- Ground-motion model is:
_{h}= 6.75 (hinge magnitude), V_{ref}= 760m∕s (specified reference velocity corresponding to the NEHRP B/C boundary), a_{1}= 0.03g (threshold for linear amplifcation), a_{2}= 0.09g (threshold for nonlinear amplification), pga_low = 0.06g (for transition between linear and nonlinear behaviour), pga4nl is predicted PGA in g for V_{ref}with F_{S}= 0, V_{1}= 180m∕s, V_{2}= 300m∕s, b_{lin}= -0.360, b_{1}= -0.640, b_{2}= -0.14, M_{ref}= 4.5, R_{ref}= 1km, c_{1}= -0.66050, c_{2}= 0.11970, c_{3}= -0.01151, h = 1.35, e_{1}= -0.53804, e_{2}= -0.50350, e_{3}= -0.75472, e_{4}= -0.50970, e_{5}= 0.28805, e_{6}= -0.10164, e_{7}= 0.0; σ = 0.502 (intra-event); τ_{U}= 0.265, τ_{M}= 0.260 (inter-event); σ_{TU}= 0.566, σ_{TM}= 0.560 (total). - Characterise sites using V
_{S30}. Believe equations applicable for 180 ≤ V_{S30}≤ 1300m∕s (state that equations should not be applied for very hard rock sites, V_{S30}≥ 1500m∕s). Bulk of data from NEHRP C and D sites (soft rock and firm soil) and very few data from A sites (hard rock). Use three equations for nonlinear amplification: to prevent nonlinear amplification increasing indefinitely as pga4nl decreases and to smooth transition from linear to nonlinear behaviour. Equations for nonlinear site amplification simplified version of those of Choi and Stewart (2005) because believe NGA database insufficient to simultaneously determine all coefficients for nonlinear site equations and magnitude-distance scaling due to trade-offs between parameters. Note that implicit trade-offs involved and change in prescribed soil response equations would lead to change in derived magnitude-distance scaling. - Focal depths between 2 and 31km with most < 20km.
- Use data from the PEER Next Generation Attenuation (NGA) Flatfile supplemented with additional data from three small events (2001 Anza M4.92, 2003 Big Bear City M4.92 and 2002 Yorba Linda M4.27) and the 2004 Parkfield earthquake, which were used only for a study of distance attenuation function but not the final regression (due to rules of NGA project).
- Use three faulting mechanism categories using P and T axes:
- SS
- Strike-slip. Plunges of T and P axes < 40
^{∘}. 35 earthquakes. Dips between 55 and 90^{∘}. 4.3 ≤ M ≤ 7.9. SS = 1, U = 0, NS = 0, RS = 0. - RS
- Reverse. Plunge of T axis > 40
^{∘}. 12 earthquakes. Dips between 12 and 70^{∘}. 5.6 ≤ M ≤ 7.6. RS = 1, U = 0, SS = 0, NS = 0. - NS
- Normal. Plunge of P axis > 40
^{∘}. 11 earthquakes. Dips between 30 and 70^{∘}. 5.3 ≤ M ≤ 6.9. NS = 1, U = 0, SS = 0, RS = 0.

Note that some advantages to using P and T axes to classify earthquakes but using categories based on rake angles with: within 30

^{∘}of horizontal as strike-slip, from 30 to 150^{∘}as reverse and from -30^{∘}to -150^{∘}as normal, gives essentially the same classification. Also allow prediction of motions for unspecified (U = 1, SS = 0, NS = 0, RS = 0) mechanism (use σs and τs with subscript U otherwise use σs and τs with subscript M). - Exclude records from obvious aftershocks because believe that spectral scaling of aftershocks could be different than that of mainshocks. Note that this cuts the dataset roughly in half.
- Exclude singly-recorded earthquakes.
- Note that possible bias due to lack of low-amplitude data (excluded due to non-triggering of instrument, non-digitisation of record or below the noise threshold used in determining low-cut filter frequencies). Distance to closest non-triggered station not available in NGA Flatfile so cannot exclude records from beyond this distance. No information available that allows exclusion of records from digital accelerograms that could remove this bias. Hence note that obtained distance dependence for small earthquakes and long periods may be biased towards a decay that is less rapid than true decay.
- Use estimated R
_{JB}s for earthquakes with unknown fault geometries. - Lack of data at close distances for small earthquakes.
- Three events (1987 Whittier Narrows, 1994 Northridge and 1999 Chi-Chi) contribute large proportion of records (7%, 10% and 24%).
- Note that magnitude scaling better determined for strike-slip events, which circumvent using common magnitude scaling for all mechanisms.
- Seek simple functional forms with minimum required number of predictor variables. Started with simplest reasonable form and added complexity as demanded by comparisons between predicted and observed motions. Selection of functional form heavily guided by subjective inspection of nonparametric plots of data.
- Data clearly show that modelling of anelastic attenuation required for distances > 80km and that effective geometric spreading is dependent on magnitude. Therefore, introduce terms in the function to model these effects, which allows model to be used to 400km.
- Do not include factors for depth-to-top of rupture, hanging wall/footwall or basin depth because residual analysis does not clearly show that the introduction of these factors would improve the predictive capabilities of model on average.
- Models are data-driven and make little use of simulations.
- Believe that models provide a useful alternative to more complicated NGA models as they are easier to implement in many applications.
- Firstly correct ground motions to obtain equivalent observations for reference velocity of 760m∕s using site
amplification equations using only data with R
_{JB}≤ 80km and V_{S30}> 360m∕s. Then regress site-corrected observations to obtain F_{D}and F_{M}with F_{S}= 0. No smoothing of coefficients determined in regression (although some of the constrained coefficients were smoothed). - Assume distance part of model applies for crustal tectonic regimes represented by NGA database. Believe that this is a reasonable initial approach. Test regional effects by examining residuals by region.
- Note that data sparse for R
_{JB}> 80km, especially for moderate events, and, therefore, difficult to obtain robust c_{1}(slope) and c_{3}(curvature) simultaneously. Therefore, use data from outside NGA database (three small events and 2004 Parkfield) to define c_{3}and use these fixed values of c_{3}within regression to determine other coefficients. To determine c_{3}and h from the four-event dataset set c_{1}equal to -0.5, -0.8 and -1.0 and c_{2}= 0 if the inclusion of event terms c_{0}for each event. Use c_{3}s when c_{1}= -0.8 since it is a typical value for this parameter in previous studies. Find that c_{3}and h are comparable to those in previous studies. - Note that desirable to constrain h to avoid overlap in curves for large earthquakes at very close distances. Do this by initially performing regression with h as free parameter and then modifying h to avoid overlap.
- After h and c
_{3}have been constrained solve for c_{1}and c_{2}. - Constrain quadratic for magnitude scaling so that maximum not reached for M < 8.5 to prevent
oversaturation. If maximum reached for M < 8.5 then perform two-segment regression hinged at M
_{h}with quadratic for M ≤ M_{h}and linear for M > M_{h}. If slope of linear segment is negative then repeat regression by constraining slope above M_{h}to 0.0. Find that data generally indicates oversaturation but believe this effect is too extreme at present. M_{h}fixed by observation that ground motions at short periods do not get significantly larger with increasing magnitude. - Plots of event terms (from first stage of regression) against M show that normal-faulting earthquakes have
ground motions consistently below those of strike-slip and reverse events. Firstly group data from all fault
types together and solved for e
_{1}, e_{5}, e_{6}, e_{7}and e_{8}by setting e_{2}, e_{3}and e_{4}to 0.0. Then repeat regression fixing e_{5}, e_{6}, e_{7}and e_{8}to values obtained in first step to find e_{2}, e_{3}and e_{4}. - Examine residual plots and find no significant trends w.r.t. M, R
_{JB}or V_{S30}although some small departures from a null residual. - Examine event terms from first stage of regression against M and conclude functional form provides reasonable fit to near-source data.
- Examine event terms from first stage of regression against M for surface-slip and no-surface-slip earthquakes. Find that most surface-slip events correspond to large magnitudes and so any reduction in motions for surface-slip earthquakes will be mapped into reduced magnitude scaling. Examine event terms from strike-slip earthquakes (because both surface- and buried-slip events in same magnitude range) and find no indication of difference in event terms for surface-slip and no-surface-slip earthquakes. Conclude that no need to include dummy variables to account for this effect.
- Examine residuals for basin depth effects. Find that V
_{S30}and basin depth are highly correlated and so any basin-depth effect will tend to be captured by empirically-determined site amplifications. To separate V_{S30}and basin-depth effects would require additional information or assumptions but since aiming for simplest equations no attempt made to break down separate effects. Examine residuals w.r.t. basin depth and find little dependence. - Chi-Chi data forms significant fraction (24% for PGA) of data set. Repeat complete analysis without these data to examine their influence. Find that predictions are not dramatically different.
- Note that use of anelastic coefficients derived using data from four earthquakes in central and southern California is not optimal and could lead to inconsistencies in hs.