Earthquake productivity within general ETAS models

Earthquake clustering is a prominent feature of seismic catalogs. Several models have been proposed to describe the triggering mechanism of earthquakes; nonetheless, some theoretical aspects have not yet been thoroughly explored. In this study we investigate the distribution of the number of triggered events as described by a branching process, which might provide useful practical constraints for empirical analysis of seismic catalogs. According to recent literature (e.g. Shebalin et al., 2020, and references therein), the productivity of a magnitude m event can be defined as the number of triggered events of magnitude above m-D, where D is a positive default value. In particular we distinguish between the number of direct descendants and the number of all descendants, denoted respectively by the random variables v(D) and V(D). In the standard Epidemic Type Aftershock Sequence (ETAS) model, the distribution of v(D) is Poissonian. However, evidence has recently emerged in favor of the discrete exponential distribution for v(D) and for V(D) with a dominant initial magnitude m (the case of a simple aftershock cluster). Therefore we consider the general ETAS model adapted to any distribution of v(D) and prove that the branching structure of the model excludes the possibility of having a common distribution type for both v(D) and V(D) at once (Molchan et al., 2022). We then investigate the features of the V(D) distribution within a wide class of ETAS models. We show that there is a fundamental difference in tail behavior of the V(D)-distributions for general-type clusters and for clusters with a dominant initial magnitude: the tail is heavy in the former case and light in the latter. The real data display similar behavior. Theoretical conclusions are also illustrated through the analysis of a synthetic earthquake catalog.