Aftershock Sequences

After a large event, the so-called main shock, aftershocks are smaller earthquakes resulting of perturbations of the state of stress. In a Limited Power Law model, the aftershock rate results from the superposition of independent exponential decay rates with different characteristic time constants: these characteristic times to failure represent the time-dependent strength of the different seismogenic domains within the aftershock zone according to a mechanism of static fatigue.

You can derive the Modified Omori law from the Limited Power Law (see the comparison here).

In the Limited Power Law model, the power-law aftershock decay rate is limited in time by two characteristic rate constants (see Narteau et al., 2002):

Over short times, a linear decay rate dominates:

Considering that aftershocks result from a steplike perturbation of stress in the neighbourhood of a triggering event, the model predicts that the amplitude of this perturbation controls the duration of the linear regime of the aftershock decay rate over short time.  We  use this time delay before the onset of the power-law aftershock decay rate to estimate variation of the differential shear stress in active tectonic settings (see Narteau et al., 2005, Narteau et al., 2008 and Narteau et al., 2009).

Over long times, an exponential decay rate dominates:

Under low stress these strengthening processes may dominate, and prevent rupture initiation. For small stress, we adopt a different time-depedent behavior by considering a fracturing threshold. Below this threshold, the time required to produce an aftershock becomes infinite. This may be described as a minimum velocity of subcritical crack growth. As a result, the power-law aftershock decay rate is remplaced by an exponential decay over long times (see Narteau et al., 2003).


A limited power-law model

for the aftershock decay rate