 12/2000 and 03/2001 (maximum \(M_{CL} 2.17\)). Supplement with data (2 records) from a \(M 4.2\) earthquake at Willow Creak mine to provide data at much higher magnitude.

Most data from \(M_w<1.7\).

Lower magnitude limit dictated by need for adequate signal-to-noise ratio.

Focal depths between \(50\) and \(720\,\mathrm{m}\) (relative to the ground surface).

Note that although data may be poorly suited to determine both \(d\) and \(k\) simultaneously they are retained because both attenuation mechanisms must be operative. State that \(d\) and \(k\) should be solely considered as empirical parameters due to trade-offs during fitting.

Do not include a quadratic \(M\) term because it is generally of little consequence.

Use \(r_{hypo}\) because earthquakes are small compared to distances so can be considered as point sources.

Selected events using these criteria:

event was recorded by \(\geq 6\) stations;

data had high signal-to-noise ratio;

to obtain the broadest \(M\)-range as possible; and

to have a broad distribution of epicentral locations.

Find that \(M_w\) (estimated for 6 events) does not significantly differ from \(M_{CL}\).

Find that constrains must be applied to coefficients. Constrain \(k\) to range \(-2\)â\(0\) because otherwise find small positive values. Believe that this is because data inadequate for independently determining \(d\) and \(k\).

Ground-motion model is: \[\ln Y=C_1+C_2 M-C_3 \ln r-C_4 r\] where \(Y\) is in \(\,\mathrm{g}\), \(C_1=-3.6\), \(C_2=0.72\), \(C_3=1.08\) and \(C_4=-0.007\) (SIC) (\(\sigma\) is not given).

Do not consider site effects but note that sediment cover is very thin.

Data from 9 stations (1 K2 and 8 Etna instruments) established by Indian Institute of Technology, Kharagpur in 1998.

Focal depths from \(3.01\) to \(34.27\,\mathrm{km}\).

Use data with signal-to-noise ratios \(\geq\) 3.

Instrument and baseline correct data and bandpass filter with cut-offs of \(0.1\) and \(30\,\mathrm{Hz}\).

Ground-motion model is: \[\ln(A)=c_1+c_2(M-6)+c_3\ln(\sqrt{\mathrm{EPD}^2+h^2})+c_4S\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(c_1=7.969\), \(c_2=1.220\), \(c_3=-1.131\), \(c_4=0.212\), \(h=10\,\mathrm{km}\) (fixed after tests) and \(\sigma=0.825\) for horizontal PGA and \(c_1=7.262\), \(c_2=1.214\), \(c_3=-1.094\)

^{20}, \(c_4=0.103\), \(h=10\,\mathrm{km}\) (fixed after tests) and \(\sigma=0.773\) for vertical PGA.Uses four site categories (\(S\) equals number of site category):

Rock. 117 records.

Alluvial. 52 records.

Gravel and sandy. 70 records.

Soft. 39 records.

Does analysis combining 1 and 2 together in a firm rock category (\(S=0\)) and 3 and 4 in a soft soil category (\(S=1\)) and for all site categories combined. Reports coefficients for these two tests.

Focal depths between \(9\) and \(73\,\mathrm{km}\). Most depths are shallow (depths fixed at \(33\,\mathrm{km}\)) and majority are about \(10\,\mathrm{km}\). Does not use depth as independent parameter due to uncertainties in depths.

Uses \(M_w\) because nearly all reported Ground-motion models use \(M_w\).

Uses macroseismic distance for three events since no \(r_{epi}\) reported.

Believes that methods other than vectorial sum of both horizontal PGAs underestimates true PGA that acts on the structure. Notes that vectorial sum ideally requires that PGAs on the two components arrive at the same time but due to unknown or inaccurate timing the occurrence time cannot be used to compute the resolved component.

Does not consider faulting mechanism due to lack of information for many events.

Most records from \(M_w\leq 5\).

Originally includes terms \(c_5(M-6)^2\) and \(c_6\mathrm{EPD}\) but finds them statistically insignificant so drops them.

Notes that all coefficients pass the \(t\)-test of significance but that the site coefficients are not highly significant, which relates to poor site classification for some stations.

Compares observed and predicted PGAs with respect to distance. Notes that match to observations is relatively good.

Compares observed PGAs during Bam 2003 earthquake to those predicted and finds good match.

Ground-motion model is: \[x=\frac{A \mathrm{e}^{BM}}{(R+C)^D}\] where \(x\) is in \(\,\mathrm{cm/s^2}\), \(A=4\), \(B=1.3\), \(C=30\) and \(D=1.43\) for horizontal PGA, hard rock sites and thrust earthquakes; \(A=2\), \(B=1.28\), \(C=30\) and \(D=1.09\) for horizontal PGA, rock and hard soil sites and thrust earthquakes; \(A=11\), \(B=1.11\), \(C=30\), \(D=1.41\) for vertical PGA, hard rock sites and thrust earthquakes; \(A=18\), \(B=1.31\), \(C=30\), \(D=1.65\) for vertical PGA, rock and hard soil sites and thrust earthquakes; \(A=3840\), \(B=1.2\), \(C=80\) and \(D=2.16\) for horizontal PGA, rock and hard soil sites and intermediate-depth earthquakes; and \(A=66687596\), \(B=1.2\), \(C=80\) and \(D=4.09\) for vertical PGA, rock and hard soil sites and intermediate-depth earthquakes.

Use two site categories:

\(V_s>1500\,\mathrm{m/s}\). 8 records.

\(360<V_s<1500\,\mathrm{m/s}\). 41 records.

Focal depths between \(28.8\) and \(50.0\,\mathrm{km}\).

Develop separate equations for interface and intraslab (intermediate-depth) events.

Baseline correct and bandpass filter (fourth-order Butterworth) with cut-offs \(0.167\) and \(25\,\mathrm{Hz}\).

8 records from between \(M_s 6.0\) and \(7.0\), 13 from between \(7.0\) and \(7.5\) and 20 from between \(7.5\) and \(8.0\).

Values of coefficient \(D\) taken from previous studies.

Ground-motion model is: \[\begin{aligned} \log_e(y)&=&aM_w+bx-\log_e(r)+e(h-h_c)\delta_h+F_R+S_I+S_S+S_{SL} \log_e(x)+C_k\\ \mbox{where} \quad r&=&x+c \exp(dM_w)\end{aligned}\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(\delta_h=1\) when \(h\geq h_c\) and \(0\) otherwise, \(a=1.101\), \(b=-0.00564\), \(c=0.0055\), \(d=1.080\), \(e=0.01412\), \(S_R=0.251\), \(S_I=0.000\), \(S_S=2.607\), \(S_{SL}=-0.528\), \(C_H=0.293\), \(C_1=1.111\), \(C_2=1.344\), \(C_3=1.355\), \(C_4=1.420\), \(\sigma=0.604\) (intra-event) and \(\tau=0.398\) (inter-event). Use \(h_c=15\,\mathrm{km}\) because best depth effect for shallow events.

Use five site classes (\(T\) is natural period of site):

NEHRP site class A, \(V_{s,30}>1100\,\mathrm{m/s}\). 93 records. Use \(C_H\).

Rock, NEHRP site classes A+B, \(600<V_{s,30}\leq 1100\,\mathrm{m/s}\), \(T<0.2\,\mathrm{s}\). 1494 records. Use \(C_1\).

Hard soil, NEHRP site class C, \(300<V_{s,30} \leq 600\,\mathrm{m/s}\), \(0.2 \leq T<0.4\,\mathrm{s}\). 1551 records. Use \(C_2\).

Medium soil, NEHRP site class D, \(200<V_{s,30} \leq 300\,\mathrm{m/s}\), \(0.4 \leq T<0.6\,\mathrm{s}\). 629 records. Use \(C_3\).

Soft soil, NEHRP site classes E+F, \(V_{s,30} \leq 200\,\mathrm{m/s}\), \(T\geq 0.6\,\mathrm{s}\). 989 records. Use \(C_4\).

Site class unknown for 63 records.

Focal depths, \(h\), between about \(0\) and \(25\,\mathrm{km}\) for crustal events, between about \(10\) and \(50\,\mathrm{km}\) for interface events, and about \(15\) and \(162\,\mathrm{km}\) for intraslab events. For earthquakes with \(h>125\,\mathrm{km}\) use \(h=125\,\mathrm{km}\).

Classify events into three source types:

Crustal.

Interface. Use \(S_I\).

Slab. Use \(S_S\) and \(S_{SL}\).

and into four mechanisms using rake angle of \(\pm 45^{\circ}\) as limit between dip-slip and strike-slip earthquakes except for a few events where bounds slightly modified:

Reverse. Use \(F_R\) if also crustal event.

Strike-slip

Normal

Unknown

Distribution of records by source type, faulting mechanism and region is given in following table.

Region Focal Mechanism Crustal Interface Slab Total Japan Reverse 250 1492 408 2150 Strike-slip 1011 13 574 1598 Normal 24 3 735 762 Unknown 8 8 Total 1285 1508 1725 4518 Iran and Western USA Reverse 123 12 135 Strike-slip 73 73 Total 196 12 208 All Total 1481 1520 1725 4726 Exclude data from distances larger than a magnitude-dependent distance (\(300\,\mathrm{km}\) for intraslab events) to eliminate bias introduced by untriggered instruments.

Only few records from \(<30\,\mathrm{km}\) and all from \(<10\,\mathrm{km}\) from 1995 Kobe and 2000 Tottori earthquake. Therefore add records from overseas from \(<40\,\mathrm{km}\) to constrain near-source behaviour. Note that could affect inter-event error but since only 20 earthquakes (out of 269 in total) added effect likely to be small.

Do not include records from Mexico and Chile because Mexico is characterised as a âweakâ coupling zone and Chile is characterised as a âstrongâ coupling zone (the two extremes of subduction zone characteristics), which could be very different than those in Japan.

Note reasonably good distribution w.r.t. magnitude and depth.

State that small number of records from normal faulting events does not warrant them between considered as a separate group.

Note that number of records from each event varies greatly.

Process all Japanese records in a consistent manner. First correct for instrument response. Next low-pass filter with cut-offs at \(24.5\,\mathrm{Hz}\) for 50 samples-per-second data and \(33\,\mathrm{Hz}\) for 100 samples-per-second data. Find that this step does not noticeably affect short period motions. Next determine location of other end of usable period range. Note that this is difficult due to lack of estimates of recording noise. Use the following procedure to select cut-off:

Visually inspect acceleration time-histories to detect faulty recordings, S-wave triggers or multiple events.

If record has relatively large values at beginning (P wave) and end of record, the record was mirrored and tapered for \(5\,\mathrm{s}\) at each end.

Append \(5\,\mathrm{s}\) of zeros at both ends and calculate displacement time-history in frequency domain.

Compare displacement amplitude within padded zeros to peak displacement within the record. If displacement in padded zeros was relatively large, apply a high-pass filter.

Repeat using high-pass filters with increasing corner frequencies, \(f_c\), until the displacement within padded zeros was âsmallâ (subjective judgement). Use \(1/f_c\) found as maximum usable period.

Verify method by using K-Net data that contains \(10\,\mathrm{s}\) pre-event portions.

Conduct extensive analysis on inter- and intra-event residuals. Find predictions are reasonably unbiased w.r.t. magnitude and distance for crustal and interface events and not seriously biased for slab events.

Do not smooth coefficients.

Do not impose constraints on coefficients. Check whether coefficient is statistically significant.

Note that the assumption of the same anelastic attenuation coefficient for all types and depths of earthquakes could lead to variation in the anelastic attenuation rate in a manner that is not consistent with physical understanding of anelastic attenuation.

Derive \(C_H\) using intra-event residuals for hard rock sites.

Residual analyses show that assumption of the same magnitude scaling and near-source characteristics for all source types is reasonable and that residuals not not have a large linear trend w.r.t. magnitude. Find that introducing a magnitude-squared term reveals different magnitude scaling for different source types and a sizable reduction in inter-event error. Note that near-source behaviour mainly controlled by crustal data. Derive correction function from inter-event residuals of each earthquake source type separately to avoid trade-offs. Form of correction is: \(\log_e (S_{MSst})=P_{st}(M_w-M_C)+Q_{st}(M_w-M_C)^2+W_{st}\). Derive using following three-step process:

Fit inter-event residuals for earthquake type to a quadratic function of \(M_w-M_C\) for all periods.

Fit coefficients \(P_{st}\) for \((M_w-M_C)\) and \(Q_{st}\) for \((M_w-M_C)^2\) (from step 1) where subscript \(st\) denotes source types, to a function up to fourth oder of \(\log_e(T)\) to get smoothed coefficients.

Calculate mean values of differences between residuals and values of \(P_{st}(M_w-M_C)+Q_{st}(M_w-M_C)^2\) for each earthquake, \(W_{st}\), and fit mean values \(W_{st}\) to a function of \(\log_e(T)\).

For PGA \(Q_C=W_C=Q_I=W_I=0\), \(\tau_C=0.303\), \(\tau_I=0.308\), \(P_S=0.1392\), \(Q_S=0.1584\), \(W_S=-0.0529\) and \(\tau_S=0.321\). Since magnitude-square term for crustal and interface is not significant at short periods when coefficient for magnitude-squared term is positive, set all coefficients to zero. Find similar predicted motions if coefficients for magnitude-squared terms derived simultaneously with other coefficients even though the coefficients are different than those found using the adopted two-stage approach.

Compare predicted and observed motions normalized to \(M_w 7\) and find good match for three source types and the different site conditions. Find model overpredicts some near-source ground motions from SC III and SC IV that is believed to be due to nonlinear effects.

Ground-motion model is: \[\begin{aligned} \log_{10}(Y)&=&B_1+B_2(M-6)-B_5 \log_{10}R\\ \mbox{where} \quad R&=&\sqrt{R_{jb}^2+6^2}\end{aligned}\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(B_1=4.037\), \(B_2=0.572\), \(B_5=1.757\) and \(\sigma=0.836\).

Ground-motion model is: \[\begin{aligned} \log Y&=&c0+c1(\mathbf{M}-5)+c2(\mathbf{M}-5)^2+c3\log R+c4 R+S_i\\ R&=&\sqrt{d^2+h^2}\end{aligned}\] where \(Y\) is in \(\,\mathrm{m/s^2}\), \(c0=2.007\), \(c1=0.567\), \(c2=0.0311\), \(c3=-1.472\), \(c4=0.00000\), \(h=5\,\mathrm{km}\) [from Boore, Joyner, and Fumal (1997)], \(\sigma(\mathrm{BJF})=0.309\), \(\sigma(\mathrm{emp-amp})=0.307\) and \(\sigma(\mathrm{NoSiteCorr})=0.305\). Individual station: with empirical-corrected amplitudes \(\sigma=0.269\) and with BJF-corrected amplitudes \(\sigma=0.268\).

Uses data from 21 TriNet stations with known \(V_{s,30}\) values. \(190\leq V_{s,30} \leq 958\,\mathrm{m/s}\). Uses two approaches for site term \(S_i\). In first method (denoted âempirically-corrected amplitudesâ, \(\mathrm{emp-amp}\)) uses empirical site amplification factors from previous study of TriNet stations (for PGA uses site factor for PSA at \(0.3\,\mathrm{s}\) because correction for PGA is unavailable). In second method [denoted âBoore-Joyner-Fumal (BJF)-corrected amplitudesâ, \(\mathrm{BJF}\)] uses amplification factors based on \(V_{s,30}\) from Boore, Joyner, and Fumal (1997) to correct observations to reference (arbitrarily selected) \(V_{s,30}=760\,\mathrm{m/s}\).

Uses only data with amplitudes \(>0.01\% \,\mathrm{g}\) (100 times greater than resolution of data, \(0.0001\% \,\mathrm{g}\)).

States that developed relations not intended for engineering applications due to lack of data from large events and from short distances. Equations developed for investigation of variability issues for which database limitations are not crucial.

Many records from Landers mainshock and aftershocks.

Uses standard linear regression since facilitates comparisons using regressions of different types of datasets, including single-station datasets.

Notes possible complications to functional form due to effects such as magnitude-dependent shape are not important due to small source size of most events.

Truncates data at \(300\,\mathrm{km}\) to get dataset that is well distributed in distance-amplitude space.

Notes that small differences between \(\sigma\)s when no site correction is applied and when site correction is applied could be due to complex site response in Los Angeles basin.

Fits trend-lines to residuals versus distance for each station and finds slope not significantly different from zero at most stations except for Osito Audit (OSI) (lying in mountains outside the geographical area defined by other stations), which has a significant positive trend.

Finds empirical-amplification factors give better estimate of average site response (average residuals per station closer to zero) than \(V_{s,30}\)-based factors at short periods but the reverse for long periods. Notes \(V_{s,30}\) gives more stable site-response estimates, with residuals for individual stations less than factor of \(1.6\) for most stations.

Finds standard deviations of station residuals not unusually large at sites with large mean residual, indicating that average site response estimates could be improved.

Plots standard deviation of station residuals using \(V_{s,30}\)-based factors and the average of these weighted by number of observations per station. Compares with standard deviation from entire databank. Finds that generally standard deviations of station residuals slightly lower (about \(10\%\)) than for entire databank.

Examines standard deviations of residuals averaged over \(0.5\)-unit magnitude bins and finds no apparent trend for \(\mathbf{M} 3.5\) to \(\mathbf{M} 7.0\) but notes lack of large magnitude data.

Restricts data by magnitude range (e.g. \(4 \leq \mathbf{M} \leq 6\)) and/or distance (e.g. \(\leq 80\,\mathrm{km}\)) and find no reduction in standard deviation.

Finds no reduction in standard deviation using one component rather than both.

Performs separate analysis of residuals for Landers events (10 stations having \(\geq 20\) observations) recorded at \(>100\,\mathrm{km}\). Notes that due to similarity of source and path effects for a station this should represent a minimum in single-station \(\sigma\). Finds \(\sigma\) of \(0.18 \pm 0.06\).

Exact functional form of Ground-motion model is not given but note includes linear and quadratic terms of magnitude and a geometric spreading term. Coefficients not given but report ratios of \(\sigma\) using different definitions w.r.t. \(\sigma\) using geometric mean.

Distribution w.r.t. NEHRP site classes is:

8 records

37 records

358 records

534 records

11 records

1 record

Use data from Next Generation Attenuation (NGA) database.

Distribution w.r.t. mechanism is:

333 records, 51 earthquakes

36 records, 12 earthquakes

329 records, 21 earthquakes

223 records, 9 earthquakes

25 records, 7 earthquakes

3 records, 3 earthquakes

Exclude records from Chi-Chi 1999 earthquake and its aftershocks to avoid bias due to over-representation of these data (\(>50\%\) of 3551 records of NGA databank).

Exclude records with PGA (defined using geometric mean) \(<0.05\,\mathrm{g}\) to focus on motions of engineering significance and to avoid problems with resolution of analogue records.

Exclude records with maximum usable period \(<0.5\,\mathrm{s}\).

Exclude records without hypocentral depth estimate since use depth in regression analysis.

Earthquakes contribute between \(1\) and \(138\) accelerograms.

Note data is from wide range of \(M\), \(d\), mechanism, site class and instrument type.

State aim was not to derive state-of-the-art Ground-motion models but to derive models with the same data and regression method for different component definitions.

Assume ratios of \(\sigma\)s from different models fairly insensitive to assumptions made during regression but that these assumptions affect \(\sigma\) values themselves.

Find ratios of \(\sigma\)s from using different definitions close to \(1\).

Note that results should be applied with caution to subduction and stable continental regions since have not been checked against these data.

Ground-motion model is for \(r_{epi}\): \[\log(y)=a+bM+c\log \sqrt{(R^2+h^2)}+e_1 S_1+e_2 S_2+e_3 S_3+e_4 S_4\] where \(y\) is in \(\,\mathrm{g}\), \(a=-2.487\), \(b=0.534\), \(c=-1.280\), \(h=3.94\), \(e_1=0\), \(e_2=0.365\), \(e_3=0.065\), \(e_4=0.053\), \(\sigma_{\mathrm{event}}=0.117\) and \(\sigma_{\mathrm{record}}=0.241\) (or alternatively \(\sigma_{\mathrm{station}}=0.145\) and \(\sigma_{\mathrm{record}}=0.232\)). For \(r_{hypo}\): \[\log(y)=a+bM+c\log R_h+e_1 S_1+e_2 S_2+e_3 S_3+e_4 S_4\] where \(y\) is in \(\,\mathrm{g}\), \(a=-2.500\), \(b=0.544\), \(c=-1.284\) and \(\sigma=0.292\) (do not report site coefficients for \(r_{hypo}\)).

Use four site classes:

Lacustrine and alluvial deposits with thickness \(>30\,\mathrm{m}\) (\(180\leq V_{s,30}<360\,\mathrm{m/s}\)). Sites in largest lacustrine plains in Umbria region. \(S_4=1\) and others are zero.

Lacustrine and alluvial deposits with thickness \(10\)â\(30\,\mathrm{m}\) (\(180\leq V_{s,30}<360\,\mathrm{m/s}\)). Sites in narrow alluvial plains or shallow basins. \(S_3=1\) and others are zero.

Shallow debris or colluvial deposits (\(3\)â\(10\,\mathrm{m}\)) overlaying rock (surface layer with \(V_s<360\,\mathrm{m/s}\)). Sites located on shallow colluvial covers or slope debris (maximum depth \(10\,\mathrm{m}\)) on gentle slopes. \(S_2=1\) and others are zero.

Rock (\(V_{s,30}>800\,\mathrm{m/s}\)). Sites on outcropping rock, or related morphologic features, such as rock crests and cliffs. \(S_1=1\) and others are zero.

Base classifications on recently collected detailed site information from site investigations, census data, topographic maps, data from previous reports on depth of bedrock, and data from public and private companies. Subscripts correspond to classification in Eurocode 8.

Focal depths between \(1.1\) and \(8.7\,\mathrm{km}\) except for one earthquake with depth \(47.7\,\mathrm{km}\).

Nearly all earthquakes have normal mechanism, with a few strike-slip earthquakes.

Select earthquakes with \(M_L\geq 4.0\) and \(d<100\,\mathrm{km}\).

Use \(M_L\) since available for all events.

Fault geometries only available for three events so use \(r_{epi}\) and \(r_{hypo}\) rather than \(r_{jb}\). Note that except for a few records differences between \(r_{epi}\) and \(r_{jb}\) are small.

Correct for baseline and instrument response and filter analogue records to remove high- and low-frequency noise by visually selecting a suitable frequency interval: average range was \(0.5\)â\(25\,\mathrm{Hz}\). Filter digital records with bandpass of, on average, \(0.3\)â\(40\,\mathrm{Hz}\).

For \(M_L<5\) no records from \(d_e>50\,\mathrm{km}\).

Use maximum-likelihood regression with event and record \(\sigma\)s and also one with station and record \(\sigma\)s. Perform each regression twice: once including site coefficients and once without to investigate reduction in \(\sigma\)s when site information is included.

Investigate difference in residuals for different stations when site coefficients are included or not. Find significant reductions in residuals for some sites, particularly for class \(\mathrm{C_E}\).

Note that some stations seem to display site-specific amplifications different than the general trend of sites within one site class. For these sites the residuals increase when site coefficients are introduced.

Find large negative residuals for records from the deep earthquake.

Find similar residuals for the four earthquakes not from the 1997â1998 Umbria-Marche sequence.

Ground-motion model is: \[\begin{aligned} \ln Y&=&f_1(M)+f_2(R)+f_3(F)+f_4(\mathrm{HW})+f_5(S)+f_6(D)\\ f_1(M)&=&\left\{ \begin{array}{l@{\quad}l} c_0+c_1 M&M\leq 5.5\\ c_0+c_1M+c_2(M-5.5)&5.5<M\leq 6.5\\ c_0+c_1M+c_2(M-5.5)+c_3(M-6.5)&M>6.5 \end{array} \right.\\ f_2(R)&=&(c_4+c_5M)\ln(\sqrt{r_{\mathrm{rup}}^2+c_6^2})\\ f_3(F)&=&c_7F_{\mathrm{RV}}f_F(H)+c_8F_N\\ f_F(H)&=&\left\{ \begin{array}{l@{\quad}l} H&H<1\,\mathrm{km}\\ 1&H\geq 1\,\mathrm{km} \end{array} \right.\\ f_4(\mathrm{HW})&=&c_9F_{\mathrm{RV}}f_{\mathrm{HW}}(M)f_{\mathrm{HW}}(H)\\ f_{\mathrm{HW}}(R)&=&\left\{ \begin{array}{l@{\quad}l} 1&r_{\mathrm{jb}}=0\,\mathrm{km}\\ 1-(r_{\mathrm{jb}}/r_{\mathrm{rup}})&r_{jb}>0\,\mathrm{km} \end{array} \right.\\ f_{\mathrm{HW}}(M)&=&\left\{ \begin{array}{l@{\quad}l} 0&M\leq 6.0\\ 2(M-6.0)&6.0<M<6.5\\ 1&M\geq 6.5 \end{array} \right.\\ f_{\mathrm{HW}}(H)&=&\left\{ \begin{array}{l@{\quad}l} 0&H\geq 20\,\mathrm{km}\\ 1-(H/20)&H<20\,\mathrm{km}\\ \end{array} \right.\\ f_5(S)&=&\left\{ \begin{array}{l@{\quad}l} c_{10}\ln\left(\frac{V_{s30}}{k_1}\right)+k_2\left\{\ln\left[\mathrm{PGA}_r+c\left(\frac{V_{s30}}{k_1}\right)^n\right]-\ln[\mathrm{PGA}_r+c]\right\}&V_{s30}<k_1\\ (c_{10}+k_2 n)\ln\left(\frac{V_{s30}}{k_1}\right)&V_{s30}\geq k_1 \end{array} \right.\\ f_6(D)&=&\left\{ \begin{array}{l@{\quad}l} c_{11}(D-1)&D<1\,\mathrm{km}\\ 0&1\leq D\leq 3\,\mathrm{km}\\ c_{12}\{k_3[0.0000454-\exp(-3.33D)]+k_4[0.472-\exp(-0.25D)]\}&D>3\,\mathrm{km} \end{array} \right.\end{aligned}\] Do not report coefficients, only display predicted ground motions. \(H\) is the depth to top of coseismic rupture in \(\,\mathrm{km}\), \(\mathrm{PGA}_r\) is the reference value of PGA on rock with \(V_{s30}=1100\,\mathrm{m/s}\), \(D\) is depth to \(2.5\,\mathrm{km/s}\) shear-wave velocity horizon (so-called sediment or basin depth) in \(\,\mathrm{km}\).

Use \(V_{s30}\) (average shear-wave velocity in top \(30\,\mathrm{m}\) in \(\,\mathrm{m/s}\)) to characterise site conditions.

Model developed as part of PEER Next Generation Attenuation (NGA) project.

State that model is not final and articles should be considered as progress reports.

NGA database only includes records that represent free-field conditions (i.e. records from large buildings are excluded).

Include earthquake if: 1) it occurred within the shallow continental lithosphere, 2) it was in a region considered to be tectonically active, 3) it had enough records to establish a reasonable source term and 4) it had generally reliable source parameters.

Exclude records from earthquakes classified as poorly recorded defined by: \(M<5.0\) and \(N<5\), \(5.0\leq M<6.0\) and \(N<3\) and \(6.0\leq M<7.0\), \(r_{\mathrm{rup}}>60\,\mathrm{km}\) and \(N<2\) where \(N\) is number of records. Include singly-recorded earthquakes with \(M\geq 7.0\) and \(r_{\mathrm{rup}}\leq 60\,\mathrm{km}\) because of importance in constraining near-source estimates.

Include records if: 1) it was from or near ground level, 2) it had negligible structural interaction effects and 3) it had generally reliable site parameters.

Find two-step regression technique was much more stable than one-step method and allows the independent evaluation and modelling of ground-motion scaling effects at large magnitudes. Find random effects regression analysis gives very similar results to two-step method.

Use classical data exploration techniques including analysis of residuals to develop functional forms. Develop forms using numerous iterations to capture observed trends. Select final forms based on: 1) their simplicity, although not an overriding factor, 2) their seismological bases, 3) their unbiased residuals and 4) their ability to be extrapolated to parameter values important for engineering applications (especially probabilistic seismic hazard analysis). Find that data did not always allow fully empirical development of functional form therefore apply theoretical constraints [coefficients \(n\) and \(c\) (period-independent) and \(k_i\) (period-dependent)].

Use three faulting mechanisms:

Reverse and reverse-oblique faulting,\(30^\circ<\lambda<150^\circ\), where \(\lambda\) is the average rake angle.

Normal and normal-oblique faulting, \(-150^\circ<\lambda<-30^\circ\).

Strike-slip, other \(\lambda\)s.

Find slight tendency for over-saturation of short-period ground motions at large magnitudes and short distances. Find other functional forms for magnitude dependence too difficult to constrain empirically or could not be reliably extrapolated to large magnitudes.

Note transition depth for buried rupture (\(1\,\mathrm{km}\)) is somewhat arbitrary.

Find weak but significant trend of increasing ground motion with dip for both reverse and strike-slip faults. Do not believe that seismological justified therefore do not include such a term.

Nonlinear site model constrained by theoretical studies since empirical data insufficient to constrain complex nonlinear behaviour.

Use depth to \(2.5\,\mathrm{km/s}\) horizon because it showed strongest correlation with shallow and deep sediment-depth residuals.

Believe that aspect ratio (ratio of rupture length to rupture width) has promise as a source parameter since it shows high correlation with residuals and could model change in ground-motion scaling at large magnitudes.

Do not find standard deviations are magnitude-dependent. Believe difference with earlier conclusions due to larger number of high-quality intra-event recordings for both small and large earthquakes.

Find standard deviation is dependent on level of ground shaking at soft sites.

Ground-motion model is: \[\log_{10}(\mathrm{PGA})=c_0+c_1 M+c_2 M^2+(c_3+c_4M)\log(\sqrt{d^2+h^2})+c_S S\] where \(\mathrm{PGA}\) is in \(\,\mathrm{g}\), \(c_0=-3.879\), \(c_1=1.178\), \(c_2=-0.068\), \(c_3=-2.063\), \(c_4=0.102\), \(c_S=0.411\), \(h=7.8\) and \(\sigma=0.3448\) (for larger horizontal component), \(c_0=-3.401\), \(c_1=1.140\), \(c_2=-0.070\), \(c_3=-2.356\), \(c_4=0.150\), \(c_S=0.415\), \(h=8.2\) and \(\sigma=0.3415\) (for horizontal component using vectorial addition), \(c_0=-3.464\), \(c_1=0.958\), \(c_2=-0.053\), \(c_3=-2.224\), \(c_4=0.147\), \(c_S=0.330\), \(h=6.1\) and \(\sigma=0.3137\) (for vertical).

Use two site classes (since do not have detailed information on geology at all considered stations):

Rock

Soil

Use selection criteria: \(3.0\leq M\leq 6.5\) and \(1\leq d_e\leq 100\,\mathrm{km}\).

Bandpass filter with cut-offs between \(0.1\) and \(0.25\,\mathrm{Hz}\) and between \(25\) and \(30\,\mathrm{Hz}\).

Compute mean ratio between recorded and predicted motions at some stations of the RAF network. Find large ratios for some stations on soil and for some on rock.

Ground-motion model is: \[\ln a=\alpha_0 +\alpha_1 M +\alpha_2 M^2+\alpha_3 \ln R+\alpha_5 R\] where \(a\) is in \(\,\mathrm{cm/s^2}\), \(\alpha_0=1.237\), \(\alpha_1=1.519\), \(\alpha_2=-0.0313\), \(\alpha_3=-0.844\), \(\alpha_5=-0.004\) and \(\sigma=0.780\).

Exclude records from soft soil sites or with previously known site effects (amplification or deamplification).

Focal depths between \(5\) and \(80\,\mathrm{km}\).

Also derive equation using functional form \(\ln a=\alpha_0+\alpha_1 M+\alpha_2 \ln R+\alpha_4 R\).

Select records from stations located along the seismically active Mexican Pacific coast.

Only use records from earthquakes with \(M>4.5\).

Exclude data from normal faulting earthquakes using focal mechanisms, focal depths, location of epicentre and characteristics of records because subduction zone events are the most dominant and frequent type of earthquakes.

Use \(M_w\) because consider best representation of energy release.

Visually inspect records to exclude poor quality records.

Exclude records from dams and buildings.

Exclude records from âslowâ earthquakes, which produce smaller short-period ground motions.

Correct accelerations by finding quadratic baseline to minimize the final velocity then filter using most appropriate bandpass filter (low cut-off frequencies between \(0.05\) and \(0.4\,\mathrm{Hz}\) and high cut-off frequency of \(30\,\mathrm{Hz}\)).

Use data from 105 stations: 7 in Chiapas, 6 in Oaxaca, 6 in Colima, 19 in Jalisco, 49 in Guerrero, 14 in MichoacÃ¡n and 6 near the MichoacÃ¡n-Guerrero border.

Ground-motion model is: \[\log (y)=a M_L-\log(X)+bX+c_j\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(a=0.41296\), \(b=0.0003\), \(c_1=0.5120\), \(c_2=0.3983\), \(c_3=0.2576\), \(c_4=0.1962\), \(c_5=0.1129\) and \(\sigma=0.2331\).

Data from ARM1 and ARM2 vertical borehole arrays of the Hualien LSST array at: surface (use \(c_1\)), \(5.3\,\mathrm{m}\) (use \(c_2\)), \(15.8\,\mathrm{m}\) (use \(c_3\)), \(26.3\,\mathrm{m}\) (use \(c_4\)) and \(52.6\,\mathrm{m}\) (use \(c_5\)). Surface geology at site is massive unconsolidated poorly bedded Pleistocene conglomerate composed of pebbles varying in diameter from \(5\) to \(20\,\mathrm{cm}\), following \(5\,\mathrm{m}\) is mainly composed of fine and medium sand followed by a gravel layer of \(35\,\mathrm{m}\).

Apply these criteria to achieve uniform data: \(M_L>5\), focal depth \(<30\,\mathrm{km}\) and \(0.42M_L-\log(X+0.025 10^{0.42M_L}-0.0033X+1.22>\log 10\) from a previous study.

Most records from \(M_L<6\).

Bandpass filter records with cut-offs at \(0.08\) and \(40\,\mathrm{Hz}\).

Propose \(M_s=1.154 M_L-1.34\).

Some comparisons between records and predicted spectra are show for four groups of records and find a good match although for the group \(M_L 6.75\) and \(X=62\,\mathrm{km}\) find a slight overestimation, which believe is due to not modelling nonlinear magnitude dependence.

Coefficients for vertical equations not reported.

Ground-motion model is: \[\ln \mathrm{Sa_{CU}}=\alpha_1+\alpha_2 (M_w-6)+\alpha_3 \ln R+\alpha_4 R\] where \(\mathrm{Sa_{CU}}\) is in \(\,\mathrm{cm/s^2}\), \(\alpha_1=5.6897\), \(\alpha_2=1.1178\), \(\alpha_3=-0.50\) and \(\alpha_4=-0.0060\) (\(\sigma\) is not reported).

Only use data from Ciudad Universitaria (CU) station, which is the reference site in the hill zone (rock) of Mexico City. Also derive models for Secretaria de Comunicaciones y Transportes (SCT) and Central de Abastos (CD), which are in lakebed zone, using same approach.

Weight data so large and small earthquakes equally represented in regression.

Use Bayesian regression where prior probability distributions of coefficients are assigned based on \(\omega^2\) model for source, frequency-dependent attenuation parameters and duration from previous studies and random vibration theory. Coefficients are updated using the records.

Compare observed and predicted spectra. Find match acceptable except for event 13 (Ometepec, 25/04/1989, \(M_w 6.9\)), which was an anomalously intense earthquake.

Derive model for comparison with less direct ways of estimating motions.

Ground-motion model is: \[\ln Y_s=C_0+C_1\{B_1+b_2 M-b_3\ln[R+b_4\exp(b_5M)]\}\] Coefficients not reported. \(\sigma=0.78\) for hard-site model.

Use data from hard sites (\(760\leq V_s \leq 1500\,\mathrm{m/s}\)) to develop hard-site model (term in curly brackets) and develop site terms for about 450 stations based on residuals w.r.t. rock model for \(>\)3000 records from \(>\)242 events.

Develop model for use in early warning system.

Use data from two networks of Central Weather Bureau. Select all data from Real-Time Digital network and, to account for lack of near-source records, data with \(r_{hypo}<25\,\mathrm{km}\) from Taiwan Strong-Motion Instrumentation Program network.

Compare observed and predicted hard-site PGAs grouped into magnitude ranges.

Ground-motion model is for \(D\leq 30\,\mathrm{km}\): \[\log \mathrm{pre}=a_1 M_w+b_1 X -\log (X+d_1 10^{0.5 M_w})+c_1\] and for \(D>30\,\mathrm{km}\): \[\log \mathrm{pre}=a_2M_w+b_2 X -\log (X)+c_2\] where \(\mathrm{pre}\) is in \(\,\mathrm{cm/s^2}\), \(a_1=0.56\), \(b_1=-0.0031\), \(c_1=0.26\), \(d_1=0.0055\), \(a_2=0.41\), \(b_2=-0.0039\), \(c_2=1.56\), \(\sigma_1=0.37\) and \(\sigma_2=0.40\).

Use \(V_{s,30}\) to characterise site effects using correction formula: \(G=\log (\mathrm{obs}/\mathrm{pre})=p \log V_{s,30} + q\). Derive \(p\) and \(q\) by regression analysis on residuals averaged at intervals of every \(100\,\mathrm{m/s}\) in \(V_{s,30}\). \(p=-0.55\) and \(q=1.35\) for PGA. Note that the equation without site correction predicts ground motions at sites with \(V_{s,30} \approx 300\,\mathrm{m/s}\).

Focal depths, \(D\), for shallow events between \(0\,\mathrm{km}\) and \(30\,\mathrm{km}\) and for deep events between \(30\,\mathrm{km}\) and about \(180\,\mathrm{km}\).

Note that it is difficult to determine a suitable model form due to large variability of strong-motion data, correlation among model variables and because of coupling of variables in the model. Therefore choose a simple model to predict average characteristics with minimum parameters.

Introduce correction terms for site effects and regional anomalies.

Originally collect 91731 records from 4967 Japanese earthquakes.

Include foreign near-source data (from California and Turkey, which are compressional regimes similar to Japan) because insufficient from Japan.

High-pass filter records with cut-off of \(0.1\,\mathrm{Hz}\). Low-pass filter analogue records using cut-offs selected by visual inspection.

Choose records where: 1) \(M_w\geq 5.5\), 2) data from ground surface, 3) two orthogonal horizontal components available, 4) at least five stations triggered and 5) the record passed this \(M_w\)-dependent source distance criterion: \(f(M_w,X)\geq \log 10\) (for data from mechanical seismometer networks) or \(f(M_w,X)\geq \log 2\) (for data from other networks) where \(f(M_w,X)=0.42 M_w-0.0033X-\log(X+0.025 10^{0.43 M_w})+1.22\) (from a consideration of triggering of instruments).

Examine data distributions w.r.t. amplitude and distance for each magnitude. Exclude events with irregular distributions that could be associated with a particular geological/tectonic feature (such as volcanic earthquakes).

Do not include data from Chi-Chi 1999 earthquake because have remarkably low amplitudes, which could be due to a much-fractured continental margin causing different seismic wave propagation than normal.

Data from 2236 different sites in Japan and 305 in other countries.

Note relatively few records from large and deep events.

Note that maybe best to use stress drop to account for different source types (shallow, interface or intraslab) but cannot use since not available for all earthquakes in dataset.

Investigate effect of depth on ground motions and find that ground-motions amplitudes from earthquakes with \(D>30\,\mathrm{km}\) are considerably different than from shallower events hence derive separate equations for shallow and deep events.

Select \(0.5\) within function from earlier study.

Weight regression for shallow events to give more weight to near-source data. Use weighting of \(6.0\) for \(X\leq 25\,\mathrm{km}\), \(3.0\) for \(25<X\leq 50\,\mathrm{km}\), \(1.5\) for \(50<X\leq 75\,\mathrm{km}\) and \(1.0\) for \(X>75\,\mathrm{km}\). Note that weighting scheme has no physical meaning.

Note that amplitude saturation at short distances for shallow model is controlled by crustal events hence region within several tens of \(\,\mathrm{km}\)s of large (\(M_w>8.0\)) interface events falls outside range of data.

Note standard deviation decreases after site correction term is introduced.

Introduce correction to model anomalous ground motions in NE Japan from intermediate and deep earthquakes occurring in the Pacific plate due to unique \(Q\) structure beneath the island arc. Correction is: \(\log (\mathrm{obs}/\mathrm{pre})=(\alpha R_{\mathrm{tr}}+\beta)(D-30)\) where \(R_{\mathrm{tr}}\) is shortest distance from site to Kuril and Izu-Bonin trenches. \(\alpha\) and \(\beta\) are derived by regression on subset fulfilling criteria: hypocentre in Pacific plate, station E of \(137^{\circ}\) E and station has \(V_{s,30}\) measurement. For PGA \(\alpha=-6.73 \times 10^{-5}\) and \(\beta=2.09 \times 10^{-2}\). Find considerable reduction in standard deviation after correction. Note that \(R_{\mathrm{tr}}\) may not be the best parameter due to observed bias in residuals for deep events.

Examine normalised observed ground motions w.r.t. predicted values and find good match.

Examine residuals w.r.t. distance and predicted values. Find residuals decrease with increasing predicted amplitude and with decreasing distance. Note that this is desirable from engineering point of view, however, note that it may be due to insufficient data with large amplitudes and from short distances.

Examine total, intra-event and inter-event residuals w.r.t. \(D\) for \(D>30\,\mathrm{km}\). When no correction terms are used, intra-event residuals are not biased but inter-event residuals are. Find mean values of total error increase up to \(D=70\,\mathrm{km}\) and then are constant. Find depth correction term reduces intra-event residuals considerably but increases inter-event error slightly. Overall bias improves for \(D<140\,\mathrm{km}\). Find site corrections have marginal effect on residuals.

Find no bias in residuals w.r.t. magnitude.

Ground-motion model is: \[\log_{10} Y=a_1 M_w-bX+c_0-\log_{10}(X+d 10^{0.5 M_w})+c_j\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(a_1=0.595\), \(b=0.00395\), \(c_0=0.03\), \(d=0.0065\), \(\sigma_{intra}=0.129\), \(\sigma_{inter}=0.110\) and \(\sigma_{total}=0.169\).

Data from stations in site classes I, II and III, for which derive correction factors w.r.t. overall model.

Also derive models using data from 136 subduction earthquakes (5882 records).

Also present models using focal depth and short period level of acceleration source spectrum as additional variables.

Find that including short period level of acceleration source spectrum significantly reduces inter-event \(\sigma\).

Ground-motion model is: \[y=c\exp(\alpha M_s)[D^k+a]^{-\beta-\gamma R}\] where \(D\) is \(r_{hypo}\) and \(R\) is \(r_{epi}\), \(y\) is in \(\,\mathrm{m/s^2}\), \(c=0.38778\), \(\alpha=0.32927\), \(k=0.29202\), \(a=1.557574\), \(\beta=1.537231\), \(\gamma=0.027024\) and \(\sigma=0.03\) (note that this \(\sigma\) is additive).

All records except one at \(13\,\mathrm{km}\) from distances of \(20\) to \(70\,\mathrm{km}\) so note that lack information from near field.

Compare predictions to records from the 2003 Boumerdes (\(M_w 6.8\)) earthquake and find that it underpredicts the recorded motions, which note maybe due to local site effects.

Ground-motion model is: \[\log_{10} Y=a+bM+c\log_{10}R+s_{1,2}\] \(Y\) is in \(\,\mathrm{g}\), \(a=-4.417\), \(b=0.770\), \(c=-1.097\), \(s_1=0\), \(s_2=0.123\), \(\sigma_{\mathrm{event}}=0.069\) and \(\sigma_{\mathrm{record}}=0.339\) (for horizontal PGA assuming intra-event \(\sigma\)), \(a=-4.367\), \(b=0.774\), \(c=-1.146\), \(s_1=0\), \(s_2=0.119\), \(\sigma_{\mathrm{station}}=0.077\) and \(\sigma_{\mathrm{record}}=0.337\) (for horizontal PGA assuming intra-station \(\sigma\)), \(a=-4.128\), \(b=0.722\) ,\(c=-1.250\), \(s_1=0\), \(s_2=0.096\), \(\sigma_{\mathrm{event}}=0.085\) and \(\sigma_{\mathrm{record}}=0.338\) (for vertical PGA assuming intra-event \(\sigma\)), \(a=-4.066\), \(b=0.729\), \(c=-1.322\), \(s_1=0\), \(s_2=0.090\), \(\sigma_{\mathrm{station}}=0.105\) and \(\sigma_{\mathrm{record}}=0.335\) (for vertical PGA assuming intra-station \(\sigma\)).

Use two site classes:

Rock, where \(V_s>800\,\mathrm{m/s}\). Use \(s_1\).

Soil, where \(V_s<800\,\mathrm{m/s}\). This includes all kinds of superficial deposits from weak rock to alluvial deposits. Use \(s_2\).

Can only use two classes due to limited information.

Use 195 accelerometric records from 51 earthquakes (\(2.5\leq M_L \leq 5.4\)) from 29 sites. Most records are from rock or stiff sites. Most data from \(r_{hypo}<50\,\mathrm{km}\) with few from \(>100\,\mathrm{km}\). Also use data from velocimeters (Lennartz \(1\) or \(5\,\mathrm{s}\) sensors and Guralp CMG-40Ts). In total 2895 records with \(r_{hypo}<50\,\mathrm{km}\) from 78 events and 22 stations available, most from \(20\leq r_{hypo}\leq 30\,\mathrm{km}\).

For records from analogue instruments, baseline correct, correct for instrument response and bandpass filter with average cut-offs at \(0.5\) and \(20\,\mathrm{Hz}\) (after visual inspection of Fourier amplitude spectra). For records from digital instruments, baseline correct and bandpass filter with average cut-offs at \(0.2\) and \(30\,\mathrm{Hz}\). Sampling rate is \(200\,\mathrm{Hz}\). For records from velocimeters, correct for instrument response and bandpass filter with average cut-offs at \(0.5\) and \(25\,\mathrm{Hz}\). Sampling rate is \(100\,\mathrm{Hz}\).

Select records from 37 stations with \(10\leq r_{hypo}\leq 50\,\mathrm{km}\).

Compare predictions and observations for \(M_L 4.4\) and find acceptable agreement. Also find agreement between data from accelerometers and velocimeters.

Ground-motion model is: \[\log (y)=a+bM+c\log(R)+dR\] where \(y\) is in \(\,\mathrm{cm/s^2}\). For horizontal PGA: \(a=1.861\), \(b=0.201\), \(c=-0.554\), \(d=-0.0091\) and \(\sigma=0.242\) (for Zagros, rock sites and \(M_s\geq 4.5\) or \(m_b\geq 5.0\)), \(a=1.831\), \(b=0.208\), \(c=-0.499\), \(d=-0.0137\) and \(\sigma=0.242\) (for Zagros, rock sites and \(3<M_s<4.6\) or \(4.0\leq m_b<5.0\)), \(a=2.058\), \(b=0.243\), \(c=-1.02\), \(d=-0.000875\) and \(\sigma=0.219\) (for central Iran and rock sites), \(a=2.213\), \(b=0.225\), \(c=-0.847\), \(d=-0.00918\) and \(\sigma=0.297\) (for Zagros and soil sites), \(a=1.912\), \(b=0.201\), \(c=-0.790\), \(d=-0.00253\) and \(\sigma=0.204\) (for central Iran and soil sites). For vertical PGA: \(a=2.272\), \(b=0.115\), \(c=-0.853\), \(d=-0.00529\) and \(\sigma=0.241\) (for Zagros, rock sites and \(M_s\geq 4.5\) or \(m_b\geq 5.0\)), \(a=2.060\), \(b=0.147\)

^{21}, \(c=-0.758\), \(d=-0.00847\) and \(\sigma=0.270\) (for Zagros, rock sites and \(M_s\geq 3.0\) or \(m_b\geq 4.0\)), \(a=1.864\), \(b=0.232\), \(c=-1.049\), \(d=-0.000372\) and \(\sigma=0.253\) (for central Iran and rock sites), \(a=2.251\), \(b=0.140\)^{22}, \(c=-0.822\), \(d=-0.00734\) and \(\sigma=0.290\)^{23}(for Zagros and soil sites) and \(a=1.76\), \(b=0.232\)^{24}, \(c=-1.013\), \(d=-0.000551\) and \(\sigma=0.229\) (for central Iran and soil sites).Uses two site classes:

Sedimentary. 55 records.

Rock. 95 records.

Bases classification on geological maps, station visits, published classifications and shape of response spectra from strong-motion records. Notes that the classification could be incorrect for some stations. Uses only two classes to reduce possible errors.

Divides Iran into two regions: Zagros and other areas.

Select data with \(M_s\) or \(m_b\) where \(m_b>3.5\). Notes that only earthquakes with \(m_b>5.0\) are of engineering concern for Iran but since not enough data (especially for Zagros) includes smaller earthquakes.

Use \(M_s\) when \(m_b\geq 4\).

Records bandpass filtered using Ormsby filters with cut-offs and roll-offs of \(0.1\)â\(0.25\,\mathrm{Hz}\) and \(23\)â\(25\,\mathrm{Hz}\).

Notes that some data from far-field.

Notes that some records do not feature the main portion of shaking.

To be consistent, calculates \(r_{hypo}\) using S-P time difference. For some records P wave arrival time is unknown so use published hypocentral locations. Assumes focal depth of \(10\,\mathrm{km}\) for small and moderate earthquakes and \(15\,\mathrm{km}\) for large earthquakes.

Does not recommend use of relation for Zagros and soil sites due to lack of data (15 records) and large \(\sigma\).

Compares recorded and predicted motions for some ranges of magnitudes and concludes that they are similar.

Ground-motion model for crustal earthquakes is: \[\begin{aligned} \ln \mathrm{SA}'_{A/B}(T)&=&C_1'(T)+C_{4AS}(M-6)+C_{3AS}(T)(8.5-M)^2+C_5'(T)r\\ &&{}+[C_8'(T)+C_{6AS}(M-6)]\ln\sqrt{r^2+C_{10AS}^2(T)}+C_{46}'(T)r_{VOL}\\ &&{}+C_{32}\mathrm{CN}+C_{33AS}(T)\mathrm{CR}+F_{HW}(M,r)\end{aligned}\] Ground-motion model for subduction earthquakes is: \[\begin{aligned} \ln \mathrm{SA}'_{A/B}(T)&=&C_{11}'(T)+\{ C_{12Y}+[C_{15}'(T)-C_{17}'(T)]C_{19Y}\} (M-6)\\ &&{}+C_{13Y}(T)(10-M)^3+C_{17}'(T) \ln[r+C_{18Y} \exp(C_{19Y}M)]+C_{20}'(T)H_c\\ &&{}+C_{24}'(T) \mathrm{SI}+C_{46}'(T) r_{VOL} (1-\mathrm{DS})\end{aligned}\] where \(C_{15}'(T)=C_{17Y}(T)\). For both models: \[\ln \mathrm{SA}'_{C,D}(T)=\ln \mathrm{SA}'_{A/B} (T)+C_{29}'(T) \delta_C+[C_{30AS}(T) \ln(\mathrm{PGA}'_{A/B}+0.03)+C_{43}'(T)] \delta_D\] where \(\mathrm{PGA}'_{A/B}=\mathrm{SA}'_{A/B}(T=0)\). Final model given by: \[\mathrm{SA}_{A/B, C, D}(T)=\mathrm{SA}'_{A/B, C, D}(T) (\mathrm{PGA}_{A/B, C, D}/\mathrm{PGA}'_{A/B, C, D})\] where \(\mathrm{SA}_{A/B, C, D}\) is in \(\,\mathrm{g}\), \(r_{VOL}\) is length in \(\,\mathrm{km}\) of source-to-site path in volcanic zone and \(F_{HW}(M,r)\) is hanging wall factor of Abrahamson and Silva (1997). Coefficients for \(\mathrm{PGA}\) (larger component) are: \(C_1=0.28815\), \(C_3=0\), \(C_4=-0.14400\), \(C_5=-0.00967\), \(C_6=0.17000\), \(C_8=-0.70494\), \(C_{10}=5.60000\), \(C_{11}=8.68354\), \(C_{12}=1.41400\), \(C_{13}=0\), \(C_{15}=-2.552000\), \(C_{17}=-2.56727\), \(C_{18}=1.78180\), \(C_{19}=0.55400\), \(C_{20}=0.01550\), \(C_{24}=-0.50962\), \(C_{29}=0.30206\), \(C_{30}=-0.23000\), \(C_{32}=0.20000\), \(C_{33}=0.26000\), \(C_{43}=-0.31769\), \(C_{46}=-0.03279\), \(\sigma_{M6}=0.4865\), \(\sigma_{slope}=-0.1261\), where \(\sigma=\sigma_{M6}+\sigma_{slope}(M_w-6)\) for \(5<M_w<7\), \(\sigma=\sigma_{M6}-\sigma_{slope}\) for \(M_w<5\) and \(\sigma=\sigma_{M6}+\sigma_{slope}\) for \(M_w>7\) (intra-event), and \(\tau=0.2687\) (inter-event). Coefficients for \(\mathrm{PGA}'\) (larger component) are: \(C_1=0.18130\), \(C_3=0\), \(C_4=-0.14400\), \(C_5=-0.00846\), \(C_6=0.17000\), \(C_8=-0.75519\), \(C_{10}=5.60000\), \(C_{11}=8.10697\), \(C_{12}=1.41400\), \(C_{13}=0\), \(C_{15}=-2.552000\), \(C_{17}=-2.48795\), \(C_{18}=1.78180\), \(C_{19}=0.55400\), \(C_{20}=0.01622\), \(C_{24}=-0.41369\), \(C_{29}=0.44307\), \(C_{30}=-0.23000\), \(C_{32}=0.20000\), \(C_{33}=0.26000\), \(C_{43}=-0.29648\), \(C_{46}=-0.03301\), \(\sigma_{M6}=0.5035\), \(\sigma_{slope}=-0.0635\) and \(\tau=0.2598\).

Use site classes (combine A and B together and do not use data from E):

Strong rock. Strong to extremely-strong rock with: a) unconfined compressive strength \(>50\,\mathrm{MPa}\), and b) \(V_{s,30}>1500\,\mathrm{m/s}\), and c) not underlain by materials with compressive strength \(<18\,\mathrm{MPa}\) or \(V_s<600\,\mathrm{m/s}\).

Rock. Rock with: a) compressive strength between \(1\) and \(50\,\mathrm{MPa}\), and b) \(V_{s,30}>360\,\mathrm{m/s}\), and c) not underlain by materials having compressive strength \(<0.8\,\mathrm{MPa}\) or \(V_s<300\,\mathrm{m/s}\).

\(\delta_C=1\), \(\delta_D=0\). Shallow soil sites. Sites that: a) are not class A, class B or class E sites, and b) have low-amplitude natural period, \(T\), \(\leq 0.6\,\mathrm{s}\), or c) have soil depths \(\leq\) these depths:

Soil type Maximum and description soil depth (\(\,\mathrm{m}\)) Cohesive soil Representative undrained shear strengths (\(\,\mathrm{kPa}\)) Very soft \(<12.5\) 0 Soft \(12.5\)â\(25\) 20 Firm \(25\)â\(50\) 25 Stiff \(50\)â\(100\) 40 Very stiff or hard \(100\)â\(200\) 60 Cohesionless soil Representative SPT N values Very loose \(<6\) 0 Loose dry \(6\)â\(10\) 40 Medium dense \(10\)â\(30\) 45 Dense \(30\)â\(50\) 55 Very dense \(>50\) 60 Gravels \(>30\) 100 \(\delta_D=1\), \(\delta_C=0\). Deep or soft soil sites. Sites that: a) are not class A, class B or class E sites, and b) have a low-amplitude \(T>0.6\,\mathrm{s}\), or c) have soil depths \(>\) depths in table above, or c) are underlain by \(<10\,\mathrm{m}\) of soils with an undrained shear-strength \(<12.5\,\mathrm{kPa}\) or soils with SPT N-values \(<6\).

Very soft soil sites. Sites with: a) \(>10\,\mathrm{m}\) of very soft soils with undrained shear-strength \(<12.5\,\mathrm{kPa}\), b) \(>10\,\mathrm{m}\) of soils with SPT N values \(<6\), c) \(>10\,\mathrm{m}\) of soils with \(V_s<150\,\mathrm{m/s}\), or d) \(>10\,\mathrm{m}\) combined depth of soils with properties as described in a), b) and c).

Categories based on classes in existing New Zealand Loadings Standard but modified following statistical analysis. Note advantage of using site categories related to those in loading standards. Site classifications based on site periods but generally categories from site descriptions.

Classify earthquakes in three categories:

Earthquakes occurring in the shallow crust of overlying Australian plate. 24 earthquakes. Classify into:

\(-33\leq \lambda \leq 33^{\circ}\), \(147 \leq \lambda \leq 180^{\circ}\) or \(-180 \leq \lambda \leq -147^{\circ}\) where \(\lambda\) is the rake. 6 earthquakes. Centroid depths, \(H_c\), \(4\leq H_c \leq 13\,\mathrm{km}\). \(5.20\leq M_w \leq 6.31\). \(\mathrm{CN}=0\), \(\mathrm{CR}=0\).

\(-146 \leq \lambda \leq -34^{\circ}\). 7 earthquakes. \(7\leq H_c \leq 17\,\mathrm{km}\). \(5.27\leq M_w \leq 7.09\). \(\mathrm{CN}=-1\), \(\mathrm{CR}=0\).

\(33 \leq \lambda \leq 66^{\circ}\) or \(124 \leq \lambda \leq 146^{\circ}\). 3 earthquakes. \(5\leq H_c \leq 19\,\mathrm{km}\). \(5.75\leq M_w \leq 6.52\). \(\mathrm{CR}=0.5\), \(\mathrm{CN}=0\).

\(67 \leq \lambda \leq 123^{\circ}\). 8 earthquakes. \(4\leq H_c \leq 13\,\mathrm{km}\). \(5.08\leq M_w \leq 7.23\). \(\mathrm{CR}=1\), \(\mathrm{CN}=0\).

Earthquake occurring on the interface between Pacific and Australian plates with \(H_c<50\,\mathrm{km}\). 5 reserve and 1 strike-slip with reverse component. Use data with \(15\leq H_c \leq 24\,\mathrm{km}\). Classify using location in 3D space. 6 earthquakes. \(5.46\leq M_w \leq 6.81\). \(\mathrm{SI}=1\), \(\mathrm{DS}=0\).

Earthquakes occurring in slab source zone within the subducted Pacific plate. Predominant mechanism changes with depth. 19 earthquakes. \(26\leq H_c \leq 149\,\mathrm{km}\). Split into shallow slab events with \(H_c \leq 50\,\mathrm{km}\) (9 normal and 1 strike-slip, \(5.17\leq M_w \leq 6.23\)) and deep slab events with \(H_c>50\,\mathrm{km}\) (6 reverse and 3 strike-slip, \(5.30\leq M_w \leq 6.69\)). \(\mathrm{SI}=0\), \(\mathrm{DS}=1\) (for deep slab events).

Note seismicity cross sections not sufficient to distinguish between interface and slab events, also require source mechanism.

Find that mechanism is not a significant extra parameter for motions from subduction earthquakes.

State that model is not appropriate for source-to-site combinations where the propagation path is through the highly attenuating mantle wedge.

Note magnitude range of New Zealand is limited with little data for large magnitudes and from short distances. Most data from \(d>50\,\mathrm{km}\) and \(M_w<6.5\).

Only include records from earthquakes with available \(M_w\) estimates because correlations between \(M_L\) and \(M_w\) are poor for New Zealand earthquakes. Include two earthquakes without \(M_w\) values (\(M_s\) was converted to \(M_w\)) since they provide important data for locations within and just outside the Central Volcanic Region.

Only include data with centroid depth, mechanism type, source-to-site distance and a description of site conditions.

Only include records with PGA above these limits (dependent on resolution of instrument):

Acceleroscopes (scratch-plates): \(0.02\,\mathrm{g}\)

Mechanical-optical accelerographs: \(0.01\,\mathrm{g}\)

Digital 12-bit accelerographs: \(0.004\,\mathrm{g}\)

Digital 16-bit accelerographs: \(0.0005\,\mathrm{g}\)

Exclude data from two sites: Athene A (topographic effect) and Hanmer Springs (site resonance at \(1.5\)â\(1.7\,\mathrm{Hz}\)) that exhibit excessive amplifications for their site class.

Exclude data from sites of class E (very soft soil sites with \(\gtrsim 10\,\mathrm{m}\) of material with \(V_s<150\,\mathrm{m/s}\)) to be consistent with Abrahamson and Silva (1997) and Youngs et al. (1997). Not excluded because of large amplifications but because spectra appear to have site-specific characteristics.

Exclude records from bases of buildings with \(>4\) storeys because may have been influenced by structural response.

Exclude data from very deep events with travel paths passing through the highly attenuating mantle were excluded.

Only use response spectral ordinates for periods where they exceed the estimated noise levels of the combined recording and processing systems.

Lack of data from near-source. Only 11 crustal records from distances \(<25\,\mathrm{km}\) with 7 of these from 3 stations. To constrain model at short distances include overseas PGA data using same criteria as used for New Zealand data. Note that these data were not intended to be comprehensive for \(0\)â\(10\,\mathrm{km}\) range but felt to be representative. Note that it is possible New Zealand earthquakes may produce PGAs at short distances different that those observed elsewhere but feel that it is better to constrain the near-source behaviour rather than predict very high PGAs using an unconstrained model.

In order to supplement limited data from moderate and high-strength rock and from the volcanic region, data from digital seismographs were added.

Data corrected for instrument response.

Derive model from âbase modelsâ (other Ground-motion models for other regions). Select âbase modelâ using residual analyses of New Zealand data w.r.t. various models. Choose models of Abrahamson and Silva (1997) for crustal earthquakes and Youngs et al. (1997). Link these models together by common site response terms and standard deviations to get more robust coefficients.

Apply constraints using âbase modelsâ to coefficients that are reliant on data from magnitude, distance and other model parameters sparsely represented in the New Zealand data. Coefficients constrained are those affecting estimates in near-source region, source-mechanism terms for crustal earthquakes and hanging-wall terms. Eliminate some terms in âbase modelsâ because little effect on measures of fit using Akaike Information Criterion (AIC).

Apply the following procedure to derive model. Derive models for PGA and SA using only records with response spectra available (models with primed coefficients). Next derive model for PGA including records without response spectra (unprimed coefficients). Finally multiply model for SA by ratio between the PGA model using all data and that using only PGA data with corresponding response spectra. Apply this method since PGA estimates using complete dataset for some situations (notably on rock and deep soil and for near-source region) are higher than PGA estimates using reduced dataset and are more in line with those from models using western US data. This scaling introduces a bias in final model. Do not correct standard deviations of models for this bias.

Use \(r_{rup}\) for 10 earthquakes and \(r_c\) for rest. For most records were \(r_c\) was used, state that it is unlikely model is sensitive to use \(r_c\) rather than \(r_{rup}\). For five records discrepancy likely to be more than \(10\%\).

Free coefficients are: \(C_1\), \(C_{11}\), \(C_8\), \(C_{17}\), \(C_5\), \(C_{46}\), \(C_{20}\), \(C_{24}\), \(C_{29}\) and \(C_{43}\). Other coefficients fixed during regression. Coefficients with subscript AS are from Abrahamson and Silva (1997) and those with subscript Y are from Youngs et al. (1997). Try varying some of these fixed coefficients but find little improvement in fits.

State that models apply for \(5.25\leq M_w \leq 7.5\) and for distances \(\leq 400\,\mathrm{km}\), which is roughly range covered by data.

Note possible problems in applying model for \(H_c>150\,\mathrm{km}\) therefore suggest \(H_c\) is fixed to \(150\,\mathrm{km}\) if applying model to deeper earthquakes.

Note possible problems in applying model for \(M_w<5.25\).

Apply constraints to coefficients to model magnitude- and distance-saturation.

Try including an anelastic term for subduction earthquakes but find insignificant.

Investigate possibility of different magnitude-dependence and attenuation rates for interface and slab earthquakes but this required extra parameters that are not justified by AIC.

Investigate possible different depth dependence for interface and slab earthquakes but extra parameters not justified in terms of AIC.

Try adding additive deep slab term but not significant according to AIC.

Cannot statistically justify nonlinear site terms. Believe this could be due to lack of near-source records.

Find that if a term is not included for volcanic path lengths then residuals for paths crossing the volcanic zone are increasingly negative with distance but this trend is removed when a volcanic path length term is included.

Compare predictions to observed ground motions in 21/08/2003 Fiordland interface (\(M_w 7.2\)) earthquake and its aftershocks. Find ground motions, in general, underestimated.

Ground-motion model is [adopted from Boore, Joyner, and Fumal (1997)]: \[\ln(Y)=\theta_1+\theta_2(M_w-6)+\theta_3(M_w-6)^2-\theta_4\ln(\sqrt{R_{jb}^2+\theta_5^2})-\theta_6 \ln(V_{s,30}/\theta_7)\]

Use \(V_{s,30}\) to characterize site.

Use data of Boore, Joyner, and Fumal (1997).

Develop Bayesian regression method to account for parameter uncertainty in measured accelerations (due to orientation of instrument) (coefficient of variation of \(\sim 0.30\), based on analysis of recorded motions) and magnitudes (coefficient of variation of \(\sim 0.10\), based on analysis of reported \(M_w\) by various agencies) to better understand sources of uncertainty and to reduce model variance.

Do not report coefficients. Only compare predictions with observations and with predictions by model of Boore, Joyner, and Fumal (1997) for \(M_w 7.5\) and \(V_{s,30}=750\,\mathrm{m/s}\). Find slightly different coefficients than Boore, Joyner, and Fumal (1997) but reduced model standard deviations.

Ground-motion model is: \[\log_{10}(\mathrm{PGA})=a_{\mathrm{PGA}}M+b_{\mathrm{PGA}}R-\log_{10}(R)+S_{\mathrm{PGA},k}, k=1, 2, \ldots, 5\] where \(\mathrm{PGA}\) is in \(\,\mathrm{cm/s^2}\), \(a_{\mathrm{PGA}}=0.4346\), \(b_{\mathrm{PGA}}=-0.002459\), \(S_{\mathrm{PGA},1}=0.9259\), \(S_{\mathrm{PGA},2}=0.9338\), \(S_{\mathrm{PGA},3}=0.9929\), \(S_{\mathrm{PGA},4}=0.9656\), \(S_{\mathrm{PGA},5}=0.9336\) and \(\sigma=0.2966\).

Use five site categories (from Eurocode 8):

\(V_{s,30}>800\,\mathrm{m/s}\). Use \(S_{\mathrm{PGA},1}\). 43 stations, 396 records.

\(360<V_{s,30}<800\,\mathrm{m/s}\). Use \(S_{\mathrm{PGA},2}\). 399 stations, 4190 records.

\(180<V_{s,30}<360\,\mathrm{m/s}\). Use \(S_{\mathrm{PGA},3}\). 383 stations, 4108 records.

\(V_{s,30}<180\,\mathrm{m/s}\). Use \(S_{\mathrm{PGA},4}\). 65 stations, 644 records.

Site D or C underlain in first \(20\,\mathrm{m}\) with a stiffer layer of \(V_s>800\,\mathrm{m/s}\). Use \(S_{\mathrm{PGA},5}\). 6 stations, 52 records.

Use statistical method of Boore (2004) with parameters derived from KiK-Net profiles in order to extend \(V_s\) profiles down to \(30\,\mathrm{m}\) depth.

Records from K-Net network whose digital stations have detailed geotechnical characterisation down to \(20\,\mathrm{m}\) depth.

Retain only records from events whose focal depths \(<25\,\mathrm{km}\).

Convert \(M_{\mathrm{JMA}}\) to \(M_w\) using empirical conversion formula to be consist with other studies.

Apply magnitude-distance cut-off to exclude distant records.

Bandpass filter all records with cut-offs \(0.25\) and \(25\,\mathrm{Hz}\). Visually inspect records for glitches and to retain only main event if multiple events recorded.

Find that one-stage maximum likelihood regression gives almost the same results.

Also derive equations for other strong-motion parameters.

Ground-motion model is: \[\log_{10}(\mathrm{PGA})=a+b M+c\log_{10} R\] where \(y\) is in \(\,\mathrm{m/s^2}\), \(a=-2.50 \pm 0.18\), \(b=0.99 \pm 0.05\) and \(c=-2.22 \pm 0.08\) when \(M=M_{\mathrm{LDG}}\) and \(a=-2.55 \pm 0.19\), \(b=1.04 \pm 0.05\) and \(c=-2.17 \pm 0.08\) when \(M=M_{\mathrm{ReNass}}\) (\(\sigma\) is not given although notes that âexplained variance is of the order of 84%â).

Focal depths between \(0\) and \(17\,\mathrm{km}\).

Most data from \(R<200\,\mathrm{km}\).

Uses PGAs from S-waves.

Finds that introducing an anelastic attenuation term does not significantly improve explained variance because term is poorly constrained by data due to trade offs with geometric term and travel paths are short. When an anelastic term is introduced finds: \(\log_{10}(\mathrm{PGA})=-3.19(\pm 0.25)+1.09 (\pm 0.05) M_{\mathrm{ReNass}}-1.83 (\pm 0.12)\log_{10} R-0.0013 (\pm 0.0004)R\).

Ground-motion model is: \[\begin{aligned} \log y&=&a+bM+c \log r+d r\\ r&=&\sqrt{r^2+h^2}\end{aligned}\] where \(y\) is in \(\,\mathrm{g}\), \(a=-1.8\), \(b=0.45\), \(c=-1.6\), \(d=-0.0013\), \(h=10\) and \(\sigma=0.426\).

Use data from networks of IGC/ICC and IGN (Spain), RAP (France), SSN-ENEL (Italy) and CRECIT (Andorra). Umbria-Marche 1997â1998 sequence contributes 144 records from 8 earthquakes, SE Spain 56 records from 13 events, France 32 records from 1 event and the Pyrenees 102 records from 9 events.

Data reasonably uniform up to about \(100\,\mathrm{km}\). Believe model can be used up to \(M 5.2\).

Records filtered with cut-offs of \(0.25\) and \(25\,\mathrm{Hz}\).

Examine residuals w.r.t. magnitude and distance.

Compare predictions and observations for Alhucemas (24/2/2004, \(M_L 6.5\)), Lourdes (17/11/2006, \(M_L 5.1\)) and San Vicente (12/2/2007, \(M_w 6.1\)) earthquakes.

Ground-motion model is: \[\log \mathrm{PGA}=\theta_0+\theta_1 M+\theta_2 M^2+\theta_3 R+\theta_4 \log(R+\theta_5 10^{\theta_6 M})\] where \(\mathrm{PGA}\) is in \(\,\mathrm{cm/s^2}\), \(\theta_0=0.4063\), \(\theta_1=0.7936\), \(\theta_2=-0.02146\), \(\theta_3=0.0004183\), \(\theta_4=-1.7056\), \(\theta_5=5.7814\), \(\theta_6=-0.05656\), \(\sigma_e=0.17561\) (inter-event), \(\sigma_s=0.17065\) (inter-site), \(\sigma_r=0.19925\) (residual) and \(\sigma_T=0.31569\) (total).

Use data from 204 sites.

Use regression approach of Y.-H. Chen and Tsai (2002) to separate variance into 3 components: inter-event, inter-site and residual.

Plot inter-event residuals (event terms) of shallow (focal depth \(\leq 30\,\mathrm{km}\)) earthquakes w.r.t. \(M_L\) and on map. Find residuals are independent of magnitude and not correlated with location.

Plot inter-site residuals (site terms) on map. Find that these are more coherent in space, which are generally consistent with local geology (e.g. positive terms for alluvium sites.).

Plot direct travel paths from all shallow earthquakes to associated sites. Expect that variability of path-to-path component of error would be similar to event terms or at least not smaller than site terms. Hence assume \(\sigma_P=0.17321\) (path component) and compute refined \(\sigma\)s.

Show that by specifying

*a priori*the variability of path-to-path component that this can be removed from residual.

Ground-motion model is: \[\log \mathrm{Sa}(T)=a_1(T)M+a_2(T)M^2+b(T)\log(R)+c_i(T)S_i\] where \(\mathrm{Sa}\) is in \(\,\mathrm{g}\), \(a_1=0.5781\), \(a_2=-0.0317\), \(b=-0.4352\), \(c_1=-2.6224\), \(c_2=-2.5154\), \(c_3=-2.4654\), \(c_4=-2.6213\) and \(\sigma=0.2768\) (for horizontal PGA), \(a_1=0.5593\), \(a_2=-0.0258\), \(b=-0.6119\), \(c_1=-2.6261\), \(c_2=-2.6667\), \(c_3=-2.5633\), \(c_4=-2.7346\) and \(\sigma=0.2961\) (for vertical PGA).

Use four site classes based on fundamental frequency, \(f\), from receiver functions:

\(f>15\,\mathrm{Hz}\). Corresponds to rock and stiff sediment sites with \(V_{s,30}>700\,\mathrm{m/s}\). 22 records. \(S_1=1\) and other \(S_i=0\).

\(5<f\leq 15\,\mathrm{Hz}\). Corresponds to stiff sediments and/or soft rocks with \(500<V_{s,30}\leq 700\,\mathrm{m/s}\). 16 records. \(S_2=1\) and other \(S_i=0\).

\(2<f\leq 5\,\mathrm{Hz}\). Corresponds to alluvial sites with \(300<V\leq 500\,\mathrm{m/s}\). 25 records. \(S_3=1\) and other \(S_i=0\).

\(f\leq 2\,\mathrm{Hz}\). Corresponds to thick soft alluvium. 26 records. \(S_4=1\) and other \(S_i=0\).

Separate records into four mechanisms: reverse (14 records), reverse/strike-slip (1 record), strike-slip (26 records) and unknown (48 records).

Select records that have PGA \(>0.05\,\mathrm{g}\) on at least one component and are of good quality in frequency band of \(0.3\,\mathrm{Hz}\) or less.

Find results using one- or two-step regression techniques are similar. Only report results from one-step regression.

\(M_w\) for earthquakes obtained directly from level of acceleration spectra plateau of records used.

\(r_{hypo}\) for records obtained from S-P time difference.

Most data from \(r_{hypo}<60\,\mathrm{km}\).

Bandpass filter records with cut-offs of between \(0.08\) and \(0.3\,\mathrm{Hz}\) and between \(16\) and \(40\,\mathrm{Hz}\).

Note that the lack of near-field data is a limitation.

Ground-motion model is: \[\log y=b_1+b_2 M+b_3 M^2+(b_4+b_5M)\log \sqrt{R_{\mathrm{jb}}^2+b_6^2}+b_7S_S+b_8S_A+b_9F_N+b_{\mathrm{10}}F_R\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(b_1=1.647\), \(b_2=0.767\), \(b_3=-0.074\), \(b_4=-3.162\), \(b_5=0.321\), \(b_6=7.682\), \(b_7=0.105\), \(b_8=0.020\), \(b_9=-0.045\), \(b_{10}=0.085\), \(\sigma_1=0.557-0.049M\) (intra-event) and \(\sigma_2=0.189-0.017M\) (inter-event) when \(b_3\) is unconstrained and \(b_1=4.185\), \(b_2=-0.112\), \(b_4=-2.963\), \(b_5=0.290\), \(b_6=7.593\), \(b_7=0.099\), \(b_8=0.020\), \(b_9=-0.034\), \(b_{10}=0.104\), \(\sigma_1=0.557-0.049M\) (intra-event) and \(\sigma_2=0.204-0.018M\) (inter-event) when \(b_3\) is constrained to zero (to avoid super-saturation of PGA).

Use three site categories:

\(S_S=1\), \(S_A=0\).

\(S_A=1\), \(S_S=0\).

\(S_S=0\), \(S_A=0\).

Use three faulting mechanism categories:

\(F_N=1\), \(F_R=0\).

\(F_N=0\), \(F_R=0\).

\(F_R=1\), \(F_N=0\).

Use same data as Akkar and Bommer (2007b), which is similar to that used by N. N. Ambraseys et al. (2005a).

Individually process records using well-defined correction procedure to select the cut-off frequencies (Akkar and Bommer 2006).

Use pure error analysis to determine magnitude dependence of inter- and intra-event variabilities before regression analysis.

Ground-motion model is: \[\ln y=C_1+C_2 M_s+C_3 \ln[R+C_4 \exp(M_s)]+C_5R\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(C_1=4.15\), \(C_2=0.623\), \(C_3=-0.96\) and \(\sigma=0.478\) for horizontal PGA, rock sites and Alborz and central Iran; \(C_1=3.46\), \(C_2=0.635\), \(C_3=-0.996\) and \(\sigma=0.49\) for vertical PGA, rock sites and Alborz and central Iran; \(C_1=3.65\), \(C_2=0.678\), \(C_2=-0.95\) and \(\sigma=0.496\) for horizontal PGA, soil sites and Alborz and central Iran; \(C_1=3.03\), \(C_2=0.732\), \(C_3=-1.03\) and \(\sigma=0.53\) for vertical PGA, soil sites and Alborz and central Iran; \(C_1=5.67\), \(C_2=0.318\), \(C_3=-0.77\), \(C_5=-0.016\) and \(\sigma=0.52\) for horizontal PGA, rock sites and Zagros; \(C_1=5.26\), \(C_2=0.289\), \(C_3=-0.8\), \(C_5=-0.018\) and \(\sigma=0.468\) for vertical PGA, rock sites and Zagros; \(C_1=5.51\), \(C_2=0.55\), \(C_3=-1.31\) and \(\sigma=0.488\) for horizontal PGA, soil sites and Zagros; and \(C_1=5.52\), \(C_2=0.36\), \(C_3=-1.25\) and \(\sigma=0.474\) for vertical PGA, soil sites and Zagros. Constrain \(C_4\) to zero for better convergence even though \(\sigma\)s are higher.

Use two site categories (derive individual equations for each):

Roughly \(V_s\geq 375\,\mathrm{m/s}\).

Roughly \(V_s<375\,\mathrm{m/s}\).

Divide Iran into two regions: Alborz and central Iran, and Zagros, based on tectonics and derive separate equations for each.

Use S-P times to compute \(r_{hypo}\) for records for which it is unknown.

Exclude data from earthquakes with \(M_s\leq 4.5\) to remove less accurate data and since larger earthquakes more important for seismic hazard assessment purposes.

Most records from \(r_{hypo}>50\,\mathrm{km}\).

Exclude poor quality records.

Instrument, baseline correct and bandpass filter records with cut-offs depending on instrument type and site class. For SSA-2 recommend: \(0.15\)â\(0.2\,\mathrm{Hz}\) and \(30-33\,\mathrm{Hz}\) for rock records and \(0.07\)â\(0.2\,\mathrm{Hz}\) and \(30\)â\(33\,\mathrm{Hz}\) for soil records. For SMA-1 recommend: \(0.15\)â\(0.25\,\mathrm{Hz}\) and \(20\)â\(23\,\mathrm{Hz}\) for rock records and \(0.15\)â\(0.2\,\mathrm{Hz}\) and \(20\)â\(23\,\mathrm{Hz}\) for soil records. Apply trial and error based on magnitude, distance and velocity time-history to select cut-off frequencies.

Test a number of different functional forms.

Often find a positive (non-physical) value of \(C_5\). Therefore, remove this term. Try removing records with \(r_{hypo}>100\,\mathrm{km}\) but find little difference and poor convergence due to limited data.

Do not include term for faulting mechanism because such information not available for Iranian events.

Ground-motion model is: \[a_{\mathrm{max}}=F(V_s) G(R, \theta ) H(M)\]

Characterises sites by \(V_s\) (shear-wave velocity).

Considers effect of faulting mechanism.

Considers angle between strike and station, \(\theta\).

Ground-motion models are: \[\log_{10} Y=a+bM+(c+dM) \log_{10} R_{\mathrm{hypo}}+s_{1,2}\] where \(Y\) is in \(\,\mathrm{m/s^2}\), \(a=-1.4580\), \(b=0.4982\), \(c=-2.3639\), \(d=0.1901\), \(s_2=0.4683\), \(\sigma_{\mathrm{eve}}=0.0683\) (inter-event), \(\sigma_{\mathrm{sta}}=0.0694\) (inter-station) and \(\sigma_{\mathrm{rec}}=0.2949\) (record-to-record) for horizontal PGA; and \(a=-1.3327\), \(b=0.4610\), \(c=-2.4148\), \(d=0.1749\), \(s_2=0.3094\), \(\sigma_{\mathrm{eve}}=0.1212\) (inter-event), \(\sigma_{\mathrm{sta}}=0.1217\) (inter-station) and \(\sigma_{\mathrm{rec}}=0.2656\) (record-to-record) for vertical PGA. \[\log_{10} Y=a+bM+(c+dM)\log_{10} (R_{\mathrm{epi}}^2+h^2)^{0.5}+s_{1,2}\] where \(Y\) is in \(\,\mathrm{m/s^2}\), \(a=-2.0924\), \(b=0.5880\), \(c=-1.9887\), \(d=0.1306\), \(h=3.8653\), \(s_2=0.4623\), \(\sigma_{\mathrm{eve}}=0.0670\) (inter-event), \(\sigma_{\mathrm{sta}}=0.0681\) (inter-station) and \(\sigma_{\mathrm{rec}}=0.2839\) (record-to-record) for horizontal PGA; and \(a=-1.8883\), \(b=0.5358\), \(c=-2.0869\), \(d=0.1247\), \(h=4.8954\), \(s_2=0.3046\), \(\sigma_{\mathrm{eve}}=0.1196\) (inter-event), \(\sigma_{\mathrm{sta}}=0.0696\) (inter-station) and \(\sigma_{\mathrm{rec}}=0.2762\) (record-to-record). Coefficients not reported in article but in electronic supplement.

Use two site categories:

Rock. Maximum amplification less than \(2.5\) (for accelerometric stations) or than \(4.5\) (for geophone stations). Amplification thresholds defined after some trials.

Soil. Maximum amplification greater than thresholds defined above.

Classify stations using generalized inversion technique.

Focal depths between \(5\) and \(15\,\mathrm{km}\).

Use aftershocks from the 1999 Kocaeli (\(M_w 7.4\)) earthquake.

Use data from 31 \(1\,\mathrm{Hz}\) 24-bit geophones and 23 12-bit and 16-bit accelerometers. Records corrected for instrument response and bandpass filtered (fourth order Butterworth) with cut-offs \(0.5\) and \(25\,\mathrm{Hz}\) for \(M_L\leq 4.5\) and \(0.1\) and \(25\,\mathrm{Hz}\) for \(M_L>4.5\). Find filters affect PGA by maximum \(10\%\).

Only 13 earthquakes have \(M_L<1.0\). Most data between have \(1.5<M_L<5\) and from \(10 \leq d_e \leq 140\,\mathrm{km}\).

Geophone records from free-field stations and accelerometric data from ground floors of small buildings.

Use \(r_{hypo}\) and \(r_{epi}\) since no evidence for surface ruptures from Turkey earthquakes with \(M_L<6\) and no systematic studies on the locations of the rupture planes for events used.

Since most earthquakes are strike-slip do not include style-of-faulting factor.

Find differences in inter-event \(\sigma\) when using \(M_L\) or \(M_w\), which relate to frequency band used to compute \(M_L\) (about \(1\)â\(10\,\mathrm{Hz}\)) compared to \(M_w\) (low frequencies), but find similar intra-event \(\sigma\)s using the two different magnitudes, which expected since this \(\sigma\) not source-related.

Investigate influence of stress drop on inter-event \(\sigma\) for horizontal PGA relations using \(r_{epi}\) and \(M_L\) or \(M_w\). Find inter-event errors range from negative (low stress drop) to positive (high stress drop) depending on stress drop.

Regress twice: firstly not considering site classification and secondly considering. Find site classification significantly reduces inter-station errors for velometric stations but inter-station errors for accelerometric stations less affected.

Ground-motion model is: \[\begin{aligned} \log_{10}[\mathrm{PSA}(T)]&=&b_1+b_2M_w+b_3M_w^2+(b_4+b_5M_w)\log_{10}\sqrt{R_{jb}^2+b_6^2}+b_7S_S+b_8S_A\\ &&{}+b_9F_N+b_{10}F_R\end{aligned}\] where \(\mathrm{PSA}(T)\) is in \(\,\mathrm{cm/s^2}\), \(b_1=0.0031\), \(b_2=1.0848\), \(b_3=-0.0835\), \(b_4=-2.4423\), \(b_5=0.2081\), \(b_6=8.0282\), \(b_7=0.0781\), \(b_8=0.0208\), \(b_9=-0.0292\), \(b_{10}=0.0963\), \(\sigma_1=0.599\pm 0.041-0.058\pm 0.008M_w\) (intra-event) and \(\sigma_2=0.323\pm 0.075-0.031\pm 0.014M_w\) (inter-event).

Use three site categories:

\(V_{s,30}<360\,\mathrm{m/s}\). \(S_S=1\), \(S_A=1\). 75 records from \(3\leq M_w<5\).

\(360<V_{s,30}<750\,\mathrm{m/s}\). \(S_A=1\), \(S_S=0\). 173 records from \(3\leq M_w<5\).

\(V_{s,30}\geq 750\,\mathrm{m/s}\). \(S_S=0\), \(S_A=0\). 217 records from \(3\leq M_w<5\).

Use three faulting mechanism categories:

\(F_N=1\), \(F_R=0\). 291 records from \(3\leq M_w<5\).

\(F_N=0\), \(F_R=0\). 140 records from \(3\leq M_w<5\).

\(F_R=1\), \(F_N=0\). 24 records from \(3\leq M_w<5\). \(12\%\) of all records. Note that reverse events poorly represented.

Investigate whether Ground-motion models can be extrapolated outside the magnitude range for which they were derived.

Extend dataset of Akkar and Bommer (2007a) by adding data from earthquakes with \(3 \leq M_w<5\). Search ISESD for records from earthquakes with \(M_w<5\), known site class and known faulting mechanism. Find one record from a \(M_w 2\) event but only 11 for events with \(M_w<3\) therefore use \(M_w 3\) as lower limit. Select 465 records from 158 events with \(3 \leq M_w<5\). Many additional records from Greece (mainly singly-recorded events), Italy, Spain, Switzerland, Germany and France. Few additional records from Iran and Turkey.

Data well distributed w.r.t. magnitude, distance and site class but for \(M_w<4\) data sparse for distances \(>40\,\mathrm{km}\).

Additional data has been uniformly processed with cut-offs at \(0.25\) and \(25\,\mathrm{Hz}\).

Use same regression technique as Akkar and Bommer (2007a).

Observe that equations predict expected behaviour of response spectra so conclude that equations are robust and reliable.

Compare predicted ground motions with predictions from model of Akkar and Bommer (2007a) and find large differences, which they relate to the extrapolation of models outside their range of applicability.

Investigate effect of different binning strategies for pure error analysis (Douglas and Smit 2001). Derive weighting functions for published equations using bins of \(2\,\mathrm{km}\times 0.2\) magnitude units and require three records per bin before computing \(\sigma\). Repeat using \(1\,\mathrm{km}\times 0.1\) unit bins. Find less bins allow computation of \(\sigma\). Also repeat original analysis but require four or five records per bin. Find more robust estimates of \(\sigma\) but note that four or five records are still small samples. Also repeating using logarithmic rather than linear distance increments for bins since ground motions shown to mainly decay geometrically. For all different approaches find differences in computed magnitude dependence depending on binning scheme. None of the computed slopes are significant at \(95\%\) confidence level.

Repeat analysis assuming no magnitude dependence of \(\sigma\). Find predictions with this model are very similar to those assuming a magnitude-dependent \(\sigma\).

Find that compared to \(\sigma\)s of Akkar and Bommer (2007a) that inter-event \(\sigma\)s has greatly increased but that intra-event \(\sigma\)s has not, which they relate to the uncertainty in the determination of \(M_w\) and other parameters for small earthquakes.

Repeat analysis exclude data from (in turn) Greece, Italy, Spain and Switzerland to investigate importance of regional dependence on results. Find that results are insensitive to the exclusion of individual regional datasets.

Compute residuals with respect to \(M_w\) for four regional datasets and find that only for Spain (the smallest set) is a significant difference to general results found.

Examine total and intra-event residuals for evidence of soil nonlinearity. Find that evidence for nonlinearity is weak although the expected negative slopes are found. Conclude that insufficient data (and too crude site classification) to adjust the model for soil nonlinearity.

Plot inter-event and intra-event residuals w.r.t. \(M_w\) and find no trend and hence conclude that new equations perform well for all magnitudes.

Do not propose model for application in seismic hazard assessments.

Ground-motion model is: \[\begin{aligned} \ln Y&=&F_M(M)+F_D(R_{JB},M)+F_S(V_{S30},R_{JB},M)\\ F_D(R_{JB},M)&=&[c_1+c_2(M-M_{ref})]\ln(R/R_{ref})+c_3(R-R_{ref})\\ R&=&\sqrt{R_{JB}^2+h^2}\\ F_M(M)&=&\left\{ \begin{array}{l@{\quad}l} e_1\mathrm{U}+e_2\mathrm{SS}+e_3\mathrm{NS}+e_4\mathrm{RS}+e_5(M-M_h)+\\ \quad e_6(M-M_h)^2 \quad \mbox{for} \quad M\leq M_h\\ e_1\mathrm{U}+e_2\mathrm{SS}+e_3\mathrm{NS}+e_4\mathrm{RS}+e_7(M-M_h) \quad \mbox{for} \quad M>M_h \end{array} \right.\\ F_S&=&F_{LIN}+F_{NL}\\ F_{LIN}&=&b_{lin}\ln(V_{S30}/V_{ref})\\ F_{NL}&=&\left\{ \begin{array}{l@{\quad}l} b_{nl}\ln(\mathrm{pga\_low}/0.1) \quad \mbox{for} \quad \mathrm{pga4nl}\leq a_1\\ b_{nl}\ln(\mathrm{pga\_low}/0.1)+c[\ln(\mathrm{pga4nl}/a_1)]^2+\\ \quad d[\ln(\mathrm{pga4nl}/a_1)]^3 \quad \mbox{for} \quad a_1<\mathrm{pga4nl}\leq a_2\\ b_{nl}\ln(\mathrm{pga4nl}/0.1) \quad \mbox{for} \quad a_2<\mathrm{pga4nl} \end{array} \right.\\ c&=&(3\Delta y-b_{nl}\Delta x)/\Delta x^2\\ d&=&-(2\Delta y-b_{nl} \Delta x)/\Delta x^3\\ \Delta x&=&\ln(a_2/a_1)\\ \Delta y&=&b_{nl} \ln(a_2/\mathrm{pga\_low})\\ b_{nl}&=&\left\{ \begin{array}{l@{\quad}l} b_1 \quad \mbox{for} \quad V_{S30}\leq V_1\\ (b_1-b_2)\ln (V_{S30}/V_2)/\ln(V_1/V_2)+b_2 \quad \mbox{for} \quad V_1<V_{S30}\leq V_2\\ b_2\ln(V_{S30}/V_{ref})/\ln(V_2/V_{ref}) \quad \mbox{for} \quad V_2<V_{S30}<V_{ref}\\ 0.0 \quad \mbox{for} \quad V_{ref}\leq V_{S30} \end{array} \right.\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(M_h=6.75\) (hinge magnitude), \(V_{ref}=760\,\mathrm{m/s}\) (specified reference velocity corresponding to the NEHRP B/C boundary), \(a_1=0.03\,\mathrm{g}\) (threshold for linear amplifcation), \(a_2=0.09\,\mathrm{g}\) (threshold for nonlinear amplification), \(\mathrm{pga\_low}=0.06\,\mathrm{g}\) (for transition between linear and nonlinear behaviour), \(\mathrm{pga4nl}\) is predicted PGA in \(\,\mathrm{g}\) for \(V_{ref}\) with \(F_S=0\), \(V_1=180\,\mathrm{m/s}\), \(V_2=300\,\mathrm{m/s}\), \(b_{lin}=-0.360\), \(b_1=-0.640\), \(b_2=-0.14\), \(M_{ref}=4.5\), \(R_{ref}=1\,\mathrm{km}\), \(c_1=-0.66050\), \(c_2=0.11970\), \(c_3=-0.01151\), \(h=1.35\), \(e_1=-0.53804\), \(e_2=-0.50350\), \(e_3=-0.75472\), \(e_4=-0.50970\), \(e_5=0.28805\), \(e_6=-0.10164\), \(e_7=0.0\); \(\sigma=0.502\) (intra-event); \(\tau_U=0.265\), \(\tau_M=0.260\) (inter-event); \(\sigma_{TU}=0.566\), \(\sigma_{TM}=0.560\) (total).

Characterise sites using \(V_{S30}\). Believe equations applicable for \(180\leq V_{S30}\leq 1300\,\mathrm{m/s}\) (state that equations should not be applied for very hard rock sites, \(V_{S30}\geq 1500\,\mathrm{m/s}\)). Bulk of data from NEHRP C and D sites (soft rock and firm soil) and very few data from A sites (hard rock). Use three equations for nonlinear amplification: to prevent nonlinear amplification increasing indefinitely as \(\mathrm{pga4nl}\) decreases and to smooth transition from linear to nonlinear behaviour. Equations for nonlinear site amplification simplified version of those of Choi and Stewart (2005) because believe NGA database insufficient to simultaneously determine all coefficients for nonlinear site equations and magnitude-distance scaling due to trade-offs between parameters. Note that implicit trade-offs involved and change in prescribed soil response equations would lead to change in derived magnitude-distance scaling.

Focal depths between \(2\) and \(31\,\mathrm{km}\) with most \(<20\,\mathrm{km}\).

Use data from the PEER Next Generation Attenuation (NGA) Flatfile supplemented with additional data from three small events (2001 Anza \(M 4.92\), 2003 Big Bear City \(M 4.92\) and 2002 Yorba Linda \(M 4.27\)) and the 2004 Parkfield earthquake, which were used only for a study of distance attenuation function but not the final regression (due to rules of NGA project).

Use three faulting mechanism categories using P and T axes:

Strike-slip. Plunges of T and P axes \(<40^{\circ}\). 35 earthquakes. Dips between \(55\) and \(90^{\circ}\). \(4.3\leq M \leq 7.9\). \(\mathrm{SS}=1\), \(\mathrm{U}=0\), \(\mathrm{NS}=0\), \(\mathrm{RS}=0\).

Reverse. Plunge of T axis \(>40^{\circ}\). 12 earthquakes. Dips between \(12\) and \(70^{\circ}\). \(5.6 \leq M \leq 7.6\). \(\mathrm{RS}=1\), \(\mathrm{U}=0\), \(\mathrm{SS}=0\), \(\mathrm{NS}=0\).

Normal. Plunge of P axis \(>40^{\circ}\). 11 earthquakes. Dips between \(30\) and \(70^{\circ}\). \(5.3\leq M \leq 6.9\). \(\mathrm{NS}=1\), \(\mathrm{U}=0\), \(\mathrm{SS}=0\), \(\mathrm{RS}=0\).

Note that some advantages to using P and T axes to classify earthquakes but using categories based on rake angles with: within \(30^{\circ}\) of horizontal as strike-slip, from \(30\) to \(150^{\circ}\) as reverse and from \(-30^{\circ}\) to \(-150^{\circ}\) as normal, gives essentially the same classification. Also allow prediction of motions for unspecified (\(\mathrm{U}=1\), \(\mathrm{SS}=0\), \(\mathrm{NS}=0\), \(\mathrm{RS}=0\)) mechanism (use \(\sigma\)s and \(\tau\)s with subscript U otherwise use \(\sigma\)s and \(\tau\)s with subscript M).

Exclude records from obvious aftershocks because believe that spectral scaling of aftershocks could be different than that of mainshocks. Note that this cuts the dataset roughly in half.

Exclude singly-recorded earthquakes.

Note that possible bias due to lack of low-amplitude data (excluded due to non-triggering of instrument, non-digitisation of record or below the noise threshold used in determining low-cut filter frequencies). Distance to closest non-triggered station not available in NGA Flatfile so cannot exclude records from beyond this distance. No information available that allows exclusion of records from digital accelerograms that could remove this bias. Hence note that obtained distance dependence for small earthquakes and long periods may be biased towards a decay that is less rapid than true decay.

Use estimated \(R_{JB}\)s for earthquakes with unknown fault geometries.

Lack of data at close distances for small earthquakes.

Three events (1987 Whittier Narrows, 1994 Northridge and 1999 Chi-Chi) contribute large proportion of records (\(7\%\), \(10\%\) and \(24\%\)).

Note that magnitude scaling better determined for strike-slip events, which circumvent using common magnitude scaling for all mechanisms.

Seek simple functional forms with minimum required number of predictor variables. Started with simplest reasonable form and added complexity as demanded by comparisons between predicted and observed motions. Selection of functional form heavily guided by subjective inspection of nonparametric plots of data.

Data clearly show that modelling of anelastic attenuation required for distances \(>80\,\mathrm{km}\) and that effective geometric spreading is dependent on magnitude. Therefore, introduce terms in the function to model these effects, which allows model to be used to \(400\,\mathrm{km}\).

Do not include factors for depth-to-top of rupture, hanging wall/footwall or basin depth because residual analysis does not clearly show that the introduction of these factors would improve the predictive capabilities of model on average.

Models are data-driven and make little use of simulations.

Believe that models provide a useful alternative to more complicated NGA models as they are easier to implement in many applications.

Firstly correct ground motions to obtain equivalent observations for reference velocity of \(760\,\mathrm{m/s}\) using site amplification equations using only data with \(R_{JB}\leq 80\,\mathrm{km}\) and \(V_{S30}>360\,\mathrm{m/s}\). Then regress site-corrected observations to obtain \(F_D\) and \(F_M\) with \(F_S=0\). No smoothing of coefficients determined in regression (although some of the constrained coefficients were smoothed).

Assume distance part of model applies for crustal tectonic regimes represented by NGA database. Believe that this is a reasonable initial approach. Test regional effects by examining residuals by region.

Note that data sparse for \(R_{JB}>80\,\mathrm{km}\), especially for moderate events, and, therefore, difficult to obtain robust \(c_1\) (slope) and \(c_3\) (curvature) simultaneously. Therefore, use data from outside NGA database (three small events and 2004 Parkfield) to define \(c_3\) and use these fixed values of \(c_3\) within regression to determine other coefficients. To determine \(c_3\) and \(h\) from the four-event dataset set \(c_1\) equal to \(-0.5\), \(-0.8\) and \(-1.0\) and \(c_2=0\) if the inclusion of event terms \(c_0\) for each event. Use \(c_3\)s when \(c_1=-0.8\) since it is a typical value for this parameter in previous studies. Find that \(c_3\) and \(h\) are comparable to those in previous studies.

Note that desirable to constrain \(h\) to avoid overlap in curves for large earthquakes at very close distances. Do this by initially performing regression with \(h\) as free parameter and then modifying \(h\) to avoid overlap.

After \(h\) and \(c_3\) have been constrained solve for \(c_1\) and \(c_2\).

Constrain quadratic for magnitude scaling so that maximum not reached for \(M<8.5\) to prevent oversaturation. If maximum reached for \(M<8.5\) then perform two-segment regression hinged at \(M_h\) with quadratic for \(M\leq M_h\) and linear for \(M>M_h\). If slope of linear segment is negative then repeat regression by constraining slope above \(M_h\) to \(0.0\). Find that data generally indicates oversaturation but believe this effect is too extreme at present. \(M_h\) fixed by observation that ground motions at short periods do not get significantly larger with increasing magnitude.

Plots of event terms (from first stage of regression) against \(M\) show that normal-faulting earthquakes have ground motions consistently below those of strike-slip and reverse events. Firstly group data from all fault types together and solved for \(e_1\), \(e_5\), \(e_6\), \(e_7\) and \(e_8\) by setting \(e_2\), \(e_3\) and \(e_4\) to \(0.0\). Then repeat regression fixing \(e_5\), \(e_6\), \(e_7\) and \(e_8\) to values obtained in first step to find \(e_2\), \(e_3\) and \(e_4\).

Examine residual plots and find no significant trends w.r.t. \(M\), \(R_{JB}\) or \(V_{S30}\) although some small departures from a null residual.

Examine event terms from first stage of regression against \(M\) and conclude functional form provides reasonable fit to near-source data.

Examine event terms from first stage of regression against \(M\) for surface-slip and no-surface-slip earthquakes. Find that most surface-slip events correspond to large magnitudes and so any reduction in motions for surface-slip earthquakes will be mapped into reduced magnitude scaling. Examine event terms from strike-slip earthquakes (because both surface- and buried-slip events in same magnitude range) and find no indication of difference in event terms for surface-slip and no-surface-slip earthquakes. Conclude that no need to include dummy variables to account for this effect.

Examine residuals for basin depth effects. Find that \(V_{S30}\) and basin depth are highly correlated and so any basin-depth effect will tend to be captured by empirically-determined site amplifications. To separate \(V_{S30}\) and basin-depth effects would require additional information or assumptions but since aiming for simplest equations no attempt made to break down separate effects. Examine residuals w.r.t. basin depth and find little dependence.

Chi-Chi data forms significant fraction (\(24\%\) for PGA) of data set. Repeat complete analysis without these data to examine their influence. Find that predictions are not dramatically different.

Note that use of anelastic coefficients derived using data from four earthquakes in central and southern California is not optimal and could lead to inconsistencies in \(h\)s.

Ground-motion model is: \[\begin{aligned} \ln \hat{Y}&=&f_{mag}+f_{dis}+f_{flt}+f_{hng}+f_{site}+f_{sed}\\ f_{mag}&=&\left\{ \begin{array}{l@{\quad}l} c_0+c_1 \quad \mbox{for} \quad M\leq 5.5\\ c_0+c_1M+c_2(M-5.5) \quad \mbox{for} \quad 5.5<M\leq 6.5\\ c_0+c_1M+c_2(M-5.5)+c_3(M-6.5) \quad \mbox{for} \quad M>6.5 \end{array} \right.\\ f_{dis}&=&(c_4+c_5M)\ln(\sqrt{R_{RUP}^2+c_6^2})\\ f_{flt}&=&c_7F_{RV}f_{flt,Z}+c_8 F_{NM}\\ f_{flt,Z}&=&\left\{ \begin{array}{l@{\quad}l} Z_{TOR} \quad \mbox{for} \quad Z_{TOR}<1\\ 1 \quad \mbox{for} \quad Z_{TOR}\geq 1 \end{array} \right.\\ f_{hng}&=&c_9 f_{hng,R} f_{hng,M} f_{hng,Z} f_{hng,\delta}\\ f_{hng,R}&=&\left\{ \begin{array}{l@{\quad}l} 1 \quad \mbox{for} \quad R_{JB}=0 \\ \{\max(R_{RUP},\sqrt{R_{JB}^2+1})-R_{JB}\}/\\ \quad \max(R_{RUP},\sqrt{R_{JB}^2+1}) \quad \mbox{for} \quad R_{JB}>0, Z_{TOR}<1\\ (R_{RUP}-R_{JB})/R_{RUP} \quad \mbox{for} \quad R_{JB}>0, Z_{TOR}\geq 1 \end{array} \right.\\ f_{hng,M}&=&\left\{ \begin{array}{l@{\quad}l} 0 \quad \mbox{for} \quad M\leq 6.0\\ 2(M-6.0) \quad \mbox{for} \quad 6.0<M<6.5\\ 1 \quad \mbox{for} \quad M\geq 6.5 \end{array} \right.\\ f_{hng,Z}&=&\left\{ \begin{array}{l@{\quad}l} 0 \quad \mbox{for} \quad Z_{TOR} \geq 20\\ (20-Z_{TOR})/20 \quad \mbox{for} \quad 0 \leq Z_{TOR}<20 \end{array} \right.\\ f_{hng,\delta}&=&\left\{ \begin{array}{l@{\quad}l} 1 \quad \mbox{for} \quad \delta \leq 70\\ (90-\delta)/20 \quad \mbox{for} \quad \delta>70 \end{array} \right.\\ f_{site}&=&\left\{ \begin{array}{l@{\quad}l} c_{10} \ln\left(\frac{V_{S30}}{k_1}\right)+k_2 \left\{ \ln\left[ A_{1100}+c\left(\frac{V_{S30}}{k_1}\right)^n\right]-\ln(A_{1100}+c)\right\} \quad \mbox{for} \quad V_{S30}<k_1\\ (c_{10}+k_2 n)\ln\left(\frac{V_{S30}}{k_1}\right) \quad \mbox{for} \quad k_1 \leq V_{S30}<1100\\ (c_{10}+k_2 n)\ln\left(\frac{1100}{k_1}\right) \quad \mbox{for} \quad V_{S30}\geq 1100 \end{array} \right.\\ f_{sed}&=&\left\{ \begin{array}{l@{\quad}l} c_{11}(Z_{2.5}-1) \quad \mbox{for} \quad Z_{2.5}<1\\ 0 \quad \mbox{for} \quad 1 \leq Z_{2.5} \leq 3\\ c_{12} k_3 \mathrm{e}^{-0.75}[1-\mathrm{e}^{-0.25(Z_{2.5}-3)}] \quad \mbox{for} \quad Z_{2.5}>3 \end{array} \right.\\ \sigma&=&\sqrt{\sigma_{\ln Y}^2+\sigma_{\ln AF}^2+\alpha^2\sigma_{\ln A_B}^2+2\alpha \rho \sigma_{\ln Y_B} \sigma_{\ln A_B}}\\ \alpha&=&\left\{ \begin{array}{l@{\quad}l} k_2A_{1100}\{[A_{1100}+c(V_{S30}/k_1)^n]^{-1}-(A_{1100}+c)^{-1}\} \quad \mbox{for} \quad V_{S30}<k_1\\ 0 \quad \mbox{for} \quad V_{S30}\geq k_1 \end{array} \right.\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(c_0=-1.715\), \(c_1=0.500\), \(c_2=-0.530\), \(c_3=-0.262\), \(c_4=-2.118\), \(c_5=0.170\), \(c_6=5.60\), \(c_7=0.280\), \(c_8=-0.120\), \(c_9=0.490\), \(c_{10}=1.058\), \(c_{11}=0.040\), \(c_{12}=0.610\), \(k_1=865\), \(k_2=-1.186\), \(k_3=1.839\), \(\sigma_{\ln Y}=0.478\) (intra-event), \(\tau_{\ln Y}=0.219\) (inter-event), \(\sigma_C=0.166\), \(\sigma_T=0.526\) (total), \(\sigma_{Arb}=0.551\) and \(\rho=1.000\) (correlation coefficient between intra-event residuals of ground-motion parameter of interest and PGA). \(\sigma_{\ln Y_B}=(\sigma_{\ln Y}^2-\sigma_{\ln AF}^2)^{1/2}\) is standard deviation at base of site profile. Assume that \(\sigma_{\ln AF}\approx 0.3\) based on previous studies for deep soil sites. \(\sigma_{Arb}=\sqrt{\sigma_T^2+\sigma_C^2}\) for estimating aleatory uncertainty of arbitrary horizontal component.

Characterise sites using \(V_{S30}\). Account for nonlinear effects using \(A_{1100}\), median estimated PGA on reference rock outcrop (\(V_{S30}=1100\,\mathrm{m/s}\)) in \(\,\mathrm{g}\). Linear part of \(f_{site}\) is consistent with previous studies but with constraint for constant site term for \(V_{S30}>1100\,\mathrm{m/s}\) (based on residual analysis) even though limited data for \(V_{S30}>1100\,\mathrm{m/s}\). When only including linear part of shallow site response term find residuals clearly exhibit bias when plotted against rock PGA, \(A_{1100}\). Find that residuals not sufficient to determine functional form for nonlinear amplification so use 1D equivalent-linear site response simulations to constrain form and coefficients. Believe model applicable for \(V_{S30}=150\)â\(1500\,\mathrm{m/s}\).

Also use depth to \(2.5\,\mathrm{km/s}\) shear-wave velocity horizon (basin or sediment depth) in \(\,\mathrm{km}\), \(Z_{2.5}\). Deep-basin term modelled based on 3D simulations for Los Angeles, San Gabriel and San Fernando basins (southern California) calibrated empirically from residual analysis, since insufficient observational data for fully empirical study. Shallow-sediment effects based on analysis of residuals. Note high correlation between \(V_{S30}\) and \(Z_{2.5}\). Provide relationships for predicting \(Z_{2.5}\) based on other site parameters. Believe model applicable for \(Z_{2.5}=0\)â\(10\,\mathrm{km}\).

Use three faulting mechanism categories based on rake angle, \(\lambda\):

Reverse and reverse-oblique. \(30<\lambda<150^{\circ}\). 17 earthquakes. \(F_{RV}=1\) and \(F_{NM}=0\).

Normal and normal-oblique. \(-150<\lambda<-30^{\circ}\). 11 earthquakes. \(F_{NM}=1\) and \(F_{RV}=0\).

Strike-slip. All other rake angles. 36 earthquakes. \(F_{RV}=0\) and \(F_{NM}=0\).

Use data from PEER Next Generation Attenuation (NGA) Flatfile.

Select records of earthquakes located within shallow continental lithosphere (crust) in a region considered to be tectonically active from stations located at or near ground level and which exhibit no known embedment or topographic effects. Require that the earthquakes have sufficient records to reliably represent the mean horizontal ground motion (especially for small magnitude events) and that the earthquake and record is considered reliable.

Exclude these data: 1) records with only one horizontal component or only a vertical component; 2) stations without a measured or estimated \(V_{S30}\); 3) earthquakes without a rake angle, focal mechanism or plunge of the P- and T-axes; 4) earthquakes with the hypocentre or a significant amount of fault rupture located in lower crust, in oceanic plate or in a stable continental region; 5) LDGO records from the 1999 DÃ¼zce earthquake that are considered to be unreliable due to their spectral shapes; 6) records from instruments designated as low-quality from the 1999 Chi-Chi earthquake; 7) aftershocks but not triggered earthquakes such as the 1992 Big Bear earthquake; 8) earthquakes with too few records (\(N\)) in relation to its magnitude, defined as: a) \(M<5.0\) and \(N<5\), b) \(5.0\leq M <6.0\) and \(N<3\), c) \(6.0\leq M<7.0\), \(R_{RUP}>60\,\mathrm{km}\) and \(N<2\) (retain singly-recorded earthquakes with \(M\geq 7.0\) and \(R_{RUP}\leq 60\,\mathrm{km}\) because of their significance); 9) records considered to represent non-free-field site conditions, defined as instrument located in a) basement of building, b) below the ground surface, c) on a dam except the abutment; and 10) records with known topographic effects such as Pacoima Dam upper left abutment and Tarzana Cedar Hill Nursery.

Functional forms developed or confirmed using classical data exploration techniques, such as analysis of residuals. Candidate functional forms developed using numerous iterations to capture the observed trends in the recorded ground motion data. Final functional forms selected according to: 1) sound seismological basis; 2) unbiased residuals; 3) ability to be extrapolated to magnitudes, distances and other explanatory variables that are important for use in engineering and seismology; and 4) simplicity, although this was not an overriding factor. Difficult to achieve because data did not always allow the functional forms of some explanatory variables to be developed empirically. Theoretical constraints were sometimes used to define the functional forms.

Use two-stage maximum-likelihood method for model development but one-stage random-effects method for final regression.

Also perform statistical analysis for converting between selected definition of horizontal component and other definitions.

Include depth to top of coseismic rupture plane, \(Z_{TOR}\), which find important for reverse-faulting events. Find that some strike-slip earthquakes with partial or weak surface expression appeared to have higher-than-average ground motions but other strike-slip events contradict this, which believe could be due to ambiguity in identifying coseismic surface rupture in NGA database. Therefore, believe additional study required before \(Z_{TOR}\) can be used for strike-slip events. Believe model applicable for \(Z_{TOR}=0\)â\(15\,\mathrm{km}\).

Include dip of rupture plane, \(\delta\). Believe model applicable for \(\delta=15\)â\(90^{\circ}\).

Assume that \(\tau\) is approximately equal to standard deviation of inter-event residuals, \(\tau_{\ln Y}\), since inter-event terms are not significantly affected by soil nonlinearity. Note that if \(\tau\) was subject to soil nonlinearity effects it would have only a relatively small effect on \(\sigma_T\) because intra-event \(\sigma\) dominates. \(\sigma\) takes into account soil nonlinearity effects. Assume that \(\sigma_{\ln Y}\) and \(\sigma_{\ln PGA}\) represent aleatory uncertainty associated with linear site response, reflecting dominance of such records in database.

Based on statistical tests on binned intra-event residuals conclude that intra-event standard deviations not dependent on \(V_{S30}\) once nonlinear site effects are taken into account.

Use residual analysis to derive trilinear functional form for \(f_{mag}\). Piecewise linear relationship allows greater control of \(M>6.5\) scaling and decouples this scaling from that of small magnitude scaling. Demonstrate using stochastic simulations that trilinear model fits ground motions as well as quadratic model for \(M\leq 6.5\). Find that large-magnitude scaling of trilinear model consistent with observed effects of aspect ratio (rupture length divided by rupture width), which was abandoned as explanatory variable when inconsistencies in NGA database for this variable found.

Original unconstrained regression resulted in prediction of oversaturation at short periods, large magnitudes and short distances. Oversaturation not statistically significant nor is this behaviour scientifically accepted and therefore constrain \(f_{mag}\) to saturate at \(M>6.5\) and \(R_{RUP}=0\) when oversaturation predicted by unconstrained regression analysis. Constraint equivalent to setting \(c_3=-c_1-c_2-c_5\ln(c_6)\). Inter- and intra-event residual plots w.r.t. \(M\) show predictions relatively unbiased, except for larger magnitudes where saturation constraint leads to overestimation of short-period ground motions.

Examine inter-event residuals w.r.t. region and find some bias, e.g. find generally positive inter-event residuals at relatively long periods of \(M>6.7\) events in California but only for five events, which believe insufficient to define magnitude scaling for this region. Note that user may wish to take these dependences into account.

Note that adopted distance-dependence term has computational advantage since it transfers magnitude-dependent attenuation term to outside square root, which significantly improves stability of nonlinear regression. Note that adopted functional form consistent with broadband simulations for \(6.5\) and \(7.5\) between \(2\) and \(100\,\mathrm{km}\) and with simple theoretical constraints. Examine intra-event residuals w.r.t. distance and find that they are relatively unbiased.

Functional form for \(f_{flt}\) determined from residual analysis. Find coefficient for normal faulting only marginally significant at short periods but very significant at long periods. Believe long-period effects due to systematic differences in sediment depths rather than source effects, since many normal-faulting events in regions with shallow depths to hard rock (e.g. Italy, Greece and Basin and Range in the USA), but no estimates of sediment depth to correct for this effect. Constrain normal-faulting factor found at short periods to go to zero at long periods based on previous studies.

Functional form for \(f_{hng}\) determined from residual analysis with additional constraints to limit range of applicability so that hanging-wall factor has a smooth transition between hanging and foot walls, even for small \(Z_{TOR}\). Include \(f_{hng,M}\), \(f_{hng,Z}\) and \(f_{hng,\delta}\) to phase out hanging-wall effects at small magnitudes, large rupture depths and large rupture dips, where residuals suggest that effects are either negligible or irresolvable from data. Include hanging-wall effects for normal-faulting and non-vertical strike-slip earthquakes even those statistical evidence is weak but it is consistent with better constrained hanging-wall factor for reverse faults and it is consistent with foam-rubber experiments and simulations.

Ground-motion model is: \[\log_{10} Y=a+bM-c \log_{10} \sqrt{R^2+h^2}+eS+fF\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(a=0.883\), \(b=0.458\), \(c=1.278\), \(h=11.515\), \(e=0.038\), \(f=0.116\), \(\tau=0.109\) (intra-event) and \(\sigma=0.270\) (inter-event).

Use three site classes:

Rock, \(V_{s,30}>800\,\mathrm{m/s}\). \(S=0\). 75 records.

Stiff soil, \(360 \leq V_s \leq 665\,\mathrm{m/s}\). \(S=1\). 197 records.

Soft soil, \(200 \leq V_s \leq 360\,\mathrm{m/s}\). \(S=2\). 63 records.

From initial analysis find that ground-motions on D sites are double those on C sites.

Use three style-of-faulting categories:

\(F=1\)

\(F=1\)

\(F=0\)

From initial analysis find that thrust and strike-slip ground motions are similar but greater than normal motions.

Focal depths between \(0\) and \(30\,\mathrm{km}\) with mean of \(10.66\,\mathrm{km}\).

Most records from earthquakes near the Ionian islands.

Use records from free-field stations and from basements of buildings with \(<2\) storeys. Note that some bias may be introduced by records from buildings but due to lack of data from free-field stations these records must be included.

Use corrected records from ISESD (bandpass filtered \(0.25\) and \(25\,\mathrm{Hz}\)).

Use epicentral distance because most earthquakes are offshore and those that are onshore do not display evidence of surface faulting and therefore cannot use a fault-based distance measure.

Data from large events recorded at intermediate and long distances and small events at small distances. Correlation coefficient between magnitude and distance is \(0.64\).

Recommend that equation not used outside range of data used.

Analyse residuals normalized to have zero mean and unity variance (only display results for PGA and SA at \(1\,\mathrm{s}\) due to similar results for all periods). Find that residuals do not show trends and are uncorrelated (at more than \(99\%\) confidence level) w.r.t. independent variables. Show normality of residuals through histograms for PGA and SA at \(1\,\mathrm{s}\).

Also derive equations for various other strong-motion parameters.

Ground-motion model is: \[\log y=a_1+a_2 M+a_3 \log \sqrt{(d^2+5^2)}+a_{3+i}S_i\] Coefficients not reported since purpose is not to develop models for seismic hazard assessments but to derive confidence limits on median PGA and thereafter to examine possible regional dependence of ground motions.

Rederives models of Joyner and Boore (1981), Boore, Joyner, and Fumal (1993, 1997), N. N. Ambraseys, Simpson, and Bommer (1996), N. N. Ambraseys et al. (2005a), Ulusay et al. (2004), Kalkan and GÃ¼lkan (2004b) and Sabetta and Pugliese (1987) to find their complete covariance matrices in order to compute confidence limits of the predicted median PGA.

Uses same site classifications as original studies. \(S_i=1\) for site class \(i\) and \(0\) otherwise.

Adopts a simple linear functional form and standard one-stage regression method so that the covariance matrices can be easily computed.

Assumes a fixed coefficient of \(5\,\mathrm{km}\) (a rough average value for this coefficient for most models using adopted functional form) inside square root to make function linear.

Examines \(95\%\) confidence limits on PGA since it is standard to use \(5\%\) significance levels when testing null hypotheses. Plots predicted median PGAs and their confidence limits for \(M_w 5\), \(6.5\) and \(8.0\) up to \(200\,\mathrm{km}\) to show effects of extrapolation outside range of applicability of models. Finds that confidence limits for models derived using limited data (Ulusay et al. 2004; Kalkan and GÃ¼lkan 2004b; Sabetta and Pugliese 1987) are wider than models derived using large well-distributed datasets (Joyner and Boore 1981; Boore, Joyner, and Fumal 1993, 1997; N. N. Ambraseys, Simpson, and Bommer 1996; N. N. Ambraseys et al. 2005a). Notes that for \(5.5<M_w<7\) and \(10\leq d_f \leq 60\,\mathrm{km}\) the \(95\%\)-confidence limits of the median are narrow and within bands \(10\)â\(30\%\) from the median but for other magnitudes and distances (away from the centroid of data) they are much wider (bands of \(100\%\) from the median). Notes that inclusion of data from large magnitude events decreases the width of the confidence limits of the model derived using the data of Boore, Joyner, and Fumal (1993, 1997) compared with that derived using the data of Joyner and Boore (1981) and similarly that derived with the data of N. N. Ambraseys et al. (2005a) compared with that derived using the data of N. N. Ambraseys, Simpson, and Bommer (1996).

Ground-motion model is: \[\log a=a_1 M+a_2h+a_3\log[\Delta+a_4\exp(a_5M)]+a_6\] where \(a\) is in \(\,\mathrm{cm/s^2}\), \(a_1=0.606\), \(a_2=0.00459\), \(a_3=-2.136\), \(a_4=0.334\), \(a_5=0.653\), \(a_6=1.730\), \(\phi=0.251\) (intra-event), \(\tau=0.192\) (inter-event) and \(\sigma=0.317\) (total).

Select K-Net and KiK-Net data within geographical region 137â142E and 34Nâ38N with \(M>5\) and focal depths \(200\,\mathrm{km}\) from 09/1996 to 07/2006. Remove inadequate records, e.g. those with small amplitudes.

\(h\) is central depth of rupture plane.

Data from 186 stations.

Ground-motion model is: \[\begin{aligned} \ln(Y)&=&\ln(A)-0.5\ln \left[\left(1-\frac{R}{R_0}\right)^2+4D_0^2\frac{R}{R_0}\right]\\ &&{}-0.5\ln\left[ \left(1-\sqrt{\frac{R}{R_1}} \right)^2+4D_1^2 \sqrt{\frac{R}{R_1}} \right]+b_v \ln \left(\frac{V_{s,30}}{V_A} \right)\\ A&=&[c_1 \arctan(M+c_2)+c_3]F\\ R_0&=&c_4 M+c_5\\ D_0&=&c_6\cos [c_7(M+c_8)]+c_9\end{aligned}\] where \(Y\) is in \(\,\mathrm{g}\), \(c_1=0.14\), \(c_2=-6.25\), \(c_3=0.37\), \(c_4=2.237\), \(c_5=-7.542\), \(c_6=-0.125\), \(c_7=1.19\), \(c_8=-6.15\), \(c_9=0.525\), \(b_v=-0.25\), \(V_A=484.5\), \(R_1=100\,\mathrm{km}\) and \(\sigma=0.552\).

Characterise sites by \(V_{s,30}\) (average shear-wave velocity in upper \(30\,\mathrm{m}\)). Note that approximately half the stations have measured shear-wave velocity profiles.

Include basin effects through modification of \(D_1\). For sediment depth (\(Z\geq 1\,\mathrm{km}\) \(D_1=0.35\); otherwise \(D_1=0.65\).

Use three faulting mechanism classes:

13 records

1120 records. \(F=1.00\).

1450 records. \(F=1.28\) (taken from previous studies).

but only retain two (strike-slip and reverse) by combining normal and strike-slip categories.

Only use earthquakes with focal depths \(<20\,\mathrm{km}\). Focal depths between \(4.6\) and \(19\,\mathrm{km}\).

Exclude data from aftershocks.

Use data from: Alaska (24 records), Armenia (1 record), California (2034 records), Georgia (8), Iran (7 records) Italy (10 records), Nevada (8 records), Taiwan (427 records), Turkey (63 records) and Uzbekistan (1 record).

Most data from \(5.5 \leq M_w \leq 7.5\).

Adopt functional form to model: a constant level of ground motion close to fault, a slope of about \(R^{-1}\) for \(>10\,\mathrm{km}\) and \(R^{-1.5}\) at greater distances (\(>100\,\mathrm{km}\)) and observation (and theoretical results) that highest amplitude ground motions do not always occur nearest the fault but at distances of \(3\)â\(10\,\mathrm{km}\).

Choose functional form based on transfer function of a SDOF oscillator since this has similar characteristics to those desired.

Note that magnitude scaling may need adjusting for small magnitudes.

Firstly regress for magnitude and distance dependency and then regress for site and basin effects.

Examine residual w.r.t. magnitude and distance and observe no significant trends.

Compare predictions to observations for 12 well-recorded events in the dataset and find that the observations are well predicted for near and far distances.

Demonstrate (for the 2004 Parkfield earthquake) that it is possible to add an additional âfilterâ term in order to predict ground motions at large distances without modifying the other terms.

Ground-motion model is: \[\ln \mathrm{PGA}=a_1+a_2M+a_3\ln r_{epi}+a_4 r_{epi}+a_5 C_1+a_6 C_2+a_7 C_3\] where \(a_1=-4.8272\), \(a_2=0.90061\), \(a_3=-0.28195\), \(a_4=-0.00831\), \(a_5=0.61098\), \(a_6=0.37342\) and \(a_7=0.2117\) (\(\sigma\) is not reported)

Use 4 site classes (ZarÃ© and Bard 2002):

Rock and hard alluvial sites, \(f_0>15\,\mathrm{Hz}\), \(V_s>800\,\mathrm{m/s}\). \(C_1=C_2=C_3=0\).

Alluvial sites, thin soft alluvium, \(5<f_0<15\,\mathrm{Hz}\), \(500<V_s<700\,\mathrm{m/s}\). \(C_1=1\), \(C_2=C_3=0\).

Soft gravel and sandy sites, \(2<f_0<5\,\mathrm{Hz}\), \(300<V_s<500\,\mathrm{m/s}\). \(C_2=1\), \(C_1=C_3=0\).

Soft soil sites, thick soft alluvia, \(f_0<2\,\mathrm{Hz}\), \(V_s<300\,\mathrm{m/s}\). \(C_3=1\), \(C_1=C_2=0\).

Derive model to compare to neural-network-based model.

Ground-motion model is: \[\ln Y=b_1+b_2 (\textbf{M}-7)+b_3(\textbf{M}-7)^2+[b_4+b_5(\textbf{M}-4.5)]\ln[(r_{\mathrm{jb}}^2+h^2)^{0.5}]+\mathrm{AF}_s\] where \(Y\) is in \(\,\mathrm{g}\), \(b_1=1.096\), \(b_2=0.444\), \(b_3=0.0\), \(b_4=-1.047\), \(b_5=0.038\), \(h=5.7\), \(\sigma_\eta=0.190\) (inter-event) and \(\sigma_\epsilon=0.464\) (intra-event) for geometric mean.

\(\mathrm{AF}_s\) is the amplification factor due to linear and nonlinear soil behaviour used by G. M. Atkinson and Boore (2006), which is a function of \(V_{s,30}\) and expected PGA at site with \(V_{s,30}=760\,\mathrm{m/s}\), \(\mathrm{PGA_{ref}}\). Derive equation for \(\mathrm{PGA_{ref}}\) of form \(\ln \mathrm{PGA_{ref}}=b_1+b_2(M-7)+b_4 \ln((r_{jb}^2+h^2)^{0.5})\), where \(b_1=0.851\), \(b_2=0.480\), \(b_4=-0.884\) and \(h=6.3\,\mathrm{km}\) for geometric mean (\(\sigma\) not reported).

Use data from the PEER Next Generation Attenuation (NGA) database.

Investigate the spatial correlation of ground motions and their variabilities.

Generate datasets using normally distributed values of \(M\) (truncated at \(\pm 2\) standard deviations that are reported in the PEER NGA database) for earthquakes and lognormally-distributed values of \(V_{s,30}\) (again using standard deviations from PEER NGA database) for stations. Repeat regression analysis and find coefficients very similar to those obtained ignoring the uncertainty in \(M\) and \(V_{s,30}\).

Ground-motion model is: \[\log_{10}(Y)=a+b M_L+c\log(R)+d S_{\mathrm{soil}}\] where \(Y\) is in \(\,\mathrm{g}\), \(a=-3.2191\pm 0.16\), \(b=0.7194 \pm 0.025\), \(c=-1.7521\pm 0.075\), \(d=0.1780\) and \(\sigma=0.282\).

Originally use three site classes based on Eurocode 8:

Rock, \(V_{s,30}>800\,\mathrm{m/s}\). Marine clay or other rocks (Lower Pleistocene and Pliocene), volcanic rock and deposits. 11 stations. 833 records.

Stiff soil, \(360<V_{s,30}<800\,\mathrm{m/s}\). Colluvial, alluvial, lacustrine, beach, fluvial terraces, glacial deposits and clay (Middle-Upper Pleistocene). Sand and loose conglomerate (Pleistocene and Pliocene). Travertine (Pleistocene and Holocene). 6 stations. 163 records.

Soft soil, \(V_{s,30}<360\,\mathrm{m/s}\). Colluvial, alluvial, lacustrine, beach and fluvial terrace deposits (Holocene). 3 stations. 67 records.

Classify stations using geological maps. Find that results obtained using this classification are not realistic because of some stations on very thick (\(>1000\,\mathrm{m}\)) sedimentary deposits whose amplification factors are small. Therefore, use two site classes using H/V ratios both using noise and earthquake records. Confirm H/V results by computing magnitude residuals at each station.

Final site classes are:

Site amplification factors \(<2\) at all considered frequencies from H/V analysis. 422 records. \(S_{\mathrm{soil}}=0\).

Site amplification factors \(>2\). 641 records. \(S_{\mathrm{soil}}=1\).

Use data from velocimeters (31 stations) and accelerometers (2 stations) from 33 sites with sampling rates of \(62.5\,\mathrm{samples/s}\).

Relocate events and calculate \(M_L\).

Exclude data from \(M_L<2.5\) and \(r_{hypo}>300\,\mathrm{km}\).

Few near-source records (\(r_{hypo}<150\,\mathrm{km}\)) from \(M_L>4\) but for \(M_L<4\) distances from \(0\) to \(300\,\mathrm{km}\) well represented.

Exclude records with signal-to-noise ratios \(<10\,\mathrm{dB}\).

Correct for instrument response and bandpass filter between \(0.5\) and \(25\,\mathrm{Hz}\) and then the velocimetric records have been differentiated to obtain acceleration.

Visually inspect records to check for saturated signals and noisy records.

Compare records from co-located velocimetric and accelerometric instruments and find that they are very similar.

Compare PGAs using larger horizontal component, geometric mean of the two horizontal components and the resolved component. Find that results are similar and that the records are not affected by bias due to orientation of sensors installed in field.

Try including a quadratic magnitude term but find that it does not reduce uncertainties and therefore remove it.

Try including an anelastic attenuation term but find that the coefficient is not statistically significant and that the coefficient is positive and close to zero and therefore remove this term.

Try using a term \(c \log_{10} \sqrt{R_{\mathrm{epi}}^2+h^2}\) rather than \(c\log_{10}(R)\) but find that \(h\) is not well constrained and hence PGAs for distances \(<50\,\mathrm{km}\) underpredicted.

Find that using a maximum-likelihood regression technique leads to very similar results to the one-stage least-squares technique adopted, which relate to lack of correlation between magnitudes and distances in dataset.

Find site coefficients via regression following the derivation of \(a\), \(b\) and \(c\) using the 422 rock records.

Compare observed and predicted ground motions for events in narrow (usually \(0.3\) units) magnitude bands. Find good match.

Examine residuals w.r.t. magnitude and distance and find no significant trends except for slight underestimation for short distances and large magnitudes. Also check residuals for different magnitude ranges. Check for bias due to non-triggering stations.

Compare predicted PGAs to observations for 69 records from central northern Italy from magnitudes \(5.0\)â\(6.3\) and find good match except for \(r_{hypo}<10\,\mathrm{km}\) where ground motions overpredicted, which relate to lack of near-source data.

Ground-motion model is: \[\log A=C_1 M_w+C_2 \log R+C_3\] where \(A\) in in \(\,\mathrm{cm/s^2}\), \(C_1=0.80 \pm 0.05\), \(C_2=-0.30 \pm 0.08\), \(C_3=-2.93\) and \(\sigma=0.314\) using \(r_{epi}\) and \(C_1=0.79 \pm 0.05\), \(C_2=-0.89 \pm 0.38\), \(C_3=-1.43\) and \(\sigma=0.341\) using \(r_{hypo}\).

Adjust observations by multiplicative factor \(S\) to account for site conditions (\(0.8\leq S \leq 1\) for hard rocks, \(0.7\leq S\leq 0.8\) for thin sedimentary layers and \(0.65\leq S \leq 0.7\) for thick sedimentary cover.

Focal depths between \(60\) and \(166\,\mathrm{km}\).

Data from digital strong-motion network (K2 instruments) from 1997 to 2000 (\(4\leq M_w \leq 6\)) plus data (SMA-1) from 30th August 1986 (\(M_w 7.1\)) and 30th and 31st May 1990 (\(M_w 6.9\) and \(6.4\)) earthquakes.

Regression in two steps: a) dependence on \(M_w\) found and then b) dependence on \(R\) is found (details on this procedure are not given).

Also regress using just K2 data (\(\log A=0.94 \pm 0.09 M_w-1.01\pm 0.42 \log R-1.84\), \(\sigma=0.343\)) and using \(r_{epi}\) (\(\log A=0.89 \pm 0.09 M_w-0.28 \pm 0.09 \log \Delta-3.35\), \(\sigma=0.322\)). Note that correlation coefficients are higher and \(\sigma\)s are lower when all data is used and that match (based on relative residuals) to data from 1986 and 1990 earthquakes is better when all data is used.

Present relative residuals for sites in epicentral area and in Bucharest. Note that for \(63\%\) of earthquakes relative errors are \(<50\%\) for at least one station; for \(43\%\) of earthquake relative errors are \(<30\%\) for at least one station; and for 9 earthquakes relative errors are smaller than \(10\%\) for at least one station (BMG, the extreme site). Based on this analysis it is concluded that predictions more reliable in far-field than in epicentral area. Also find that largest absolute residuals are for MLR (stiff rock).

Note largest relative errors are for \(4\leq M_w \leq 4.5\).

Ground-motion model is: \[\log y=a_1+a_2 M_w+(a_3+a_4Mw) \log \sqrt{r_{jb}^2+a_5^2}+a_6S_S+a_7S_A+a_8F_N+a_9F_T+a_{10}F_O\] where \(y\) is in \(\,\mathrm{m/s^2}\), \(a_1=-0.703\), \(a_2=0.392\), \(a_3=-0.598\), \(a_4=-0.100\), \(a_5=-7.063\), \(a_6=0.186\), \(a_7=0.125\), \(a_8=0.082\), \(a_9=0.012\) and \(a_{10}=-0.038\) (do not report \(\sigma\) but unbiased mean square error) for horizontal PGA; and \(a_1=0.495\), \(a_2=0.027\), \(a_3=-2.83\), \(a_4=0.235\), \(a_5=7.181\), \(a_6=1.150\), \(a_7=1.103\), \(a_8=-0.074\), \(a_9=0.065\) and \(a_{10}=-0.170\) (do not report \(\sigma\) but unbiased mean square error).

Use three site categories:

\(S_S=1\), \(S_A=0\).

\(S_A=1\), \(S_S=0\).

\(S_S=0\), \(S_A=0\).

Use four faulting mechanisms:

\(F_N=1\), \(F_T=0\), \(F_O=0\).

\(F_N=0\), \(F_T=0\), \(F_O=0\).

\(F_T=1\), \(F_N=0\), \(F_O=0\).

\(F_O=1\), \(F_N=0\), \(F_T=0\).

Use same data and functional form as N. N. Ambraseys et al. (2005a) and N. N. Ambraseys et al. (2005b) but exclude six records that were not available.

Use genetic (global optimization) algorithm to find coefficients so as to find the global (rather than a local) minimum. Use the unbiased mean square error as the error (cost or fitness) function in the algorithm. Use 20 chromosomes as initial population, best-fitness selection for offspring generation, uniform random selection for mutation of chromosomes and heuristic crossover algorithm for generation of new offspring.

Find smaller (by \(26\%\) for horizontal and \(16.66\%\) for vertical) unbiased mean square error than using standard regression techniques.

Ground-motion model is: \[\log_{10} y=\theta_1+\theta_2 M+\theta_3 M^2+\theta_4 R+\theta_5 \log_{10}(R+\theta_6 10^{\theta_7 M})\] where \(y\) is in \(\,\mathrm{cm/s^2}\), \(\theta_1=-3.4712\), \(\theta_2=2.2639\), \(\theta_3=-0.1546\), \(\theta_4=0.0021\), \(\theta_5=-1.8011\), \(\theta_6=0.0490\), \(\theta_7=0.2295\), \(\sigma_r=0.2203\) (intra-event) and \(\sigma_e=0.2028\) (inter-event).

All records from rock sites.

Strong correlation between magnitude and distance in dataset.

Use a derivative-free approach based on a hybrid genetic algorithm to derive the model. Use a simplex search algorithm to reduce the search domain to improve convergence speed. Then use a genetic algorithm to obtain the coefficients and uncertainties using one-stage maximum-likelihood estimation. Believe that approach is able to overcome shortcomings of previous methods in providing reliable and stable solutions although it is slower.

In hybrid genetic algorithm an initial population of possible solutions is constructed in a random way and represented as vectors called strings or chromosomes of length determined by number of regression coefficients and variance components. Population size is usually more than twice string length. Each value of population array is encoded as binary string with known number of bits assigned according to level of accuracy or range of each variable. Use three operations (reproduction/selection, crossover and mutation) to conduct directed search. In reproduction phase each string assigned a fitness value derived from its raw performance measure given by objective function. Probabilities of choosing a string is related to its fitness value. Crossover or mating combines pairs of strings to create improved strings in next population. In mutation one or more bits of every string are altered randomly. The process is then repeated until a termination criterion is met. Demonstrate approach using test function and find small maximum bias in results. Conclude that method is reliable.

Use Taiwanese dataset of Y.-H. Chen and Tsai (2002) to demonstrate method.

Compare results with those obtained using methods of Brillinger and Preisler (1985), Joyner and Boore (1993) and Y.-H. Chen and Tsai (2002). Find differences in coefficients (although predictions are very similar except at edges of dataspace) and standard deviations (slightly lower for proposed method).

Compare predicted motions for \(M_L 5.5\) with observations for \(M_L 5\)â\(6\). Find good fit.

Plot total residuals against magnitude and distance and find no trends.

Note that residuals show that model is satisfactory up to \(100\,\mathrm{km}\) but for larger distances assumption of geometric spreading of body waves in not appropriate due to presence of waves reflected off Moho.

Note that near-source saturation should be included. Apply proposed method using a complex functional form with different equations for three distance ranges and compare results to those using simple functional form. Find differences at short and large distances.

Ground-motion model is: \[\ln A=c_1+c_2M-c_3 \ln h-c_4 \ln R\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(c_1=-0.5342\), \(c_2=2.1380\), \(c_3=0.4440\), \(c_4=1.4821\) and \(\sigma=0.28\) for horizontal PGA and \(c_1=-0.5231\), \(c_2=1.9876\), \(c_3=0.5502\), \(c_4=1.4038\) and \(\sigma=0.27\) for vertical PGA.

Most stations on rock or firm ground. 4 instruments (from close to coast) installed on sandy or silty-sandy soils. Not enough data to correct for site effects or derive site coefficients. Check residuals (not shown) for each station and find no systematic bias.

Focal depths \(h\) between \(3.4\) and \(76.0\,\mathrm{km}\) (most \(<40\,\mathrm{km}\)). No correlation between \(h\) and \(r_{epi}\).

Use data from 12 (5 Etnas and 7 GSR-18s) temporary and 5 permanent strong-motion stations.

Since data from digital instruments only apply baseline correction.

Exclude data from 3 events only recorded at 3 stations.

Relocate earthquakes because of poor locations given by agencies. Recompute \(M_L\) from accelerograms.

Inclusion of \(h\) leads to less scatter but note need for larger database to better understand effect of \(h\).

Examine residuals w.r.t. distance and find no trend or bias.

Ground-motion model is: \[\begin{aligned} \ln \mathrm{Sa}(\,\mathrm{g})&=&f_1(M,R_{rup})+a_{12}F_{RV}+a_{13}F_{NM}+a_{15}F_{AS}+f_5(\widehat{\mathrm{PGA_{1100}}},V_{S30})\\ &&+F_{HW}f_4(R_{jb},R_{rup},R_x,W,\delta,Z_{TOR},M)+f_6(Z_{TOR})+f_8(R_{rup},M)\\ &&+f_{10}(Z_{1.0},V_{S30})\\ f_1(M,R_{rup})&=&\left\{ \begin{array}{l@{\quad}l} a_1+a_4(M-c_1)+a_8(8.5-M)^2+[a_2+a_3(M-c_1)]\ln(R) \quad \mbox{for} \quad M\leq c_1\\ a_1+a_5(M-c_1)+a_8(8.5-M)^2+[a_2+a_3(M-c_1)]\ln(R) \quad \mbox{for} \quad M>c_1 \end{array} \right.\\ R&=&\sqrt{R_{rup}^2+c_4^2}\\ f_5(\widehat{\mathrm{PGA_{1100}}},V_{S30})&=&\left\{ \begin{array}{l@{\quad}l} a_{10}\ln\left(\frac{V^*_{S30}}{V_{LIN}}\right)-b\ln(\widehat{\mathrm{PGA_{1100}}}+c)\\ +b\ln\left(\widehat{\mathrm{PGA_{1100}}}+c\left(\frac{V^*_{S30}}{V_{LIN}}\right)^n\right) \quad \mbox{for} \quad V_{S30}<V_{LIN}\\ (a_{10}+bn)\ln\left(\frac{V^*_{S30}}{V_{LIN}}\right) \quad \mbox{for} \quad V_{S30}\geq V_{LIN} \end{array} \right.\\ \mbox{where} \quad V^*_{S30}&=&\left\{ \begin{array}{l@{\quad}l} V_{S30} \quad \mbox{for} \quad V_{S30}<V_1\\ V_1 \quad \mbox{for} \quad V_{S30}\geq V_1 \end{array} \right.\\ \mbox{and} \quad V_1&=&\left\{ \begin{array}{l@{\quad}l} 1500 \quad \mbox{for} \quad T\leq 0.50\,\mathrm{s}\\ \exp[8.0-0.795\ln(T/0.21)] \quad \mbox{for} \quad 0.50<T\leq 1\,\mathrm{s}\\ \exp[6.76-0.297\ln(T)] \quad \mbox{for} \quad 1<T<2\,\mathrm{s}\\ 700 \quad \mbox{for} \quad T\geq 2\,\mathrm{s} \end{array} \right.\\ f_4(R_{jb},R_{rup},\delta,Z_{TOR},M,W)&=&a_{14}T_1(R_{jb})T_2(R_x,W,\delta)T_3(R_x,Z_{TOR})T_4(M)T_5(\delta)\\ \mbox{where} \quad T_1(R_{jb})&=&\left\{ \begin{array}{l@{\quad}l} 1-\frac{R_{jb}}{30} \quad \mbox{for} \quad R_{jb}<30\,\mathrm{km}\\ 0 \quad \mbox{for} \quad R_{jb}\geq 30\,\mathrm{km} \end{array} \right.\\ T_2(R_x,W,\delta)&=&\left\{ \begin{array}{l@{\quad}l} 0.5+\frac{R_x}{2W \cos(\delta)} \quad \mbox{for} \quad R_x \leq W \cos(\delta)\\ 1 \quad \mbox{for} \quad R_x>W \cos(\delta) \quad \mbox{or} \quad \delta=90^\circ \end{array} \right.\\ T_3(R_x,Z_{TOR})&=&\left\{ \begin{array}{l@{\quad}l} 1 \quad \mbox{for} \quad R_x\geq Z_{TOR}\\ \frac{R_x}{Z_{TOR}} \quad \mbox{for} \quad R_x <Z_{TOR} \end{array} \right.\\ T_4(M)&=&\left\{ \begin{array}{l@{\quad}l} 0 \quad \mbox{for} \quad M\leq 6\\ M-6 \quad \mbox{for} \quad 6<M<7\\ 1 \quad \mbox{for} \quad M \geq 7 \end{array} \right.\\ T_5(\delta)&=&\left\{ \begin{array}{l@{\quad}l} 1-\frac{\delta-30}{60} \quad \mbox{for} \quad \delta \geq 30\\ 1 \quad \mbox{for} \quad \delta<30 \end{array} \right.\\ f_6(Z_{TOR})&=&\left\{ \begin{array}{l@{\quad}l} \frac{a_{16} Z_{TOR}}{10} \quad \mbox{for} \quad Z_{TOR}<10\,\mathrm{km}\\ a_{16} \quad \mbox{for} \quad Z_{TOR}\geq 10\,\mathrm{km} \end{array} \right.\\ f_8(R_{rup},M)&=&\left\{ \begin{array}{l@{\quad}l} 0 \quad \mbox{for} \quad R_{rup}<100\,\mathrm{km}\\ a_{18}(R_{rup}-100)T_6(M) \quad \mbox{for} \quad R_{rup} \geq 100\,\mathrm{km} \end{array} \right.\\ \mbox{where} \quad T_6(M)&=&\left\{ \begin{array}{l@{\quad}l} 1 \quad \mbox{for} \quad M<5.5\\ 0.5(6.5-M)+0.5 \quad \mbox{for} \quad 5.5\leq M \leq 6.5\\ 0.5 \quad \mbox{for} \quad M>6.5 \end{array} \right.\\ f_{10}(Z_{1.0},V_{S30})&=&a_{21}\ln \left(\frac{Z_{1.0}+c_2}{\hat{Z}_{1.0}(V_{S30})+c_2}\right)+\left\{ \begin{array}{l@{\quad}l} a_{22}\ln \left( \frac{Z_{1.0}}{200} \right) \quad \mbox{for} \quad Z_{1.0} \geq 200\\ 0 \quad \mbox{for} \quad Z_{1.0}<200 \end{array} \right.\\ \mbox{where} \quad \ln[\hat{Z}_{1.0}(V_{S30})]&=&\left\{ \begin{array}{l@{\quad}l} 6.745 \quad \mbox{for} \quad V_{S30}<180\,\mathrm{m/s}\\ 6.745-1.35 \ln \left(\frac{V_{S30}}{180}\right) \quad \mbox{for} \quad 180\leq V_{S30} \leq 500\,\mathrm{m/s}\\ 5.394-4.48 \ln \left(\frac{V_{S30}}{500}\right) \quad \mbox{for} \quad V_{S30}>500\,\mathrm{m/s} \end{array} \right.\\ a_{21}&=&\left\{ \begin{array}{l@{\quad}l} 0 \quad \mbox{for} \quad V_{S30}\geq 1000\\ \frac{-(a_{10}+bn)\ln \left(\frac{V^*_{S30}}{\min(V_1,1000)}\right)}{\ln\left(\frac{Z_{1.0}+c_2}{\hat{Z}_{1.0}+c_2}\right)} \quad \mbox{for} \quad (a_{10}+bn)\ln\left(\frac{V^*_{S30}}{\min(V_1,1000)}\right)+e_2\ln\left(\frac{Z_{1.0}+c_2}{\hat{Z}_{1.0}+c_2}\right)<0\\ e_2 \quad \mbox{otherwise} \end{array} \right.\\ e_2&=&\left\{ \begin{array}{l@{\quad}l} 0 \quad \mbox{for} \quad T<0.35\,\mathrm{s}\quad \mbox{or} \quad V_{S30}>1000\\ -0.25 \ln\left(\frac{V_{S30}}{1000}\right)\ln\left(\frac{T}{0.35}\right) \quad \mbox{for} \quad 0.35\leq T\leq 2\,\mathrm{s}\\ -0.25 \ln\left(\frac{V_{S30}}{1000}\right)\ln\left(\frac{2}{0.35}\right) \quad \mbox{for} \quad T>2\,\mathrm{s} \end{array} \right.\\ a_{22}&=&\left\{ \begin{array}{l@{\quad}l} 0 \quad \mbox{for} \quad T<2\,\mathrm{s}\\ 0.0625(T-2) \quad \mbox{for} \quad T\geq 2\,\mathrm{s} \end{array} \right.\end{aligned}\] The model for the standard deviation is: \[\begin{aligned} \sigma_B(M,T)&=&\sqrt{\sigma_0^2(M,T)-\sigma_{Amp}^2(T)}\\ \sigma(T,M,\widehat{\mathrm{PGA_{1100}}},V_{S30})&=&\left[\begin{array}{l@{\quad}l} \sigma_B^2(M,T)+\sigma_{Amp}^2(T)\\ +\left(\frac{\partial \ln \mathrm{Amp}(T,\widehat{\mathrm{PGA_{1100}}},V_{S30})}{\partial \ln \mathrm{PGA_{1100}}}\right)^2 \sigma_B^2(M,\mathrm{PGA})\\ +2\left(\frac{\partial \ln \mathrm{Amp}(T,\widehat{\mathrm{PGA_{1100}}},V_{S30})}{\partial \ln \mathrm{PGA_{1100}}}\right)\\ \times \sigma_B(M,T)\sigma_B(M,\mathrm{PGA})\rho_{\epsilon/\sigma}(T,\mathrm{PGA}) \end{array} \right]^{1/2}\\ \frac{\partial \ln \mathrm{Amp}(T,\widehat{\mathrm{PGA_{1100}}},V_{S30})}{\partial \ln \mathrm{PGA_{1100}}}&=&\left\{\begin{array}{l@{\quad}l} 0 \quad \mbox{for} \quad V_{S30}\geq V_{LIN}\\ \frac{-b(T)\widehat{\mathrm{PGA_{1100}}}}{\widehat{\mathrm{PGA_{1100}}}+c}+\frac{-b(T)\widehat{\mathrm{PGA_{1100}}}}{\widehat{\mathrm{PGA}_{1100}}+c\left(\frac{V_{S30}}{V_{LIN}}\right)^n} \quad \mbox{for} \quad V_{S30}<V_{LIN} \end{array} \right.\\ \sigma_0(M)&=&\left\{\begin{array}{l@{\quad}l} s_1\quad \mbox{for} \quad M<5\\ s_1+\left(\frac{s_2-s_1}{2}\right)(M-5) \quad \mbox{for} \quad 5\leq M \leq 7\\ s_2\quad \mbox{for} \quad M>7 \end{array} \right.\\ \tau_0(M)&=&\left\{\begin{array}{l@{\quad}l} s_3\quad \mbox{for} \quad M<5\\ s_3+\left(\frac{s_4-s_3}{2}\right)(M-5) \quad \mbox{for} \quad 5 \leq M \leq 7\\ s_4\quad \mbox{for} \quad M>7 \end{array} \right.\end{aligned}\] where \(\mathrm{Sa}\) is in \(\,\mathrm{g}\), \(\hat{\mathrm{PGA_{1100}}}\) is median peak acceleration for \(V_{S30}=1100\,\mathrm{m/s}\), \(\sigma_B\) and \(\tau_B\) (\(=\tau_0(M,T)\)) are intra-event and inter-event standard deviations, \(\sigma_0\) and \(\tau_0\) are intra-event and inter-event standard deviations of the observed ground motions for low levels of outcrop rock motions (directly from regression), \(\sigma_{amp}\) is intra-event variability of the site amplification factors (assumed equal to \(0.3\) for all periods based on 1D site response results), \(c_1=6.75\), \(c_4=4.5\), \(a_3=0.265\), \(a_4=-0.231\), \(a_5=-0.398\), \(N=1.18\), \(c=1.88\), \(c_2=50\), \(V_{LIN}=865.1\), \(b=-1.186\), \(a_1=0.804\), \(a_2=-0.9679\), \(a_8=-0.0372\) ,\(a_{10}=0.9445\), \(a_{12}=0.0000\), \(a_{13}=-0.0600\), \(a_{14}=1.0800\), \(a_{15}=-0.3500\), \(a_{16}=0.9000\), \(a_{18}=-0.0067\), \(s_1=0.590\) and \(s_2=0.470\) for \(V_{S30}\) estimated, \(s_1=0.576\) and \(s_2=0.453\) for \(V_{S30}\) measured, \(s_3=0.470\), \(s_4=0.300\) and \(\rho(T,\mathrm{PGA})=1.000\).

Characterise sites using \(V_{S30}\) and depth to engineering rock (\(V_s=1000\,\mathrm{m/s}\)), \(Z_{1.0}\). Prefer \(V_{s,30}\) to generic soil/rock categories because it is consistent with site classification in current building codes. Note that this does not imply that \(30\,\mathrm{m}\) is key depth range for site response but rather that \(V_{s,30}\) is correlated with entire soil profile.

Classify events in three fault mechanism categories:

\(F_{RV}=1\), \(F_{NM}=0\). Earthquakes defined by rake angles between \(30\) and \(150^{\circ}\).

\(F_{RV}=0\), \(F_{NM}=1\). Earthquakes defined by rake angles between \(-60\) and \(-120^{\circ}\).

\(F_{RV}=0\), \(F_{NM}=0\). All other earthquakes.

Believe that model applicable for \(5\leq M_w \leq 8.5\) (strike-slip) and \(5\leq M_w \leq 8.0\) (dip-slip) and \(0\leq d_r \leq 200\,\mathrm{km}\).

Use simulations for hard-rock from 1D finite-fault kinematic source models for \(6.5 \leq M_w \leq 8.25\), 3D basin response simulations for sites in southern California and equivalent-linear site response simulations to constrain extrapolations beyond the limits of the empirical data.

Select data from the Next Generation Attenuation (NGA) database (flat-file version 7.2). Include data from all earthquakes, including aftershocks, from shallow crustal earthquakes in active tectonic regions under assumption that median ground motions from shallow crustal earthquakes at \(d_r<100\,\mathrm{km}\) are similar. This assumes that median stress-drops are similar between shallow crustal events in: California, Alaska, Taiwan, Japan, Turkey, Italy, Greece, New Zealand and NW China. Test assumption by comparing inter-event residuals from different regions to those from events in California. Since aim is for model for California and since difference in crustal structure and attenuation can affect ground motions at long distances exclude data from \(d_r>100\,\mathrm{km}\) from outside western USA.

Also exclude these data: events not representative of shallow crustal tectonics, events missing key source metadata, records not representative of free-field motion, records without a \(V_{s,30}\) estimate, duplicate records from co-located stations, records with missing horizontal components or poor quality accelerograms and records from western USA from \(d_r>200\,\mathrm{km}\).

Classify earthquakes by event class: AS (aftershock) (\(F_{AS}=1\)); MS (mainshock), FS (foreshock) and swarm (\(F_{AS}=0\)). Note that classifications not all unambiguous.

Use depth-to-top of rupture, \(Z_{TOR}\), fault dip in degrees, \(\delta\) and down-dip rupture width, \(W\).

Use \(r_{jb}\) and \(R_x\) (horizontal distance from top edge of rupture measured perpendicular to fault strike) to model hanging wall effects. For hanging wall sites, defined by vertical projection of the top of the rupture, \(F_{HW}=1\). \(T_1\), \(T_2\) and \(T_3\) constrained by 1D rock simulations and the Chi-Chi data. \(T_4\) and \(T_5\) constrained by well-recorded hanging wall events. Only \(a_{14}\) was estimated by regression. State that hanging-wall scaling is one of the more poorly-constrained parts of model

^{25}.Records well distributed w.r.t. \(M_w\) and \(r_{rup}\).

For four Chi-Chi events show steep distance decay than other earthquakes so include a separate coefficient for the \(\ln(R)\) term for these events so they do not have a large impact on the distance scaling. Retain these events since important for constraining other aspects of the model, e.g. site response and intra-event variability.

Only used records from \(5 \leq M \leq 6\) to derive depth-to-top of rupture (\(Z_{TOR}\)) dependence to limit the effect on the relation of the positive correlation between \(Z_{TOR}\) and \(M\).

Constrain (outside the main regression) the large distance (\(R_{rup}>100\,\mathrm{km}\)) attenuation for small and moderate earthquakes (\(4\leq M \leq 5\)) using broadband records of 3 small (\(M 4\)) Californian earthquakes because limited data for this magnitude-distance range in NGA data set.

Note difficult in developing model for distinguishing between shallow and deep soil sites due to significant inconsistencies between \(V_{S30}\) and depth of soil (\(Z_{1.0}\)), which believe to be unreliable in NGA Flat-File. Therefore, develop soil-depth dependence based on 1D (for \(Z_{1.0}<200\,\mathrm{m}\)) and 3D (for \(Z_{1.0}>200\,\mathrm{m}\)) site response simulations. Motion for shallow soil sites do not fall below motion for \(V_{S30}=1000\,\mathrm{m/s}\).

\(T_D\) denotes period at which rock (\(V_{S30}=1100\,\mathrm{m/s}\)) spectrum reaches constant displacement. Using point-source stochastic model and 1D rock simulations evaluate magnitude dependence of \(T_D\) as \(\log_{10}(T_D)=-1.25+0.3M\). For \(T>T_D\) compute rock spectral acceleration at \(T_D\) and then scale this acceleration at \(T_D\) by \((T_D/T)^2\) for constant spectral displacements. The site response and soil depth scaling is applied to this rock spectral acceleration, i.e. \(\mathrm{Sa}(T_D,V_{S30}=1100)\frac{T_D^2}{T^2}+f_5(\hat{\mathrm{PGA}_{1100}},V_{S30},T)+f_{10}(Z_{1.0},V_{S30},T)\).

Reduce standard deviations to account for contribution of uncertainty in independent parameters \(M\), \(R_{rup}\), \(Z_{TOR}\) and \(V_{S30}\).

Note that regression method used prevents well-recorded earthquakes from dominating regression.

Examine inter-event residuals and find that there is no systemic trend in residuals for different regions. Find that residuals for \(M>7.5\) are biased to negative values because of full-saturation constraint. Examine intra-event residuals and find no significant trend in residuals.

Although derive hanging-wall factor only from reverse-faulting data suggest that it is applied to normal-faulting events as well.

State that should use median \(\mathrm{PGA_{1100}}\) for nonlinear site amplification even if conducting a seismic hazard analysis for above median ground motions.

State that if using standard deviations for estimated \(V_{S30}\) and \(V_{S30}\) is accurate to within \(30\%\) do not need to use a range of \(V_{S30}\) but if using measured-\(V_{S30}\) standard deviations then uncertainty in measurement of \(V_{S30}\) should be estimated by using a range of \(V_{S30}\) values.

State that if do not know \(Z_{1.0}\) then use median \(Z_{1.0}\) estimated from equations given and do not adjust standard deviation.

Ground-motion models are: \[\log_{10}(\mathrm{acceleration})=a \log_{10}(R)+b \log_{10}(M)+c\] where \(\mathrm{acceleration}\) is in \(\,\mathrm{m/s^2}\), \(a=-1.95600\), \(b=9.59878\), \(c=-4.87778\) and \(\sigma=0.4591\), and: \[\log_{10}(\mathrm{acceleration})=a \log_{10}(R)+b M+c\] where \(a=-1.96297\), \(b=0.89343\), \(c=-2.65660\) and \(\sigma=0.4596\).

Select data from SIL database with \(M_{Lw}>3.5\) in latitude range 63.5 to 64.3\(^\circ\)N and longitude range 18 to 23.5\(^\circ\)W between July 1992 and April 2007.

Exclude data where several earthquakes are superimposed and retain only âcleanâ waveforms.

Most data from \(5\,\mathrm{Hz}\) Lennarz seismometers. Some from \(1\,\mathrm{Hz}\) and long-period instruments. Sampling frequency is \(100\,\mathrm{Hz}\).

Use data from SW Iceland plus data from Reykjanes Ridge and Myrdalsjokull volcano.

Investigate decay in several individual earthquakes and fit equations of form \(\log y=a \log R+b\). Note that relations are well behaved so fit entire dataset.

Ground-motion model is: \[\begin{aligned} \ln y&=&c_1+f_1(M_w)+f_2(M_w)f_3(R)+f_4(F)+\mathrm{FR}f_5(Z_{FR})+\\ &&\mathrm{FS}f_6(Z_{FR})+f_7(\mathrm{HW},R_{JB},M_w,\mathrm{DIP})+\\ &&f_8(V_{s,30},V_{lin},\mathrm{PGA}_{non-lin},\mathrm{PGA}_{rock})+f_9(V_{s,30},Z_{1.5})\\ \mbox{where for} \quad M_w\leq c_0&&\\ f_1(M_w)&=&c_3(M_w-c_0)+c_8(T)(8.5-M_w)^n\\ f_2(M_w)&=&c_2(T)+c_4(M_w-c_0)\\ \mbox{and for} \quad M_w>c_0&&\\ f_1(M_w)&=&c_5 (M_w-c_0)+c_8(T)(8.5-M_w)^n\\ f_2(M_w)&=&c_2(T)+c_6(M_w-c_0)\\ f_3(R)&=&\ln\sqrt{R_{rup}^2+c_7(T)^2}\\ f_4(F)&=&c_9(T)\mathrm{FR}+c_{10}(T)\mathrm{FS}+c_{11}(T)\mathrm{FN}\\ f_5(Z_{\mathrm{FR}})&=&\left\{ \begin{array}{l@{\quad}l} 0&Z_{top}\leq 2\,\mathrm{km}\\ c_{12}(T)(Z_{top}-2)/3&2<Z_{top} \leq 5\,\mathrm{km}\\ c_{12}(T)&5<Z_{top} \leq 10\,\mathrm{km}\\ c_{12}(T)[1-(Z_{top}-10)/5]&5<Z_{top} \leq 10\,\mathrm{km}(sic)\\ 0&Z_{top}>10\,\mathrm{km} \end{array} \right.\\ f_6(Z_{\mathrm{FS}})&=&\left\{ \begin{array}{l@{\quad}l} c_{13}(T) Z_{top}/2&0<Z_{top}\leq 2\,\mathrm{km}\\ c_{13}(T)&2<Z_{top} \leq 4\,\mathrm{km}\\ c_{13}(T)[1-(Z_{top}-4)/2]&4<Z_{top} \leq 6\,\mathrm{km}\\ 0&Z_{top}>6\,\mathrm{km} \end{array} \right.\\ g_1(R_{JB})&=&\left\{ \begin{array}{l@{\quad}l} 1-R_{JB}/45&0\leq R_{JB}<15\,\mathrm{km}\\ \frac{2}{3}(2-R_{JB}/15)&15\leq R_{JB}<30\,\mathrm{km}\\ 0&R_{JB}\geq 30\,\mathrm{km}\\ \end{array} \right.\\ g_2(M_w)&=&\left\{ \begin{array}{l@{\quad}l} 0&M_w<6.0\\ 2(M_w-6)&6.0\leq M_w<6.5\\ 1&M_w\geq 6.5\\ \end{array} \right.\\ g_3(\mathrm{DIP})&=&\left\{ \begin{array}{l@{\quad}l} 1-(\mathrm{DIP}-70)/20&\mathrm{DIP}\geq 70\\ 1&\mathrm{DIP}<70\\ \end{array} \right.\\ f_7(\mathrm{HW},R_{JB},M_w,\mathrm{DIP})&=&c_{14}(T)\mathrm{HW}g_1(R_{JB})g_2(M_w)g_3(\mathrm{DIP})\\ f_8(V_{s,30},V_{lin},\mathrm{PGA}_{non-lin},\mathrm{PGA}_{rock})&=&g_4(V_{s,30},V_{lin})+g_5(\mathrm{PGA}_{non-lin},\mathrm{PGA}_{rock})\\ g_4(V_{s,30},V_{lin})&=&c_{15}(T)\ln(V_{s,30}/V_{lin})\\ g_5(\mathrm{PGA}_{non-lin},\mathrm{PGA}_{rock})&=&\left\{ \begin{array}{l@{\quad}l} c_{16}(T)\ln (\mathrm{PGA}_{min}/0.1)&\mathrm{PGA}_{non-lin}<a_1\\ c_{16}(T)[\ln (\mathrm{PGA}_{min}/0.1)&\\ +a\ln(\mathrm{PGA}_{non-lin}/a_1)&\\ +b(\ln(\mathrm{PGA}_{non-lin}/a_1))^2]&a_1\leq \mathrm{PGA}_{non-lin}\leq a_2\\ c_{16}(T)\ln (\mathrm{PGA}_{non-lin}/0.1)&\mathrm{PGA}_{non-lin}\geq a_2\\ \end{array} \right.\\ f_9(V_{s,30},Z_{1.5})&=&g_6(V_{s,30},Z_{1.5},\hat{Z})+g_7(Z_D,Z_{1.5})\\ g_6(V_{s,30},Z_{1.5},\hat{Z})&=&c_{17}(T)(1/\hat{Z})\ln(V_{s,30}/1500)\ln(Z_{1.5})\\ g_7(Z_{1.5},Z_D)&=&Z_D c_{18}(T) K_1 (1-\exp(-(Z_{1.5}-200)/300))+\\ &&Z_D c_{19}(T) K_2 (1-\exp(-(Z_{1.5}-200)/4000))\end{aligned}\] where \(y\) is in \(\,\mathrm{g}\), \(c_1=1.81\), \(c_2=-1.18\), \(c_7=8.647\), \(c_8=-0.028\), \(c_9=-0.176\), \(c_{10}=-0.266\), \(c_{11}=-0.476\), \(c_{12}=0.52\), \(c_{13}=-0.32\), \(c_{14}=0.4\), \(c_{15}=-0.36\), \(c_{17}=0\), \(c_{18}=0\), \(c_{19}=0\), \(c_{20}=0.496\), \(c_{21}=0.427\), \(K_1=2.260\), \(K_2=1.04\), \(V_{lin}=760\), \(\sigma=c_{20}(T)+[c_{21}(T)-c_{20}(T)]M_w\) for \(5.0\leq M_w<7.0\) and \(\sigma=c_{21}(T)\) for \(M_w\geq 7.0\).

Use \(V_{s,30}\) to characterize site conditions.

Characterize basin by depth to \(V_s=1500\,\mathrm{m/s}\), \(Z_{1.5}\), since more likely to be obtained for engineering projects.

Use three mechanism classes:

Normal. 34 records. \(\mathrm{FN}=1\), \(\mathrm{FS}=\mathrm{FR}=0\).

Strike-slip. 184 records. \(\mathrm{FS}=1\), \(\mathrm{FN}=\mathrm{FR}=0\).

Reverse. Originally classify as thrust, reverse and reverse oblique but combine. 423 records. \(\mathrm{FR}=1\), \(\mathrm{FN}=\mathrm{FS}=0\).

Note lack of records from normal earthquakes.

Use data from earthquakes with focal depths \(\leq 15\,\mathrm{km}\).

Only use data from instrument shelters, non-embedded buildings with \(<3\) stories (\(<7\) if located on firm rock) and dam abutments (to enhance database even though could be some interaction with dam).

Not sufficient data to investigate effect of tectonic environment. Exclude data from subduction zones because that is different tectonic regime than for shallow crustal earthquakes.

Data well distributed in magnitude-distance space so do not use special statistical procedures to decouple source and path effects. Do not use weights due to uniform distribution w.r.t. \(M_w\) and distance.

Exclude data from \(>60\,\mathrm{km}\) to avoid records with multiple reflections from lower crust.

Vast majority of data from western USA. Some from Alaska, Canada, Greece, Iran, Italy, Japan, Mexico, New Zealand and Turkey.

Constrain \(c_7(T)\) to be monotonically varying with period because otherwise can have large changes in spectral shape at very short distances.

Note that for \(M_w<5.8\) magnitude dependence may be due to depth-to-top (\(Z_{\mathrm{FR}}\) and \(Z_{\mathrm{FS}}\)) effects since small earthquakes have on average larger depth-to-top than larger earthquakes. Inter-event residuals from preliminary regression are functions of rake and depth-to-top (stronger than rake dependency) particularly for reverse earthquakes. These observations influence functional form of \(f_5(Z)\).

Use residuals from 1D simulations to define functional form for hanging wall effect (\(\mathrm{HW}=1\)).

Coefficients for nonlinear soil effects determined from analytical results because of correlations between other parameters and nonlinearity and since analytical results better constrained at high amplitudes than empirical data. Set \(a_1=0.04\,\mathrm{g}\), \(a_2=0.1\,\mathrm{g}\) and \(\mathrm{PGA}_{min}=0.06\,\mathrm{g}\). \(\mathrm{PGA}_{non-lin}\) is expected PGA on rock (\(V_{s,30}=760\,\mathrm{m/s}\)). \(c_{15}(T)\), \(c_{16}(T)\) and \(V_{lin}\) taken from Choi and Stewart (2005) and are not determined in regression.

Applied limited smoothing (using piecewise continuous linear fits on log period axis) to avoid variability in predicted spectral ordinates for neighbouring periods particularly at large magnitudes and short distances.

Examine normalized inter- and intra-event residuals w.r.t. \(M_w\) and distance (shown). Find no bias nor trends. Also plot against mechanism, site and other parameters and find no bias nor trends (not shown).

Ground-motion model is: \[\log(y)=c_1+c_2M+c_3\log(R)+c_4 R\] where \(y\) is in \(\,\mathrm{g}\)

^{26}, \(c_1=-3.45092\), \(c_2=0.49802\), \(c_3=-0.38004\), \(c_4=-0.00253\) and \(\sigma=0.313\).Uses data from strong-motion networks in Israel and Jordan.

Records from analogue, PDR-1, SSA-2 and Etna instruments.

21 earthquakes were recorded by only one station.

Ground-motion model is: \[\log_{10} y=a_1+a_2 M_w+a_3\log_{10} R+a_B S_B+a_C S_C+a_D S_D\] where \(y\) is in \(\,\mathrm{m/s^2}\), \(a_1=-1.296\), \(a_2=0.556\), \(a_3=-1.582\), \(a_B=0.22\), \(a_C=0.304\), \(a_D=0.332\) and \(\sigma=0.344\) for horizontal PGA.

Use four site categories based on Eurocode 8:

Rock-like. \(V_{s,30}\geq 800\,\mathrm{m/s}\). \(S_B=S_C=S_D=0\).

Stiff ground. \(360\leq V_{s,30}<800\,\mathrm{m/s}\). \(S_B=1\), \(S_C=S_D=0\).

\(180\leq V_{s,30}<360\,\mathrm{m/s}\). \(S_C=1\), \(S_B=S_D=0\).

Very soft ground. \(V_{s,30}<180\,\mathrm{m/s}\). \(S_D=1\), \(S_B=S_C=0\).

Try to retain only records from stations of known site class but keep records from stations of unknown class (\(4\%\) of total), which assume are either B or C classes. Use various techniques to extend \(20\,\mathrm{m}\) profiles of K-Net down to \(30\,\mathrm{m}\). Vast majority of data with \(V_{s,30}\leq 500\,\mathrm{m/s}\).

Use mechanism classification scheme of Boore and Atkinson (2007) based on plunges of P-, T- and B-axes:

16 earthquakes. \(5\leq M_w \leq 6.9\).

32 earthquakes. \(5 \leq M_w \leq 7.2\).

12 earthquakes. \(5.3 \leq M_w \leq 6.6\).

Develop for use in displacement-based design.

Select records with minimal long-period noise so that the displacement ordinates are reliable. Restrict selection to digital records because their displacement spectra are not significantly affected by correction procedure and for which reliable spectral ordinates up to at least \(10\,\mathrm{s}\) are obtainable. Include 9 analogue records from 1980 Irpinia (\(M_w 6.9\)) earthquake after careful scrutiny of long-period characteristics.

Use approach of Paolucci et al. (2008) to estimate cut-off frequencies for bandpass filtering. Compute noise index \(I_V\) for each record based on PGV and average value computed from coda of velocity time-history. Compare \(I_V\) with curves representing as a function of \(M_w\) the probability \(P\) that the long-period errors in the displacement spectrum are less than a chosen threshold. Use probability \(P\geq 0.9\) and drifts in displacement spectrum \(<15\%\) using \(I_V\) from geometric mean. Rejections closely correlated with instrument type (less data from high-bit instruments rejected than from low-bit instruments). Process records by removing pre-even offset from entire time-history. Following this \(57\%\) of records satisfied criterion of Paolucci et al. (2008). Remaining records filtered using fourth-order acausal filter with cut-off \(0.05\,\mathrm{Hz}\) after zero padding and cosine tapering. After this step records pass criterion of Paolucci et al. (2008). Note that filtering of \(43\%\) of records may affect reliability beyond \(15\,\mathrm{s}\).

Use data from K-Net and Kik-Net (Japan) (\(84\%\)); California (\(5\%\)); Italy, Iceland and Turkey (\(5\%\)); and Iran (\(6\%\)). Try to uniformly cover magnitude-distance range of interest. All data from \(M>6.8\) are from events outside Japan.

Exclude data from \(M_w<5\) because probabilistic seismic hazard deaggregation analyses show contribution to spectral displacement hazard from small events is very low.

Exclude data from \(M_w>7.2\) because \(7.2\) is representative of the largest estimated magnitude in historical catalogue of Italy. Most records from \(M_w\leq 6.6\).

Exclude data from subduction zone events.

Focal depths between \(2\) and \(22\,\mathrm{km}\). Exclude earthquakes with focal depth \(>22\,\mathrm{km}\) to be in agreement with focal depths of most Italian earthquakes.

Use \(r_{hypo}\) for greater flexibility in seismic hazard analyses where source zones have variable depth. Exclude data from \(r_{hypo}>150\,\mathrm{km}\) based on deaggregation results.

Test regional dependence of ground motions using analysis of variance. Divide dataset into intervals of \(10\,\mathrm{km}\times 0.3 M_w\) units and consider only bins with \(\geq 3\) records. Apply analysis for 18 bins on logarithmically transformed ground motions. Transform observed motions to site class A by dividing by site amplification factor derived by regression. Find no strong evidence for regional dependence.

Apply pure error analysis to test: i) standard logarithmic transformation, ii) magnitude-dependence of scatter and iii) lower bound on standard deviation using only \(M\) and \(r_{hypo}\). Divide dataset into bins of \(2\,\mathrm{km}\times 0.2 M_w\) units and consider only bins with \(\geq 2\) records (314 in total). Compute mean and standard deviation of untransformed ground motion and calculate coefficient of variation (COV). Fit linear equation to plots of COV against mean. Find no significant trend for almost all periods so conclude logarithmic transformation is justified for all periods. Compute standard deviation of logarithmically-transformed ground motions and fit linear equations w.r.t. \(M_w\). Find that dependence of scatter on magnitude is not significant. Compute mean standard deviation of all bins and find limit on lowest possible standard deviation using only \(M_w\) and \(r_{hypo}\).

Aim for simplest functional form and add complexity in steps, checking the statistical significance of each modification and its influence on standard error. Try including an anelastic term, quadratic \(M_w\) dependence and magnitude-dependent decay term but find none of these is statistically significant and/or leads to a reduction in standard deviation.

Try one-stage maximum likelihood regression but find higher standard deviation so reject it. Originally use two-stage approach of Joyner and Boore (1981).

Find that coefficients closely match a theoretical model at long periods.

Consider style-of-faulting by adding terms: \(a_N E_N+a_R E_R+a_S E_S\) where \(E_x\) are dummy variables for normal, reverse and strike-slip mechanisms. Find that reduction in standard deviation is only appreciable for limited period ranges but keep terms in final model.

Replace terms: \(a_B S_B+a_C S_C+a_D S_D\) by \(b_V \log_{10}(V_{s,30}/V_a)\) so that site amplification factor is continuous. \(V_{s,30}\) available for about \(85\%\) of records. To be consistent between both approaches constrain \(V_a\) to equal \(800\,\mathrm{m/s}\). Find \(b_V\) closely matches theoretical values \(1\) close to resonance period and \(0.5\) at long periods.

Examine residuals w.r.t. \(r_{hypo}\) and \(M_w\). Find no trends.

Ground-motion model is: \[\log_{10} Y=a+bM+c\log_{10} \sqrt{R^2+h^2}+eS\] where \(Y\) is in \(\,\mathrm{cm/s^2}\); when using \(r_{epi}\): \(a=1.0028\), \(b=0.3330\), \(c=-0.8842\), \(h=2.4\), \(e=0.1717\) and \(\sigma=0.3574\) (for larger), \(a=0.8947\), \(b=0.3410\), \(c=-0.8834\), \(h=2.6\), \(e=0.1725\) and \(\sigma=0.3428\) (for geometric mean) and \(a=0.9050\), \(b=0.3485\), \(c=-1.0803\), \(h=3.2\), \(e=0.1596\) and \(\sigma=0.3240\) (for vertical); and when using \(r_{hypo}\) (\(h\) is constrained to zero): \(a=1.1317\), \(b=0.3282\), \(c=-0.9297\), \(e=0.1860\) and \(\sigma=0.3680\) (for larger), \(a=1.0335\), \(b=0.3365\), \(c=-0.9352\), \(e=0.1865\) and \(\sigma=0.3527\) (for geometric mean) and \(a=1.0706\), \(b=0.3430\), \(c=-1.1418\), \(e=0.1767\) and \(\sigma=0.3392\) (for vertical).

Uses 2 site classes:

Granite, diorite, gneiss, sandstone, limestone or siltstone. Roughly \(V_s>360\,\mathrm{m/s}\). 82 records. \(S=0\).

Alluvium, sand, gravel, clay, sandy clay, silt, sandy silt or fill. 167 records. \(S=1\).

Cannot use more complex approach because of insufficient information for Chinese sites.

Selects 129 from China (mainly SMA-1A and GDQJ-1A instruments) and Taiwan and, because there are insufficient to develop model, adds 120 records from Japan.

Data from China corrected using unknown technique. Data from Japan uncorrected.

Plot residuals w.r.t. \(r\) and \(M\) and find no significant trends.

Ground-motion model is: \[\begin{aligned} \ln(y)&=&\ln(y_{ref})+\phi_1 \min\left[\ln\left(\frac{V_{S30}}{1130}\right),0\right]\\ &&{}+\phi_2\{\mathrm{e}^{\phi_3[\min(V_{S30},1130)-360]}-\mathrm{e}^{\phi_3(1130-360)}\}\ln \left(\frac{y_{ref}\mathrm{e}^{\eta}+\phi_4}{\phi_4}\right)\\ &&{}+\phi_5\left\{1-\frac{1}{\cosh[\phi_6 \max(0,Z_{1.0}-\phi_7)]}\right\}\\ &&{}+\frac{\phi_8}{\cosh[0.15 \max(0,Z_{1.0}-15)]}\\ \ln(y_{ref})&=&c_1+[c_{1a}F_{RV}+c_{1b}F_{NM}+c_7(Z_{TOR}-4)](1-\mathrm{AS})\\ &&{}+[c_{10}+c_{7a}(Z_{TOR}-4)]\mathrm{AS}+c_2(M-6)+\frac{c_2-c_3}{c_n} \ln[1+\mathrm{e}^{c_n(c_M-M)}]\\ &&{}+c_4\ln\{R_{RUP}+c_5 \cosh[c_6 \max(M-c_{HM},0)]\}\\ &&{}+(c_{4a}-c_4)\ln(\sqrt{R_{RUP}^2+c_{RB}^2})\\ &&{}+\left\{c_{\gamma 1}+\frac{1}{\cosh[\max(M-c_{\gamma 3},0)]}\right\} R_{RUP}\\ &&{}+c_9 F_{HW} \tanh \left(\frac{R_X \cos^2 \delta}{c_{9a}}\right) \left(1-\frac{\sqrt{R_{JB}^2+Z_{TOR}^2}}{R_{RUP}+0.001}\right)\\ \tau&=&\tau_1+\frac{\tau_2-\tau_1}{2} \times [\min\{\max(M,5),7\}-5]\\ \sigma&=&\left\{\sigma_1+\frac{\sigma_2-\sigma_1}{2}[\min(\max(M,5),7)-5]+\sigma_4 \times \mathrm{AS}\right\}\\ &&\quad \times \sqrt{(\sigma_3 F_{Inferred}+0.7F_{Measured})+(1+\mathrm{NL})^2}\\ \mbox{where} \quad \mathrm{NL}&=&\left(b\frac{y_{ref}\mathrm{e}^\eta}{y_{ref}\mathrm{e}^\eta+c}\right)\\ \sigma_T^2&=&(1+\mathrm{NL}_0)^2\tau^2+\sigma_{\mathrm{NL}_0}^2\end{aligned}\] where \(y\) is in \(\,\mathrm{g}\), \(c_2=1.06\), \(c_3=3.45\), \(c_4=-2.1\), \(c_{4a}=-0.5\), \(c_{RB}=50\), \(c_{HM}=3\), \(c_{\gamma 3}=4\), \(c_1=-1.2687\), \(c_{1a}=0.1\), \(c_{1b}=-0.2550\), \(c_n=2.996\), \(c_M=4.1840\), \(c_5=6.1600\), \(c_6=0.4893\), \(c_7=0.0512\), \(c_{7a}=0.0860\), \(c_9=0.7900\), \(c_{9a}=1.5005\), \(c_{10}=-0.3218\), \(c_{\gamma 1}=-0.00804\), \(c_{\gamma 2}=-0.00785\), \(\phi_1=-0.4417\), \(\phi_2=-0.1417\), \(\phi_3=-0.007010\), \(\phi_4=0.102151\), \(\phi_5=0.2289\), \(\phi_6=0.014996\), \(\phi_7=580.0\), \(\phi_8=0.0700\), \(\tau_1=0.3437\), \(\tau_2=0.2637\), \(\sigma_1=0.4458\), \(\sigma_2=0.3459\), \(\sigma_3=0.8\) and \(\sigma_4=0.0663\) (\(\eta\) is the inter-event residual). \(\sigma_T\) is the total variance for \(\ln(y)\) and is approximate based on the Taylor series expansion of the sum of the inter-event and intra-event variances. \(\sigma_{\mathrm{NL}_0}\) is the equation for \(\sigma\) evaluated for \(\eta=0\). Check approximate using Monte Carlo simulation and find good (within a few percent) match to exact answer.

Characterise sites using \(V_{S30}\). \(F_{Inferred}=1\) if \(V_{S30}\) inferred from geology and \(0\) otherwise. \(F_{Measured}=1\) if \(V_{S30}\) is measured and \(0\) otherwise. Believe model applicable for \(150\leq V_{S30}\leq 1500\,\mathrm{m/s}\).

Use depth to shear-wave velocity of \(1.0\,\mathrm{km/s}\), \(Z_{1.0}\), to model effect of near-surface sediments since \(1\,\mathrm{km/s}\) similar to values commonly used in practice for rock, is close to reference \(V_{S30}\) and depth to this velocity more likely to be available. For stations without \(Z_{1.0}\) use this empirical relationship: \(\ln(Z_{1.0})=28.5-\frac{3.82}{8}\ln(V_{S30}^8+378.7^8)\).

Use PEER Next Generation Attenuation (NGA) database supplemented by data from TriNet system to provide additional guidance on functional forms and constraints on coefficients.

Consider model to be update of Sadigh et al. (1997).

Focal depths less than \(20\,\mathrm{km}\) and \(Z_{TOR}\leq 15\,\mathrm{km}\). Therefore note that application to regions with very thick crusts (e.g. \(\gg 20\,\mathrm{km}\)) is extrapolation outside range of data used to develop model.

Develop model to represent free-field motions from shallow crustal earthquakes in active tectonic regions, principally California.

Exclude data from earthquakes that occurred in oceanic crust offshore of California or Taiwan because these data have been found to be more consistent with ground motions from subduction zones. Include data from 1992 Cape Mendocino earthquakes because source depth places event above likely interface location. Exclude data from four 1997 NW China earthquakes because of large depths (\(\geq 20\,\mathrm{km}\)) and the very limited information available on these data. Exclude data from the 1979 St Elias earthquake because believe it occurred on subduction zone interface. Include data from the 1985 Nahanni and 1992 Roermond because believe that they occurred on boundary of stable continental and active tectonic regions.

Assume that ground motions from different regions are similar and examine this hypothesis during development.

Include data from aftershocks, because they provide additional information on site model coefficients, allowing for systematic differences in ground motions with mainshock motions. \(\mathrm{AS}=1\) if event aftershock and \(0\) otherwise.

Exclude data from large buildings and at depth, which removes many old records. Include sites with known topographic effects since the effect of topography has not been systematically studied for all sites so many other stations may be affected by such effects. Topographic effects are considered to be part of variability of ground motions.

Exclude records with only a single horizontal component.

Exclude records from more than \(70\,\mathrm{km}\) (selected by visual inspection) to remove effects of bias in sample.

To complete missing information in the NGA database estimate strike, dip (\(\delta\)) and rake (\(\lambda\)) and/or depth to top of rupture, \(Z_{TOR}\), from other associated events (e.g. mainshock or other aftershock) or from tectonic environment. For events unassociated to other earthquake \(\delta\) assigned based on known or inferred mechanisms: \(90^{\circ}\) for strike-slip, \(40^{\circ}\) for reverse and \(55^{\circ}\) for normal. For events without known fault geometries \(R_{RUP}\) and \(R_{JB}\) estimated based on simulations of earthquake ruptures based on focal mechanisms, depths and epicentral locations.

Use \(M_w\) since simplest measure for correlating the amount of energy released in earthquake with ground motions. Develop functional form and constrain some coefficients for magnitude dependence based on theoretical arguments on source spectra and some previous analyses. Note that data are not sufficient to distinguish between various forms of magnitude-scaling.

Exploratory analysis indicates that reverse faulting earthquakes produce larger high-frequency motions than strike-slip events. It also shows that style-of-faulting effect is statistically significant (p-values slightly less than \(0.05\)) only when normal faulting was restricted to \(\lambda\) in range \(-120\) to \(60^{\circ}\) with normal-oblique in strike-slip class. Find style-of-faulting effect weaker for aftershocks than main shocks hence effect not included for aftershocks.

Preliminary analysis indicates statistically-significant dependence on depth to top of rupture, \(Z_{TOR}\) and that effect stronger for aftershocks therefore model different depth dependence for aftershocks and main shocks. Find that aftershocks produce lower motions than main shocks hence include this in model.

Examine various functional forms for distance-scaling and find all provide reasonable fits to data since to discriminate between them would require more data at distances \(<10\,\mathrm{km}\). Find that data shows magnitude-dependence in rate of attenuation at all distances but that at short distances due to effect of extended sources and large distances due to interaction of path \(Q\) with differences in source Fourier spectra as a function of magnitude. Choose functional form to allow for separation of effect of magnitude at small and large distances.

Examine distance-scaling at large distances using 666 records from 3 small S. Californian earthquakes (2001 Anza, \(M 4.92\); 2002 Yorba Linda, \(M 4.27\); 2003 Big Bear City, \(M 4.92\)) by fitting ground motions to three functional forms. Find that two-slope models fit slightly better than a one-slope model with break point between \(40\) and \(60\,\mathrm{km}\). Other data and simulations also show this behaviour. Prefer a smooth transition over broad distance range between two decay rates since transition point may vary from earthquake to earthquake. Constrain some coefficients based on previous studies.

Initially find that anelastic attenuation coefficient, \(\gamma\), is \(50\%\) larger for Taiwan than other areas. Believe this (and other similar effects) due to missing data due to truncation at lower amplitudes. Experiments with extended datasets for 21 events confirm this. Conclude that regression analyses using NGA data will tend to underestimate anelastic attenuation rate at large distances and that problem cannot be solved by truncated regression. Develop model for \(\gamma\) based on extended data sets for 13 Californian events.

To model hanging-wall effect, use \(R_X\), site coordinate (in \(\,\mathrm{km}\)) measured perpendicular to the fault strike from the surface projection of the updip edge of the fault rupture with the downdip direction being positive and \(F_{HW}\) (\(F_{HW}=1\) for \(R_X\geq 0\) and \(0\) for \(R_X<0\). Functional form developed based on simulations and empirical data.

Choose reference site \(V_{S30}\) to be \(1130\,\mathrm{m/s}\) because expected that no significant nonlinear site response at that velocity and very few records with \(V_{S30}>1100\,\mathrm{m/s}\) in NGA database. Functional form adopted for nonlinear site response able to present previous models from empirical and simulation studies.

Develop functional form for \(Z_{1.0}\)-dependence based on preliminary analyses and residual plots.

Model variability using random variables \(\eta_i\) (inter-event) and \(\epsilon_{ij}\) (intra-event). Assume inter-event residuals independent and normally distributed with variance \(\tau^2\). Assume intra-event error components independent and normally distributed with variances \(\sigma_P^2\) (path), \(\sigma_S^2\) (site) and \(\sigma_X^2\) (remaining). Assume total intra-event variance to be normally distributed with variance \(\sigma^2\). Show that \(\sigma^2\) is function of soil nonlinearity. Note that complete model difficult to use in regression analysis due to lack of repeatedly sampled paths and limited repeatedly sampled sites and unavailability of inference method capable of handling complicated data structure introduced by path error being included as predictor of soil amplification. Therefore apply simplification to solve problem.

Find inter-event residuals do not exhibit trend w.r.t. magnitude. Residuals for Californian and non-Californian earthquakes do not show any trends so both sets of earthquakes consistent with model. Note that inter-event term for Chi-Chi approximately \(2\tau\) below population mean.

Find intra-event residuals do not exhibit trends w.r.t. \(M\), \(R_{RUP}\), \(V_{S30}\) or \(y_{ref}\). Note that very limited data suggests slight upward trend in residuals for \(V_{S30}>1130\,\mathrm{m/s}\), which relate to lower kappa attenuation for such sites.

Preliminary analyses based on visual inspection of residuals suggested that standard errors did not depend on \(M\) but statistical analysis indicated that significant (p-values \(<0.05\)) magnitude dependence is present [using test of Youngs et al. (1995)]. Find that magnitude dependence remains even when accounting for differences in variance for aftershocks and main shocks and for nonlinear site amplification.

Note that in regions where earthquakes at distances \(>50\,\mathrm{km}\) are major contribution to hazard adjustments to \(c_{\gamma 1}\) and \(c_{\gamma 2}\) may be warranted.

Ground-motion model is: \[\log[\mathrm{PSA}(f)]=a(f)+b(f)M_w+c(f)M^2+d(f)R-\log_{10}[R+e(f)\times 10^{0.42M_w}]+S_i(f)\] where \(\mathrm{PSA}(f)\) is in \(\,\mathrm{m/s^2}\), \(a=-5.08210\), \(b=2.06210\), \(c=-0.11966\), \(d=-0.00319\), \(e=0.00488\), \(S=-0.01145\) and \(\sigma=0.32257\) for borehole stations (\(S\) applies for stations at \(200\,\mathrm{m}\)) and \(a=-4.884\), \(b=2.18080\), \(c=-0.12964\), \(d=-0.00397\), \(e=0.01226\), \(S_B=0.16101\), \(S_C=0.27345\), \(S_D=0.45195\) and \(\sigma=0.35325\) for surface stations.

Experiments on magnitude dependency of decay and \(\sigma\) reported below conducted using: \[\log_{10}[\mathrm{SA}_{i,j}(f)]=a(f)M_i+b(f)R_{\mathrm{rup},j}-\log_{10}(R_{\mathrm{rup},j})+S(f)\] Do not report coefficients of these models.

Use four site classes (based on Eurocode 8) for surface stations:

\(V_{s,30}>800\,\mathrm{m/s}\).

\(360<V_{s,30}<800\,\mathrm{m/s}\). Use coefficient \(S_B\).

\(180<V_{s,30}<360\,\mathrm{m/s}\). Use coefficient \(S_C\).

\(V_{s,30}<180\,\mathrm{m/s}\). Use coefficient \(S_D\).

Use data from boreholes to reduce influence of nonlinear site effects for investigating magnitude-dependent decay. Also derive models using surface records.

Only use data from \(<100\,\mathrm{km}\).

Only retain events with depth \(<25\,\mathrm{km}\) to exclude subduction earthquakes.

Note relatively good magnitude-distance coverage.

Visually inspect records to retain only main event if multiple events recorded and to check for glitches. Bandpass Butterworth (four poles and two passes) filter records with cut-offs \(0.25\) and \(25\,\mathrm{Hz}\). Longest usable period of model is less than \(3\,\mathrm{s}\) due to filtering.

Derive equations using data from small (\(M_w\leq 5\)) earthquakes (3376 records from 310 events) and large (\(M_w\geq 5\)) earthquakes (518 records from 27 events) to examine ability of models to predict ground motions outside their magnitude range of applicability. Find ground motions from small events attenuate faster than from large events. Predict ground motions for \(M_w\) \(4.0\), \(5.0\) and \(6.5\) and \(10\), \(30\) and \(99\,\mathrm{km}\). Find overestimation of ground motions for \(M_w 4.0\) using model derived using data from \(M_w\geq 5\) and overestimation of ground motions for \(M_w 6.5\) using model derived using data from \(M_w\leq 5\). Predictions for \(M_w 5.0\) are similar for both models. Also compare predictions from both models and observations for \(M_w 4.1\), \(4.6\), \(5.2\), \(5.7\), \(6.5\) and \(7.3\) and find similar results.

Also derive models for 11 magnitude ranges: \(4.0\)â\(4.2\), \(4.2\)â\(4.4\), \(4.4\)â\(4.6\), \(4.6\)â\(4.8\), \(4.8\)â\(5.0\), \(5.0\)â\(5.2\), \(5.2\)â\(5.4\), \(5.6\)â\(5.8\), \(5.8\)â\(6.8\) and \(6.8\)â\(7.3\). Compare predictions with observations for each magnitude range and find good match. Find that decay rate depends on \(M_w\) with faster decay for small events. Plot \(\sigma\)s from each model w.r.t. \(M_w\) and find that it has a negative correlation with \(M_w\).

Examine residuals w.r.t. distance. Find slight increase at large distances, which relate to magnitude dependency of attenuation.

Note that goal of analysis was not to compete with existing models but to compare magnitude dependency of ground motions at depth and surface.

Examine residuals w.r.t. distance and magnitude of final model. Find no trends.

Find that \(\sigma\)s for surface motions are larger (by about \(9\%\)) than those for motions at depth.

Ground-motion model is: \[\ln \mathrm{PGA}=C_0+C_1 M_w+C_2 \ln R+C_3 R+C_4 S\] where \(\mathrm{PGA}\) is in \(\,\mathrm{gal}\), \(C_0=0.192\), \(C_1=0.867\), \(C_2=-0.294\), \(C_3=-0.008\), \(C_4=0.113\) and \(\sigma=0.903\).

Use 2 site classes:

Class 1 (rock and hard alluvial with \(V_{s,30}>800\,\mathrm{m/s}\)) and class 2 (thin soft alluvial with \(500\leq V_{s,30} \leq 700\,\mathrm{m}\)). \(S=0\).

Class 3 (soft gravel and sandy sites with \(300 \leq V_{s,30} leq 500\,\mathrm{m/s}\)) and class 4 (soft soil with \(V_{s,30}<300\,\mathrm{m/s}\)). \(S=1\).

Use data of ZarÃ© and Bard (2002).

Select records with PGA of any component \(>0.05\,\mathrm{m/s^2}\).

Choose functional form because it is simple and hence avoids computational difficulties.

Use \(r_{epi}\) rather than \(r_{hypo}\) because of uncertainties in focal depth estimates and because almost all earthquakes have depths \(\leq 35\,\mathrm{km}\).

Originally use coefficients for each of the 4 site classes of ZarÃ© and Bard (2002) but find that the order of predicted PGAs is not as expected (e.g. PGAs on site class 4 are smaller than those on site class 2). Therefore, combine the original classes 1 and 2 together and classes 3 and 4 together.

Plot residuals w.r.t. predicted PGA and find no trend.

Ground-motion model is: \[\log(\mathrm{PGA})=aM+bR-\log(R)+c\] where \(\mathrm{PGA}\) is in \(\,\mathrm{cm/s^2}\), \(a=0.31\), \(b=-0.00091\), \(c=1.57\) and \(\sigma=0.23\).

Use data of Berge-Thierry et al. (2003).

Focal depths between \(0\) and \(30\,\mathrm{km}\).

Plot \(r_{hypo}\), epicentral location and \(M_s\) from ISC against those used by Berge-Thierry et al. (2003). Derive standard deviation, skewness and kurtosis based on these plots.

Account for estimated uncertainties of \(M\) and \(R\) in fuzzy regression and find same coefficients as standard regression but with estimated uncertainties and lower \(\sigma\) than in standard regression.

Find that epistemic uncertainties increase at edge of magnitude-distance space.

Ground-motion model is: \[\ln[\mathrm{PSA}(T)]=\alpha_1(T)+\alpha_2(T)M-[\beta_1(T)+\beta_2(T)M]\ln(R_{rup}+10)+\gamma(T)R_{rup}+\phi(T)F\] where \(\mathrm{PSA}\) is in \(\,\mathrm{g}\), \(\alpha_1=3.7066\) and \(\alpha_2=-0.1252\) for \(M\leq 6.75\), \(\alpha_1=5.6315\) and \(\alpha_2=-0.4104\) for \(6.75<M\leq 8.5\), \(\beta_1=2.9832\), \(\beta_2=-0.2339\), \(\gamma=0.00047\), \(\phi=0.12\) and \(\sigma=1.28+0.05\ln(T)-0.08M\). \(\sigma\) for \(M<5\) equals \(\sigma\) at \(M 5\) and \(\sigma\) for \(M>7.5\) equals \(\sigma\) at \(M 7.5\). \(\sigma\) for \(T<0.05\) equals \(\sigma\) for \(T=0.05\,\mathrm{s}\). Correction factor for \(V_{S30}>900\,\mathrm{m/s}\) \(\Delta \alpha_1(T)=\ln[(1+11T+0.27T^2)/(1+16T+0.08T^2)]\) for \(0.05\leq T \leq 10\,\mathrm{s}\) [\(\Delta \alpha_1(T)\) for \(T<0.05\,\mathrm{s}\) equals \(\Delta \alpha_1(0.05)\)].

Use two site classes (may derive model for \(180\leq V_{S30}<450\,\mathrm{m/s}\) in future):

\(V_{S30}>900\,\mathrm{m/s}\). 45 records. Since not enough records from stations with \(V_{S30}>900\,\mathrm{m/s}\) derive correction factor, \(\Delta \alpha_1(T)\), to \(\alpha_1\) based on residuals for these 45 records. Find no trends in residuals w.r.t. \(M\), \(R\) or \(V_{S30}\).

\(450\leq V_{S30}\leq 900\,\mathrm{m/s}\). 942 records (333 from stations with measured \(V_{S30}\)).

Notes that only 29% of stations have measured \(V_{S30}\); the rest have inferred \(V_{S30}\)s. Examine distributions of measured and inferred \(V_{S30}\)s and concluded no apparent bias by using inferred values of \(V_{S30}\).

Uses two mechanism categories:

Rake within \(30^{\circ}\) of horizontal. Includes records from normal events (rake within \(30^{\circ}\) of vertical downwards) because insufficient data to retain as separate category. \(F=0\).

Rake within \(30^{\circ}\) of vertical upwards. Includes records from reverse oblique and normal oblique events (remaining rake angles) because insufficient data to retain as separate categories. \(F=1\).

Uses the PEER Next Generation Attenuation (NGA) database (Flat-File version 7.2).

Excludes (to retain only free-field records): i) records from basements of any building; ii) records from dam crests, toes or abutments; and iii) records from first floor of buildings with \(\geq 3\) storeys.

Excludes records from âdeepâ events, records from distances \(>200\,\mathrm{km}\) and records from co-located stations.

Only retains records with \(450\leq V_{S30}\leq 900\,\mathrm{m/s}\) for regression. Notes that initial analysis indicated that ground motions not dependent on value of \(V_{S30}\) in this range so do not include a dependency on \(V_{S30}\).

Uses 187 records from California (42 events), 700 records from Taiwan (Chi-Chi, 152 records, and 5 aftershocks, 548 records) and 55 records from 24 events in other regions (USA outside California, Canada, Georgia, Greece, Iran, Italy, Mexico and Turkey).

Only 17 records from \(R\leq 5\,\mathrm{km}\) and 33 from \(R\leq 10\,\mathrm{km}\) (for \(M\leq 7\) only 3 records from California for these distance ranges) (all site classes). Therefore, difficult to constrain predictions at short distances, particularly for large magnitudes.

States that, from a geotechnical engineering perspective, use of \(V_{S30}\) bins is more appropriate than use of \(V_{S30}\) as an independent parameter.

Does not investigate the influence of other parameters within the NGA Flat-File on ground motions.

Uses \(\mathrm{PSA}\) at \(0.01\,\mathrm{s}\) for PGA (checked difference and generally less than \(2\%\)).

Divides data into magnitude bins \(0.5\) units wide and conducts one-stage regression analysis for each. Compares observed and predicted PGAs at distances of \(3\), \(10\), \(30\) and \(100\,\mathrm{km}\) against magnitude. Find that results for each magnitude bin generally well represent observations. Find oversaturation for large magnitudes due to presence of many records (152 out of 159 records for \(M>7.5\)) from Chi-Chi. Does not believe that this is justified so derive \(\alpha_1\) and \(\alpha_2\) for \(M>6.75\) by regression using the expected magnitude dependency based on previous studies and 1D simulations.

Examines residuals w.r.t. \(M\), \(R\) and \(V_{S30}\) and concludes that for \(5.2\leq M\leq 7.2\) model provides excellent representation of data. Examine residuals for 5 Chi-Chi aftershocks and find that for \(R>15\,\mathrm{km}\) there is no bias but for shorter distances some negative bias.

Compares predictions to observations for Hector Mine (\(M 7.1\)), Loma Prieta (\(M 6.9\)), Northridge (\(M 6.7\)) and San Fernando (\(M 6.6\)) events w.r.t. \(R\). Finds good match.

Comments on the insufficiency of \(V_{S30}\) as a parameter to characterise site response due to soil layering and nonlinear effects.

Ground-motion model is: \[\ln(y)=C_1+C_2M+C_3\ln(R+C_4\mathrm{e}^{C_5M})+C_6H+C_7Z_t\] where \(y\) is in \(\,\mathrm{g}\), \(C_1=-2.5\), \(C_2=1.205\), \(C_3=-1.905\), \(C_4=0.516\), \(C_5=0.6325\), \(C_6=0.0075\), \(C_7=0.275\) and \(\sigma=0.5268\) for rock sites and \(C_1=-0.9\), \(C_2=1.00\), \(C_3=-1.90\), \(C_4=0.9918\), \(C_5=0.5263\), \(C_6=0.004\), \(C_7=0.31\) and \(\sigma=0.6277\) for soil sites.

Use two site categories (separate equations for each):

B and C type sites

D and E type sites

Use two earthquake types:

Shallow angle thrust events occurring at interface between subducting and over-riding plates. Classified events using \(50\,\mathrm{km}\) maximum focal depth for interface events. 12 events from Taiwan (819 records) and 5 from elsewhere (54 records). \(Z_t=0\).

Typically high-angle normal-faulting events within the subducting oceanic plate. 32 events from Taiwan (3865 records) and 5 from elsewhere (85 records). \(Z_t=1\).

Focal depths, \(H\), between \(3.94\) and \(30\,\mathrm{km}\) (for interface) and \(43.39\) and \(161\,\mathrm{km}\) (for intraslab).

Develop separate \(M_L\)-\(M_w\) conversion formulae for deep (\(H>50\,\mathrm{km}\)) and shallow events.

Use data from TSMIP and the SMART-1 array.

Lack data from large Taiwanese earthquake (especially interface events). Therefore, add data from foreign subduction events (Mexico, western USA and New Zealand). Note that future study should examine suitability of adding these data.

Exclude poor-quality records by visual screening of available data. Baseline correct records.

Weight data given the number of records from different sources (Taiwan or elsewhere). Focus on data from foreign events since results using only Taiwanese data are not reliable for large magnitudes. Note that should use maximum-likelihood regression method.

Compare predicted and observed PGAs for the two best recorded events (interface \(M_w 6.3\) \(H=6\,\mathrm{km}\) and intraslab \(M_w 5.9\) \(H=39\,\mathrm{km}\)) and find good fit.

Examine residuals and find that a normal distribution fits them very well using histograms.

From limited analysis find evidence for magnitude-dependent \(\sigma\) but do not give details.

Note that some events could be mislocated but that due to large distances of most data this should not have big impact on results.

Ground-motion model is: \[\log_{10}(Y)=a+bM+c\log(R^2+h^2)^{1/2}+s_1 S_A+s_2 S_{(B+C)}\] where \(Y\) is in \(\,\mathrm{g}\); \(a=-2.66\), \(b=0.76\), \(c=-1.97\), \(d=10.72\), \(s_1=0\), \(s_2=0.13\), \(\sigma_{eve}=0.09\) (inter-event) and \(\sigma_{rec}=0.27\) (intra-event) for horizontal PGA and \(M_L\); \(a=-2.66\), \(b=0.76\), \(c=-1.97\), \(d=10.72\), \(s_1=0\), \(s_2=0.13\), \(\sigma_{sta}=0.09\) (inter-site) and \(\sigma_{rec}=0.28\) (intra-site) for horizontal PGA and \(M_L\); \(a=-2.59\), \(b=0.69\), \(c=-1.95\), \(d=11.16\), \(s_1=0\), \(s_2=0.12\), \(\sigma_{eve}=0.09\) (inter-event) and \(\sigma_{rec}=0.26\) (intra-event) for vertical PGA and \(M_L\); \(a=-2.59\), \(b=0.69\), \(c=-1.95\), \(d=11.16\), \(s_1=0\), \(s_2=0.12\), \(\sigma_{eve}=0.08\) (inter-site) and \(\sigma_{rec}=0.26\) (intra-site) for vertical PGA and \(M_L\); \(a=-3.62\), \(b=0.93\), \(c=-2.02\), \(d=11.71\), \(s_1=0\), \(s_2=0.12\), \(\sigma_{eve}=0.10\) (inter-event) and \(\sigma_{rec}=0.28\) (intra-event) for horizontal PGA and \(M_w\); \(a=-3.62\), \(b=0.93\), \(c=-2.02\), \(d=11.71\), \(s_1=0\), \(s_2=0.12\), \(\sigma_{sta}=0.11\) (inter-site) and \(\sigma_{rec}=0.29\) (intra-site) for horizontal PGA and \(M_w\); \(a=-3.49\), \(b=0.85\), \(c=-1.99\), \(d=11.56\), \(s_1=0\), \(s_2=0.11\), \(\sigma_{eve}=0.09\) (inter-event) and \(\sigma_{rec}=0.29\) (intra-event) for vertical PGA and \(M_w\); \(a=-3.49\), \(b=0.85\), \(c=-1.99\), \(d=11.56\), \(s_1=0\), \(s_2=0.11\), \(\sigma_{eve}=0.12\) (inter-site) and \(\sigma_{rec}=0.30\) (intra-site) for vertical PGA and \(M_w\).

Also use functional form: \(\log_{10}(Y)=a+bM+(c+eM)\log(R^2+h^2)^{1/2}+s_1 S_A+s_2 S_{(B+C)}\) but do not report coefficients since find small values for \(e\).

Use three site classifications based on Eurocode 8 for the 77 stations:

Rock, \(V_{s,30}>800\,\mathrm{m/s}\): marine clay or other rocks (Lower Pleistocene and Pliocene) and volcanic rock and deposits. 49 stations. \(S_A=1\) and \(S_{(B+C)}=0\).

Stiff soil, \(360<V_{s,30}<800\,\mathrm{m/s}\): colluvial, alluvial, lacustrine, beach, fluvial terraces, glacial deposits and clay (Middle-Upper Pleistocene); sand and loose conglomerate (Pleistocene and Pliocene); and travertine (Pleistocene and Holocene). 19 stations. \(S_{(B+C)}=1\) and \(S_A=0\).

Soft soil, \(V_s<360\,\mathrm{m/s}\): colluvial, alluvial, lacustrine, beach and fluvial terraces deposits (Holocene). 9 stations. \(S_{(B+C)}=1\) and \(S_A=0\).

Because of limited records from class C combine classes B and C in regression. Note that the classification of some stations in class A could not be appropriate due to site amplification due to structure-soil interaction and topographic effects. Also note that class C is not appropriate for some stations on Po Plain due to deep sediments but that there are few data from these sites so no bias.

Use data from various analogue and digital strong-motion (Episensor, K2, Etna, SSA-1 or SMA-1 instruments) and digital velocimetric (Mars-Lite, Mars88-MC, Reftek 130 or other instruments) networks in northern Italy, western Slovenia and southern Switzerland.

Originally collect about 10 000 records but reduce by careful selection. Exclude data with \(d_e>100\,\mathrm{km}\) and with \(M_L<3.5\). Consider earthquakes down to \(M_L 3.5\) because such earthquakes could damage sensitive equipment in industrial zones.

216 components (both horizontal and vertical combined) from earthquakes with \(M_L>4.5\).

Focal depths between \(1.9\) and \(57.9\,\mathrm{km}\). Most less than \(15\,\mathrm{km}\).

Bandpass filter using fourth-order acausal Butterworth filter with cut-offs of \(0.4\) and \(25\,\mathrm{Hz}\) for \(M_L\leq 4.5\) and \(0.2\) and \(25\,\mathrm{Hz}\) for \(M_L>4.5\). Check using some records that PGA is not affected by filtering nor are spectral accelerations in the period range of interest. Check filtering of analogue records by visually examining Fourier amplitude spectra. Check conversion of velocimetric records to acceleration is correct by examining records from co-located instruments of different types. Exclude clipped records or records affected by noise.

Try including a quadratic magnitude term but find that the coefficient is not statistically significant.

Try including an anelastic attenuation term but find that coefficient is not statistically significant.

Do not use \(r_{jb}\) since not sufficient information on rupture locations. Do not use \(r_{hypo}\) so as not to introduce errors due to unreliable focal depths.

Do not include style-of-faulting terms because most data from reverse-faulting earthquakes (often with strike-slip component).

Apply simple tests to check regional dependence and do not find significant evidence for regional differences in ground motions. Since records from similar earthquakes of similar mechanisms conclude that models appropriate for whole of northern Italy (\(6^{\circ}\)â\(15^{\circ}\)E and \(43^{\circ}\)â\(47^{\circ}\)N).

Examine residuals (against earthquake and station indices, as box and whisker plots and against distance and magnitude) for sites A and sites B & C and for \(M_L\leq 4.5\) and \(M_L>4.5\). Also compare predicted and observed ground motions for various magnitudes and events. Find good results.

Suggest that for \(d_e<10\,\mathrm{km}\) and \(M_L>5.5\) \(10\,\mathrm{km}\) is considered the distance at which distance saturation starts (since little data with \(d_e<10\,\mathrm{km}\) to constrain curves and predictions for shorter distances unrealistically high).

Also derive equations for other strong-motion intensity parameters.

Ground-motion model is: \[\ln Y=C_1+C_2 M+C_3 \ln R\] where \(Y\) is in \(\,\mathrm{cm/s^2}\), \(C_1=0.125\), \(C_2=1.286\), \(C_3=-1.133\) and \(\sigma=0.69\). Only derive equation for firm soil sites due to insufficient data for other classes. For compact rock sites propose using ratio between PGA on firm soil and rock derived by Campbell (1997).

Use three site classifications:

Compact rock. Crystalline rocks (granite and basalt), metamorphic rocks (e.g. marble, gneiss, schist and quartzite) and Cretaceous and older sedimentary deposits following criteria of Campbell (1997). Similar to Spanish building code classes I and II with \(400\leq V_s \leq 750\,\mathrm{m/s}\). 23 stations.

Alluvium or firm soil. Quaternary consolidated deposits. Similar to Spanish building code class III with \(200\leq V_s \leq 400\,\mathrm{m/s}\). 29 stations.

Soft sedimentary deposits. 52 stations.

Classify using crude qualitative descriptions.

Most stations in basements of small buildings (e.g. city council offices) and therefore records are not truly free-field.

Only consider data with \(5\leq d_e \leq 100\,\mathrm{km}\) and \(M\geq 3\).

Focal depths between \(1\) and \(16\,\mathrm{km}\).

Most data from \(3\leq M \leq 4\) and \(d_e \leq 50\,\mathrm{km}\). Only one record with \(M>5\) and \(d_e<20\,\mathrm{km}\).

Use hypocentral distance because no information on locations of rupture planes and since using hypocentral distance automatically limits near-source ground motions.

Do not consider style-of-faulting since no reported mechanisms are available for most events.

Compare predicted PGA for \(M_w 5\) with observations for \(4.9 \leq M_w \leq 5.1\). Find reasonable fit.

Ground-motion model is: \[\log_{10} Y=a+bM+c\log_{10}R+s_{1,2}\] where \(Y\) is in \(\,\mathrm{g}\), \(a=-4.417\), \(b=0.770\), \(c=-1.097\), \(D=0\), \(D_1=0.123\), \(\sigma_{\mathrm{eve}}=0.069\) and \(\sigma_{\mathrm{rec}}=0.339\) for horizontal PGA and intra-event sigma; \(a=-4.128\), \(b=0.722\), \(c=-1.250\), \(D=0\), \(D_1=0.096\), \(\sigma_{\mathrm{eve}}=0.085\) and \(\sigma_{\mathrm{rec}}=0.338\) for vertical PGA and intra-event sigma; \(a=-4.367\), \(b=0.774\), \(c=-1.146\), \(D=0\), \(D_1=0.119\), \(\sigma_{\mathrm{sta}}=0.077\) and \(\sigma_{\mathrm{rec}}=0.337\) for horizontal PGA and intra-station sigma; and \(a=-4.066\), \(b=0.729\), \(c=-1.322\), \(D=0\), \(D_1=0.090\), \(\sigma_{\mathrm{sta}}=0.105\) and \(\sigma_{\mathrm{rec}}=0.335\).

Use two site categories (\(s_{1,2}\)) because insufficient information to use more:

Rock. Average \(V_s>800\,\mathrm{m/s}\). 10 stations.

Soil. Average \(V_s<800\,\mathrm{m/s}\). Includes all kinds of superficial deposits, from weak rocks to alluvial deposits although they are mainly shallow alluvium and soft rock (\(600\)-\(700\,\mathrm{m/s}\)) sites. 27 stations.

Use data from the 2002â2003 Molise sequence from various agencies.

Use data from accelerometers (SMA-1, 3 stations; RFT-250, 2 stations; Episensor, 10 stations) and velocimeters (CMG-40T, 4 stations; Lennartz \(1\,\mathrm{s}\), 5 stations; Lennartz \(5\,\mathrm{s}\), 13 stations).

Select data with \(M>2.7\).

Baseline and instrument correct records from analogue accelerometric instruments and filter in average bandpass \(0.5\)â\(20\,\mathrm{Hz}\) after visual inspection of the Fourier amplitude spectra. Baseline correct records from digital accelerometric instruments and filter in average bandpass \(0.2\)â\(30\,\mathrm{Hz}\) after visual inspection of the Fourier amplitude spectra. Instrument correct records from digital velocimetric instruments and filter in average bandpass \(0.5\)â\(25\,\mathrm{Hz}\) after visual inspection of the Fourier amplitude spectra.

Most data from \(r_{hypo}<40\,\mathrm{km}\) and almost all velocimetric data from \(20\)-\(30\,\mathrm{km}\).

Most focal depths between \(10\) and \(30\,\mathrm{km}\).

Relocate events using manual picks of P and S phases and a local velocity model.

Compute \(M_L\)s using velocimetric data.

Note that small value of \(\sigma_{\mathrm{eve}}\) suggests that the calibrated local magnitudes and relocated hypocentral locations are accurate.

Note that small value of \(\sigma_{\mathrm{sta}}\) suggests that the site classification is correct.

Note that records from accelerometric and velocimetric instruments are similar.

Ground-motion model is: \[\begin{aligned} \log_{10} PGA&=&a+(b+cM_s)M_s+(d+eM_s)\log_{10}r\\ \mbox{where} \quad r^2&=&D^2+h^2\end{aligned}\] where \(PGA\) is in \(\,\mathrm{g}\), \(a=-2.14\), \(b=0.98\), \(c=-0.06\), \(d=-1.88\), \(e=0.0009\), \(h=13.4\) and \(\sigma=0.35\).

Only use data for \(d_e<100\,\mathrm{km}\) because data from larger distances only available for large earthquakes.

Only eight records have PGA\(<0.005\,\mathrm{g}\) (standard trigger level).

Use truncated regression analysis (Bragato 2004) to account for bias due to non-triggering stations.

Ground-motion model is: \[\log(A)=c_1+c_2M-b \log(X+\mathrm{e}^{c_3M})\] where \(A\) is in \(\,\mathrm{cm/s^2}\), \(c_1=-1.3489\), \(c_2=1.0095\), \(b=0.1956\), \(c_3=0.1272\) and \(\sigma=0.20\).

Use data from one station.

Data from rockbursts in mines in the Kolar Gold Fields.

Exclude records with \(r_{hypo}<1\,\mathrm{km}\) due to large change in PGAs in near-source region.

Regress data using \(\log(A)=-b\log(X)+c\) for data binned in 5 \(0.2\) magnitude unit bins from \(2.0\) upwards.

Also regress data using \(\log(A)=aM-b\log(X)+c\).

Also regress using \(\log(A)=c_1+c_2M-b c_4 \log(X+\mathrm{e}^{c_3M})\) (sic) but find \(c_4\) has a very large standard error so remove it.

Compare predictions and observations for \(M 2.1\), \(2.3\), \(2.5\), \(2.7\) and \(2.9\).

Ground-motion model is: \[\ln Y=C_1+C_2M+C_3M^C_4+C_5\ln[R+C_6\exp(C_7M)]+C_8H\] where \(Y\) is in \(\,\mathrm{gal}\), \(C_1=-0.469151\), \(C_2=7.108251\times 10^{-4}\), \(C_3=0.456626\), \(C_4=-0.032769\), \(C_5=2.122059\times 10^{-3}\), \(C_6=235088.506\), \(C_7=0.664657\) and \(C_8=-2.860212 \times 10^{-7}\) (\(\sigma\) is not given).

Focal depths \(H\) between \(16.2\) and \(576\,\mathrm{km}\).

Data from about 11 stations of Malaysia Meteorological Department.

Also derive various models using worldwide data.

Records from subduction earthquakes on Sumatran arc: 3 are deep events (\(289.2\leq H \leq 576\,\mathrm{km}\)) and 11 are shallow (\(16.2\leq H \leq 35\,\mathrm{km}\)).

Plot residuals w.r.t. \(r_{epi}\), \(M\) and \(H\).

Ground-motion model is: \[\begin{aligned} \ln y&=&c_1+f_1(M_w)+f_2(M_w)f_3(R)+f_4(F)+\mathrm{FR}f_5(Z_{FR})+\\ &&\mathrm{FS}f_6(Z_{FR})+f_7(\mathrm{HW},R_{JB},M_w,\mathrm{DIP})+\\ &&f_8(V_{s,30},V_{lin},\mathrm{PGA}_{non-lin},\mathrm{PGA}_{rock})+f_9(V_{s,30},Z_{1.5})\\ \mbox{where for} \quad M_w\leq c_0&&\\ f_1(M_w)&=&c_3(M_w-c_0)+c_8(T)(8.5-M_w)^n\\ f_2(M_w)&=&c_2(T)+c_4(M_w-c_0)\\ \mbox{and for} \quad M_w>c_0&&\\ f_1(M_w)&=&c_5 (M_w-c_0)+c_8(T)(8.5-M_w)^n\\ f_2(M_w)&=&c_2(T)+c_6(M_w-c_0)\\ f_3(R)&=&\ln\sqrt{R_{rup}^2+c_7(T)^2}\\ f_4(F)&=&c_9(T)\mathrm{FR}+c_{10}(T