Renormalization of earthquake aftershocks

TL;DR

This paper develops a theoretical model explaining the observed dual power-law regimes in earthquake aftershock decay, linking crustal temperature and heat flow to variations in Omori's law exponent.

Contribution

It introduces a renormalization approach for aftershock sequences, revealing two distinct regimes with different decay exponents based on earthquake triggering dynamics.

Findings

Two power-law regimes in aftershock decay with exponents p_- and p_+

The crossover time t^* depends on the fraction of triggered earthquakes

Higher crustal heat flow correlates with larger Omori's law exponent

Abstract

Together with the Gutenberg-Richter distribution of earthquake magnitudes, Omori's law is the best established empirical characterization of earthquake sequences and states that the number of smaller earthquakes per unit time triggered by a main shock decays approximately as the inverse of the time ($1/t^p$, with $p \approx 1$) since the main shock. Based on these observations, we explore the theoretical hypothesis in which each earthquake can produce a series of aftershock independently of its size according to its ``local'' Omori's law with exponent $p=1+\theta$. In this scenario, an aftershock of the main shock produces itself other aftershocks which themselves produce aftershocks, and so on. The global observable Omori's law is found to have two distinct power law regimes, the first one with exponent $p_-=1 - \theta$ for time $t < t^* \sim \kappa^{-1/\theta}$, where $0<1-\kappa <1$ measures the fraction of triggered earthquakes per triggering earthquake, and the second one with exponent $p_+=1 + \theta$ for larger times. The existence of these two regimes rationalizes the observation of Kisslinger and Jones [1991] that the Omori's exponent $p$ seems positively correlated to the surface heat flow: a higher heat flow is a signature of a higher crustal temperature, which leads to larger strain relaxation by creep, corresponding to fewer events triggered per earthquake, i.e. to a larger $\kappa$, and thus to a smaller $t^*$, leading to an effective measured exponent more heavily weighted toward $p_+>1$.