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Diffusion Model-Based Posterior Sampling in Full Waveform Inversion

Mohammad H. Taufik, Tariq Alkhalifah

TL;DR

This work addresses uncertainty quantification in large-scale full waveform inversion by coupling an unconditional diffusion prior with encoded-shot likelihoods to enable posterior sampling with substantially fewer wave-equation solves. It introduces a decoupled sampling workflow that, at each diffusion level, first predicts a clean model, then performs a short stochastic Langevin refinement under the wave-equation data term, and finally re-noises to the next level, preserving mixing across levels. The method leverages encoded simultaneous-source data to reduce forward/adjoint solves while using a diffusion prior to suppress source-related artefacts, and it trains priors in 2D patches and 3D cubes for scalability. Compared to a strong SVGD baseline, the approach yields lower model and data misfits, better posterior calibration, and markedly lower computational cost, with demonstrated robustness across 2D synthetics and 3D upscaling. The framework enables calibrated posterior inference on shot records for large-scale 2D and 3D FWI and suggests avenues for efficiency gains and broader guidance integration.

Abstract

Bayesian full waveform inversion (FWI) offers uncertainty-aware subsurface models; however, posterior sampling directly on observed seismic shot records is rarely practical at the field scale because each sample requires numerous wave-equation solves. We aim to make such sampling feasible for large surveys while preserving calibration, that is, high uncertainty in less illuminated areas. Our approach couples diffusion-based posterior sampling with simultaneous-source FWI data. At each diffusion noise level, a network predicts a clean velocity model. We then apply a stochastic refinement step in model space using Langevin dynamics under the wave-equation likelihood and reintroduce noise to decouple successive levels before proceeding. Simultaneous-source batches reduce forward and adjoint solves approximately in proportion to the supergather size, while an unconditional diffusion prior trained on velocity patches and volumes helps suppress source-related numerical artefacts. We evaluate the method on three 2D synthetic datasets (SEG/EAGE Overthrust, SEG/EAGE Salt, SEAM Arid), a 2D field line, and a 3D upscaling study. Relative to a particle-based variational baseline, namely Stein variational gradient descent without a learned prior and with single-source (non-simultaneous-source) FWI, our sampler achieves lower model error and better data fit at a substantially reduced computational cost. By aligning encoded-shot likelihoods with diffusion-based sampling and exploiting straightforward parallelization over samples and source batches, the method provides a practical path to calibrated posterior inference on observed shot records that scales to large 2D and 3D problems.

Diffusion Model-Based Posterior Sampling in Full Waveform Inversion

TL;DR

This work addresses uncertainty quantification in large-scale full waveform inversion by coupling an unconditional diffusion prior with encoded-shot likelihoods to enable posterior sampling with substantially fewer wave-equation solves. It introduces a decoupled sampling workflow that, at each diffusion level, first predicts a clean model, then performs a short stochastic Langevin refinement under the wave-equation data term, and finally re-noises to the next level, preserving mixing across levels. The method leverages encoded simultaneous-source data to reduce forward/adjoint solves while using a diffusion prior to suppress source-related artefacts, and it trains priors in 2D patches and 3D cubes for scalability. Compared to a strong SVGD baseline, the approach yields lower model and data misfits, better posterior calibration, and markedly lower computational cost, with demonstrated robustness across 2D synthetics and 3D upscaling. The framework enables calibrated posterior inference on shot records for large-scale 2D and 3D FWI and suggests avenues for efficiency gains and broader guidance integration.

Abstract

Bayesian full waveform inversion (FWI) offers uncertainty-aware subsurface models; however, posterior sampling directly on observed seismic shot records is rarely practical at the field scale because each sample requires numerous wave-equation solves. We aim to make such sampling feasible for large surveys while preserving calibration, that is, high uncertainty in less illuminated areas. Our approach couples diffusion-based posterior sampling with simultaneous-source FWI data. At each diffusion noise level, a network predicts a clean velocity model. We then apply a stochastic refinement step in model space using Langevin dynamics under the wave-equation likelihood and reintroduce noise to decouple successive levels before proceeding. Simultaneous-source batches reduce forward and adjoint solves approximately in proportion to the supergather size, while an unconditional diffusion prior trained on velocity patches and volumes helps suppress source-related numerical artefacts. We evaluate the method on three 2D synthetic datasets (SEG/EAGE Overthrust, SEG/EAGE Salt, SEAM Arid), a 2D field line, and a 3D upscaling study. Relative to a particle-based variational baseline, namely Stein variational gradient descent without a learned prior and with single-source (non-simultaneous-source) FWI, our sampler achieves lower model error and better data fit at a substantially reduced computational cost. By aligning encoded-shot likelihoods with diffusion-based sampling and exploiting straightforward parallelization over samples and source batches, the method provides a practical path to calibrated posterior inference on observed shot records that scales to large 2D and 3D problems.

Paper Structure

This paper contains 24 sections, 13 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Forward model, data, and likelihood
  4. Bayesian formulation of FWI.
  5. Encoded (simultaneous-sources) shots.
  6. Diffusion prior and reverse parameterization
  7. Decoupled diffusion inference with encoded shots data
  8. (i) Predict a clean model.
  9. (ii) Clean-space Langevin refinement.
  10. (iii) Re-noise to the next level.
  11. Computational cost
  12. Numerical experiments
  13. Diffusion model training
  14. Posterior sampling using 2D ocean-bottom node synthetic data
  15. Posterior sampling using 2D towed-streamer field data
  16. ...and 9 more sections

Figures (12)

  • Figure 1: Samples from the training (Realistic2D) dataset for the 2D diffusion model.
  • Figure 2: Samples from the training (Realistic3D) dataset for the 3D diffusion model.
  • Figure 3: SEAM Arid synthetic (2D OBN). (a) Initial model; (b) ground truth; (c) diffusion posterior mean with multi-source FWI guidance; (d) SVGD mean with conventional FWI gradients; (e–f) corresponding posterior standard deviations, respectively.
  • Figure 4: SEG/EAGE Salt synthetic (2D OBN). (a) Initial model; (b) ground truth; (c) diffusion posterior mean with multi-source FWI guidance; (d) SVGD mean with conventional FWI gradients; (e–f) corresponding posterior standard deviations, respectively.
  • Figure 5: SEG/EAGE Overthrust synthetic (2D OBN). (a) Initial model; (b) ground truth; (c) diffusion posterior mean with multi-source FWI guidance; (d) SVGD mean with conventional FWI gradients; (e–f) corresponding posterior standard deviations, respectively.
  • ...and 7 more figures