Number of Aftershocks in Epidemic-type Seismicity Models

Abstract

Summary The Epidemic Type Aftershock Sequence (ETAS) model describes how an earthquake generates its own aftershocks. The regular ETAS model assumes that distribution F of the number of direct aftershocks is Poissonian, however there is evidence suggesting that a geometric distribution might be more adequate. Let ${V}_M({m}_ \bullet )$ be the number of $m > M$ aftershocks generated by ${m}_ \bullet $event. In this study we consider the ${V}_M({m}_ \bullet )$ distribution within Epidemic-type Seismicity models, ETAS(F). These models include the Gutenberg-Richter law for magnitude and Utsu law for average ${m}_ \bullet $- productivity, but differ in the type of F distribution for the number $v({m}_ \bullet )$ of direct aftershocks. The class of F is quite broad and includes both the Poisson distribution, which is the basis for the regular ETAS model, and its possible alternative, the Geometric distribution. We replace the traditional $M = {m}_ \bullet - \Delta $ threshold in $\Delta $-analysis with $M = {m}_a - \Delta $ where ${m}_a$ is the distribution mode of the strongest aftershocks. Under these conditions we find the limit ${V}_M({m}_ \bullet )$ distribution at ${m}_ \bullet > > 1$. In the subcritical case, the limit distribution is extremely simple and identical to the $v({m}_\Delta )$ distribution with a suitable magnitude ${m}_\Delta $. This result allows us to validate both the priority of the geometric distribution of F for direct aftershocks and the very concept of epidemic-type clustering on global seismicity data.