Multi-level elastic full waveform inversion in isotropic media via quantitative Lipschitz stability estimates
Résumé
We study the inverse boundary value problem for the time-harmonic elastic wave equation and recovery of the Lam\'e parameters with an unknown fixed density model. The data represents a Neumann-to-Dirichlet map, which is a Hilbert-Schmidt operator and consequently is the residual. Thus the discretization is consistent with the use of the Least-Squares misfit function. The stability result for the elastic inverse problem shows that the Fr\'{e}chet derivative has a strictly positive `lower bound'. We estimate the lower bound via the computation of condition number and smallest eigenvalue of the Gauss-Newton Hessian. This provides a stability estimate to decide on the model parametrization and its representation during our multi-frequency algorithm. Additionally, we extend the adjoint method with free surface boundary condition to higher order to provide an option to deal with nonlinearities. We also demonstrate that adjoint equation is a boundary value problem when we restrict the receivers to the surface while it remains as an interior body force problem when we consider receivers in the domain (where we extend our domain via some type of absorbing boundary condition such as Perfect Match Layer). We eventually design a hierarchical compressed reconstruction in a multi-level scheme for the inverse problem associated with the time-harmonic elastic isotropic wave equation, at selected frequencies of the data. The compression is based on a domain partitioning of the subsurface, while the hierarchy is established through refinement. The coefficients are assumed to be piecewise constant functions following the domain partitioning. The partitioning is naturally defined with the Gauss-Newton Hessian information. It allows us to start inversion with minimal prior model information. We carry out numerical experiments in two and three dimensions.