Robust Physics-Guided Diffusion for Full-Waveform Inversion
Authors: Jishen Peng, Enze Jiang, Zheng Ma, Xiongbin Yan
Organizations: School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China · Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, China · Qing Yuan Research Institute, Shanghai Jiao Tong University, Shanghai, China · CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, China · School of Mathematics and Statistics, Lanzhou University, Lanzhou, China.
We develop a robust physics-guided diffusion framework for full-waveform inversion that combines a score-based generative prior with likelihood guidance computed through wave-equation simulations. We adopt a transport-based data-consistency potential (Wasserstein-2), incorporating wavefield enhancement via bounded weighting and observation-dependent normalization, thereby improving robustness to amplitude imbalance and time/phase misalignment. On the inference side, we introduce a preconditioned guided reverse-diffusion scheme that adapts the guidance strength and spatial scaling throughout the reverse-time dynamics, yielding a more stable and effective data-consistency guidance step than standard diffusion posterior sampling (DPS). Numerical experiments on OpenFWI datasets demonstrate improved reconstruction quality over deterministic optimization baselines and standard DPS under comparable computational budgets.
Figures & tables
Figure 1: Comparison of the original and weighted wavefield traces at Receiver 1 for the first three source–receiver pairs. The original recordings are dominated by early large-amplitude arrivals, whereas the proposed bounded weighting compresses the dynamic range and renders later weak-amplitude events more comparable in scale.
Table 1: Hyperparameters used in each dataset. Here γ is the exponent in the diagonal preconditioner Di for OT-WE-PDPS ( 27 ). The symbol ρ0 denotes the base guidance strength in the noise-aware schedule ρi for OT-WE-PDPS ( 26 ), and it also denotes the (constant) step size used in the first-order solver for the W2+TV baseline. The parameter α is the TV regularization weight used in the TV-regularized baselines (Baseline 1 and Baseline 2), with TV defined in ( 25 ).
Table 2: Quantitative comparison of inversion methods on OpenFWI velocity models. Metrics are the relative ℓ2 error eℓ2 , PSNR, and SSIM for the reconstructed velocity field (arrows indicate the preferred direction). Methods include W2+TV (Baseline 1), OT-WE +TV (Baseline 2), DPS (Baseline 3), and the proposed method.
Figure 2: Inversion results across different datasets. The first row displays the true velocity field vtrue . Rows 2 to 5 show the inverted velocity fields vrec obtained using different algorithms. Rows 6 to 9 present the difference between the true and inverted velocity fields, vrec−vtrue . The observed wavefield has been perturbed with additive noise of intensity σ=0.05 .
Figure 3: Relative l2 -error of the reconstructed velocity field v versus the TV regularization parameter α for the two deterministic methods: (a) W2+TV and (b) OT-WE +TV .
Figure 4: Relative l2 -error of the reconstructed velocity field v versus the iteration number for the two deterministic methods with their optimal TV regularization parameters α : (a) W2+TV and (b) OT-WE +TV .
Figure 5: Ablation study on CurveVel-B. The top panel shows the ground-truth velocity model. The second and third panels report reconstructions obtained with W2 -based and MSE-based guidance, respectively. From left to right, the columns correspond to Only (baseline DPS), Step (preconditioned guidance update Pi=ρiDi ), Enh (wavefield enhancement: amplitude-adaptive weighting and misfit-scale normalization), and All ( Step + Enh ). The bottom two panels show the corresponding error maps (reconstruction minus ground truth).
Method
eℓ2↓
PSNR ↑
SSIM ↑
MSE guidance
Only
18.21%
16.45
0.3827
Step
12.52%
19.70
0.5082
Enh
9.20%
22.37
0.6578
All
5.10%
27.50
0.8226
W2 guidance
Table 3: Ablation results on CurveVel-B: relative ℓ2 error ( eℓ2 ; lower is better), PSNR and SSIM (higher is better). Here Only denotes the baseline DPS update with the specified misfit, Enh enables wavefield enhancement (amplitude-adaptive data weighting and misfit-scale normalization), Step enables the preconditioned guidance update Pi=ρiDi , and All enables both Enh and Step .
Figure 6: Evolution of the statistical metrics over reverse-time steps for the ablation settings Enh , Step , Only and All . Panels (a)–(c) show the relative ℓ2 -error, PSNR and SSIM of the intermediate iterate vi under W2 -based guidance; panels (d)–(f) show the corresponding metrics for the denoised estimate v^0(i) under W2 -based guidance. Panels (g)–(i) and (j)–(l) report the analogous quantities under MSE-based guidance for vi and v^0(i) , respectively.
Figure 7: Qualitative robustness to observation noise across the six datasets. Each column corresponds to one dataset. The first row shows the true velocity field, and the remaining rows show the reconstructed velocity fields under progressively increasing noise levels σ∈{0,10−4,10−3,10−2,5×10−2} . The reconstructions remain visually stable as the noise level increases, with the main interfaces and fault structures largely preserved.
Table 4: Quantitative robustness to observation noise across the six datasets. Relative ℓ2 -error ( eℓ2 ; lower is better), PSNR, and SSIM (higher is better) of the reconstructed velocity fields under different noise levels σ .
Figure 8: Inversion results under different forward-operator configurations (FlatFault-B). From left to right: Init (baseline setup), Shot17 (17 uniformly distributed sources), Depth2 (sources and receivers shifted downward by 2dx ), Freq10 (Ricker wavelet peak frequency fp=10,Hz ), and AllChange (all modifications applied). The reconstructions remain visually consistent across operator changes, indicating that the proposed method generalizes well to perturbed forward operators without retraining the prior.
Operator
ρ0
eℓ2↓
PSNR ↑
SSIM ↑
Init
1.45
6.13%
28.48
0.9272
Shot17
2.25
5.24%
29.84
0.9399
Depth2
1.60
5.37%
29.62
0.9343
Freq10
1.00
5.85%
28.87
0.9160
AllChange
1.30
5.82%
28.92
0.9168
Table 5: Reconstruction performance on FlatFault-B under different forward-operator configurations. Reported are the tuned values of ρ0 , together with the relative ℓ2 -error ( eℓ2 ; lower is better), PSNR, and SSIM (higher is better).
Figure 9: Comparison of inversion results produced by the mixed and separate models for CurveVel-A, FlatFault-A, and CurveFault-B. The first row shows the true velocity models vtrue . The second and third rows display the reconstructed velocity models vrec obtained by the mixed model and the separate model, respectively. The fourth and fifth rows show the reconstruction errors, vrec−vtrue , corresponding to the mixed and separate models, respectively.
Dataset
eℓ2↓
PSNR ↑
SSIM ↑
CurveVel-A (separate model)
3.50%
31.42
0.9039
CurveVel-A (mixed model)
3.87%
30.54
0.8906
FlatFault-A (separate model)
2.91%
32.74
0.9646
FlatFault-A (mixed model)
3.28%
31.72
0.9179
CurveFault-B (separate model)
5.32%
27.95
0.7956
CurveFault-B (mixed model)
5.91%
27.04
0.7079
Table 6: Quantitative comparison of inversion results on OpenFWI velocity models using the separate and mixed models. Metrics include the relative ℓ2 error eℓ2 , PSNR, and SSIM for the reconstructed velocity model (arrows indicate the preferred direction).
Figure 10: Inversion results for the four local velocity models extracted from Marmousi2, denoted by Marmousi2-P1–P4. The first row shows the true velocity models vtrue . The second row presents the reconstructed velocity models vrec obtained by the proposed method with the mixed model. The third row shows the corresponding reconstruction errors vrec−vtrue .
Local model
Physical size
dx (m)
dz (m)
ρ0
Marmousi2-P1
700m×175m
10
2.5
3.9
Marmousi2-P2
700m×263m
10
3.75
17.9
Marmousi2-P3
1000m×175m
7100
2.5
6.1
Marmousi2-P4
1500m×175m
7150
2.5
5.0
Table 7: Experimental settings for the four local velocity models extracted from Marmousi2.
Local model
eℓ2↓
PSNR ↑
SSIM ↑
Marmousi2-P1 ( 700m×175m )
1.80%
35.95
0.9206
Marmousi2-P2 ( 700m×263m )
3.57%
30.49
0.7553
Marmousi2-P3 ( 1000m×175m )
2.81%
33.21
0.8975
Marmousi2-P4 ( 1500m×175m )
5.31%
26.96
0.8536
Table 8: Quantitative comparison of inversion results obtained by the mixed model for the four local velocity models extracted from Marmousi2. Metrics include the relative ℓ2 error eℓ2 , PSNR, and SSIM for the reconstructed velocity model (arrows indicate the preferred direction).
Full waveform inversion (FWI) estimates subsurface velocity from seismic recordings, but its ill-posedness and nonlinearity make accurate reconstruction strongly dependent on initialization and prior information. Diffusion posterior sampling provides a learned geological prior, yet directly coupling its denoiser to the nonlinear wave solver can yield unreliable physical guidance. We propose Physical-State-Guided Diffusion Sampling (PSG), which couples a persistent physical velocity to the diffusion prior through a Gaussian bridge. The physical state is refined by waveform fitting regularized by the denoised velocity, and in turn guides the reverse diffusion process. This formulation separates the wave-equation and denoiser gradients while preserving conventional FWI initialization and accumulated optimization history. On four OpenFWI families, PSG's terminal denoised estimates outperform classical and diffusion-based baselines under clean and missing-trace acquisitions and maintain strong structural recovery under measurement noise. Repeated stochastic runs preserve the dominant geological structures, with ensemble variability concentrated near geological interfaces and positively associated with local inversion error. A frozen OpenFWI-trained prior further supports inversion of the larger Marmousi, Overthrust, and BP2004 Salt models, recovering complex geological structures without retraining.
Chen Min, Haowen Jiang, Zheng Ma +1
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China · CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, China · School of Mathematics and Statistics, Lanzhou University, Lanzhou, China
Full waveform inversion (FWI) recovers subsurface velocity from seismic recordings by solving a severely ill-posed, nonconvex PDE-constrained optimization. Classical regularizers stabilize the inversion but fail to reproduce realistic geological structures; recent diffusion-prior methods improve realism at the cost of a fragile trade-off between data fidelity and prior consistency. We propose Decoupled Latent Optimization (DLO), which relaxes the standard latent-optimization formulation into a quadratic-penalty objective over an auxiliary physical variable and a latent variable. The data-fidelity gradient acts in physical space, the diffusion sampler contributes only through a decoded prior sample, and the standard smoothed-velocity initialization of classical FWI is preserved. On the OpenFWI benchmark, DLO outperforms classical regularizers and existing diffusion-based methods under clean, noisy, and missing-trace acquisitions. The prior, trained on 70*70 OpenFWI models, transfers directly to the Marmousi and Overthrust benchmarks, where DLO recovers intricate fault structures and remains robust to initialization smoothing and measurement noise.
Chen Min, Zheng Ma
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China · CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, China
Seismic full-waveform inversion is a core technology for obtaining high-resolution subsurface model parameters. However, its highly nonlinear characteristics and strong dependence on the initial model often lead to the inversion process getting trapped in local minima. In recent years, generative diffusion models have provided a way to regularize full-waveform inversion by learning implicit prior distributions. However, existing methods mostly use unconditional diffusion processes, ignoring the inherent physical coupling relationship between velocity and density and other physical properties. This paper proposes a full-waveform inversion method based on conditional diffusion model regularization. By improving the backbone network structure of the diffusion model, two-dimensional density information is introduced as a conditional input into the U-Net network. Experimental results show that the full-waveform inversion method based on the conditional diffusion model significantly improves the resolution and structural fidelity of the inversion results, and exhibits stronger stability and robustness when dealing with complex situations. This method effectively utilizes density information to constrain the inversion and has good practical application value. Keywords: Deep learning; Diffusion model; Full waveform inversion.
Caiyun Liu, Siyang Pei, Qingfeng Yu +1
School of Information and Mathematics, Yangtze University, Jingzhou, Hubei Province 434023, China · School of Electronic Information and Electrical Engineering, Yangtze University, Jingzhou, Hubei Province 434023, China