On the convolution of stochastic processes for modelling strong earthquake occurrences: a multi-rupture model driven by a self-correcting model

Two widely noted features of earthquake generation are the following: - earthquakes tend to occur in clusters, sometimes, but not only, referred as "swarms", "foreshocks activity" and "aftershocks activity"; - the fault ruptures that generate earthquakes decrease the amount of strain present at the locations along the fault where rupture occurs. Two different classes of models: self-exciting models and self-correcting models correspond respectively to the two features and have been widely studied separately in the literature. Models that try to capture both these diametrically opposed features should reconcile contrasting trends. The simplest solution would be to mix stochastic models of the two classes: trigger and strain-release models (Schoenberg and Bolt, 2000); in this way, since it is unknown who belongs to what (which events are triggered and which trigger), each event is meant to be generated by both models and the normalised estimate of the conditional intensities lambda_i / (lambda_1+lambda_2), i=1,2, indicates the percentage of events belonging to each class. The large difference between the scales, at which the triggering and strain-release mechanisms appear to operate, may be a misleading element. To overcome this issue we can assume that the different behaviours correspond to different phases of the seismic activity and the dynamics of their activation times is driven by an unobserved pure Jump Markov process; in this perspective a seismic sequence can be considered as a realization of a series of three marked point processes: Poisson, stress release and trigger models (Varini, 2005; Varini and Rotondi, 2006). The comparison on simulated datasets shows that about 70% of the events are correctly classified but the model is hardly able to fit the abrupt changes of state. This leads to think that it is more reasonable to assume that the different behavioural trends (models) are superimposed rather than consecutive. In this perspective we consider a sequence of strong earthquakes { t_i, M_i }, i=1,...,n, where ti indicates the occurrence time and Mi the magnitude. Among these events we distinguish the leaders, with higher magnitude (exceeding a fixed threshold) from the subordinates, with lower magnitude. The leaders follow a stress release model; conditioned on their occurrence, the remaining events constitute a set of ordered times of minor ruptures occurring in the time interval between two consecutive leader-events. In other words, the events of I level (leaders) match the elastic rebound theory, while the events of II level (subordinates) depend on the previous ones and take charge of the other trends. A preliminary application to data from an Italian seismogenic source is shown.