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. In this model, each earthquake of magnitude m triggers aftershocks with a rate proportional to 10. 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) with p = 1 + θ. Contrary to the usual definition, the ETAS model does not impose an aftershock to have a magnitude smaller than the mainshock. Starting with a mainshock at time t = 0 that triggers aftershocks according to the local Omori law, that in turn trigger their own aftershocks and so on, we study the seismicity rate of the global aftershock sequence composed of all the secondary and subsequent aftershock sequences. 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. A characteristic time t, function of all the ETAS parameters, marks the transition from the early time behavior to the large time behavior. In the sub-critical regime, we recover and document the crossover from an Omori exponent 1 − θ for t < t to 1 + θ 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 θ > 0, we find a novel transition from an Omori decay law with exponent 1− θ fot t < t to an explosive exponential increase of the seismicity rate fot t > t. The case θ < 0 yields an infinite n-value. In this case, we find another characteristic time τ controlling the crossover from an Omori law with exponent 1 − |θ| for t < τ , 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 [Liu, 1984; Bowman, 1997] 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.