We propose a model for earthquake afterslip based on rate and state variable friction laws. In the model, afterslip is attributed to the interaction of a velocity-weakening region at depth (within which earthquakes nucleate) with an upper region of velocity-strengthening frictional behavior. The existence of this upper region is supported by independent seismologic observations and the results of laboratory friction experiments. In our model, afterslip is the result of relaxation of a stress perturbation within the velocity-strengthening region, which arises when an earthquake propagates into that region from below. We derive the stress perturbation and its decay from the friction constitutive law using a simple, 1 degree-of-freedom approximation for the elastic interaction between the fault and its surroundings. This approximation is based on thickness-averaged displacements and slip velocities within the velocity-strengthening region, which is assumed to slip as a rigid block. Coseismic and postseismic slip are coupled through the thickness-averaged stiffness k of the velocity-strengthening region. We assume k to be inversely proportional to the thickness of this region, which means that thicker velocity strengthening regions have a greater tendency to arrest coseismic slip. We model the afterslip-time histories of the 1966 Parkfield and 1987 Superstition Hills earthquakes and relate the model parameters to physical parameters which may govern the rheologic behavior of the faults. In accord with field observations, our model predicts (1) that afterslip on some faults scales with the thickness of the (unconsolidated) sedimentary cover and (2) that proportionally more afterslip occurs for earthquakes in which coseismic surface slip is small compared with coseismic slip at depth. Velocity-strengthening fricitional behavior is to be expected for faults within poorly consolidated sediments and for those that contain significant gouge zones (about > 500 m) within their shallow regions (< 3-5 km). Combining our results with those of recent laboratory friction studies indicates that relatively young faults with little accumulated fault gouge should exhibit little afterslip.
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