Using the standard ETAS model of triggered seismicity, we present a rigorous theoretical analysis of the main statistical properties of temporal clusters, defined as the group of events triggered by a given main shock of fixed magnitude m that occurred at the origin of time, at times larger than some present time t. Using the technology of generating probability function (GPF), we derive the explicit expressions for the GPF of the number of future offsprings in a given temporal seismic cluster, defining, in particular, the statistics of the clusters duration and the clusters offsprings maximal magnitudes. We find the remarkable result that the magnitude difference between the largest and second largest event in the future temporal cluster is distributed according to the regular Gutenberg-Richer law that controls the unconditional distribution of earthquake magnitudes. For earthquakes obeying the Omori-Utsu law for the distribution of waiting times between triggering and triggered events, we show that the distribution of the durations of temporal clusters of events of magnitudes above some detection threshold u has a power law tail that is fatter in the non-critical regime $n<1$ than in the critical case n=1. This paradoxical behavior can be rationalised from the fact that generations of all orders cascade very fast in the critical regime and accelerate the temporal decay of the cluster dynamics.
Download