Topological Properties of Epidemic Aftershock Processes

Abstract

Earthquakes in seismological catalogs and acoustic emission events in lab experiments can be statistically described as point events in linear Hawkes processes, where the spatiotemporal rate is a linear superposition of background intensity and aftershock clusters triggered by preceding activity. Traditionally, statistical seismology interpreted these models as the outcome of epidemic branching processes, where one-to-one causal links can be established between mainshocks and aftershocks. Declustering techniques are used to infer the underlying triggering trees and relate their topological properties with epidemic branching models. Here, we review how the standard Epidemic Type Aftershock Sequence (ETAS) model extends from the Galton-Watson branching processes and bridges two extreme cases: Poisson and scale-free power law trees. We report the statistical laws expected in triggering trees regarding some topological properties. We find that the statistics of such topological properties depend exclusively on two parameters of the standard ETAS model: the average branching ratio nb and the ratio between exponents α and b characterizing the production of aftershocks and the distribution of magnitudes, respectively. In particular, the classification of clusters into bursts and swarms proposed by Zaliapin and Ben-Zion (2013b, https://doi.org/10.1002/jgrb.50178) appears naturally in the aftershock sequences of the standard ETAS model depending on nb and α/b. On the other hand swarms can also appear by false causal connections between independent events in nontectonic seismogenic episodes. From these results, one can use the memory-less Galton-Watson as a null model for empirical triggering processes and assess the validity of the ETAS hypothesis to reproduce the statistics of natural and artificial catalogs.