The stretched exponential function as an alternative model for aftershock decay rate

Abstract

The stretched exponential (Williams‐Watts) relaxation function is N*(t)=N*(0)exp[−(t/t0)q], 0<q≤1. As applied to aftershocks, N* (0) is the number of aftershocks that have not yet occurred at time t, starting with a finite number of potential events, N* (0). This function describes slow relaxation in a number of distinct physical systems. Whether it is the appropriate function for aftershocks, as an alternative to the modified Omori power law function, depends on the long‐time behavior of the mean interevent time. The behavior of this function has been tested on synthetic sequences in comparison with the Omori relation to determine the feasibility of determining which is the better model of aftershock rate decay. The differences are practically indistinguishable for very slowly decaying sequences but are measurable for normally and rapidly decaying sequences if excellent data are available. Maximum likelihood estimates of the parameters in the stretched exponential function were determined for 29 sequences and the fit of the model to the data was compared with the modified Omori fit by the Akaike Information Criterion. If the time of the rnainshock is taken as the start time for the sequence, the Omori fit is almost always better. If a later start time is used, 15 min to 2.5 hours for the cases tested, the stretched exponential fit is better in about half the cases. The selected start time is the time at which the rate of occurrence begins to decrease, after an initial apparent increase probably due to missed events during the very active interval. As expected, the relaxation time t0 is inversely related to the Omori p value and may prove to be more easily interpretable in terms of the physics of the aftershock‐generating process.

Mainshocks are aftershocks of conditional foreshocks: How do foreshock statistical properties emerge from aftershock laws

“…Omori law describing the power law decay ∼ 1/(t − t c ) p of aftershock rate with time from a mainshock that occurred at t c has been proposed more than one century ago [Omori, 1894], and has since been verified by many studies [Kagan and Knopoff, 1978;Davis and Frohlich, 1991;Kisslinger and Jones, 1991;Utsu et al, 1995]. See however [Kisslinger, 1993;Gross and Kisslinger, 1994] for alternative decay laws such as the stretched exponential and its possible explanation [Helmstetter and Sornette, 2002a].…”

Onset of power law aftershock decay rates in southern California

“…[3] In the last decades, the aftershock decay rate has been put on more solid physical ground as it has been related to fluid migration [Nur and Booker, 1972], rate-and-state friction [Dieterich, 1994], viscous rheology [Deng et al, 1999] or alternative relaxation mechanisms [Kisslinger, 1993]. Following this trend, our objective is to capture the main features of aftershock sequences through a limited number of parameters, which may be correlated with independent data sets [Narteau et al, 2003].…”

Comparing different models of aftershock rate decay: The role of catalog incompleteness in the first times after main shock

“…On the basis of physical considerations, originally made by Shlesinger and Montroll (1984), for a physical problem -the dielectric relaxation-that presents many analogies with aftershock decay, Kisslinger (1993) proposed the stretched exponential law (STREXP)…”