GNGTS 2018 - 37° Convegno Nazionale

GNGTS 2018 S essione 2.1 359 A MEASURE OF THE CRITICALITY OF A SEISMOGENIC SYSTEM FROM THE VIEWPOINT OF STATISTICAL PHYSICS R. Rotondi 1 , E. Varini 2 1 Istituto di Matematica Applicata e Tecnologie Informatiche “Enrico Magenes”, Italy 2 Consiglio Nazionale delle Ricerche, Milano, Italy Introduction. Despite the extreme complexity that characterizes the mechanism of the earthquake generation process, some phenomenological features seem to be identifiable in the collective properties of seismicity. It follows that the physics of many earthquakes has to be studied with a different approach than the physics of one earthquake and in this sense the use of statistical physics can be considered as necessary to understand the collective properties of earthquakes and to bridge the gap between physics-based models of individual events and statistics-based models of event populations. In this context the first question that arises is what type of statistical physics is appropriate to describe effects that affect different scales, variables with fractal distributions and long-range interactions. A possible answer could be non-extensive statistical physics (NESP), a generalization of the Boltzmann-Gibbs (BG) statistical physics that is based on the generalized entropic form, proposed by Tsallis in 1988, which recovers the BG entropy as a particular case. Following this approach, over the last two decades a series of studies has been performed on the power-law statistical distributions of energy release, inter-event time and spatial distances (Vallianatos et al ., 2016). In this work we focus on the distribution of the magnitude and deal with some inferential aspects of the so- called q-exponential distribution in the Bayesian framework. The same distribution has been examined by Telesca (2012) according to the frequentist approach. NESP formulation of a probability distribution. Let us consider a continuous variable X with a probability distribution f(X) ; in geophysics this variable could be seismic moment, inter-event time or distance between successive earthquakes. Tsallis entropy of f(X) is given by: where q is the entropic index. For q= 1 it reduces to the well-known Shannon entropy, whereas for q 1 it is not additive, that is, we have (A+B) = (A) + (B) + (1-q) (A) (B) for any two independent systems; in particular, if q<1 we speak of super-additivity, and of sub-additivity when q . The probability distribution f(X) is defined by maximizing the Tsallis entropy under appropriate constraints about, e.g., the generalized expectation value and the normalization constant. In this way f(X) turns out to be a q-exponential distribution, a special case of the generalized Pareto distribution. If we assume M 2/3 E, then we have: (1) being the magnitude threshold. In the Bayesian perspective the two parameters a and q are considered as random variables; for computational reasons we reparametrize the distribution by setting = (2-q)/(q-1). To estimate the parameters a and in the Bayesian framework we assign their prior distributions, both LogNormal with hyperparameters such that ( a )= 25, ( a ) = 64 and = 2, ()= 4 respectively, borrowing these values from the literature. The posterior distributions are obtained as the equilibrium distributions of Markov chains generated by the Metropolis- Hastings algorithm with LogNormal proposal distributions depending on the current value of the chain. When the number of data is particularly large (some thousands) the Markov chain