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.

翻译：我们提出了一种基于伴随的优化方法，用于反演地震建模中使用的应力与摩擦参数。正问题为线性弹性动力学，包含非线性速率-状态断层摩擦模型。失配函数量化了接收器位置处模拟与实测质点位移或速度之间的差异，并可包含时窗或滤波算子。我们推导了相应的伴随问题，即具有线性化速率-状态摩擦的线性弹性问题，其时间依赖系数由正问题解导出。通过断层上正演与伴随变量的卷积，可高效计算失配函数的梯度。该方法将全波形反演框架拓展至包含速率-状态摩擦的摩擦断层系统。此外，我们针对反平面剪切中具有粗糙断层的动态破裂问题，提出了一种时空对偶相容离散格式：采用高阶精度求和-分块有限差分与显式龙格-库塔时间积分相结合。该离散格式的对偶相容性保证了离散伴随梯度既是离散失配函数的精确梯度，又是连续梯度的一致近似。理论结果通过合成数据反演得到了验证。我们预计，基于伴随的地震和/或大地测量数据反演将成为研究地震源过程的强大工具，也可用于解释实验室摩擦实验。