Number of aftershocks in epidemic-type seismicity models

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 VM(m•) be the number of m > M aftershocks generated by m•event. In this study we consider the VM(m•) distribution within Epidemic-type Seismicity models, ETAS(F). These models include the Gutenberg–Richter law for magnitude and Utsu law for average m•-productivity, but differ in the type of F distribution for the number v(m•) 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• − ⃤ threshold in ⃤-analysis with M = ma − ⃤ where ma is the distribution mode of the strongest aftershocks. Under these conditions we find the limit VM(m•) distribution at m• >> 1. In the subcritical case, the limit distribution is extremely simple and identical to the v(m⃤) distribution with a suitable magnitude m⃤. 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.