Eur. Phys. J. B 49, 377-401 (2006)
https://doi.org/10.1140/epjb/e2006-00075-3

Power law distribution of seismic rates: theory and data analysis

¹ Mathematical Department, Nizhny Novgorod State University, Gagarin Prosp. 23, Nizhny Novgorod, 603950, Russia
² Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA, 90095, USA
³ Institute of Geophysics and Planetary Physics and Department of Earth and Space Sciences, University of California, Los Angeles, CA, 90095, USA
⁴ Laboratoire de Physique de la Matière Condensée, CNRS UMR 6622 and Université de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France

Corresponding author: ^a sornette@moho.ess.ucla.edu

Received: 28 September 2005
Revised: 1 January 2006
Published online: 10 March 2006

Abstract

We report an empirical determination of the probability density functions P_data(r) (and its cumulative version) of the number r of earthquakes in finite space-time windows for the California catalog, over fixed spatial boxes 5 ×5 km², 20 ×20 km² and 50 ×50 km² and time intervals and 1000 days. The data can be represented by asymptotic power law tails together with several cross-overs. These observations are explained by a simple stochastic branching process previously studied by many authors, the ETAS (epidemic-type aftershock sequence) model which assumes that each earthquake can trigger other earthquakes (“aftershocks”). An aftershock sequence results in this model from the cascade of aftershocks of each past earthquake. We develop the full theory in terms of generating functions for describing the space-time organization of earthquake sequences and develop several approximations to solve the equations. The calibration of the theory to the empirical observations shows that it is essential to augment the ETAS model by taking account of the pre-existing frozen heterogeneity of spontaneous earthquake sources. This seems natural in view of the complex multi-scale nature of fault networks, on which earthquakes nucleate. Our extended theory is able to account for the empirical observation but some discrepancies, especially for the shorter time windows, point to limits of both our theoretical approach and of the ETAS model.