Estimation of Tail Distribution of the Annual Maximum Earthquake Magnitude Using Extreme Value Theory

The goal of many earthquake engineering analyses is to ensure that a structure can withstand a given level of ground shaking while maintaining a desired level of performance. But there is a great deal of uncertainty about the numerous factors that impact the level of ground shaking. probabilistic seismic hazard analyses (PSHA) aim to quantify these uncertainties and combine them to produce an explicit description of the distribution of future shaking that may occur at a site. Among others, the distribution of the magnitudes is required by PSHA. This paper presents a new approach to estimate both the tail distribution of earthquake magnitudes and the tail distribution of the maximum earthquake magnitude over a year. This approach uses extreme value models based on Poisson process. The main innovations consist of combining the use of the generalized Pareto distribution (GPD) to model the tail distribution of earthquake magnitudes and the exact relation leading to the tail distribution of the annual maximum earthquake magnitude. We also propose a statistical inference that takes into account the growing incompleteness of the data in earlier times, which enables us to use as much available data as we can. This paper also provides a sensitivity analysis, which quantifies the robustness of the results with respect to two types of uncertainty: the statistical uncertainty due to the limited number of data used for the statistical inference and the uncertainty related to the determination of the complete observation periods of time by the seismologists. We detail validation tests to check the main assumptions underlying the probabilistic model: the Poisson occurrences of earthquakes and the GPD behavior of the tail distribution. The methodology is applied to an area located in the south of France, which provides estimates of some extreme quantiles of the annual maximum earthquake magnitude.