On failure processes driven by a self-correcting model. Application to sequences of seismic events

The long-term recurrence of strong earthquakes is often modelled by the stationary Poisson process for the sake of simplicity, although renewal and self-correcting point processes (with non-decreasing hazard functions) are more appropriate. Short-term models mainly fit earthquake clusters due to the tendency of an earthquake to trigger other earthquakes; in this case, self-exciting point processes with non-increasing hazard are especially suitable. In order to provide a unified framework for analyzing earthquake catalogs, Schoenberg and Bolt proposed the SELC model (BSSA, 2000) and Varini (PhD thesis, 2005) employed a state-space model for estimating different phases of a seismic cycle. Both attempts are combinations of long-and short-term models, but results are not completely satisfactory, due to the different scales at which these models appear to operate. In this study, we split a seismic sequence in two groups: the leader events, whose magnitude exceeds a threshold magnitude, and the remaining ones considered as subordinates. The leaders are assumed to follow a well-known self-correcting point process named stress release model (Vere-Jones, 1978). In the interval between two subsequent leaders, subordinates are expected to cluster at the beginning (aftershocks) and at the end (foreshocks); hence, they are modelled by a failure process that allows bathtub-shaped hazard function. In particular, we have examined generalized Weibull distributions, a large family that contain distributions with different bathtub-shaped hazard as well as the standard Weibull distribution (Lai, 2014). The model is fitted to a set of Italian historical earthquakes and the results of Bayesian inference are shown.