The Centenary of the Omori Formula for a Decay Law of Aftershock Activity.

Abstract

The Omori formula n(t)=K(t+c)-1 and its modified form n(t)=K(t+c)-P have been successfully applied to many aftershock sequences since the former was proposed just 100 years ago. This paper summarizes studies using these formulae. The problems of fitting these formulae and related point process models to observational data are discussed mainly. Studies published during the last 1/3 century confirmed that the modified Omori formula generally provides an appropriate representation of the temporal variation of aftershock activity. Although no systematic dependence of the index p has been found on the magnitude of the main shock and on the lowest limit of magnitude above which aftershocks are counted, this index (usually p = 0.9-1.5) differs from sequence to. sequence. This variability may be related to the tectonic condition of the region such as structural heterogeneity, stress, and temperature, but it is not clear which factor is most significant in controlling the p value. The constant c is a controversial quantity. It is strongly influenced by incomplete detection of small aftershocks in the early stage of sequence. Careful analyses indicate that c is positive at least for some sequences. Point process models for the temporal pattern of shallow seismicity must include the existence of aftershocks, most suitably expressed by the modified Omori law. Among such models, the ETAS model seems to best represent the main features of seismicity with only five parameters. An anomalous decrease in aftershock activity below the level predicted by the modified Omori formula sometimes precedes a large aftershock. An anomalous decrease in seismic activity of a region below the level predicted by the ETAS model is sometimes followed by a large earthquake in the same or in a neighboring region.

Apparent criticality and calibration issues in the Hawkes self-excited point process model: application to high-frequency financial data

“…which describes the modified Omori-Utsu law of aftershock rates [16,17]. The time constant c regularizes the behavior of the power law kernel at very short times.…”

Source parameters and stress release of seismic sequences occurred in the Friuli-Venezia Giulia region (Northeastern Italy) and in Western Slovenia

“…The decay of the aftershock activity is characterized by p=0.8 (Tab. 2), a low value if compared to the p values (0.6-1.55) of well documented sequences quoted by Utsu et al (1995).…”

Comparison of postseismic afterslip models with aftershock seismicity for three subduction-zone earthquakes: Nias 2005, Maule 2010 and Tohoku 2011

“…K S is a measure of the aftershock productivity of the sequence and the exponent p is usually between 0.9 and 1.5 with a median of about 1.1 (Reasenberg & Jones 1989;Utsu et al 1995). The variable C S is often a fraction of a day (e.g.…”

“…The variable C S is often a fraction of a day (e.g. Utsu et al 1995;Enescu et al 2009) but it is unclear whether this has a physical reason or simply reflects decreased catalogue completeness early in the aftershock sequence (Utsu 2002). Integrating eq.…”