Paper detail

Sub-critical and Super-critical Regimes in Epidemic Models of Earthquake Aftershocks

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. The occurrence rate of aftershocks triggered by a single mainshock decreases with the time from the mainshock according to the modified Omori law K/(t+c)^p with p=1+theta. A mainshock at time t=0 triggers aftershocks according to the local Omori law, that in turn trigger their own aftershocks and so on. The effective branching parameter n, defined as the mean aftershock number triggered per event, controls the transition between a sub-critical regime n<1 to a super-critical regime n>1. In the sub-critical regime, we recover and document the crossover from an Omori exponent 1-theta for t<t* to 1+theta for t<t* found previously in [Sornette and Sornette, 1999a] for a special case of the ETAS model. In the super-critical regime n>1 and theta>0, we find a novel transition from an Omori decay law with exponent 1-theta fot t<t* to an explosive exponential increase of the seismicity rate fot t>t*. The case theta<0 yields an infinite n-value. In this case, we find another characteristic time tau controlling the crossover from an Omori law with exponent 1-theta for t<tau, similar to the local law, to an exponential increase at large times. 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, (iii) the positive correlation between b and p-values, (iv) the observation that great earthquakes are sometimes preceded by a decrease of b-value and (v) the acceleration of the seismicity preceding great earthquakes.