- Ground-motion model is: where y is in g, c2 = 1.06, c3 = 3.45, c4 = -2.1, c4a = -0.5, cRB = 50, cHM = 3, cγ3 = 4, c1 = -1.2687,
c1a = 0.1, c1b = -0.2550, cn = 2.996, cM = 4.1840, c5 = 6.1600, c6 = 0.4893, c7 = 0.0512, c7a = 0.0860,
c9 = 0.7900, c9a = 1.5005, c10 = -0.3218, cγ1 = -0.00804, cγ2 = -0.00785, ϕ1 = -0.4417, ϕ2 = -0.1417,
ϕ3 = -0.007010, ϕ4 = 0.102151, ϕ5 = 0.2289, ϕ6 = 0.014996, ϕ7 = 580.0, ϕ8 = 0.0700, τ1 = 0.3437,
τ2 = 0.2637, σ1 = 0.4458, σ2 = 0.3459, σ3 = 0.8 and σ4 = 0.0663 (η is the inter-event residual). σT 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. σNL0 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. FInferred = 1 if V S30 inferred from geology and 0 otherwise. FMeasured = 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, Z1.0, 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 Z1.0 use this empirical relationship:
ln(Z1.0) = 28.5 - ln(V S308 + 378.78).
- 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 ZTOR ≤ 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,
- 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
- 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, ZTOR, 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 RRUP and RJB estimated based on simulations of earthquake ruptures based on focal
mechanisms, depths and epicentral locations.
- Use Mw 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, ZTOR 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 RX, 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 FHW (FHW = 1 for RX ≥ 0 and 0 for RX < 0. Functional form developed based on simulations and
- 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 Z1.0-dependence based on preliminary analyses and residual plots.
- Model variability using random variables ηi (inter-event) and ϵij (intra-event). Assume inter-event residuals
independent and normally distributed with variance τ2. Assume intra-event error components independent
and normally distributed with variances σP 2 (path), σS2 (site) and σX2 (remaining). Assume total
intra-event variance to be normally distributed with variance σ2. Show that σ2 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, RRUP , V S30 or yref. Note that very limited data
suggests slight upward trend in residuals for V S30 > 1130m∕s, which relate to lower kappa attenuation for
- 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 cγ1 and cγ2 may be warranted.