Stability of earthquake clustering models: Criticality and branching ratios

Jiancang Zhuang, Maximilian J. Werner, and David S. Harte
Phys. Rev. E 88, 062109 – Published 4 December 2013

Abstract

We study the stability conditions of a class of branching processes prominent in the analysis and modeling of seismicity. This class includes the epidemic-type aftershock sequence (ETAS) model as a special case, but more generally comprises models in which the magnitude distribution of direct offspring depends on the magnitude of the progenitor, such as the branching aftershock sequence (BASS) model and another recently proposed branching model based on a dynamic scaling hypothesis. These stability conditions are closely related to the concepts of the criticality parameter and the branching ratio. The criticality parameter summarizes the asymptotic behavior of the population after sufficiently many generations, determined by the maximum eigenvalue of the transition equations. The branching ratio is defined by the proportion of triggered events in all the events. Based on the results for the generalized case, we show that the branching ratio of the ETAS model is identical to its criticality parameter because its magnitude density is separable from the full intensity. More generally, however, these two values differ and thus place separate conditions on model stability. As an illustration of the difference and of the importance of the stability conditions, we employ a version of the BASS model, reformulated to ensure the possibility of stationarity. In addition, we analyze the magnitude distributions of successive generations of the BASS model via analytical and numerical methods, and find that the compound density differs substantially from a Gutenberg-Richter distribution, unless the process is essentially subcritical (branching ratio less than 1) or the magnitude dependence between the parent event and the direct offspring is weak.

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  • Received 27 December 2012

DOI:https://doi.org/10.1103/PhysRevE.88.062109

©2013 American Physical Society

Authors & Affiliations

Jiancang Zhuang*

  • The Institute of Statistical Mathematics, Research Organization of Information and Systems, Tokyo 190-8562, Japan and Department of Statistical Science, The Graduate University for Advanced Studies, Tokyo 190-8562, Japan

Maximilian J. Werner

  • Department of Geosciences, Princeton University, New Jersey 08544, USA

David S. Harte

  • GNS Science, Wellington, New Zealand

  • *zhuangjc@ism.ac.jp
  • mwerner@princeton.edu
  • d.harte@gns.cri.nz

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Issue

Vol. 88, Iss. 6 — December 2013

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