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 time 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 impedance. 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.

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