stat.MLOct 13, 2026

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

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

Organizations: The University of Texas at Austin

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Abstract

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.

Figures & tables

Figure 1: Representative samples from the GeoFWI dataset, including folding structures, faults, and salt bodies with varying numbers of layers.
Figure 2: A sketch of diffusion model guided annealed SVGD FWI, where a diffusion prior is introduced for improved geological plausibility.
Figure 3: Illustration of the structure-preserving behavior of the learned diffusion prior. (a) Clean velocity model. (b) Perturbed velocity model with random perturbation. (c) Filtered result using Gaussian smoothing. (d) Filtered result using the learned diffusion prior. (e) Filtering difference corresponding to Gaussian smoothing. (f) Filtering difference corresponding to the diffusion prior. Compared with Gaussian smoothing, the diffusion prior suppresses random perturbations while better preserving coherent geological structures and sharp interfaces.
Figure 4: (a) Marmousi model (ground truth). (b) The initial Marmousi model was used for inversion.
Figure 5: Observed data generated using real Marmousi model (DSNR=15.7 dB).
Figure 6: Nine randomly selected initial particles generated by perturbing the smoothed Marmousi model with GRFs.
Figure 7: Estimated mean value of Marmousi model using (a) classic ASVGD method and (b) diffusion-regularized ASVGD method. Estimated std value of the Marmousi model using (c) classic ASVGD method and (d) diffusion-regularized ASVGD method.
Metricclassic ASVGDdiffusion-guided ASVGD
MAE ↓\downarrow94.8092.68
SSIM ↑\uparrow0.690.72
MSNR (dB) ↑\uparrow21.1921.48
Table 1: Performance statistics of the inverted Marmousi model using classic ASVGD and diffusion-guided ASVGD using different metrics. The symbol ↓\downarrow means the smaller value is better, and ↑\uparrow means the larger value is better. The better results are marked in red.
Figure 8: Verticle profile comparison of the Marmousi model. (a) Vertical profile showing mean and three confidence intervals for the experiment using classic ASVGD. (b) Vertical profile showing mean and three confidence intervals for the experiment using diffusion-guided ASVGD.
Figure 9: (a) Overthrust model (ground truth). (b) Initial Overthrust model used for inversion.
Figure 10: Observed data generated using real Overthrust model (DSNR=16.9 dB).
Figure 11: Nine randomly selected initial particles generated by perturbing the smoothed Overthrust model with GRFs.
Figure 12: Estimated mean value of the Overthrust model using (a) the classic ASVGD method and (b) the diffusion-guided ASVGD method. Estimated standard deviation value of the Overthrust model using (c) the classic ASVGD method and (d) the diffusion-guided ASVGD method.
Metricclassic ASVGDdiffusion-guided ASVGD
MAE ↓\downarrow173153
SSIM ↑\uparrow0.620.64
SNR (dB) ↑\uparrow24.725.3
Table 2: Performance statistics of the inverted Overthrust model using classic ASVGD and diffusion-guided ASVGD using different metrics. The symbol ↓\downarrow means the smaller value is better, and ↑\uparrow means the larger value is better. The better results are marked in red.
Figure 13: Verticle profile comparison of the Overthrust model. (a) Vertical profile showing mean and three confidence intervals for the experiment using classic ASVGD. (b) Vertical profile showing mean and three confidence intervals for the experiment using diffusion-guided ASVGD.
Figure 14: (a) Initial model corresponding to the field data. (b) Partial field shot gathers.
Figure 15: Nine randomly selected initial particles for the field data case, generated by perturbing the initial model using GRFs.
Figure 16: Field-data inversion results. Estimated mean velocity models obtained using (a) the classic ASVGD method and (b) the diffusion-regularized ASVGD method. Estimated standard deviation maps obtained using (c) the classic ASVGD method and (d) the diffusion-regularized ASVGD method. The red arrows highlight the artifacts may be caused by insufficient illumination, acquisition limitations, or noise in the field data.
Figure 17: Migration profile obtained using the velocity model of (a) classic ASVGD method and (b) diffusion-reggularized ASVGD method. (c) ADCIG at different locations corresponding to Fig 17 a. (d) ADCIG at different locations corresponding to Fig 17 b. The black arrows highlight the visible improvements in the image profile and ADCIG gathers, indicating enhanced image resolution.
Figure 18: Evolution of the velocity mean at different iterations: (a) 5, (b) 25, (c) 45, and (d) 85 iterations. The results demonstrate that the inversion process progressively incorporates structural information, evolving from a smooth large-scale background model to a geologically consistent model with coherent layers, sharper velocity contrasts, and more detailed fault-related structures.
Figure 19: Comparison of Gaussian smoothing and diffusion-based regularization for suppressing GRF-induced perturbations. (a) Reference velocity model. (b) GRF-perturbed velocity model. (c) Result after conventional Gaussian smoothing. (d) Result after diffusion-based regularization. (e) Difference between the Gaussian-smoothed model and the reference model. (f) Difference between the diffusion-regularized model and the reference model. The diffusion prior more effectively suppresses GRF-induced perturbations while preserving the layer boundaries and structural features.
Figure 20: (a) Estimated velocity mean value using 100 different initial particles. (b) Estimated std map using 100 different initial particles.
Figure 21: (a) Estimated velocity mean value under a DSNR of 2 dB. (b) Estimated std map under a DSNR of 2 dB.
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