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.

翻译：扩散模型为深度生成建模任务提供了稳定的训练和最先进的性能。本文探讨其在多元地下建模与概率反演中的应用。我们首先证明，相较于变分自编码器和生成对抗网络，扩散模型增强了多元建模能力。在扩散建模中，生成过程涉及相对较多的时步，其更新规则可被修改以纳入条件数据。针对Chung等人（2023）提出的经典扩散后验采样方法，我们提出了多种修正方案，特别引入了一种考虑扩散模型固有噪声污染的似然近似方法。我们在包含岩相与相关声阻抗的多元地质场景中评估了模型性能，并分别利用局部硬数据（测井曲线）和非线性地球物理数据（全叠加地震数据）展示了条件建模效果。测试结果表明：相较于原始方法，新方法显著提升了统计稳健性，增强了对后验概率密度函数的采样效率，并降低了计算成本。该方法可单独或同时使用硬数据与间接条件数据。由于反演过程内嵌于扩散流程中，其计算速度优于需要在生成模型外部进行循环迭代的方法（如马尔可夫链蒙特卡洛）。