- Ground-motion model is:
_{2}= 1.06, c_{3}= 3.45, c_{4}= -2.1, c_{4a}= -0.5, c_{RB}= 50, c_{HM}= 3, c_{γ3}= 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_{γ1}= -0.00804, c_{γ2}= -0.00785, ϕ_{1}= -0.4417, ϕ_{2}= -0.1417, ϕ_{3}= -0.007010, ϕ_{4}= 0.102151, ϕ_{5}= 0.2289, ϕ_{6}= 0.014996, ϕ_{7}= 580.0, ϕ_{8}= 0.0700, τ_{1}= 0.3437, τ_{2}= 0.2637, σ_{1}= 0.4458, σ_{2}= 0.3459, σ_{3}= 0.8 and σ_{4}= 0.0663 (η is the inter-event residual). σ_{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. σ_{NL0}is the equation for σ evaluated for η = 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 ≤ V_{S30}≤ 1500m∕s. - Use depth to shear-wave velocity of 1.0km∕s, Z
_{1.0}, to model effect of near-surface sediments since 1km∕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 - 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 20km and Z
_{TOR}≤ 15km. Therefore note that application to regions with very thick crusts (e.g. ≫ 20km) 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 (≥ 20km) 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. 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 70km (selected by visual inspection) to remove effects of bias in sample.
- To complete missing information in the NGA database estimate strike, dip (δ) and rake (λ) 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 δ assigned based on known or inferred mechanisms: 90^{∘}for strike-slip, 40^{∘}for reverse and 55^{∘}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 λ in range -120 to 60
^{∘}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 < 10km. 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, M4.92; 2002 Yorba Linda, M4.27; 2003 Big Bear City, M4.92) 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 60km. 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, γ, 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 γ based on extended data sets for 13 Californian events.
- To model hanging-wall effect, use R
_{X}, site coordinate (in 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}≥ 0 and 0 for R_{X}< 0. Functional form developed based on simulations and empirical data. - Choose reference site V
_{S30}to be 1130m∕s because expected that no significant nonlinear site response at that velocity and very few records with V_{S30}> 1100m∕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 η
_{i}(inter-event) and ϵ_{ij}(intra-event). Assume inter-event residuals independent and normally distributed with variance τ^{2}. Assume intra-event error components independent and normally distributed with variances σ_{P }^{2}(path), σ_{S}^{2}(site) and σ_{X}^{2}(remaining). Assume total intra-event variance to be normally distributed with variance σ^{2}. Show that σ^{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 2τ 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}> 1130m∕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 > 50km are major contribution to hazard adjustments
to c
_{γ1}and c_{γ2}may be warranted.