Source Estimation by Full Wave Form Inversion

We consider the inverse problem of estimating the parameters describing a seismic source, based on time-dependent ground motion recordings at a number of receiver stations. The source is modeled as a point moment tensor forcing, characterized by its location, moment tensor components, start time, and frequency parameter in the time function. In total, there are 11 unknown parameters. We use a non-linear conjugate gradient algorithm to minimize the full waveform misfit between observed and computed ground motions at the receiver stations. An important underlying assumption of the minimization problem is that seismic wave propagation can be accurately modeled by the elastic wave equation in a heterogeneous isotropic material. We use a fourth order accurate finite difference method to evolve the seismic waves in time. The discretization satisfies a summation by parts property that guarantees stability of the explicit time-stepping scheme. The adjoint of the discretized elastic wave equation is used to compute the gradient of the misfit, which is needed by the non-linear conjugated gradient minimization algorithm. A new moment tensor source discretization is derived that is twice continuously differentiable with respect to the source location. It guarantees that the Hessian of the misfit is a continuous function of the source parameters. We show how the Hessian can be calculated by solving 11 elastic wave equations and one adjoint wave equation. Because Hessian of the unscaled problem has a very large condition number, a preconditioner must be used to scale the parameters in the non-linear conjugated gradient algorithm. Compared to several other scaling approaches, we find that the diagonal of the Hessian provides the most reliable alternative. Numerical experiments are presented for estimating the source parameters from synthetic data in a layer over half-space problem (LOH.1), demonstrating good convergence properties of the proposed approach.