The Applicability of Point-Source Models in Ground-Motion Prediction

Abstract

Point-source simulations with simple functional shapes of radiated Fourier spectra are widely used in earthquake-hazard assessments. Such an approximation is based on two physical assumptions: that (1) all near-field phenomena and (2) the wave-interference effects, caused by fault finiteness, are negligibly small (the far-field and the point-source approximations, respectively). The limits of applicability of these assumptions can be deduced from the complete theoretical description of the seismic field radiated by a fault rupture, expressed in the representation integral of elasticity. The far-field condition, deduced directly from the representation integral, is controlled by the slip and the slip rate on the fault; for a M_w 4 earthquake (M_w is the moment magnitude), it is reasonably satisfied at the distance of a few hundred meters. The point-source approximation is not satisfied even for the smallest earthquakes considered in seismic hazards: for a M_w 4 earthquake, the radiated finite-fault spectra significantly deviate from the commonly postulated omega-square shapes already at the frequencies around 1 Hz and above. The interference phenomena caused by fault finiteness act as a high-cut filter, creating the observed deficit in the high-frequency energy not accounted for by point-source spectra. To correct, the point-source models apply ad-hoc filtering, such as the kappa operator, acting as a substitute for the filtering naturally created by the fault itself. The finite-fault spectra without additional filtering can be formally explained by an equivalent point source with the kappa operator applied. The κ values determined from the equivalent point-source spectrum are in the same range as those empirically observed. However, if a finite-fault spectrum is interpreted as a point-source one with kappa, the values of the maximum slip velocity, an influential physical parameter of rupture, are recovered incorrectly. The kappa filtering can be fully explained by the finite-fault effects always present in all earthquakes of practical significance.