Variational inference full-waveform inversion using geostatistical prior particles

ABSTRACT Bayesian full-waveform inversion (FWI) addresses the problem of the nonuniqueness of solutions in traditional deterministic FWI by quantifying the model uncertainties, which can be realized by a variational inference (VI) approach. As an efficient VI algorithm, the Stein variational gradient descent (SVGD) has been used to develop a VI-based FWI method, which approximates the posterior probability density function using the distribution of a particle set. However, the SVGD-based FWI method reported in the existing literature usually uses some weak priors, such as a uniform distribution, to generate prior particles (or models, a term commonly used in the geophysics community). The particles generated from such priors exhibit random structures. Although this maximizes the retention of all possible solutions, it usually requires a large number of iterations to ensure convergence to the results with clear geologic implications. To address this, a geostatistical method was introduced to extract geologic structure information from seismic images and this information was used to generate prior particles. Specifically, the particles were generated by perturbing a smooth model with products of a pattern-feature correlation (PFC) matrix and random vectors. The elements of the PFC matrix, quantitatively determined as correlation coefficients of the pattern score vectors at each point, represent similarities of the geologic patterns at different positions. To reduce the storage amount of the PFC matrix and eliminate the spurious spatial correlations, which typically occur between two spatially distant points, the variogram function in geostatistics was adopted to determine the maximum correlation radius and the PFC matrix was sparsified according to this radius. The sparsified PFC matrix was then used to generate informed prior particles for SVGD-based FWI. Numerical examples demonstrate clearly that using the geostatistical prior particles as initial particles enhances the convergence of SVGD-based FWI visibly and yields an accurate characterization of the posterior distribution of the velocity model.