The model

To model afterslip at Parkfield we discretize a 65-km-long segment of the San Andreas fault into rectangular patches of uniform slip dislocations in a homogeneous elastic half-space. We assume the slip rate distribution on the fault prior to the earthquake is that estimated by Murray et al. (2001) who inverted GPS velocities averaged over 1991-1998 for the interseismic slip rate. Also following Murray et al. (2001), we assume the fault is locked down to 15 km depth to the southeast of Parkfield, creeping at 25 mm/yr down to 15 km depth northwest of Parkfield, and creeping at 32 mm/yr everywhere below 15 km depth. These large creeping sections of the fault are assumed to extend infinitely along the strike of the San Andreas fault. We envision that the area of the fault bounded by micro-seismicity (Nadeau et al., 2005) is locked between earthquakes and is surrounded by aseismic fault creep. The locked part of the fault ruptures during the earthquake and creep in the surrounding areas accelerates to relax the coseismic stress load. We assume the coseismic stresses relax during afterslip according to the following Dieterich-Ruina formulation of rate-state friction, equations (1) and (2), together with the equation relating stress on the fault to slip
\begin{displaymath}
\tau=\sigma \left\{\mu + A \ln(V/V^*) + B \ln(V^*\theta/d_c)
\right\}
\end{displaymath} (18.1)


\begin{displaymath}
{d \theta \over dt}= 1 - {V \theta \over d_c }
\end{displaymath} (18.2)

where $\tau$ is shear stress on the fault, $\sigma$ is the normal stress, $V$ is sliding velocity, $V^*$ is a reference velocity, $\mu$ is the nominal coefficient of friction at the steady reference velocity, $\theta$ is a state variable that evolves with time, $A$ and $B$ are laboratory-derived constants, and $d_c$ is the so-called critical slip distance. $d_c$ is interpreted as an indication of the size of asperity contacts and is thought of as the slip necessary to renew surface contacts. In this formulation, the state, $\theta$, can be interpreted as the average asperity contact time because it increases linearly with time at zero slip velocity. The conditions on stress and state before the earthquake are obtained from the interseismic slip rate distribution (Murray et al., 2001) assuming steady state conditions ( $d \theta / dt =
0$). The initial condition on stress immediately after the earthquake (the beginning of the afterslip period) is the pre-earthquake stress plus the stress change caused by the earthquake. The objective is to invert GPS data with a forward model of the coseismic and postseismic processes to obtain estimates of the frictional parameters, $A$, $B$, and $d_c$. The two-step forward model produces a coseismic slip distribution and the resulting afterslip distribution that is driven by the coseismic stress change. In the first step, we perform a linear slip inversion of the coseismic GPS data for the coseismic slip distribution. In the second step, we specify the rate-state friction parameters and initial conditions and solve for the evolution of afterslip as. $\sigma A$, $\sigma B$, and $D_c$ vary linearly with depth.

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