Eigenvector models for solving the seismic inverse problem for the Helmholtz equation

Abstract

We study the seismic inverse problem for the recovery of subsurface properties in acoustic media. In order to reduce the ill-posedness of the problem, the heterogeneous wave speed parameter is represented using a limited number of coefficients associated with a basis of eigenvectors of a diffusion equation, following the regularization by discretization approach. We compare several choices for the diffusion coefficient in the partial differential equations, which are extracted from the field of image processing.We first investigate their efficiency for image decomposition (accuracy of the representation with respect to the number of variables). Next, we implement the method in the quantitative reconstruction procedure for seismic imaging, following the full waveform inversion method, where the difficulty resides in that the basis is defined from an initial model where none of the actual structures is known. In particular, we demonstrate that the method may be relevant for the reconstruction of media with salt-domes.We use themethod in 2-D and 3-D experiments, and showthat the eigenvector representation compensates for the lack of low-frequency information, it eventually serves us to extract guidelines for the implementation of the method.