Comparison Analysis of the ETAS Model with Gutenberg–Richter (GR), Tapered-GR and Characteristic Magnitude Distributions

SUMMARY In this paper, we carry out a comparison analysis of the Epidemic Type Aftershock Sequence (ETAS) model for the earthquake process, embedded with the three main exponential-type distributions adopted in practical applications to describe the magnitudes of seismic events, that are, the Gutenberg–Richter (GR), the tapered Gutenberg–Richter (TGR) and the CHaracteristic (CH) frequency–magnitude distributions (FMDs). The first law is a pure-power decreasing function, while both the other two introduce a more rapid decay in the tail of the distribution: a soft taper in the TGR model and a sharp cut-off in the CH one. To perform the comparison, we first investigate some theoretical features of the ETAS model with CH-distributed magnitudes (ETAS-CH), which have not been deeply analysed in the literature as much as for ETAS-TGR and ETAS-GR. In particular, we explicitly compute the branching ratio, we analyse its asymptotics in relation to its parameters, and we derive the proper stability conditions. We then move to the comparison among the three ETAS-GR, ETAS-TGR and ETAS-CH processes, to highlight differences and similarities. This is done by carrying out both a theoretical analysis, mainly focused on the three models’ branching ratios and the relative sensitivity, and a simulation analysis of realistic synthetic catalogues to compare the processes’ numbers, events’ magnitude distribution and temporal evolution. The results we obtained show that the ETAS-TGR and ETAS-CH processes have very similar features. They both have also less restrictive non-explosion conditions than for ETAS-GR; in fact, differently from this latter case, their branching ratios exist for any value of the parameters and are lower than the one of ETAS-GR, to which they converge for large magnitudes.