We study a reconstruction algorithm for the general inverse scattering problem based on the estimate of not only medium properties, as in more conventional approaches, but also wavefields propagating inside the computational domain. This extended set of unknowns is justified as a way to prevent local minimum stagnation, which is a common issue for standard methods. At each iteration of the algorithm, (i) the model parameters are obtained by solution of a convex problem, formulated from a special bilinear relationship of the data with respect to properties and wavefields (where the wavefield is kept fixed), and (ii) a better estimate of the wavefield is calculated, based on the previously reconstructed properties. The resulting scheme is computationally convenient since step (i) can greatly benefit from parallelization and the wavefield update (ii) requires modeling only in the known background model, which can be sped up considerably by factorization-based direct methods. The inversion method is successfully tested on synthetic elastic datasets.

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Full-waveform inversion is a promising tool for a wide range of imaging scenario, in that it has the potential to harness the non-linear relationship between model parameters and data (as opposed to traditional methodologies), in order to produce truly quantitative results. Non-linearity represents an opportunity, in this sense, but it also begets local minimum issues when gradientbased optimization is employed. In this thesis, we are particularly interested in the quantitative estimation of the elastic parameters of the earth, such as compressibility, shear modulus, and density. If successful, this procedure brings geophysical imaging closer to the ultimate step of seismic exploration: the retrieval of pore pressure and rock properties. This would be of direct use for, say, the oil and gas industry. Other interesting applications are the description of the near surface/near ocean bottom, as a way to reduce drilling hazards, or non-destructive inspection of defects in oil and gas pipes, when ultrasounds are employed. @en

We investigate the numerical performance of an iterative solver for a frequency-domain finite-difference discretization of the isotropic elastic wave equation. The solver is based on the ‘shifted-Laplacian’ preconditioner, originally designed for the acoustic wave equation. This preconditioner represents a discretization of a heavily damped wave equation and can be efficiently inverted by a multigrid iteration. However, the application of multigrid to the elastic case is not straightforward because standard methods, such as point-Jacobi, fail to smooth the S-wave wavenumber components of the error when high P-to-S velocity ratios are present. We consider line smoothers as an alternative and apply local-mode analysis to evaluate the performance of the various components of the multigrid preconditioner. Numerical examples in 2-D demonstrate the efficacy of our method.@en