dorsal/arxiv
View SchemaDistribution of the Largest Aftershocks in Branching Models of Triggered Seismicity: Theory of the Universal Bath's law
| Authors | A. Saichev, D. Sornette |
|---|---|
| Categories | |
| ArXiv ID | physics/0501102 |
| URL | https://arxiv.org/abs/physics/0501102 |
| DOI | 10.1103/PhysRevE.71.056127 |
| Journal | Phys. Rev. E 71, 056127 (2005) |
Abstract
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.
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"abstract": "Using the ETAS branching model of triggered seismicity, we apply the\nformalism of generating probability functions to calculate exactly the average\ndifference between the magnitude of a mainshock and the magnitude of its\nlargest aftershock over all generations. This average magnitude difference is\nfound empirically to be independent of the mainshock magnitude and equal to\n1.2, a universal behavior known as Bath\u0027s law. Our theory shows that Bath\u0027s law\nholds only sufficiently close to the critical regime of the ETAS branching\nprocess. Allowing for error bars +- 0.1 for Bath\u0027s constant value around 1.2,\nour exact analytical treatment of Bath\u0027s law provides new constraints on the\nproductivity exponent alpha and the branching ratio n: $0.9 \u003c= alpha \u003c= 1$ and\n0.8 \u003c= n \u003c= 1. We propose a novel method for measuring alpha based on the\npredicted renormalization of the Gutenberg-Richter distribution of the\nmagnitudes of the largest aftershock. We also introduce the ``second Bath\u0027s law\nfor foreshocks: the probability that a main earthquake turns out to be the\nforeshock does not depend on its magnitude.",
"arxiv_id": "physics/0501102",
"authors": [
"A. Saichev",
"D. Sornette"
],
"categories": [
"physics.geo-ph",
"physics.gen-ph"
],
"doi": "10.1103/PhysRevE.71.056127",
"journal_ref": "Phys. Rev. E 71, 056127 (2005)",
"title": "Distribution of the Largest Aftershocks in Branching Models of Triggered Seismicity: Theory of the Universal Bath\u0027s law",
"url": "https://arxiv.org/abs/physics/0501102"
},
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