Bayesian analysis of temporal variations of seismicity based on non-extensive statistical mechanics

In 1988, Tsallis introduced a non-additive entropy that has found applications in various fields such as physics, chemistry, medicine, informatics, linguistics, and economics. The concept also gained attention in statistical seismology due to the evidence that Earth's crust behaves as a complex, non-linear dynamic system characterized by long-range correlations. By addressing the problem in the context of non-extensive statistical mechanics, the classical methods of analysis are unsuitable: the Boltzmann-Gibbs (BG) entropy is appropriate to study systems with short range interactions, while new entropies that generalize some of the properties of BG entropy are better suited to describe systems with long range interactions. In this respect a new probability distribution of the magnitude was derived by maximizing a non-extensive generalization of the Boltzmann-Gibbs entropy, namely the Tsallis entropy (Tsallis, 2009). The result is the q-exponential distribution, whose shape parameter q is called the entropic index and is found to characterize the subadditive (q>1) and superadditive (q<1) regimes; the exponential distribution is recovered as q tends to 1. Two active seismogenic areas in central Italy are analyzed, the former including the L'Aquila sequence that occurred in 2009 and the latter the Amatrice-Norcia sequence that occurred in 2016. Parameter estimation is performed by following the Bayesian approach in order to exploit the prior knowledge on the phenomenon (Vallianatos et al., 2016) and to assess the uncertainties. A detailed analysis of the variations of both q index and Tsallis entropy is performed by estimating them over time windows of a fixed number of events that shift at each new event. Both the q index and the Tsallis entropy show signicant and lasting decreases before the first strong earthquake in the sequences, and sudden increases after them. This indicates that these quantities offer clues on the activation state of the systems (Rotondi et al., 2022). In the literature the q-exponential distribution has been used to study other seismic parameters like seismic moment, inter-event times, Euclidean distance between successive events. Following the example of Schoenberg et al. (2009), we have thought that the q-exponential distribution could describe, in addition to earthquake magnitude, also the spatial properties of seismicity such as the area of the Voronoi cells generated by the epicentral locations. We extend our analysis to the areas of Voronoi cells generated by the recent seismicity recorded in the study regions. Then we compare the performance of the estimated q-distribution with that of some promising probability distributions of the Voronoi cell area, namely the tapered Pareto distribution and the generalized Gamma distribution. Also in this case the analysis of the time variations of the probability model that provides the best tting to the datasets provides us information on the phases of the seismic cycle. Future work aims to analyze different seismic areas in order to test and consolidate the results.