A numerical study of multi-parameter full waveform inversion with iterative regularization using multi-frequency vibroseis data

Abstract

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.

Identification of Cavities and Inclusions in Linear Elasticity with a Phase-Field Approach

“…Identification of defects from boundary measurements plays an important role in non-destructive testing for damage assessment of mechanical specimens, which are possibly defective due to the presence of interior voids or cavities appearing during the manufacturing process, see, for instance, [33,47,55,63] for possible applications to 3D-printing and additive manufacturing. This kind of inverse problems has application also in medical imaging and in particular in elastography, a modality mapping the elastic properties and stiffness of soft tissue, [6-8, 31, 59, 60, 64] (to cite a few), and in reflection seismology [20,62], a non invasive technique used by the oil and gas industry to map petroleum deposits in the Earth's upper crust and based on seismic data from land acquisition, see for example [61]. We also mention some applications in volcanology, see for example [9,10,58] and references therein.…”

“…For other reconstruction approaches we refer to the review paper [17] and references therein. Identification of cavities and elastic inclusions could be interpreted as a special case of the determination of Lamé parameters from boundary measurements, see for example [7,41] and [61].…”

Identification of cavities and inclusions in linear elasticity with a phase-field approach

“…For other reconstruction approaches we refer to the review paper [14] and references therein. Identification of cavities and elastic inclusions could be read as a special case of determination of Lamé parameters from boundary measurements, see for example [6,36] and [53].…”