Ground motion prediction equations 1964–2019

John Douglas
Department of Civil and Environmental Engineering
University of Strathclyde
James Weir Building
75 Montrose Street
Glasgow
G1 1XJ
United Kingdom
john.douglas@strath.ac.uk
https://www.strath.ac.uk/staff/douglasjohndr/

Synopsis

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(2008), concerning GMPEs published in 2007 and 2008 (plus some earlier models); and the report of Douglas (2011), which superseded all these reports and covered the period up to 2010.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 1964–2019, http://www.gmpe.org.uk.

Introduction

ESEE Report 01-1 ‘A comprehensive worldwide summary of strong-motion attenuation relationships for peak ground acceleration and spectral ordinates (1969 to 2000)’ (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 2004–2006 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 LaTeXTeX 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 models1 (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 — 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.

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

  2. 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).

  3. 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.

  4. The model has been superseded by a more recent publication.

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

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

  7. 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.

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

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

  10. 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: ‘[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.’

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).

Other summaries and reviews of GMPEs

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 available2. For example: Bommer and Martı́nez-Pereira (1999), Alarcón (2007) and Bommer, Stafford, and Alarcón (2009) review predictive equations for strong-motion duration; Tromans (2004) summarizes equations for the prediction of PGV and displacement (PGD); Bommer and Alarcón (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.

GMPEs summarised here

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 ’source energy logarithms’ 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 Chapter 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. Güllü and Erçelebi 2007) are only listed.

GMPEs for the prediction of PGA are summarized in Chapters 2 and 3 and those for spectral ordinates are summarized in Chapters 4 and 5. Chapter 6 lists other ground-motion models that are not detailed in the previous chapters. The final chapter (Chapter 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 (Illustrations [tab:pga] & [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 (Chapters 4 and 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 ‘Note that …’ should be read ‘The authors [of the original model] note that …’. 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 missed3. 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 (2000–2019), 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 Tables [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 ‘attenuation relation(ships)’ is used for the models reported. The current de facto standard is to refer to such models as ‘ground motion prediction equations’ (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. ‘ground-motion model’ or ‘ground motion estimation equations’) and some people disagree with the use of the word ‘prediction’ 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 — the reader is encouraged to consult the original reference if a model is to be used.

Summary of published GMPEs for PGA

Esteva and Rosenblueth (1964)

Kanai (1966)

Milne and Davenport (1969)

Esteva (1970)

Denham and Small (1971)

Davenport (1972)

Denham, Small, and Everingham (1973)

Donovan (1973)

Esteva and Villaverde (1973) & Esteva (1974)

Katayama (1974)

McGuire (1974) & McGuire (1977)

Orphal and Lahoud (1974)

Ahorner and Rosenhauer (1975)

N. N. Ambraseys (1975), N. Ambraseys (1975) & N. N. Ambraseys (1978a)

Shah and Movassate (1975)

Trifunac and Brady (1975a), Trifunac (1976a) & Trifunac and Brady (1976)

Blume (1977)

Milne (1977)

Saeki, Katayama, and Iwasaki (1977)

N. N. Ambraseys (1978b)

Donovan and Bornstein (1978)

Faccioli (1978)

Goto et al. (1978)

R. K. McGuire (1978b)

A. Patwardhan, K. Sadigh, I.M. Idriss, R. Youngs (1978) reported in Idriss (1978)

Cornell, Banon, and Shakal (1979)

Faccioli (1979)

Faccioli and Agalbato (1979)

Aptikaev and Kopnichev (1980)

Blume (1980)

Iwasaki, Kawashima, and Saeki (1980)

Matuschka (1980)

Ohsaki, Watabe, and Tohdo (1980)

TERA Corporation (1980)

Campbell (1981)

Chiaruttini and Siro (1981)

Goto, Kameda, and Sugito (1981)

Joyner and Boore (1981)

Bolt and Abrahamson (1982)

Joyner and Boore (1982b) & Joyner and Boore (1988)

PML (1982)

Schenk (1982)

Brillinger and Preisler (1984)

Campbell (1984) & K.W. Campbell (1988) reported in Joyner and Boore (1988)

Joyner and Fumal (1984), Joyner and Fumal (1985) & Joyner and Boore (1988)

Kawashima, Aizawa, and Takahashi (1984) & Kawashima, Aizawa, and Takahashi (1986)

McCann Jr. and Echezwia (1984)

Schenk (1984)

Xu, Shen, and Hong (1984)

Brillinger and Preisler (1985)

Kawashima, Aizawa, and Takahashi (1985)

Makropoulos and Burton (1985) & Makropoulos (1978)

K.-Z. Peng, Wu, and Song (1985)

K. Peng et al. (1985)

PML (1985)

McCue (1986)

C.B. Crouse (1987) reported in Joyner and Boore (1988)

Krinitzsky, Chang, and Nuttli (1987) & Krinitzsky, Chang, and Nuttli (1988)

Sabetta and Pugliese (1987)

K. Sadigh (1987) reported in Joyner and Boore (1988)

Singh et al. (1987)

Algermissen, Hansen, and Thenhaus (1988)

Annaka and Nozawa (1988)

Fukushima, Tanaka, and Kataoka (1988) & Fukushima and Tanaka (1990)

Gaull (1988)

McCue, Gibson, and Wesson (1988)

Petrovski and Marcellini (1988)

PML (1988)

Tong and Katayama (1988)

Yamabe and Kanai (1988)

Youngs, Day, and Stevens (1988)

Abrahamson and Litehiser (1989)

Campbell (1989)

Huo (1989)

Ordaz, Jara, and Singh (1989)

Alfaro, Kiremidjian, and White (1990)

Ambraseys (1990)

Campbell (1990)

Dahle, Bungum, and Kvamme (1990) & Dahle, Bugum, and Kvamme (1990)

Jacob et al. (1990)

Sen (1990)

Sigbjörnsson (1990)

Tsai, Brady, and Cluff (1990)

Ambraseys and Bommer (1991) & N. N. Ambraseys and Bommer (1992)

Crouse (1991)

Garcı̀a-Fernàndez and Canas (1991) & Garcia-Fernandez and Canas (1995)

Geomatrix Consultants (1991), Sadigh et al. (1993) & Sadigh et al. (1997)

Huo and Hu (1991)

I.M. Idriss (1991) reported in Idriss (1993)

Loh et al. (1991)

Matuschka and Davis (1991)

Niazi and Bozorgnia (1991)

Rogers et al. (1991)

Stamatovska and Petrovski (1991)

Abrahamson and Youngs (1992)

N. N. Ambraseys, Bommer, and Sarma (1992)

J. Huo and Hu (1992)

Kamiyama, O’Rourke, and Flores-Berrones (1992) & Kamiyama (1995)

Sigbjörnsson and Baldvinsson (1992)

Silva and Abrahamson (1992)

Taylor Castillo et al. (1992)

Tento, Franceschina, and Marcellini (1992)

Theodulidis and Papazachos (1992)

Abrahamson and Silva (1993)

Boore, Joyner, and Fumal (1993), Boore, Joyner, and Fumal (1997) & Boore (2005)

Campbell (1993)

Dowrick and Sritharan (1993)

Gitterman, Zaslavsky, and Shapira (1993)

McVerry et al. (1993) & McVerry, Dowrick, and Zhao (1995)

Midorikawa (1993a)

Quijada et al. (1993)

Singh et al. (1993)

Steinberg et al. (1993)

Sun and Peng (1993)

Ambraseys and Srbulov (1994)

Boore, Joyner, and Fumal (1994a) & Boore, Joyner, and Fumal (1997)

El Hassan (1994)

Fat-Helbary and Ohta (1994)

Fukushima, Gariel, and Tanaka (1994) & Fukushima, Gariel, and Tanaka (1995)

Lawson and Krawinkler (1994)

Lungu et al. (1994)

Musson, Marrow, and Winter (1994)

Radu et al. (1994), Lungu, Coman, and Moldoveanu (1995) & Lungu et al. (1996)

Ramazi and Schenk (1994)

Xiang and Gao (1994)

Aman, Singh, and Singh (1995)

N. N. Ambraseys (1995)

Dahle et al. (1995)

V. W. Lee, Trifunac, Todorovska, et al. (1995)

Lungu et al. (1995)

Molas and Yamazaki (1995)

Sarma and Free (1995)

N. N. Ambraseys, Simpson, and Bommer (1996) & Simpson (1996)

N. N. Ambraseys and Simpson (1996) & Simpson (1996)

Aydan, Sedaki, and Yarar (1996) & Aydan (2001)

Bommer et al. (1996)

Crouse and McGuire (1996)

Free (1996) & Free, Ambraseys, and Sarma (1998)

Inan et al. (1996)

Ohno et al. (1996)

Romeo, Tranfaglia, and Castenetto (1996)

Sarma and Srbulov (1996)

Singh, Aman, and Prasad (1996)

Spudich et al. (1996) & Spudich et al. (1997)

Stamatovska and Petrovski (1996)

Ansal (1997)

Campbell (1997), Campbell (2000), Campbell (2001) & Campbell and Bozorgnia (1994)

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.

Munson and Thurber (1997)

Pancha and Taber (1997)

Rhoades (1997)

Schmidt, Dahle, and Bungum (1997)

Youngs et al. (1997)

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.

Zhao, Dowrick, and McVerry (1997)

Baag et al. (1998)

Bouhadad et al. (1998)

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Bommer et al. (2007)

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