Number of Aftershocks in Epidemic-type Seismicity Models

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 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_ • - Δ threshold in Δ-analysis with M = mₐ - Δ 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(m_Δ ) distribution with a suitable magnitude m_Δ. 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.