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.