A reduced-order variational Bayesian approach for efficient subsurface imaging

This work considers the reconstruction of a subsurface model from seismic observations, which is known to be a high-dimensional and ill-posed inverse problem. Two approaches are combined to tackle this problem: the discrete cosine transform (DCT) approach, used in the forward modelling step, and the variational Bayesian (VB) approach, used in the inverse reconstruction step. VB can provide not only point estimates but also closed forms of the full posterior probability distributions. To efficiently compute such estimates of the full joint posterior distributions of large-scale seismic inverse problems, we resort to a DCT order-reduction scheme with a VB approximation of the posteriors, avoiding the need for costly Bayesian sampling methods. More specifically, we first reduce the model parameters through truncation of their DCT coefficients. This helps regularizing our seismic inverse problem and alleviates its computational complexity. Then, we apply a VB inference in the reduced-DCT space to estimate the dominant (retained) DCT coefficients together with the variance of the observational noise. We also present an efficient implementation of the derived VB-based algorithm for further cost reduction. The performances of the proposed scheme are evaluated through extensive numerical experiments for both linear and nonlinear forward models. In the former, the subsurface reflectivity model was reconstructed at a comparable estimation accuracy as the optimal weighted-regularized-least-squares solution. In the latter, the main structural features of the squared slowness model were well reconstructed.