We study the inverse boundary value problem for time-harmonic elastic waves, for the recovery of P– and S-wave speeds from vibroseis data or the Neumann-to-Dirichlet map. Our study is based on our recent result pertaining to the uniqueness and a conditional Lipschitz stability estimate for parametrizations on unstructured tetrahedral meshes of this inverse boundary value problem. With the conditional Lipschitz stability estimate, we design a procedure for full waveform inversion (FWI) with iterative regularization. The iterative regularization is implemented by projecting gradients, after scaling, onto subspaces associated with the mentioned parametrizations yielding Lipschitz stability. The procedure is illustrated in computational experiments using the continuous Galerkin finite element method of recovering the rough shapes and wave speeds of geological bodies from simple starting models, near and far from the boundary, that is, the free surface.

Shi, J., Beretta, E., Maarten, V., Francini, E., & Vessella, S. (2019). A numerical study of multi-parameter full waveform inversion with iterative regularization using multi-frequency vibroseis data. Computational Geosciences, 1-19. https://link.springer.com/article/10.1007/s10596-019-09897-6