Bulk-friction modeling of afterslip and the modified Omori law
Afterslip data from the Superstition Hills fault in southern California, a creep event on the same fault, the modified Omori law, and cumulative moments from aftershocks of the 1957 Aleutian Islands earthquake all indicate that the original formulation by Dieterich (1981) [Constitutive properties of faults with simulated gouge. AGU, Geophys. Monogr. 24, 103–120] for friction evolution is more appropriate for systems far from instability than the commonly used approximation developed by Ruina (1983) [Slip instability and state variable friction laws. J. Geophys. Res. 88, 10359–10370] to study instability. The mathematical framework we use to test the friction models is a one-dimensional, massless spring-slider under the simplifying assumption, proposed by Scholz (1990) [The Mechanics of Earthquakes and Faulting. Cambridge University Press] and used by Marone et al. (1991) [On the mechanics of earthquake afterslip. J. Geophys. Res., 96: 8441–8452], that the state variable takes on its velocity-dependent steady-state value throughout motion in response to a step in stress. This assumption removes explicit state-variable dependence from the model, obviating the need to consider state-variable evolution equations. Anti-derivatives of the modified Omori law fit our data very well and are very good approximate solutions to our model equations. A plausible friction model with Omori-law solutions used by Wesson (1988) [Dynamics of fault creep. J. Geophys. Res. 93, 8929–8951] to model fault creep and generalized by Rice (1983) [Constitutive relations for fault slip and earthquake instabilities. Pure Appl. Geophys. 121, 443–475] to a rate-and-state variable friction model yields exactly Omori's law with exponents greater than 1, but yields unstable solutions for Omori exponents less than 1. We estimate from the Dieterich formulation the dimensionless parameter a∗ which is equal to the product of the nominal coefficient of friction and the more commonly reported friction parameter a . We find that a∗ is typically positive, qualitatively consistent with laboratory observations, although our observations are considerably larger than laboratory values. However, we also find good model fits for a∗ < 0 when data correspond to Omori exponents less than 1. A modification of the stability analysis by Rice and Ruina (1983) [Stability of steady frictional slipping. J. Appl. Mech. 50, 343–349] indicates that a∗ < 0 is not a consequence of our assumption regarding state-variable evolution. A consistent interpretation of a∗ < 0 in terms of laboratory models appears to be that the data are from later portions of processes better characterized by two-state-variable friction models. a∗ < 0 is explained by assuming that our data cannot resolve the co-seismic evolution of a short-length-scale state variable to a velocity-weakening state; our parameterization leads to an apparent negative instantaneous viscosity. We estimate the largest critical slip distance associated with afterslip to be ∼1–10 cm, consistent with other estimates for near-surface materials. We assume that our observed large values for a∗ reflect the fact that our model ignores the geometrical complexities of three-dimensional stresses in fractured crustal materials around a fault zone with frictional stresses that vary on a fault surface. Our one-dimensional model parameters reflect spatially averaged, bulk , stress and frictional properties of a fault zone, where we clearly cannot specify the details of the averaging process. Our analysis of Omori's law suggests that bulk-frictional properties of a fault zone are well described by our simple laboratory-based models, but they would need to change during the seismic cycle for a mainshock instability to recur, unless a mainshock-aftershock sequence were characterized by a process similar to the arrested instabilities possible in two-state-variable systems.