Abstract
The long-term recurrence of strong earthquakes is often modeled according to stationary Poisson processes for the sake of simplicity. However, renewal and self-correcting point processes (with nondecreasing 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 nonincreasing hazard are especially suitable. To provide a unified framework for analysis of earthquake catalogs, Schoenberg and Bolt proposed the short-term exciting long-term correcting model in 2000, and in 2005, Varini used a state-space model to estimate the different phases of a seismic cycle. Both of these analyses are combinations of long-term and short-term models, and the results are not completely satisfactory, due to the different scales at which these models appear to operate. In this study, we propose alternative modeling. First, we split a seismic sequence into two groups: the leader events, non-secondary events the magnitudes of which exceed a fixed threshold; and the remaining events, which are considered as subordinate. The leader events are assumed to follow the well-known self-correcting point process known as the stress-release model. 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 modeled by a failure process that allows bathtub-shaped hazard functions. In particular, we examined generalized Weibull distributions, as a large family that contains distributions with different bathtub-shaped hazards, as well as the standard Weibull distribution. The model is fit to a dataset of Italian historical earthquakes, and the results of Bayesian inference based on the Metropolis–Hastings algorithm are shown.
Highlights
Earthquakes are expressions of complex systems in which many components interact with each other
We propose a new stochastic model for earthquake occurrences, hereinafter denoted as the compound model, which takes into account the following points: (a) the benefit of exploiting a stochastic model inspired by elastic rebound theory; (b) the need to consider jointly the opposite trends that characterize self-exciting and self-correcting models; and (c) the idea to superimpose behaviors characteristic of different time-scales in a single hierarchical model
We consider the occurrence times of these events as ordered failure times in the time interval limited by the two leaders, and we model them through distributions belonging to the family of the generalized Weibull distributions with a bathtub-shaped hazard function, so as to match the clustering trend close to the extremes of the interval
Summary
Earthquakes are expressions of complex systems in which many components interact with each other. Models of the self-correcting class are the only ones, on large space–time scales, that attempt to incorporate physical conjecture into the probabilistic framework They are inspired by the elastic rebound theory by Reid (1911), which was transposed into the framework of stochastic point processes by Vere-Jones (1978), through the first version of the stress-release model. Subsequent versions of this model express the presence of clusters of even large earthquakes, in terms of possible interactions among neighboring fault segments (Bebbington and Harte 2003). We compare our model with the stress-release and ETAS models on the basis of two validation criteria: the Bayes factor and the information criterion by Ando and Tsay
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