- Ground-motion model is:
_{rx}is predicted PGA on NEHRP B sites, c_{1}= 2.991, c_{2}= 0.03525, c_{3}= 0.00759, c_{4}= -0.00206, σ_{1}= 0.20 (intra-event) and σ_{2}= 0.11 (inter-event) for interface events and c_{1}= -0.04713, c_{2}= 0.6909, c_{3}= 0.01130, c_{4}= -0.00202, σ_{1}= 0.23 and σ_{2}= 0.14 for in-slab events and c_{5}= 0.19, c_{6}= 0.24, c_{7}= 0.29 for all events. g = 10^{1.2-0.18M}for interface events and g = 10^{0.301-0.01M}for in-slab events. Recommended revised c_{1}for interface events in Cascadia is 2.79 and in Japan 3.14, recommended revised c_{1}for in-slab events in Cascadia is -0.25 and in Japan 0.10. - Use four site categories:
- B
- NEHRP site class B, V
_{s,30}> 760m∕s. S_{C}= 0, S_{D}= 0 and S_{E}= 0. - C
- NEHRP site class C, 360 < V
_{s,30}≤ 760m∕s. S_{C}= 1, S_{D}= 0 and S_{E}= 0. - D
- NEHRP site class D, 180 ≤ V
_{s,30}≤ 360m∕s. S_{D}= 1, S_{C}= 0 and S_{E}= 0. - E
- NEHRP site class E, V
_{s,30}< 180m∕s. S_{E}= 1, S_{C}= 0 and S_{D}= 0.

Stations in KNET were classified using shear-wave velocity profiles using an statistical method to extrapolate measured shear-wave velocities to depths up to 10–20m to 30m. Stations in Guerrero array assumed to be on rock, i.e. site class B. Broadband stations in Washington and British Columbia sited on rock (V

_{s,30}≈ 1100m∕s), i.e. site class B. Strong-motion stations in Washington classified using map of site classes based on correlations between geology and V_{s,30}in Washington, and verified at 8 stations using actual borehole measurements. Converted Youngs et al. (1997) Geomatrix classifications by assuming Geomatrix A=NEHRP B, Geomatrix B=NEHRP C, Geomatrix C/D=NEHRP D and Geomatrix E=NEHRP E using shear-wave velocity and descriptions of Geomatrix classification. - Note that cannot develop equations using only Cascadia data because not enough data. Combine data of Crouse (1991) and Youngs et al. (1997) with additional data from Cascadia (strong-motion and broadband seismographic records), Japan (KNET data), Mexico (Guerrero array data) and El Salvador data.
- Classify event by type using focal depth and mechanism as:
- In-slab
- All earthquakes with normal mechanism. Earthquakes with thrust mechanism at depths > 50km or if occur on steeply dipping planes.
- Interface
- Earthquakes with thrust mechanism at depths < 50km on shallow dipping planes.

Exclude events of unknown type.

- Exclude events with focal depth h > 100km.
- Exclude events that occurred within crust above subduction zones.
- Use many thousands of extra records to explore various aspects of ground motion scaling with M and
D
_{fault}. - Data relatively plentiful in most important M-D
_{fault}ranges, defined according to deaggregations of typical hazard results. These are in-slab earthquakes of 6.5 ≤ M ≤ 7.5 for 40 ≤ D_{fault}≤ 100km and interface earthquakes of M ≥ 7.5 for 20 ≤ D_{fault}≤ 200km. - Data from KNET from moderate events at large distances are not reliable at higher frequencies due to
instrumentation limitations so exclude KNET data from M < 6 at D
_{fault}> 100km and for M ≥ 6 at D_{fault}> 200km. Excluded data may be reliable at low frequencies. - Estimate D
_{fault}for data from Crouse (1991) and for recent data using fault length versus M relations of Wells and Coppersmith (1994) to estimate size of fault plane and assuming epicentre lies above geometric centre of dipping fault plane. Verified estimates for several large events for which fault geometry is known. - Perform separate regressions for interface and in-slab events because analyses indicated extensive differences in amplitudes, scaling and attenuation between two types.
- Experiment with a variety of functional forms. Selected functional form allows for magnitude dependence of geometrical spreading coefficient, g; the observed scaling with magnitude and amplitude-dependent soil nonlinearity.
- For h > 100km use h = 100km to prevent prediction of unrealistically large amplitudes for deeper earthquakes.
- R is approximately equal to average distance to fault surface. Δ is defined from basic fault-to-site geometry.
For a fault with length and width given by equations of Wells and Coppersmith (1994), the average
distance to the fault for a specified D
_{fault}is calculated (arithmetically averaged from a number of points distributed around the fault), then used to determine Δ. Magnitude dependence of R arises because large events have a large spatial extent, so that even near-fault observation points are far from most of the fault. Coefficients in Δ were defined analytically, so as to represent average fault distance, not be regression. Although coefficients in Δ were varied over a wide range but did not improve accuracy of model predictions. - Determine magnitude dependence of g by preliminary regressions of data for both interface and in-slab
events. Split data into 1 magnitude unit increments to determine slope of attenuation as a function of
magnitude using only 1 and 2s data and records with 50 ≤ D
_{fault}≤ 300km (50km limit chosen to avoid near-source distance saturation effects). Within each bin regression was made to a simple functional form: log Y ′ = a_{1}+ a_{2}M - g log R + a_{3}S where Y ′ = Y exp(0.001R), i.e. Y corrected for curvature due to anelasticity, and S = 0 for NEHRP A or B and 1 otherwise. g is far-field slope determined for each magnitude bin. - Nonlinear soil effects not strongly apparent in database on upon examination of residuals from preliminary
regressions, as most records have PGA < 200cm∕s2, but may be important for large M and small D
_{fault}. To determine linear soil effects perform separate preliminary regressions for each type of event to determine c_{5}, c_{6}and c_{7}assuming linear response. Smooth these results (weighted by number of observations in each subset) to fix c_{5}, c_{6}and c_{7}(independent of earthquake type) for subsequent regressions. sl was assigned by looking at residual plots and from consideration of NEHRP guidelines. Conclude that there is weak evidence for records with PGA_{rx}> 100cm∕s2, for NEHRP E sites at periods < 1s. Use these observations to fix sl for final regression. - Final regression needs to be iterated until convergence because of use of PGA
_{rx}in definition of dependent variable. - To optimize fit for M-D
_{fault}range of engineering interest limit final regression to data within: 5.5 ≤M < 6.5 and D_{fault}≤ 80km, 6.5 ≤M < 7.5 and D_{fault}≤ 150km and M ≥ 7.5 and D ≤ 300km for interface events and 6.0 ≤M < 6.5 and D_{fault}≤ 100km and M ≥ 6.5 and D_{fault}≤ 200km for in-slab events. These criteria refined by experimentation until achieved an optimal fit for events that are important for seismic hazard analysis. Need to restrict M-D_{fault}for regression because set dominated by records from moderate events and from intermediate distances whereas hazard is from large events and close distances. - Lightly smooth coefficients (using a weighted 3-point scheme) over frequency to get smooth spectral shape and allows for reliable linear interpolation of coefficients for frequencies not explicitly used in regression.
- In initial regressions, use a M
^{2}term as well as a M term leading to a better fit over a linear magnitude scaling but lead to a positive sign of the M^{2}rather than negative as expected. Therefore to ensure the best fit in the magnitude range that is important for hazard and constrained by data quadratic source terms refit to linear form. Linear model constrained to provide same results in range 7.0 ≤M ≤ 8.0 for interface events and 6.5 ≤M ≤ 7.5 for in-slab events. To ensure that non-decreasing ground motion amplitudes for large magnitudes: for M > 8.5 use M = 8.5 for interface events and for M > 8.0 use M = 8.0 for in-slab events. - Calculate σ based on records with M ≥ 7.2 and D
_{fault}≤ 100km for interface events and M ≥ 6.5 and D_{fault}≤ 100km for in-slab events. These magnitude ranges selected to obtain the variability applicable for hazard calculations. Do not use KNET data when computing σ because data appear to have greater high-frequency site response than data from same soil class from other regions, due to prevalence of sites in Japan with shallow soil over rock. - Determine σ
_{1}using data for several well-recorded large events and determining average value. Then calculate σ_{2}assuming σ = . - Examine residuals w.r.t. D
_{fault}using all data from M ≥ 5.5 and D_{fault}≤ 200km and M ≥ 6.5 and D_{fault}≤ 300km. Find large variability but average residuals near 0 for D_{fault}≤ 100km. - Find significantly lower variability for M ≥ 7.2 events (σ = 0.2–0.35 for larger events and σ = 0.25–0.4 for smaller events).
- Examine graphs and statistics of subsets of data broken down by magnitude, soil type and region. Find significant positive residuals for M < 6.6 due to use of linear scaling with magnitude. Accept positive residuals because small magnitudes do not contribute strongly to hazard.
- Find large positive residuals for class C sites for interface events (most records are from Japan) whereas
residuals for class C sites for in-slab events (which are from both Japan and Cascadia) do not show trend.
No other overwhelming trends. Differences in residuals for Japan and Cascadia class C sites likely due
to differences in typical soil profiles in the two regions within the same NEHRP class. Sites in Japan are
typically shallow soil over rock, which tend to amplify high frequencies, whereas in Cascadia most soil sites
represent relatively deep layers over rock or till. Provide revised c
_{1}coefficients for Japan and Cascadia to model these differences. - Note that debate over whether 1992 Cape Mendocino earthquake is a subduction zone or crustal earthquake. Excluding it from regressions has a minor effect on results, reducing predictions for interface events for M < 7.5.