Physics – Geophysics
Scientific paper
2005-01-19
Phys. Rev. E 71, 056127 (2005)
Physics
Geophysics
25 pages with 8 figures
Scientific paper
10.1103/PhysRevE.71.056127
Using the 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: $0.9 <= alpha <= 1$ and 0.8 <= n <= 1. We propose a novel 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's law for foreshocks: the probability that a main earthquake turns out to be the foreshock does not depend on its magnitude.
Saichev Alexander
Sornette Didier
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