3.2.1 Stress-driven Afterslip Simulation
We carried out stress-driven afterslip simulations using the open-source
software RELAX, which solves for the nonlinear time-dependents (x ,t ) in the Fourier domain under the assumption
of rate-strengthening friction on faults (equation 1) (Barbot et al.,
2009a). The afterslip evolution history on a given patch of the fault is
controlled by the rate-strengthening constitutive law (Barbot et al.,
2009a).
\(s\left(t\right)=\frac{\Delta\tau_{0}}{G^{*}}[1-\frac{2}{k}\coth^{-1}(e^{\frac{t}{t_{0}}}\coth\coth\ \frac{k}{2}\ )]\)(1)
In equation 1, \(k=\frac{\Delta\tau_{0}}{\text{aσ}}\) is the
dimensionless ratio that controls the nonlinearity during the slip, and
the time evolution is controlled by k along with the relaxation
time \(t_{0}=\frac{1}{2V_{0}}\frac{\text{aσ}}{G^{*}}\text{\ \ }\).
Note that the parameter a in the equations of Barbot et al.
(2009), and as used here, is more commonly identified as
(a -b ) in the context of full rate and state friction.
Larger values of k result in models that are more strongly
non-linear, with a more rapid decay in slip velocity early in the
postseismic period (these models also require shorter time step and thus
result in much longer program execution time). \(\Delta\tau_{0}\) refers
to the shear stress perturbation due to the earthquake, σ refers to the
effective normal stress on fault, and G* is the effective elastic
constant per unit area determined by the linear dimension L and the
shear modulus. The relaxation time\(t_{0}=\frac{1}{2V_{0}}\frac{\text{aσ}}{G^{*}}\text{\ \ }\)depends on
both aσ and the reference sliding velocity on the fault \(V_{0}\). Total
slip as goes to infinity is limited to \(\frac{{}_{0}}{{}^{*}}\).
Thus, there are 2 unknown parameters to search for to solve this
problem: aσ and \(V_{0}\). Many studies assume a value ofaσ and search only for \(V_{0}\), due to the strong parameter
tradeoff between the two values when only one time period is considered
(e.g., Tian et al., 2020). We first performed a 2-d grid search foraσ and \(V_{0}\) over a relatively large range of parameter
values to find the best fit values. We calculated the reduced\(\chi^{2}\) using the three sites on the Alaska Peninsula (AC40, AB13
and AB21) that are most sensitive to the downdip afterslip.
When we consider only one time interval, for example three months, then
a very wide range of aσ values, varying by orders of magnitude,
yield models that fit the data equally well. Large values of aσ(such as 3MPa suggested by Tian et al. (2020)) produce an afterslip
evolution history at GPS sites like the orange curve in Figure 3,
showing a low degree of nonlinearity, while small values of aσ(similar to those used by Wang and Bürgmann (2020) or Zhao et al.
(2022)) produce models like the blue curve in Figure 3, showing a higher
degree of nonlinearity. Because the observations at 3 weeks more closely
align with the curve produced by smaller values for aσ (Figure 3
gray star), we limit the range of parameter values to those similar to
those of Zhao et al. (2022) and consider displacement predictions for
two time windows, 0-3 weeks after the mainshock and 0-3 months after.
Based on the total misfit and given the nonlinear nature of the very
early afterslip evolution, we fix the value of aσ to be 0.6 MPa.
Given that the two time windows we have used are short, using a
different value of aσ in our models would produce an equally good
fit, with a correspondingly different \(V_{0}\) value. In this study, we
vary the \(V_{0}\) value for each different model scenario that we
consider in the following sections, and we leave the question of whether
it is possible to determine an optimal value of aσ to a future
study with a longer time span.