Sub-critical and Super-critical Regimes in Epidemic Models of Earthquake Aftershocks
Friday, October 12, 2001 - 11:00am - 12:00pm
Agnes Helmstetter (University of California, Los Angeles)
We present an analytical solution and numerical tests of the epidemic type aftershock (ETAS) model for aftershocks, which describes foreshocks, aftershocks and mainshocks on the same footing. In this model, each earthquake of magnitude M triggers aftershocks with a rate proportional to 10(AM). The occurrence rate of aftershocks decreases with the time from the mainshock according to the modified Omori law K/(t+c)p with p=1+theta. The background seismicity rate is modeled by a stationary Poisson process with a constant occurrence rate. Contrary to the usual definition, the ETAS model does not impose an aftershock to have a magnitude smaller than the mainshock. We find two differents regimes depending on the branching ratio N, defined as the mean aftershock number triggered per event. In the sub-critical regime (N1 and theta>0), we find a novel transition from an Omori decay law with exponent 1-theta to an explosive exponential increase of the seismicity rate. These results can rationalize many of the stylized facts reported for aftershock and foreshock sequences, such as (i) the suggestion that a small p-value may be a precursor of a large earthquake, (ii) the relative seismic quiescence sometimes observed before large aftershocks, and (iii) the increase of seismic activity preceding large earthquakes.