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.

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的摩擦参数。正演问题为非线性速率-状态摩擦断层条件下的线弹性力学问题。失配泛函量化了接收位置处模拟与实测质点位移或速度的差异，且失配项可包含时窗或滤波算子。我们推导了相应的伴随问题，该问题为线性速率-状态摩擦条件下的线弹性力学问题。通过将正演变量与伴随变量在断层面上进行卷积运算，可高效计算失配梯度。因此，该方法将全波形反演框架扩展至包含速率-状态摩擦断层的场景。此外，我们针对反平面剪切中粗糙断层的动态破裂问题，提出了一种时空对偶一致的离散格式，采用高阶求和-分部有限差分结合显式龙格-库塔时间积分方法。该离散格式的对偶一致性确保了离散伴随梯度既是离散代价泛函的精确梯度，又是连续梯度的一致逼近。我们的理论结果通过合成数据反演得到了验证。