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

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 alpha and the branching ratio n: 0.9 approximately < alpha < or =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 Båth law for foreshocks:" the probability that a main earthquake turns out to be the foreshock does not depend on its magnitude rho.