Characterization of the tail of the distribution of earthquake magnitudes by combining the GEV and GPD descriptions of Extreme Value Theory

• Main Authors: Pisarenko, V. F., Sornette, A., Sornette, D., Rodkin, M. V.
• Format: Report
• Language: unknown
• Published: arXiv 2008
• Subjects: Geophysics physics.geo-ph; Data Analysis, Statistics and Probability physics.data-an; FOS Physical sciences; Fennoscandia
• Online Access: https://dx.doi.org/10.48550/arxiv.0805.1635; https://arxiv.org/abs/0805.1635

We present a generic and powerful approach to study the statistics of extreme phenomena (meteorology, finance, biology...) that we apply to the statistical estimation of the tail of the distribution of earthquake sizes. The chief innovation is to combine the two main limit theorems of Extreme Value Theory (EVT) that allow us to derive the distribution of T-maxima (maximum magnitude occurring in sequential time intervals of duration T) for arbitrary T. We propose a method for the estimation of the unknown parameters involved in the two limit theorems corresponding to the Generalized Extreme Value distribution (GEV) and to the Generalized Pareto Distribution (GPD). We establish the direct relations between the parameters of these distributions, which permit to evaluate the distribution of the T-maxima for arbitrary T. The duality between the GEV and GPD provides a new way to check the consistency of the estimation of the tail characteristics of the distribution of earthquake magnitudes for earthquake occurring over arbitrary time interval. We develop several procedures and check points to decrease the scatter of the estimates and to verify their consistency. We test our full procedure on the global Harvard catalog (1977-2006) and on the Fennoscandia catalog (1900-2005). For the global catalog, we obtain the following estimates: Mmax = 9.53 +- 0.52; quantile(0.97)==9.21 +- 0.20. For Fennoscandia, we obtain Mmax = 5.76 +- 0.165; quantile(0.97) =5.44 +- 0.073. The estimates of all related parameters for the GEV and GPD, including the most important form parameter, are also provided. : 40 pages including 17 figures