Stability of the epidemic-type aftershock-sequence model with tapered Gutenberg-Richter distributed seismic moments

The Epidemic Type Aftershock Sequence model with Tapered Gutenberg-Richter
distributed seismic moments (ETAS-TGR) 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 paper I analyze the stability of
the ETAS-TGR model by explicitly computing the relative branching ratio
$\eta_{\scriptscriptstyle TGR}$ : 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 $\beta_k=\frac{2}{3\ln10}\beta$ is
larger than the fertility parameter $\alpha_k=\frac{2}{3\ln10}\alpha$ , 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 non-explosion
conditions than in the ETAS-GR case. Furthermore, differently from the latter case in
which it must hold $\beta>\alpha$ for $\eta_{\scriptscriptstyle GR}$ to exist finite, any
order relation for $\beta_k$ and $\alpha_k$ (equivalently, for $\beta,\,\alpha$ ) is
admissible for the stability of the ETAS-TGR process, indeed $\eta_{\scriptscriptstyle
TGR}$ is well-defined and finite for any $\beta_k,\, \alpha_k$ . This theoretical result is
strengthened by a simulation analysis I performed to compare three ETAS-TGR
synthetic catalogs generated with $\beta_k\lesseqgtr\alpha_k$ . The branching ratio
$\eta_{\scriptscriptstyle TGR}$ is shown to decrease as the above parameter
difference increases, reflecting: i) a lower number of aftershocks, among which a lower
percentage of first generation shocks, ii) a lower corner seismic moment for the
moment-frequency distribution, iii) 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.