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Structure-aware Bayesian full waveform inversion via diffusion-regularized Stein variational gradient descent

Chao Li, Mrinal K. Sen, Sergey Fomel, Yangkang Chen

arXiv:2610.19187v1stat.MLcs.LGphysics.geo-ph

TL;DR

FWI can produce erroneous models when initial conditions are inaccurate, and conventional formulations may not represent complex geological structure or quantify uncertainty. This paper integrates a learned diffusion prior into ASVGD particle evolution; synthetic and field experiments report more geologically reasonable results and reduced posterior uncertainty.

  • Problem

    FWI is nonlinear and can converge incorrectly from inaccurate initial models, while conventional ASVGD-FWI lacks explicit structural plausibility constraints for noisy, incomplete, or poorly illuminated data.

    Source: §1 · p. 1, §2.3 · p. 5

  • Method

    The method incorporates a learned diffusion prior into ASVGD particle evolution, guiding posterior approximation with geological information alongside the data likelihood.

    Source: §1 · p. 2, §2.3 · p. 6

  • Results

    Synthetic and field experiments report that diffusion-regularized ASVGD-FWI reduces posterior uncertainty and produces more geologically reasonable inversion results.

    Source: §1 · p. 2, §5 · p. 22

  • Takeaways & Limitations

    Diffusion guidance provides a regularization mechanism for suppressing non-geological perturbations while preserving large-scale background velocity trends and coherent structures.

    Source: §5 · p. 22

    Generalization, particularly to field data, depends on training with sufficiently diverse and geologically plausible velocity-model datasets.

    Source: §4.4 · p. 22

Abstract

from arXiv · show

Full waveform inversion (FWI) is a powerful tool for high-resolution subsurface model reconstruction, but it remains highly nonlinear and ill-posed, especially when the initial model is inaccurate or the acquired data are noisy and incomplete. Annealed Stein variational gradient descent (ASVGD) provides a particle-based Bayesian framework for FWI by approximating the posterior distribution with an ensemble of velocity models, thereby enabling uncertainty quantification beyond a single deterministic solution. However, conventional ASVGD-FWI commonly relies on simple handcrafted priors, such as Gaussian or smoothing assumptions, which are often insufficient to represent complex geological structures. To address this limitation, we introduce a learned diffusion model (DM) as an implicit regularization mechanism within the ASVGD-FWI framework. Instead of explicitly evaluating the diffusion prior density, the trained diffusion model acts as a learned geological projection operator that guides updated particles toward the distribution of plausible velocity models. In this way, the proposed method combines data consistency from FWI, posterior diversity from ASVGD, and structural realism from the diffusion prior. Numerical experiments demonstrate that the diffusion-regularized ASVGD-FWI produces more geologically consistent mean models, suppresses nonphysical artifacts, and yields uncertainty estimates that are more concentrated around structurally complex and poorly constrained regions. These results suggest that learned diffusion priors can effectively enhance Bayesian FWI by improving both inversion stability and the interpretability of posterior uncertainty.

1 Introduction

FWI can converge to erroneous models when its initial model is inaccurate, while deterministic approaches do not quantify inversion uncertainty. The proposed framework integrates a learned diffusion prior into particle updates to guide Bayesian FWI toward geologically plausible models.

1 Introduction

  • An inaccurate initial model can make FWI converge to an erroneous solution, degrading subsequent seismic imaging and interpretation.FWI iteratively minimizes differences between observed and predicted data.Source: §1 · p. 1
  • Data-driven FWI estimates subsurface models from shot gathers without an initial model, but the proposed work targets uncertainty evaluation and structural prior integration in particle-based inversion.Prior approaches also seek to stabilize FWI through improved initial models, robust objective functions, or reconstructed low-frequency information.Source: §1 · pp. 1–2
  • Deterministic inversion typically yields one solution without quantitative reliability or uncertainty estimates, while Bayesian inference estimates posterior means and variances.Conventional MCMC can be computationally demanding for high-dimensional problems.Source: §1 · p. 2
  • The method incorporates a pretrained diffusion prior directly into SVGD particle updates, jointly guiding particle evolution with data likelihood and learned geological information.The prior is trained on GeoFWI, a dataset of more geologically realistic structural patterns.Source: §1 · p. 2
  • Experiments on synthetic and field data report reduced posterior uncertainty and more geologically reasonable inversion results with diffusion-regularized SVGD-FWI.The authors connect the approach to improved structural fidelity, inversion stability, and interpretability of posterior uncertainty.Source: §1 · p. 2

2 THEORY

This section develops a diffusion prior from diverse GeoFWI velocity models and integrates it with annealed SVGD for Bayesian FWI. In a smoothing test, the diffusion prior better preserves geological structures and yields lower MSE than Gaussian filtering.

2.1 GeoFWI dataset and diffusion prior

  • GeoFWI contains approximately 50,000 velocity models on 100 × 100 grids, spanning 1500–4500 m/s and 2–11 layers across folds, faults, and salt bodies.The varied examples include large-scale background changes and local high-contrast anomalies.Source: §2.1 · p. 3, Figure 1 · p. 3
  • A DDPM learns an implicit structural prior by corrupting clean models with Gaussian noise and training a reverse denoiser; inference uses 250 DDIM sampling steps.The diffusion process has 1000 forward steps and is trained with a noise-prediction objective.Source: §2.1 · pp. 3–4

2.2 Theory of annealed SVGD FWI

  • SVGD transports particles toward the posterior by minimizing KL divergence, with a data-driven term guiding updates and kernel repulsion preserving particle diversity.The study uses an RBF kernel to govern particle interactions.Source: §2.2 · pp. 4–5
  • Annealing lets particles explore a broader model-space region early, then concentrate in high-posterior regions, mitigating premature collapse and preserving diversity.A heuristic temperature balances driving and repulsive forces across iterations.Source: §2.2 · p. 5

2.3 Annealed SVGD for FWI with implicit diffusion prior

  • MSE was 12554.57 for diffusion-prior smoothing versus 17590.99 with Gaussian filtering, which blurred sharp interfaces and geological structures.The diffusion-prior result better preserved the main layers and salt-body boundary while reducing random perturbations.Source: §2.3 · p. 7
  • The method first applies an annealed SVGD update, then uses a pretrained diffusion model as an implicit structural-shaping operator to guide particles toward plausible geological models.The diffusion regularizer is incorporated into the variational objective to constrain the posterior approximation toward geologically plausible solutions.Source: §2.3 · pp. 5–7

3 EXAMPLES

The examples apply diffusion-regularized ASVGD-FWI to synthetic Marmousi and Overthrust models and offshore China field data. Across these cases, the method yields more coherent velocity structures and more localized, interpretable uncertainty than classic ASVGD.

3 EXAMPLES

  • For inversion models that differ from the 100 × 100 training-patch size, the method regularizes overlapping patches independently and merges them to reconstruct the full model.The patch shift is 10 pixels to mitigate boundary artifacts.Source: §3 · p. 7

3.1 Synthetic data

  • Marmousi initial particles are generated by adding GRF perturbations to a smoothed initial model, preserving its large-scale trend while introducing local variation for posterior exploration.The ensemble contains 50 particles.Source: §3.1 · p. 8
  • On Marmousi and Overthrust, diffusion-regularized ASVGD improves structural continuity and interface sharpness, reduces artifacts, and concentrates uncertainty around complex or weakly constrained regions.The Marmousi profiles show narrower uncertainty envelopes, while Overthrust improvements are especially visible below approximately 2 km.Source: §3.1 · pp. 10, 12, Figure 7 · p. 11, Figure 8 · p. 11

3.2 Field data

  • In offshore China field data, diffusion-regularized ASVGD suppresses non-geological anomalies and produces a more coherent velocity model; RTM images and ADCIGs also show better focus and kinematic consistency.The classic result contains localized anomalies that may reflect insufficient illumination, acquisition limitations, or noise.Source: §3.2 · p. 15

4 Discussion

The discussion examines how diffusion guidance incorporates structural information and evaluates the method’s behavior across particle counts, noise levels, regularization choices, and computational cost.

4.1 Role of the diffusion prior in structural information incorporation

  • Across iterations, the inversion evolves from a large-scale velocity trend toward coherent layers, sharper contrasts, and fault-related details.The early update is primarily data-driven; diffusion guidance progressively enhances structural features while suppressing unstable artifacts, and the final model retains the background trend.Source: §4.1 · pp. 16–17, Figure 18 · p. 19
  • In field data, diffusion-regularized ASVGD yields mean and standard-deviation maps with artifacts attributed to limited illumination, acquisition, or noise; its migration profile and ADCIGs show visible image-resolution improvements over classic ASVGD.The figure caption identifies the artifacts with red arrows and the imaging improvements with black arrows.Source: Figure 16 · p. 17, Figure 17 · p. 18
  • Diffusion regularization suppresses GRF perturbations more effectively than Gaussian smoothing while preserving sharp layer boundaries and structural details.Gaussian smoothing blurs interfaces and produces larger deviations from the reference model.Source: §4.1 · p. 18, Figure 19 · p. 19

4.2 Influence of the number of initial particles

  • Using 100 rather than 50 particles yields a similar posterior mean but more continuous, stable uncertainty patterns, especially near complex structures and deep interfaces.The reported computational time roughly doubles, so 50 particles balance efficiency and inversion stability in these experiments.Source: §4.2 · pp. 18–19, Figure 20 · p. 20

4.3 Influence of noise level on the final uncertainty evaluation

  • At a DSNR of 2 dB, the method recovers main velocity structures, but the mean model becomes less continuous and more locally perturbed, especially at depth and near model boundaries.The passage attributes less stable particle evolution to noise contaminating the data-misfit gradient; uncertainty increases with higher noise.Source: §4.3 · p. 20, Figure 21 · p. 21

4.4 Key factors in diffusion regularization and computational cost analysis

  • For the Overthrust example, classic and diffusion-regularized ASVGD take approximately 870 s and 964 s, respectively; timestep choice also affects denoising stochasticity.Earlier starting timesteps are preferred as the estimated model becomes more accurate, to reduce unnecessary perturbations.Source: §4.4 · p. 21
  • Diffusion-model performance and field-data generalizability depend on training data that are geologically plausible and sufficiently diverse.Once trained, the prior is incorporated as an adaptive data-driven constraint without extra parameter selection or local slope estimation.Source: §4.4 · p. 22

5 Conclusion

The conclusion presents diffusion-regularized ASVGD as a way to improve the stability and structural reliability of particle-based Bayesian FWI.

5 Conclusion

  • Diffusion guidance combines data-likelihood updates with learned geological information to suppress non-geological perturbations while preserving background trends and coherent structures.Numerical and field-data examples support improvements in posterior estimation, uncertainty characterization, and imaging consistency, with potential relevance to noise- and illumination-related artifacts.Source: §5 · p. 22

8 Open Research Section

The paper identifies an open-source repository for its diffusion-guided ASVGD FWI code and examples.

8 Open Research Section

  • The code and examples are available through the open-source repository cited as.Source: §8 · p. 22
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