Characterization of the Frequency of Extreme Events by the Generalized Pareto Distribution

Recent results in extreme value theory suggest a new technique for statistical estimation of distribution tails (Embrechts et al., 1997), based on a limit theorem known as the GnedenkoPickands-Balkema-de Haan theorem. This theorem gives a natural limit law for peak-overthreshold values in the form of the Generalized Pareto Distribution (GPD), which is a family of distributions with two parameters. The GPD is successfully applied in a number of statistical problems related to finance, insurance, hydrology, and other domains. Here, we apply the GPD approach to the well-known seismological problem of earthquake energy distribution described by the Gutenberg-Richter seismic moment-frequency law. We analyze shallow earthquakes (depth h < 70 km) in the Harvard catalog over the period 1977-2000 in 18 seismic zones. The whole GPD is found to approximate the tails of the seismic moment distributions quite well above M > 10 24 dyne-cm (i.e., moment-magnitudes larger than mW=5.3) and no statistically significant regional difference is found for subduction and transform seismic zones. We confirm that the b-value is very different (b=1.50±0.09 corresponding to a power law exponent close to 1) in mid-ocean ridges compared to other zones (b=1.00±0.05 corresponding to a power law exponent close to 2/3) with a very high statistical confidence. We propose a physical mechanism for this, contrasting slow healing ruptures in mid-ocean ridges with fast healing ruptures in other zones. The GPD can as well be applied in many problems of seismic risk assessment on a regional scale. However, in certain cases, deviations from the GPD at the very end of the tail may occur, in particular for large samples signaling a novel regime. Such deviations are detected here in the sample containing earthquakes from all major subduction zones (sample size of 4985 events). We propose a new statistical test of significance of such deviations based on the bootstrap method. The number of events deviating from the tails of GPD in the studied data sets (15-20 at most) is not sufficient for determining the functional form of those deviations. Thus, it is practically impossible to give preference to one of the previously suggested parametric families describing the ends of tails of seismic moment distributions: Gamma (Kagan, 1997), modified Gutenberg-Richter (Bird et al., 2000), Pareto with a crossover point (Pacheco et al., 1992; Sornette et al., 1996), or stretched exponential (Laherrere and Sornette, 1998). Possible statistical inference in such situation is discussed.