Adjoint-based inversion of frictional parameters in earthquake modeling

We present an adjoint-based optimization method to invert for frictional parameters used in earthquake modeling. The forward problem is linear elasticity with nonlinear rate-and-state frictional faults. The misfit functional quantifies the difference between simulated and measured particle displacements or velocities at receiver locations. The misfit may include windowing or filtering operators. We derive the corresponding adjoint problem, which is linear elasticity with linear rate-and-state friction. The gradient of the misfit is efficiently computed by convolving forward and adjoint variables on the fault. The method thus extends the framework of full-waveform inversion to include frictional faults with rate-and-state friction. In addition, we present a space-time dual-consistent discretization of a dynamic rupture problem with a rough fault in antiplane shear, using high-order accurate summation-by-parts finite differences in combination with explicit Runge-Kutta time integration. The dual consistency of the discretization ensures that the discrete adjoint-based gradient is the exact gradient of the discrete cost functional as well as a consistent approximation of the continuous gradient. Our theoretical results are corroborated by inversions with synthetic data.