Statistics of seismic cluster durations

TL;DR

This paper provides a rigorous theoretical analysis of seismic cluster durations using the ETAS model, revealing statistical properties, magnitude distributions, and power-law behaviors in different regimes.

Contribution

It introduces explicit generating probability functions for seismic cluster statistics, including durations and magnitudes, and uncovers novel insights into their distributional behaviors.

Findings

Magnitude difference follows Gutenberg-Richter law

Cluster durations have power-law tails with regime-dependent behavior

Criticality affects the cascade speed of seismic events

Abstract

Using the standard ETAS model of triggered seismicity, we present a rigorous theoretical analysis of the main statistical properties of temporal clusters, defined as the group of events triggered by a given main shock of fixed magnitude m that occurred at the origin of time, at times larger than some present time t. Using the technology of generating probability function (GPF), we derive the explicit expressions for the GPF of the number of future offsprings in a given temporal seismic cluster, defining, in particular, the statistics of the cluster's duration and the cluster's offsprings maximal magnitudes. We find the remarkable result that the magnitude difference between the largest and second largest event in the future temporal cluster is distributed according to the regular Gutenberg-Richer law that controls the unconditional distribution of earthquake magnitudes. For earthquakes obeying the Omori-Utsu law for the distribution of waiting times between triggering and triggered events, we show that the distribution of the durations of temporal clusters of events of magnitudes above some detection threshold \nu has a power law tail that is fatter in the non-critical regime $n<1$ than in the critical case n=1. This paradoxical behavior can be rationalised from the fact that generations of all orders cascade very fast in the critical regime and accelerate the temporal decay of the cluster dynamics.