BULK-FRICTION MODELING OF AFTERSLIP AND THE MODIFIED OMORI LAW

Afterslip data from the Superstition Hills fault in southern Californi a, a creep event on the same fault, the modified Omori law, and cumula tive moments from aftershocks of the 1957 Aleutian Islands earthquake all indicate that the original formulation by Dieterich (1981) [Consti tutive 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. G eophys. Res. 88, 10359-10370] to study instability The mathematical fr amework we use to test the friction models is a one-dimensional, massl ess spring-slider under the simplifying assumption, proposed by Scholz (1990) [The Mechanics of Earthquakes and Faulting. Cambridge Universi ty Press] and used by Marone et al. (1991) [On the mechanics of earthq uake afterslip. J. Geophys. Res., 96: 8441-8452], that the state varia ble takes on its velocity-dependent steady-state value throughout moti on in response to a step in stress. This assumption removes explicit s tate-variable dependence from the model, obviating the need to conside r state-variable evolution equations. Anti-derivatives of the modified Omori law fit our data very well and are very good approximate soluti ons 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 fricti on model yields exactly Omori's law with exponents greater than 1, but yields unstable solutions for Omori exponents less than 1. We estimat e 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 fmd that a, is typica lly positive, qualitatively consistent with laboratory observations, a lthough our observations are considerably larger than laboratory value s. However, we also find good model fits for a, < 0 when data correspo nd to Omori exponents less than 1. A modification of the stability ana lysis by Rice and Ruina (1983) [Stability of steady frictional slippin g. J. Appl. Mech. 50, 343-349] indicates that a < 0 is not a conseque nce 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 characterize d by two-state-variable friction models. a < 0 is explained by assumi ng that our data cannot resolve the co-seismic evolution of a short-le ngth-scale state variable to a velocity-weakening state; our parameter ization leads to an apparent negative instantaneous viscosity. We esti mate the largest critical slip distance associated with afterslip to b e similar to 1-10 cm, consistent with other estimates for near-surface materials. We assume that our observed large values for a reflect th e fact that our model ignores the geometrical complexities of three-di mensional stresses in fractured crustal materials around a fault zone with frictional stresses that vary on a fault surface. Our one-dimensi onal model parameters reflect spatially averaged, bulk, stress and fri ctional properties of a fault zone, where we clearly cannot specify th e details of the averaging process. Our analysis of Omori's law sugges ts that bulk-frictional properties of a fault zone are well described by our simple laboratory-based models, but they would need to change d uring 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.