Distribution of the Largest Aftershocks in Branching Models of Triggered Seismicity: Theory of the Universal Bath Law

Alexander I. Saichev, & Didier Sornette

Published May 2005, SCEC Contribution #863

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 Bath's law. Our theory shows that Bath's law holds only sufficiently close to the critical regime of the ETAS branching process. Allowing for error bars +/- 0.1 for Bath's constant value around 1.2, our exact analytical treatment of Bath's law provides new constraints on the productivity exponent alpha and the branching ratio n: 0.9 less than or similar to alpha <= 1 and 0.8 less than or similar to n <= 1. We propose a method for measuring alpha based on the predicted renormalization of the Gutenberg-Richter distribution of the magnitudes of the largest aftershock. We also introduce the "second Bath law for foreshocks:" the probability that a main earthquake turns out to be the foreshock does not depend on its magnitude rho.

Citation
Saichev, A. I., & Sornette, D. (2005). Distribution of the Largest Aftershocks in Branching Models of Triggered Seismicity: Theory of the Universal Bath Law. Physical Review E, 71(5), 056127. doi: 10.1103/PhysRevE.71.056127.