Diffusion of Earthquake Aftershock Epicenters, Omori's Law and Generalized Continuous-Time Random Walk Models

TL;DR

This paper models earthquake aftershock distributions using a stochastic process linked to CTRW models, revealing how seismic diffusion depends on the Omori law exponent and providing a framework for understanding aftershock spatial-temporal patterns.

Contribution

It establishes an exact mapping between the ETAS seismic model and CTRW, enabling analysis of seismic diffusion regimes and the distinction between bare and renormalized Omori laws.

Findings

Seismic diffusion occurs only when the Omori exponent is less than 1.

The model predicts sub-diffusive behavior in aftershock spatial distribution.

Numerical simulations confirm the theoretical predictions.

Abstract

The epidemic-type aftershock sequence model (ETAS) is a simple stochastic process modeling seismicity, based on the two best-established empirical laws, the Omori law (power law decay ~1/t^{1+\theta} of seismicity after an earthquake) and Gutenberg-Richter law (power law distribution of earthquake energies). In order to describe also the space distribution of seismicity, we use in addition a power law distribution ~1/r^{1+\mu} of distances between triggered and triggering earthquakes. We present an exact mapping between the ETAS model and a class of CTRW (continuous time random walk) models, based on the identification of their corresponding Master equations. This mapping allows us to use the wealth of results previously obtained on anomalous diffusion of CTRW. We provide a classification of the different regimes of diffusion of seismic activity triggered by a mainshock. Specifically, we derive the relation between the average distance between aftershocks and the mainshock as a function of the time from the mainshock and of the joint probability distribution of the times and locations of the aftershocks. Our predictions are checked by careful numerical simulations. We stress the distinction between the ``bare'' Omori law describing the seismic rate activated directly by a mainshock and the ``renormalized'' Omori law taking into account all possible cascades from mainshocks to aftershocks of aftershock of aftershock, and so on. In particular, we predict that seismic diffusion or sub-diffusion occurs and should be observable only when the observed Omori exponent is less than 1, because this signals the operation of the renormalization of the bare Omori law, also at the origin of seismic diffusion in the ETAS model.