Stability of the Epidemic‐Type Aftershock Sequence Model with Tapered Gutenberg–Richter Distributed Seismic Moments

摘要

The epidemic‐type aftershock sequence model with tapered Gutenberg–Richter (ETAS‐TGR)‐distributed seismic moments is a modification of the classical ETAS‐GR (without tapering) proposed by Kagan in 2002 to account for the finiteness of the deformational energy in the earthquake process. In this article, I analyze the stability of the ETAS‐TGR model by explicitly computing the relative branching ratio ηTGR?: it has to be set less than 1 for the process not to explode, in fact in the ETAS‐TGR model, the critical parameter equals the branching ratio as it happens for the ETAS‐GR, due to the rate separability in the seismic moments component. When the TGR parameter βk=23ln10β is larger than the fertility parameter αk=23ln10α?, respectively obtained from the GR and the productivity laws by translating moment magnitudes into seismic moments, the ETAS‐TGR model results to have less restrictive nonexplosion conditions than in the ETAS‐GR case. Furthermore, differently from the latter case in which it must hold β>α for ηGR to exist finite, any order relation for βk and αk (equivalently, for β,α?) is admissible for the stability of the ETAS‐TGR process; indeed ηTGR is well defined and finite for any βk,αk?. This theoretical result is strengthened by a simulation analysis I performed to compare three ETAS‐TGR synthetic catalogs generated with βk?αk?. The branching ratio ηTGR is shown to decrease as the previous parameter difference increases, reflecting: (1)?a lower number of aftershocks, among which a lower percentage of first generation shocks; (2)?a lower corner seismic moment for the moment–frequency distribution; and (3)?a longer temporal window occupied by the aftershocks. The less restrictive conditions for the stability of the ETAS‐TGR seismic process represent a further reason to use this more realistic model in forecasting applications.