A finite-fault statistical model of the earthquake source is used to confirm observed magnitude and distance saturation scaling in a large peak-acceleration data set. This model allows us to determine the form of peak-acceleration attenuation curves without a priori assumptions about their shape or scaling properties. The source is composed of patches having uniform size and statistical properties. The primary source parameters are the patch peak-acceleration distribution mean, the distribution standard deviation, the patch size, and patch-rupture duration. Although our model assumes no scaling of peak acceleration with magnitude at the patch, the peak-acceleration attenuation curves, nevertheless, strongly scale with magnitude (dap/dM) ≠ 0, and the scaling is distance dependent (dap/dM) ∝ f(r).
The distance-dependent magnitude scaling arises from two principal sources in the model. For a propagating rupture, loci exist on the fault from which radiated energy arrives at a particular station at the same time. These loci are referred to as isochrones. As fault size increases, the length of the isochrones and, hence, the number of additive pulses increase. Thus, peak accelerations increase with magnitude. The second effect, which arises in a completely different manner, is due to extreme-value properties. That is, as the fault size increases, the number of patches on the fault and the number of peak values at the station increase. Because these attenuated pulses are produced by a statistical distribution at the patch, the largest value will depend on the total number of peak values available on the seismogram. We refer to this result as the extremal effect, because it is predicted by the theory of extreme values. Both the extremal and isochrone effects are moderated by attenuation and distance to the fault, leading to magnitude- and distance-dependent peak-acceleration scaling. Remarkably, the scaling produced by both effects is very similar, although the underlying mechanisms are completely different.
Because this model approximates data characteristics we have observed in an earlier study, we adjusted the parameters of the model to fit a set of smoothed peak accelerations from earthquakes worldwide. These data have not been preselected for particular magnitude or distance ranges and contain earthquake records for magnitudes ranging from about M 3 to M 8 and distance ranging from a few kilometers to about 400 km. In fitting the data, we use a trial-and-error procedure, varying the mean and standard deviation of the patch peak-acceleration distribution, the patch size, and the pulse duration. The model explicitly includes triggering bias, and the triggering threshold is also a model parameter. The data can be approximated equally well by a model that includes the isochrone effect alone, the extremal effect alone, or both effects. Inclusion of both effects is likely to be closest to reality, but because both effects produce similar results, it is not possible to determine the relative contribution of each one. In any case, the model approximates the complex features of the observed data, including a decrease in magnitude scaling with increasing magnitude at short distances and increase in magnitude scaling with magnitude at large distances.