Bayesian inference and predictive accuracy of failure models driven by a self-correcting point process for long-term recurrence of strong earthquakes

The process by which earthquakes are generated is still unknown. Its understanding is aimed at reducing and managing the seismic risk as well as at being a challenging research field. The fragmentary knowledge on seismic phenomena and the availability of irremediably incomplete databases make the probabilistic approach to earthquake predictability more promising (feasible) with respect to the deterministic approach. A variety of stochastic models concerns the way in which earthquakes recur on a particular fault or in a seismic region. These models span typically different forecasting time scales: long-term (centuries to decades), medium-term (years to months), short-term (weeks to hours). For the sake of simplicity the long-term recurrence of strong earthquakes is often modelled by the stationary Poisson process, although renewal and self-correcting point processes (with non-decreasing hazard functions) are more appropriate. Short-term models fit mainly 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 this study we attempt a combination of long-term and short-term models by splitting a seismic sequence in two groups: the leader events, whose magnitude exceeds a threshold magnitude, and the remaining ones considered as subordinate events. The leader events are assumed to follow a well-known self-correcting point process named stress release model, proposed first by David Vere-Jones in 1978. In the interval between two subsequent leader events, subordinate events are expected to cluster at the beginning (aftershocks) and at the end (foreshocks) of that interval; hence, they are modelled by a failure process whose hazard function can be bathtub-shaped. In particular, we have examined the generalized Weibull distributions, a large family that contains distributions with different bathtub-shaped hazard functions as well as the standard Weibull distribution. Choosing the hazard function in this family of distributions, some different models are obtained and then fitted to a dataset of Italian historical earthquakes by Bayesian inference. The problem of model selection is addressed according to whether the primary goal is explanation (Bayes Factor) or, alternatively, prediction (Ando and Tsay criterion and a retrospective predictive analysis for each event in the dataset). In the former case, the criteria for model selection are defined over the parameter space whose physical interpretation can be difficult; in the latter case, they are defined over the space of the observations, which has a more direct physical meaning.