Posterior sampling with convolutional neural network-based plug-and-play regularization with applications to poststack seismic inversion

Abstract

Uncertainty quantification is a crucial component in any geophysical inverse problem, as it provides decision-makers with valuable information about the inversion results. Seismic inversion is a notoriously ill-posed inverse problem, due to the band-limited and noisy nature of seismic data; as such, quantifying the uncertainties associated with the ill-posed nature of this inversion process is essential for qualifying the subsequent interpretation and decision-making processes. Selecting appropriate prior information is a crucial—yet, non-trivial—step in probabilistic inversion since it influences the ability of sampling-based inference algorithms to provide geologically-plausible posterior samples. However, the necessity to encapsulate prior knowledge into a probability distribution can greatly limit our ability to define expressive priors. To address this limitation, and following in the footsteps of the Plug-and-Play methodology for deterministic inversion, we present a regularized variational inference framework that performs posterior inference by implicitly regularizing the Kullback-Leibler divergence loss—a measure of the distance between the approximated and target probabilistic distributions—with a CNN-based denoiser. We call this new algorithm Plug-and-Play Stein Variational Gradient Descent (PnP-SVGD) and demonstrate its ability to produce high-resolution, trustworthy samples that realistically represent subsurface structures. The proposed method is validated on both synthetic and field post-stack seismic data.