Stochastic summation of empirical Green's functions

Two simple strategies are presented that use random delay times for repeatedly summing the record of a relatively small earthquake to simulate the effects of a larger earthquake. The simulations do not assume any fault plane geometry or rupture dynamics, but realy only on the ω−2 spectral model of an earthquake source and elementary notions of source complexity. The strategies simulate ground motions for all frequencies within the bandwidth of the record of the event used as a summand. The first strategy, which introduces the basic ideas, is a single-stage procedure that consists of simply adding many small events with random time delays. The probability distribution for delays has the property that its amplitude spectrum is determined by the ratio of ω−2 spectra, and its phase spectrum is identically zero. A simple expression is given for the computation of this zero-phase scaling distribution. The moment rate function resulting from the single-stage simulation is quite simple and hence is probably not realistic for high-frequency (>1 Hz) ground motion of events larger than ML ∼ 4.5 to 5. The second strategy is a two-stage summation that simulates source complexity with a few random subevent delays determined using the zero-phase scaling distribution, and then clusters energy around these delays to get an ω−2 spectrum for the sum. Thus, the two-stage strategy allows simulations of complex events of any size for which the ω−2 spectral model applies. Interestingly, a single-stage simulation with too few ω−2 records to get a good fit to an ω−2 large-event target spectrum yields a record whose spectral asymptotes are consistent with the ω−2 model, but that includes a region in its spectrum between the corner frequencies of the larger and smaller events reasonably approximated by a power law trend. This spectral feature has also been discussed as reflecting the process of partial stress release (Brune, 1970), an asperity failure (Boatwright, 1984), or the breakdown of ω−2 scaling due to rupture significantly longer than the width of the seismogenic zone (Joyner, 1984).