Geophysical inverse problems with measurement-guided diffusion models

TL;DR

The paper tackles geophysical inverse problems by using measurement-guided diffusion models as powerful priors to generate realistic, uncertainty-quantified solutions from incomplete data. It analyzes two practical samplers, DPS and PGDM, and compares them with MCG, highlighting the role of the likelihood guidance and a learned prior in shaping posterior samples. By applying these methods to seismic interpolation and post-stack seismic inversion, the work shows that PGDM generally outperforms DPS at similar costs, with results strongly influenced by the chosen training data; proper reparameterization to enforce a [-1,1] bound is essential. The findings demonstrate that diffusion-based priors can produce diverse, geologically plausible realizations and provide meaningful uncertainty estimates, suggesting a scalable path for uncertainty quantification in large-scale geophysical problems.

Abstract

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.

Figures (35)

• Figure 1: Top) Training samples, and bottom) unconditionally generated samples for the Volve synthetic training dataset.
• Figure 2: Mean and standard deviation of the reconstructed wavefield for different reverse diffusion processes in the case of jittered subsampling. a) True data (sampled every 50m), b) Subsampled data, c-d) MCG, e-f) DPS, g-h) PGDM with Jacobian, j-k) PGDM without Jacobian, l-m) PGDM without Jacobian and gapped reverse process.
• Figure 3: Four realizations of the wavefield reconstructed from irregularly sampled data using 100 steps of the PGDM sampler (without Jacobian and with a jump of 10 between each step).
• Figure 4: Signal-to-noise ratio comparison between each individual realization (dashed black line), average of the SNRs of each realization (black line), SNR of the wavefield obtained by averaging all of the individual realizations (red line).
• Figure 5: a) True data (sampled every 50m), b) Subsampled data (regular subsampling with a distance of 100m between traced), c-d) mean and standard deviation of the reconstructed wavefield (sampled every 25m) for the PGDM sampler.