A fractional approach to study the pure-temporal Epidemic Type Aftershock Sequence (ETAS) process for earthquakes modeling

TL;DR

This paper introduces a fractional differential equation approach to modeling the pure temporal ETAS process in earthquakes, providing analytical solutions that enhance understanding of seismicity patterns over time.

Contribution

It presents a novel fractional calculus framework for the ETAS model, simplifying classical methods and offering new insights into earthquake intensity decay.

Findings

Analytical solutions for the fractional ETAS model are derived.

The fractional approach captures decay behavior of seismic activity.

Application to Japanese earthquake catalog demonstrates model relevance.

Abstract

In statistical seismology, the Epidemic Type Aftershocks Sequence (ETAS) model is a branching process used world-wide to forecast earthquake intensity rates and reproduce many statistical features observed in seismicity catalogs. In this paper, we describe a fractional differential equation that governs the earthquake intensity rate of the pure temporal ETAS model by using the Caputo fractional derivative and we solve it analytically. We highlight that the tools and special functions of fractional calculus simplify the classical methods employed to obtain the intensity rate and let us describe the change of solution decay for large times. We also apply and discuss the theoretical results to the Japanese catalog in the period 1965-2003.