Anomalous power law distribution of total lifetimes of branching processes: Application to earthquake aftershock sequences

Abstract

We consider a general stochastic branching process, which is relevant to earthquakes, and study the distributions of global lifetimes of the branching processes. In the earthquake context, this amounts to the distribution of the total durations of aftershock sequences including aftershocks of arbitrary generation number. Our results extend previous results on the distribution of the total number of offspring (direct and indirect aftershocks in seismicity) and of the total number of generations before extinction. We consider a branching model of triggered seismicity, the epidemic-type aftershock sequence model, which 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 ("productivity" or "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 where the distribution of fertilities mu is characterized by a power law approximately 1/ mu(1+gamma) and the bare Omori law for the memory of previous triggering mothers decays slowly as approximately 1/ t(1+theta;) , with 0<theta;<1 relevant for earthquakes. Using the tool of generating probability functions and a quasistatic approximation which is shown to be exact asymptotically for large durations, we show that the density distribution of total aftershock lifetimes scales as approximately 1/ t(1+theta;/gamma) when the average branching ratio is critical (n=1) . The coefficient 1<gamma=b/alpha<2 quantifies the interplay between the exponent b approximately 1 of the Gutenberg-Richter magnitude distribution approximately 10(-bm) and the increase approximately 10(alpham) of the number of aftershocks with mainshock magnitude m (productivity), with 0.5<alpha<1 . The renormalization of the bare Omori decay law approximately 1/ t(1+theta;) into approximately 1/ t(1+theta;/gamma) stems from the nonlinear amplification due to the heavy-tailed distribution of fertilities and the critical nature of the branching cascade process. In the subcritical case n<1 , the crossover from approximately 1/ t(1+theta;/gamma) at early times to approximately 1/ t(1+theta;) at longer times is described. More generally, our results apply to any stochastic branching process with a power-law distribution of offspring per parent and a long memory.