  We present an adjoint-based optimization method to invert for stress and
frictional parameters used in earthquake modeling. The forward problem is
linear elastodynamics 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 linearized rate-and-state friction
with time-dependent coefficients derived from the forward solution. 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 misfit functional as well as a consistent
approximation of the continuous gradient. Our theoretical results are
corroborated by inversions with synthetic data. We anticipate that
adjoint-based inversion of seismic and/or geodetic data will be a powerful tool
for studying earthquake source processes; it can also be used to interpret
laboratory friction experiments.
