- Ground-motion model is:
_{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, σ_{ln Y }= 0.478 (intra-event), τ_{ln Y }= 0.219 (inter-event), σ_{C}= 0.166, σ_{T }= 0.526 (total), σ_{Arb}= 0.551 and ρ = 1.000 (correlation coefficient between intra-event residuals of ground-motion parameter of interest and PGA). σ_{ln Y B}= (σ_{ln Y }^{2}-σ_{ln AF }^{2})^{1∕2}is standard deviation at base of site profile. Assume that σ_{ln AF }≈ 0.3 based on previous studies for deep soil sites. σ_{Arb}= 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}= 1100m∕s) in g. Linear part of f_{site}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, 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}= 150–1500m∕s. - Also use depth to 2.5km∕s shear-wave velocity horizon (basin or sediment depth) in 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}= 0–10km. - Use three faulting mechanism categories based on rake angle, λ:
- RV
- Reverse and reverse-oblique. 30 < λ < 150
^{∘}. 17 earthquakes. F_{RV }= 1 and F_{NM}= 0. - NM
- Normal and normal-oblique. -150 < λ < -30
^{∘}. 11 earthquakes. F_{NM}= 1 and F_{RV }= 0. - SS
- 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 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, R_{RUP }> 60km and N < 2 (retain singly-recorded earthquakes with M ≥ 7.0 and R_{RUP }≤ 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 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}= 0–15km. - 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, τ
_{ln 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 σ_{T }because intra-event σ dominates. σ takes into account soil nonlinearity effects. Assume that σ_{ln Y }and σ_{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 ≤ 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 100km 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,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.