Estimating stress heterogeneity from aftershock rate

We estimate the rate of aftershocks triggered by a heterogeneous stress change, using the rate-and-state model of Dieterich [1994]. We show than an exponential stress distribution P (τ) ∼ exp(−τ/τ0) gives an Omori law decay of aftershocks with time ∼ 1/t, with an exponent p = 1−Aσn/τ0, where A is a parameter of the rate-and-state friction law, and σn the normal stress. Omori exponent p thus decreases if the stress ”heterogeneity” τ0 decreases. We also invert the stress distribution P (τ) from the seismicity rate R(t), assuming that the stress does not change with time. We apply this method to a synthetic stress map, using the (modified) scale invariant ”k” slip model [Herrero and Bernard, 1994]. We generate synthetic aftershock catalogs from this stress change. The seismicity rate on the rupture area shows a huge increase at short times, even if the stress decreases on average. Aftershocks are clustered in the regions of low slip, but the spatial distribution is more diffuse than for a simple slip dislocation. Because the stress field is very heterogeneous, there are many patches of positive stress changes everywhere on the fault. This stochastic slip model gives a Gaussian stress distribution, but nevertheless produces an aftershock rate which is very close to Omori’s law, with an effective p ≤ 1, which increases slowly with time. The inversion of the full stress distribution P (τ) is badly constrained for negative stress values, and for very large positive values, if the time interval of the catalog is limited. However, constraining P (τ) to be a Gaussian distribution allows a good estimation of P (τ) for a limited number of events and catalog duration. We show that stress shadows are very difficult to observe in a heterogeneous stress context.