Diffusion models for multivariate subsurface generation and efficient probabilistic inversion

TL;DR

It is demonstrated that diffusion models enhance multivariate modeling capabilities compared to variational autoencoders and generative adversarial networks, and a likelihood approximation accounting for the noise-contamination that is inherent in diffusion modeling is introduced.

Abstract

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.

Figures (16)

• Figure 1: Prior distribution as represented in the training dataset: (a) Facies and $I_P$ realizations; (b) $I_P$ distribution per facies; (c) probability of sand distribution; (d) point-wise average and standard deviation of $I_P$.
• Figure 2: Summary of unconditional modeling performances of (a, b) DM, (c, d) GAN, and (e, f) VAE. Subplots (a), (c), (e): realizations and average distribution of facies and $I_P$ for (a) DM, (c) GAN and (e) VAE; subplots (b), (d), (f): joint and $I_P$ distributions for (b) DM, (d) GAN and (f) VAE.
• Figure 3: Test model used in inverse modeling: (a) real subsurface distribution ("True") and location of the two wells used for the linear inversion; (b) conditioning well log data for the linear inversion; (c) conditioning seismic data for the nonlinear inversion.
• Figure 4: Modeling error: (a) examples of errors at specific noise levels; (b) standard deviation of the homoscedastic error considered. The values refer to the raw network output, of range $[0,1]$.
• Figure 5: Evolution of prior (dashed) and likelihood (continuous) score functions generative steps, expressed as L2-norm, for (a) linear conditioning and (b) nonlinear conditioning case studies; and WRMSE of 100 samples generated as solutions of the (c) linear and (d) nonlinear inverse problems. The values of magnitude of the three data noise assumed are indicated in Table \ref{['tab:Table2']}.