Diffusion models for multivariate subsurface generation and efficient probabilistic inversion

Diffusion models offer stable training and state-of-the-art performance for deep generative modeling tasks. Here, we consider their use in the context of multivariate subsurface modeling and probabilistic inversion. We first demonstrate that diffusion models enhance multivariate modeling capabilities compared to variational autoencoders and generative adversarial networks. In diffusion modeling, the generative process involves a comparatively large number of generative steps with update rules that can be modified to account for conditioning data. We propose different corrections to the popular Diffusion Posterior Sampling approach by Chung et al. (2023). In particular, we introduce a likelihood approximation accounting for the noise-contamination that is inherent in diffusion modeling. We assess performance in a multivariate geological scenario involving facies and correlated acoustic impedances. Conditional modeling is demonstrated using both local hard data (well logs) and nonlinear geophysics (fullstack seismic data). Our tests show significantly improved statistical robustness, enhanced sampling of the posterior probability density function and reduced computational costs, compared to the original approach. The method can be used with both hard and indirect conditioning data, individually or simultaneously. As the inversion is included within the diffusion process, it is faster than other methods requiring an outer-loop around the generative model, such as Markov chain Monte Carlo. • Diffusion models enable accurate multivariate modeling of hydro-geological properties. • They significantly outperform variational autoencoders and generative adversarial networks. • We adapt diffusion posterior sampling for Bayesian inference, modifying the generative diffusion by an approximate likelihood score. • The inversion method performs well for both local and indirect geophysical data, in linear and nonlinear settings. • Compared to benchmarks, we demonstrate improved accuracy, robustness and reduced computational costs.