Vere-Jones' self-similar branching model
- Mathematical Department, Nizhny Novgorod State University, Gagarin prosp. 23, Nizhny Novgorod, 603950 (Russian Federation)
- Institute of Geophysics and Planetary Physics and Department of Earth and Space Sciences, University of California, Los Angeles, California 90095 (United States)
Motivated by its potential application to earthquake statistics as well as for its intrinsic interest in the theory of branching processes, we study the exactly self-similar branching process introduced recently by Vere-Jones. This model extends the ETAS class of conditional self-excited branching point-processes of triggered seismicity by removing the problematic need for a minimum (as well as maximum) earthquake size. To make the theory convergent without the need for the usual ultraviolet and infrared cutoffs, the distribution of magnitudes m{sup '} of daughters of first-generation of a mother of magnitude m has two branches m{sup '}<m with exponent {beta}-d and m{sup '}>m with exponent {beta}+d, where {beta} and d are two positive parameters. We investigate the condition and nature of the subcritical, critical, and supercritical regime in this and in an extended version interpolating smoothly between several models. We predict that the distribution of magnitudes of events triggered by a mother of magnitude m over all generations has also two branches m{sup '}<m with exponent {beta}-h and m{sup '}>m with exponent {beta}+h, with h=d{radical}(1-s), where s is the fraction of triggered events. This corresponds to a renormalization of the exponent d into h by the hierarchy of successive generations of triggered events. For a significant part of the parameter space, the distribution of magnitudes over a full catalog summed over an average steady flow of spontaneous sources (immigrants) reproduces the distribution of the spontaneous sources with a single branch and is blind to the exponents {beta},d of the distribution of triggered events. Since the distribution of earthquake magnitudes is usually obtained with catalogs including many sequences, we conclude that the two branches of the distribution of aftershocks are not directly observable and the model is compatible with real seismic catalogs. In summary, the exactly self-similar Vere-Jones model provides an attractive new approach to model triggered seismicity, which alleviates delicate questions on the role of magnitude cutoffs in other non-self-similar models. The new prediction concerning two branches in the distribution of magnitudes of aftershocks could be tested with recently introduced stochastic reconstruction methods, tailored to disentangle the different triggered sequences.
- OSTI ID:
- 20709837
- Journal Information:
- Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, Vol. 72, Issue 5; Other Information: DOI: 10.1103/PhysRevE.72.056122; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); ISSN 1063-651X
- Country of Publication:
- United States
- Language:
- English
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