Sparse Variational Bayesian Inversion for Subsurface Seismic Imaging

This article considers the Bayesian inversion of subsurface models from seismic observations, a well-known high-dimensional and ill-posed problem. A reduction of the parameters' space is considered following a truncated discrete cosine transform (DCT), as a first regularization of such a problem. The (reduced) problem is then formulated within a Bayesian framework, with a second regularization based on Laplace priors to account for sparsity. Two Laplace-based penalizations are applied: one for the DCT coefficients and one for the spatial variations of the subsurface model, in order to enhance the structure of the cross correlations of the DCT coefficients. In terms of modeling, the Laplace priors are represented by hierarchical forms, suitable for the derivation of efficient inversion algorithms. The derivation is based on the variational Bayesian (VB) approach, which approximates the joint posterior probability density function (pdf) of the target parameters together with the observation noise variance and the hyperparameters of the introduced priors by a separable product of their marginal pdfs under the Kullback-Leibler divergence (KLD) minimization criterion. The proposed VB inversion scheme is iterative, endowed with a computational complexity that scales linearly with the number of retained DCT coefficients. Its performances are evaluated and compared against a recently introduced Gaussian prior-based method through extensive numerical experiments for both linear and nonlinear forward modeling. In particular, the imposed sparsity through Laplace priors has been found to improve the reconstruction of subsurface models, as long as subsurface structures lend themselves to sparse model parameterization.