Theoretical analysis of the productivity of seismic events

We investigate earthquake clustering, a prominent feature of seismic catalogs, in terms of distribution of the number of triggered events as described by a branching process (Kagan and Knopoff, Phys. Earth Planet. Inter., 1976; Saichev et al., Pure Appl. Geophys., 2005, and references therein). According to recent literature (e.g. Shebalin et al., Geophys. J. Int., 2020, and references therein), the productivity of a magnitude m event is defined as the number of triggered events of magnitude above m-?, where ? is a positive default value. For a magnitude m event, we distinguish between the number of its direct descendants and the total number of its descendants, denoted respectively by the random variables v and V, both depending on ?. Empirical analysis often testifies in favor of the identity of the type of distribution of both quantities (v and V) associated with the main event, and hypothetically is exponential. The testing or substantiation of this hypothesis is important for modeling seismicity and presents a serious challenge for seismic statistics. In the standard Epidemic Type Aftershock Sequence - ETAS - model (Ogata, Ann. Inst. Stat. Math., 1998), the distribution of v is Poissonian. Therefore we consider the general ETAS model adapted to any distribution of v and prove that the branching structure of the model excludes the possibility of having a common distribution type (for example, Poisson or exponential) for both v and V at once (Molchan et al., Geophys. J. Int., 2022). The second theoretical result relates to the behaviour of the tails of the productivity distribution. We show that there is a fundamental difference in tail behavior of the V-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.