Two-dimensional elastic full waveform inversion using Born and Rytov formulations in the frequency domain

SUMMARY We perform the full elastic waveform inversion in the frequency domain in a 2-D geometry. This method allows imaging of two physical seismic parameters, using vertical and horizontal field components. The forward problem is discretized using finite difference, allowing to simulate the full elastic wavefield propagation. Moreover, it is solved in the frequency domain, a fast approach for multisource and multireceiver acquisition. The non-linear inversion is based on a pre-conditioned gradient method, where Born and Rytov formulations are used to compute Frechet derivatives. Parameter perturbations linearly depend on fields perturbations in the Born kernel, and on the generalized complex phases of fields in the Rytov kernel, giving different Frechet derivatives. The gradient is pre-conditioned with the diagonal part of the inverse Hessian matrix, allowing to better estimate the stepping in the optimization direction. Non-linearity is taken into account by updating parameters at each iteration and proceeding from low to high frequencies. The latter allows as well to progressively introduce smaller wavelengths in parameter images. On a very simple synthetic example, we examine the way the inversion determines the Vp (P-wave velocity) and Vs (S-wave velocity) images. We highlight that, with a transmission acquisition, final parameter images weakly depend on the chosen formulation to compute Frechet derivatives and on the inverted parameters choice. Of course, convergence strongly depends on the medium wavenumber illumination which is related somehow to the acquisition geometry. With a reflection acquisition, the Born formulation allows to better recover scatterers. Moreover, the medium anomalies are not well reconstructed when surface waves propagate in the medium. This may be due to the evanescent nature of surface waves. By selecting first body waves and then surface waves, we improve the convergence and properly reconstruct anomalies. This shows us that preparation of the seismic data before the inversion is as critical as the initial model selection.