Geophysical inverse problems with measurement-guided diffusion models

Solving inverse problems with the reverse process of a diffusion model represents an appealing avenue to produce highly realistic, yet diverse solutions from incomplete and possibly noisy measurements, ultimately enabling uncertainty quantification at scale. However, because of the intractable nature of the score function of the likelihood term (i.e., $\nabla_{\mathbf{x}_t} p(\mathbf{y} | \mathbf{x}_t)$), various samplers have been proposed in the literature that use different (more or less accurate) approximations of such a gradient to guide the diffusion process towards solutions that match the observations. In this work, I consider two sampling algorithms recently proposed under the name of Diffusion Posterior Sampling (DPS) and Pseudo-inverse Guided Diffusion Model (PGDM), respectively. In DSP, the guidance term used at each step of the reverse diffusion process is obtained by applying the adjoint of the modeling operator to the residual obtained from a one-step denoising estimate of the solution. On the other hand, PGDM utilizes a pseudo-inverse operator that originates from the fact that the one-step denoised solution is not assumed to be deterministic, rather modeled as a Gaussian distribution. Through an extensive set of numerical examples on two geophysical inverse problems (namely, seismic interpolation and seismic inversion), I show that two key aspects for the success of any measurement-guided diffusion process are: i) our ability to re-parametrize the inverse problem such that the sought after model is bounded between -1 and 1 (a pre-requisite for any diffusion model); ii) the choice of the training dataset used to learn the implicit prior that guides the reverse diffusion process. Numerical examples on synthetic and field datasets reveal that PGDM outperforms DPS in both scenarios at limited additional cost.