Variability of seismic source spectra derived from cohesive-zone models of symmetrical and asymmetrical ruptures

Abstract:

Earthquake stress drops are often estimated from far-field body-wave spectra using measurements of seismic moment, corner frequency, and a specific theoretical model of rupture behavior. Perhaps, the most widely-used model is from Madariaga (1976), who performed finite-difference calculations for a singular crack radially expanding at a constant speed and showed that $fc = k \beta/a$, where $fc$ is spherically averaged corner frequency, $\beta$ is the shear-wave speed, $a$ is the radius of the circular source, and $k$ = 0.32 and 0.21 for P and S waves, respectively, assuming the rupture speed $Vr$ = 0.9$\beta$. Since stress in the Madariaga model is singular at the rupture front, the finite mesh size and smoothing procedures may have affected the resulting corner frequencies. Here we investigate the behavior of source spectra derived from dynamic models of radially expanding rupture with a cohesive zone that prevents a stress singularity at the rupture front. We find that in the small-scale yielding limit where the cohesive-zone size becomes much smaller than the source dimension, P- and S-wave corner frequencies of far-field body-wave spectra are systematically larger than those predicted by Madariaga (1976). In particular, the model with rupture speed $Vr$ = 0.9$\beta$ shows that $k$ = 0.38 for P waves and $k$ = 0.26 for S waves, which are 19 and 24 percent larger, respectively, than those of Madariaga (1976). Thus for these ruptures, the application of the Madariaga model overestimates stress drops by a factor of 1.7. We further address the validity of a standard assumption on a symmetrical circular source applied to real earthquakes. Our results suggest that up to a factor of two differences in the spherical average of corner frequencies are expected simply from the variability in source geometry and rupture styles, translating into a factor of eight differences in estimated stress drops. In addition, the large dependence of corner frequency on take-off angle relative to the source suggests that measurements from a small number of seismic stations are unlikely to produce unbiased estimates of spherically averaged corner frequency.