Objective Estimation of Spatially Variable Parameters of Epidemic Type Aftershock Sequence Model: Application to California

2 Abstract

The ETAS model is widely employed to model the spatio-temporal distribution of earthquakes, generally using spatially invariant parameters. We propose an efficient method for the estimation of spatially varying parameters, using the Expectation Maximization (EM) algorithm and spatial Voronoi tessellation ensembles. We use the Bayesian Information Criterion (BIC) to rank inverted models given their likelihood and complexity, and select the best models to finally compute an ensemble model at any location. Using a synthetic catalog, we also check that the proposed method correctly inverts the known parameters.
We apply the proposed method to earthquakes included in the ANSS catalog that occurred within the time period 1981-2015 in a spatial polygon around California.
The results indicate significant spatial variation of the ETAS parameters. We find that the efficiency of earthquakes to trigger future ones (quantified by the branching ratio) positively correlates with surface heat flow. In contrast, the rate of earthquakes triggered by far-field tectonic loading or background seismicity rate shows no such correlation, suggesting the relevance of triggering possibly through fluid-induced activation. Furthermore, the branching ratio and background seismicity rate are found to be uncorrelated with hypocentral depths, indicating that the seismic coupling remains invariant of hypocentral depths in the study region.
Additionally, triggering seems to be mostly dominated by small earthquakes. Consequently, the static stress change studies should not only focus on the Coulomb stress changes caused by specific moderate to large earthquakes, but also account for the secondary static stress changes caused by smaller earthquakes.

3 Introduction:

The Epidemic Type Aftershock Sequence (ETAS) model [Kagan and Knopoff, 1981, 1987; Ogata, 1988, 1998] is a widely used statistical model to describe the occurrence of earthquakes in space, time and magnitude. In this model, any earthquake irrespective of its size can trigger other (larger or smaller) earthquakes, which in turn can trigger more earthquakes and so on, leading to a cascade of triggering. The key feature of the ETAS model is the apparent lack of traditional labels such as foreshock, mainshock and aftershock [Helmstetter and Sornette, 2003a; Helmstetter et al., 2003], which are often used for earthquakes by seismologists [see, for instance, Gardner and Knopoff, 1974; Reasenberg, 1985; Zaliapin et al., 2008], based on the parsimonious assumption that the same physical mechanisms give rise to all earthquakes.

Notwithstanding its simplicity, the ETAS model has been exceptionally successful in describing the numerous statistical properties associated with earthquakes [see for e.g. Helmstetter and Sornette, 2002a, 2002b, 2003a and 2003b; Helmstetter et al., 2003]. However, it fails to account for several key properties of seismicity such as existence of stress shadow regions (where seismicity rate following an earthquake is suppressed) [see for e.g. Nandan et al., 2016; Meier et al., 2014]; multifractal nature of spatial distribution of earthquakes [Kamer et al., 2013]; magnitude dependent exponent of Omori law [Ouillon and Sornette, 2005] and so on. Despite these failures, it has been very successful (relative to other models) in forecasting the rates of future events, to the extent that it easily outperforms the physics based models of seismicity and ranks among the best models of earthquake forecasting developed to date [Werner et al., 2011; Console et al., 2007; Iwata, 2010; Dieterich, 1994].

4

Considering that the parameters of the ETAS model are the manifestations of the physical properties of the crust, which exhibit spatial variability, investigating the possible existence of spatial variability of ETAS parameters is justified. In fact, numerous case studies [see Utsu and Ogata, 1995 for list of references; Wiemer and Katsumata, 1999] have documented the variability of several ETAS parameters. For instance, the exponent of

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