Ground-motion relations are developed for California using a stochastic simulation method that exploits the equivalence between finite-fault models and a two-corner point-source model of the earthquake spectrum. First, stochastic simulations are generated for finite-fault ruptures, in order to define the average shape and amplitude level of the radiated spectrum at near-source distances as a function of earthquake size. The length and width of the fault plane are defined based on the moment magnitude of the earthquake and modeled with an array of subfaults. The radiation from each subfault is modeled as a Brune point source using the stochastic model approach; the subfault spectrum has a single-corner frequency. An earthquake rupture initiates at a randomly chosen subfault (hypocenter), and propagates in all directions along the fault plane. A subfault is triggered when rupture propagation reaches its center. Simulations are generated for an observation point by summing the subfault time series, appropriately lagged in time. Fourier spectra are computed for records simulated at many azimuths, placed at equidistant observation points around the fault. The mean Fourier spectrum for each magnitude, at a reference near-source distance, is used to define the shape and amplitude levels of an equivalent point-source spectrum that mimics the salient finite-fault effects. The functional form for the equivalent point-source spectrum contains two corner frequencies.
Stochastic point-source simulations, using the derived two-corner source spectrum, are then performed to predict peak-ground-motion parameters and response spectra for a wide range of magnitudes and distances, for generic California sites. The stochastic ground-motion relations, given in the Appendix for rock and soil sites, are in good agreement with the empirical strong-motion database for California; the average ratio of observed to simulated amplitudes is near unity over all frequencies from 0.2 to 12 Hz. The stochastic relations agree well with empirical regression equations (e.g., Abrahamson and Silva, 1997; Boore et al., 1997; Sadigh et al., 1997) in the magnitude-distance ranges well represented by the data, but are better constrained at large distances, due to the use of attenuation parameters based on regional seismographic data. The stochastic ground-motion relations provide a sound basis for estimation of ground motions for earthquakes of magnitude 4 through 8, at distances from 1 to 200 km.