Earthquake Recurrence Model Based on the Generalized Pareto Distribution for Unequal Observation Periods and Imprecise Magnitudes

Abstract

Seismic risk analyses derive from earthquake catalogs a recurrence relation linking earthquake activity rate to magnitude. The most widely employed model is the log-linear Gutenberg–Richter relation (Gutenberg and Richter, Science, 83, 183–185, 1936; Gutenberg and Richter, Bulletin of the Seismological Society of America, 46(3), 105–145, 1945), with modifications at larger magnitudes (Cosentino et al., Bulletin of the Seismological Society of America, 67, 1615–1623, 1977; Kijko and Sellevoll, Bulletin of the Seismological Society of America, 79(3), 644–654, 1989; Page, Bulletin of the Seismological Society of America, 58, 1131–1168, 1968; Pisarenko and Sornette, Pure and Applied Geophysics, 160, 2343–2364, 2003; Turcotte, Physics of the Earth and Planetary Interiors, 111, 275–293, 1999). This relation leads to exponentially distributed magnitudes truncated to a maximum magnitude, a priori fixed under geophysical considerations. In this paper, we assume seismic events occur according to a Poisson distribution, but we propose to model the tail distribution of magnitudes with a generalized Pareto distribution (GPD). The GPD parameters are estimated with a maximum likelihood procedure. This GPD-based model gives rise to a new recurrence model that differs from the Gutenberg–Richter Law. It eliminates the need to introduce a maximum magnitude in the analysis that is difficult to determine. This paper details the expression of the estimators of the GPD parameters and the asymptotic normal distribution when the shape parameter $\xi >-1$. This asymptotic distribution yields confidence intervals for all parameters. The GPD parameter estimators account for the following features of the data set: (a) seismic events are collected on periods whose span depends on their magnitudes; (b) magnitudes are imprecisely known: each magnitude is supposed to uniformly belong to an interval of length 0.5. Our new model is estimated from information coming from the FCAT17 catalog. This catalog collects seismic events from the Alps region in France. We conduct an uncertainty analysis, and we quantify the impact of estimation uncertainty on the recurrence model.