Constitutive Law for Earthquake Production Based on Rate-and-State Friction: Dieterich 1994 Revisited

Dieterich (1994, https://doi.org/10.1029/93JB02581) derived a constitutive law for earthquake production based on rate-and-state friction, which has since been applied widely to earthquake triggering in various tectonic settings. Here this influential work is revisited and rederived in a more straightforward manner. Our derivation is based on computing the time to instability for a population of sources, and eschews the seismicity state variable. We demonstrate the validity of the Dieterich (1994) model for arbitrary shear stressing history at constant normal stress; however, our results differ slightly if fault normal stress changes with time. We provide simple integral expressions for the cumulative number of events and seismicity rate for arbitrary stressing history. These expressions have no explicit dependence on the time derivative of the stressing history. The simpler derivation makes it easier to assess and generalize various model assumptions in the original formulation. A principal success of the Dieterich (1994) theory is that it predicts and gives physical meaning to the Omori decay of seismicity rate following a stress step. We analyze the assumption that sources are well above steady state and find that Omori decay is produced by sources that either start or end up above steady state following the stress step. If no sources are brought above steady state by the perturbation then the time to reach steady state must be considered, and there will be no Omori sequence.