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A method is proposed for accurately describing arbitrary-shaped free boundaries in finite-difference schemes for elastodynamics, in a time-domain velocity–stress framework. The basic idea is as follows: fictitious values of the solution are built in vacuum, and injected into the numerical integration scheme near boundaries. The most original feature of this method is the way in which these fictitious values are calculated. They are based on boundary conditions and compatibility conditions satisfied by the successive spatial derivatives of the solution, up to a given order that depends on the spatial accuracy of the integration scheme adopted. Since the work is mostly done during the pre-processing step, the extra computational cost is negligible. Stress-free conditions can be designed at any arbitrary order without any numerical instability, as numerically checked. Using 10 grid nodes per minimal S-wavelength with a propagation distance of 50 wavelengths yields highly accurate results. With 5 grid nodes per minimal S-wavelength, the solution is less accurate but still acceptable. A subcell resolution of the boundary inside the Cartesian meshing is obtained, and the spurious diffractions induced by staircase descriptions of boundaries are avoided. Contrary to what occurs with the vacuum method, the quality of the numerical solution obtained with this method is almost independent of the angle between the free boundary and the Cartesian meshing. [ABSTRACT FROM AUTHOR]

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 Fréchet 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 Fréchet 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 ( P-wave velocity) and V s ( S-wave velocity) images. We highlight that, with a transmission acquisition, final parameter images weakly depend on the chosen formulation to compute Fréchet 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. [ABSTRACT FROM AUTHOR]