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.

Random Self-Similar Trees: Emergence of Scaling Laws

“…For example, this sequence is necessary to generate tree distributions that are time-invariant, critical, and having i.i.d. edge lengths (Kovchegov and Zaliapin 2018, 2019, 2020Kovchegov et al 2021).…”

“…Hierarchical representation of seismicity by branching processes is also well explored, starting from the pioneering works of Kagan (1973), Kagan andKnopoff (1976, 1981), and Vere-Jones (1976). A very popular epidemic-type aftershock sequence (ETAS) model of earthquake dynamics (Ogata 1998) is a Galton-Watson branching process with a power-law offspring distribution and space-time-magnitude marks (Saichev et al 2005;Baro ´2020). A trajectory of this process is a tree graph whose vertices represent individual earthquakes and edges-triggering processes.…”

“…For the HBP model, this assumption is satisfied only for a particular one-parameter class of trees, called critical Tokunaga process, that we describe in this section. The critical Tokunaga trees enjoy many additional symmetries as discussed by Kovchegov and Zaliapin (2018, 2019, 2020. The class is sufficiently broad and includes the critical binary Galton-Watson process with exponential edge lengths as a special case.…”

Topological Comparison Between the Stochastic and the Nearest‐Neighbor Earthquake Declustering Methods Through Network Analysis

“…This study focuses on the nearest-neighbor and the stochastic declustering algorithms because they can be used not only to identify background seismicity but also to investigate the connections between events forming seismic clusters (Baró, 2020;Guo et al, 2015Guo et al, , 2017Peresan & Gentili, 2018;Zaliapin & Ben-Zion, 2013;Zhuang et al, 2004). The aim is to compare the features of clusters identified by the two algorithms exploiting tools and measurements from network analysis.…”