Distribution of the largest aftershocks in branching models of triggered seismicity: Theory of the universal Båth law

A. Saichev and D. Sornette
Phys. Rev. E 71, 056127 – Published 31 May 2005

Abstract

Using the epidemic-type aftershock sequence (ETAS) branching model of triggered seismicity, we apply the formalism of generating probability functions to calculate exactly the average difference between the magnitude of a mainshock and the magnitude of its largest aftershock over all generations. This average magnitude difference is found empirically to be independent of the mainshock magnitude and equal to 1.2, a universal behavior known as Båth’s law. Our theory shows that Båth’s law holds only sufficiently close to the critical regime of the ETAS branching process. Allowing for error bars ±0.1 for Båth’s constant value around 1.2, our exact analytical treatment of Båth’s law provides new constraints on the productivity exponent α and the branching ratio n: 0.9α1 and 0.8n1. We propose a method for measuring α based on the predicted renormalization of the Gutenberg-Richter distribution of the magnitudes of the largest aftershock. We also introduce the “second Båth law for foreshocks:” the probability that a main earthquake turns out to be the foreshock does not depend on its magnitude ρ.

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  • Received 20 January 2005

DOI:https://doi.org/10.1103/PhysRevE.71.056127

©2005 American Physical Society

Authors & Affiliations

A. Saichev1,2 and D. Sornette3,4,*

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

  • *Electronic address: sornette@moho.ess.ucla.edu

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Vol. 71, Iss. 5 — May 2005

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