Power-law Distributions of Offspring and Generation Numbers in Branching Models of Earthquake Triggering

Weconsider a general stochastic branching process,which is relevant to earthquakes as well as to many other systems, and we study the distributions of the total number of offsprings (direct and indirect aftershocks in seismicity) and of the total number of generations before extinction.We apply our results to a branching model of triggered seismicity, the ETAS (epidemic-type aftershock sequence) model. The ETAS model assumes that each earthquake can trigger other earthquakes (‘‘aftershocks’’). An aftershock sequence results in this model from the cascade of aftershocks of each past earthquake. Due to the large fluctuations of the number of aftershocks triggered directly by any earthquake (‘‘fertility’’), there is a large variability of the total number of aftershocks from one sequence to another, for the same mainshock magnitude. We study the regime in which the distribution of fertilities l is characterized by a power law 1/l1+c. For earthquakes we expect such a power-distribution of fertilities with c=b/a based on the Gutenberg-Richter magnitude distribution 10)bm and on the increase 10am of the number of aftershocks with the mainshock magnitude m.We derive the asymptotic distributions pr(r) and pg(g) of the total number r of offsprings and of the total number g of generations until extinction following a mainshock. In the regime c<2 for which the distribution of fertilities has an infinite variance, we find prðrÞ 1=r1þ1c and pgðgÞ 1=g1þ 1 c 1: This should be compared with the distributions pgðgÞ 1=g1þ1 2 pgðgÞ obtained for standard branching processes with finite variance. These predictions are checked by numerical simulations. Our results apply directly to the ETAS model whose preferred values a=0.8–1 and b=1 puts it in the regime where the distribution of fertilities has an infinite variance. More generally, our results apply to any stochastic branching process with a power-law distribution of offsprings per mother.