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 nontrivial — step in probabilistic inversion because 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 (PnP) methodology for deterministic inversion, we develop 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 convolutional neural network-based denoiser. We call this new algorithm PnP Stein variational gradient descent and determine its ability to produce high-resolution trustworthy samples that realistically represent subsurface structures. Our method is validated on synthetic and field poststack seismic data.