Bayesian extreme values distribution for seismicity parameters assessment in South America

O. Ch. Galanis^{1}, T. M. Tsapanos^{1}, G. A. Papadopoulos^{2} and A. A. Kiratzi^{1}

1 Aristotle University of Thessaloniki, School of Geology, Geophysical Laboratory, 54006 Thessaloniki, Greece.

**Abstract**: Seismicity parameters in South America are estimated with the application of Bayesian statistics. The distribution is named “distribution of extreme values”, because in order to compute the parameters of seismicity, the distribution of maximum magnitude (i.e. an extreme value of magnitude in a given time span) is used. A concise analysis of the theory applied is given. Some simple and generally accepted hypotheses, like the Poisson model for the temporal distribution of earthquakes, as well as the exponential one for the distribution of the magnitudes of the earthquakes are adopted. Starting with these initial hypotheses and by means of Bayesian statistics, the distribution of the maximum expected magnitude for a given time span and the expected mean return period for a given magnitude are assessed. This is done in two steps. Firstly the temporal distribution of earthquake is found, while in the second step their magnitude’s distribution is determined. Then the parameters of the latter distributions are estimated. According to the methodology of Bayesian statistics, in order to start with, prior values are assigned to the parameters. Utilising the conjugate distribution theory, the posterior values are estimated using the prior values and the observational data. As these posterior parameters do not have an obvious physical meaning, they are transformed to four familiar parameters of seismicity. These are the rate of occurrence of earthquakes, l
, the parameter of the distribution of magnitude, b
, and their respective standard deviations. Seismicity parameters estimated and are expressed in terms of the probability of exceeding of a given magnitude in a given time span, and the mean return period of a given magnitude. These parameters are useful for any seismic hazard assessment in this area.

**Keywords:** Extreme-values, Bayesian Estimators, Seismicity Parameters, South America.