GNGTS 2018 - 37° Convegno Nazionale

360 GNGTS 2018 S essione 2.1 tends to get stuck around the maximum likelihood value of parameter; to overcome this issue we replace by ξ= ln[(2-q)/(q-1)]/ln 10, and we adopt the Normal distribution both as its prior and proposal distribution. Application. We applied the q-exponential distribution to the earthquakes recorded into the ISIDe (Italian Seismological Instrumental and parametric Data-base) catalog for the period 1 January 2014 through 30 June 2018, in the central Italy region included between the longitude values (12.7, 13.5), and latitude (42.3, 43.2) that was hit by the Amatrice-Norcia seismic sequence. Most of the magnitude values are expressed in , the remaining in and only few events are given in . To homogenize the data set we applied the orthogonal regressions indicated in Gasperini et al. (2013), and we analyzed the events of magnitude by partitioning them into five groups, each corresponding to a different year. We estimated the parameters of the q-exponential distribution (1) and of the classical exponential distribution to each data set and obtained as parameter estimates the posterior means reported in Table 1. Then we performed nonparametric tests to evaluate whether the various distributions estimated are significantly different; that is, we tested the null hypothesis that two independent samples were selected from populations having the same continuous distribution (or distributions with equal median) without assuming them to follow the normal distribution. To this end it is appropriate to use signed rank tests like the Mann–Whitney test or the Kruskal–Wallis test; the latter extends the former test when there are more than two groups. The samples may have different sizes but the results are more reliable if the sample sizes are similar; therefore we tested both samples of fixed size simulated, by the inverse method, from the estimated q-exponential and exponential distributions in each time interval, and the data sets of extremely different size. 2014 2015 2016 2017 2018 2014-18 1.397 1.371 1.482 1.457 1.441 1.470 2.822 3.069 2.098 2.339 2.472 2.177 Tab. 1 - Posterior means of the entropic index and of the parameter of the exponential distribution for the different data sets. Comments. The results support the conclusion that the analyzed seismic activity is generated by a critical process, in which long-range interactions in non-equilibrium states are expected. Consequently, the rate of decay of statistical dependence of two points with increasing time interval or spatial distance between the points should be slower that an exponential decay, typically a power-like decay. As for the magnitude distribution, the tests agree on accepting the null hypothesis (: distributions with equal median) for the pairs of years 2014-2018 and 2017- 2018, and on rejecting (significant differences) in the other pairs of years with the exception of the pair 2014-2015 for which the conclusion is uncertain. As expected, the magnitude distribution for the global data set is dominated by the data collected in 2016. References Gasperini P., Lolli B., Vannucci G. (2013) Empirical calibration of local magnitude data sets versus moment magnitude in Italy, BSSA, 103, 4, 2227-2246, doi: 10.1785/012012-356 Telesca L. (2012) Maximum likelihood estimation of the nonextensive parameters of the earthquake cumulative magnitude distribution, BSSA, 102, 2, 886-891 Tsallis C. (1988) Possible generalization of Boltzmann-Gibbs statistics, Journal of Statistical Physics, 52, 479–487, doi: 10.1007/BF01016429 Vallianatos F., Michas G., Papadakis G. (2016) A description of seismicity based on non-extensive statistical physics: a review , from: Earthquakes and Their Impact on Society, S. D’Amico (eds.), Springer Natural Hazards, doi: 10.1007/978-3-319-21753-6_1