- Ground-motion model is: where Y is in g, c0 = -1.715, c1 = 0.500, c2 = -0.530, c3 = -0.262, c4 = -2.118, c5 = 0.170,
c6 = 5.60, c7 = 0.280, c8 = -0.120, c9 = 0.490, c10 = 1.058, c11 = 0.040, c12 = 0.610, k1 = 865,
k2 = -1.186, k3 = 1.839, σln Y = 0.478 (intra-event), τln Y = 0.219 (inter-event), σC = 0.166, σT = 0.526
(total), σArb = 0.551 and ρ = 1.000 (correlation coefficient between intra-event residuals of ground-motion
parameter of interest and PGA). σln Y B = (σln Y 2 -σln AF 2)1∕2 is standard deviation at base of site profile.
Assume that σln AF ≈ 0.3 based on previous studies for deep soil sites. σArb = for estimating
aleatory uncertainty of arbitrary horizontal component.
- Characterise sites using V S30. Account for nonlinear effects using A1100, median estimated PGA on
reference rock outcrop (V S30 = 1100m∕s) in g. Linear part of fsite is consistent with previous studies
but with constraint for constant site term for V S30 > 1100m∕s (based on residual analysis) even though
limited data for V S30 > 1100m∕s. When only including linear part of shallow site response term find
residuals clearly exhibit bias when plotted against rock PGA, A1100. 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 = 150–1500m∕s.
- Also use depth to 2.5km∕s shear-wave velocity horizon (basin or sediment depth) in km, Z2.5. 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 Z2.5. Provide relationships for predicting Z2.5 based on other site parameters. Believe model
applicable for Z2.5 = 0–10km.
- Use three faulting mechanism categories based on rake angle, λ:
- Reverse and reverse-oblique. 30 < λ < 150∘. 17 earthquakes. FRV = 1 and FNM = 0.
- Normal and normal-oblique. -150 < λ < -30∘. 11 earthquakes. FNM = 1 and FRV = 0.
- Strike-slip. All other rake angles. 36 earthquakes. FRV = 0 and FNM = 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
- 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 Düzce 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 ≤ M < 6.0 and N < 3, c) 6.0 ≤ M < 7.0,
RRUP > 60km and N < 2 (retain singly-recorded earthquakes with M ≥ 7.0 and RRUP ≤ 60km 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
- 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
- Include depth to top of coseismic rupture plane, ZTOR, 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 ZTOR can be used for strike-slip events. Believe model applicable for ZTOR = 0–15km.
- Include dip of rupture plane, δ. Believe model applicable for δ = 15–90∘.
- Assume that τ is approximately equal to standard deviation of inter-event residuals, τln Y , since inter-event
terms are not significantly affected by soil nonlinearity. Note that if τ was subject to soil nonlinearity
effects it would have only a relatively small effect on σT because intra-event σ dominates. σ takes into
account soil nonlinearity effects. Assume that σln Y and σln 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 fmag. 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 ≤ 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 fmag to saturate at M > 6.5 and RRUP = 0 when oversaturation predicted by
unconstrained regression analysis. Constraint equivalent to setting c3 = -c1 - c2 - c5 ln(c6). 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
- 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 100km and with simple theoretical constraints. Examine intra-event residuals w.r.t.
distance and find that they are relatively unbiased.
- Functional form for fflt 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 fhng 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 ZTOR. Include fhng,M, fhng,Z and fhng,δ 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.