Robust Physics-Guided Diffusion for Full-Waveform Inversion

Jishen Peng; Enze Jiang; Zheng Ma; Xiongbin Yan

math.NAMar 17, 2026

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 ·

Abstract

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

Panel of Figure 1 — 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

\gamma

is the exponent in the diagonal preconditioner

D_{i}

for OT-WE-PDPS ( 27 ). The symbol

\rho_{0}

denotes the base guidance strength in the noise-aware schedule

\rho_{i}

for OT-WE-PDPS ( 26 ), and it also denotes the (constant) step size used in the first-order solver for the

W_{2}+\mathrm{TV}

baseline. The parameter

\alpha

is the TV regularization weight used in the TV-regularized baselines (Baseline 1 and Baseline 2), with

\mathrm{TV}

defined in ( 25 ).

Table 2: Quantitative comparison of inversion methods on OpenFWI velocity models. Metrics are the relative

\ell_{2}

error

e_{\ell_{2}}

, PSNR, and SSIM for the reconstructed velocity field (arrows indicate the preferred direction). Methods include

W_{2}+\mathrm{TV}

(Baseline 1), OT-WE

+\mathrm{TV}

(Baseline 2), DPS (Baseline 3), and the proposed method.

Panel of Figure 2 — Figure 2: Inversion results across different datasets. The first row displays the true velocity field $v_{\mathrm{true}}$ . Rows 2 to 5 show the inverted velocity fields $v_{\mathrm{rec}}$ obtained using different algorithms. Rows 6 to 9 present the difference between the true and inverted velocity fields, $v_{\mathrm{rec}}-v_{\mathrm{true}}$ . The observed wavefield has been perturbed with additive noise of intensity $\sigma=0.05$ .

Panel of Figure 3 — Figure 3: Relative $l_{2}$ -error of the reconstructed velocity field $v$ versus the TV regularization parameter $\alpha$ for the two deterministic methods: (a) $W_{2}+\mathrm{TV}$ and (b) OT-WE $+\mathrm{TV}$ .

Panel of Figure 4 — Figure 4: Relative $l_{2}$ -error of the reconstructed velocity field $v$ versus the iteration number for the two deterministic methods with their optimal TV regularization parameters $\alpha$ : (a) $W_{2}+\mathrm{TV}$ and (b) OT-WE $+\mathrm{TV}$ .

Panel of Figure 5 — 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 $W_{2}$ -based and MSE-based guidance, respectively. From left to right, the columns correspond to Only (baseline DPS), Step (preconditioned guidance update $P_{i}=\rho_{i}D_{i}$ ), 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	$e_{\ell_{2}}\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$
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
$W_{2}$ guidance

Table 3: Ablation results on CurveVel-B: relative

\ell_{2}

error (

e_{\ell_{2}}

; 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

P_{i}=\rho_{i}D_{i}

, and All enables both Enh and Step .

Panel of Figure 6 — 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 $\ell_{2}$ -error, PSNR and SSIM of the intermediate iterate $v_{i}$ under $W_{2}$ -based guidance; panels (d)–(f) show the corresponding metrics for the denoised estimate $\hat{v}_{0}^{(i)}$ under $W_{2}$ -based guidance. Panels (g)–(i) and (j)–(l) report the analogous quantities under MSE-based guidance for $v_{i}$ and $\hat{v}_{0}^{(i)}$ , respectively.

Panel of Figure 7 — 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 $\sigma\in\{0,10^{-4},10^{-3},10^{-2},5\times 10^{-2}\}$ . 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

\ell_{2}

-error (

e_{\ell_{2}}

; lower is better), PSNR, and SSIM (higher is better) of the reconstructed velocity fields under different noise levels

\sigma

Panel of Figure 8 — 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 $2\,dx$ ), Freq10 (Ricker wavelet peak frequency $f_{p}=10,\mathrm{Hz}$ ), 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	$\rho_{0}$	$e_{\ell_{2}}\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$
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

\rho_{0}

, together with the relative

\ell_{2}

-error (

e_{\ell_{2}}

; lower is better), PSNR, and SSIM (higher is better).

Panel of Figure 9 — 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 $v_{\mathrm{true}}$ . The second and third rows display the reconstructed velocity models $v_{\mathrm{rec}}$ obtained by the mixed model and the separate model, respectively. The fourth and fifth rows show the reconstruction errors, $v_{\mathrm{rec}}-v_{\mathrm{true}}$ , corresponding to the mixed and separate models, respectively.

Dataset	$e_{\ell_{2}}\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$
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

\ell_{2}

error

e_{\ell_{2}}

, PSNR, and SSIM for the reconstructed velocity model (arrows indicate the preferred direction).

Panel of Figure 10 — 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 $v_{\mathrm{true}}$ . The second row presents the reconstructed velocity models $v_{\mathrm{rec}}$ obtained by the proposed method with the mixed model. The third row shows the corresponding reconstruction errors $v_{\mathrm{rec}}-v_{\mathrm{true}}$ .

Local model	Physical size	$dx$ (m)	$dz$ (m)	$\rho_{0}$
Marmousi2-P1	$700\,\mathrm{m}\times 175\,\mathrm{m}$	10	2.5	3.9
Marmousi2-P2	$700\,\mathrm{m}\times 263\,\mathrm{m}$	10	3.75	17.9
Marmousi2-P3	$1000\,\mathrm{m}\times 175\,\mathrm{m}$	$\frac{100}{7}$	2.5	6.1
Marmousi2-P4	$1500\,\mathrm{m}\times 175\,\mathrm{m}$	$\frac{150}{7}$	2.5	5.0

Table 7: Experimental settings for the four local velocity models extracted from Marmousi2.

Local model	$e_{\ell_{2}}\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$
Marmousi2-P1 ( $700\,\mathrm{m}\times 175\,\mathrm{m}$ )	1.80%	35.95	0.9206
Marmousi2-P2 ( $700\,\mathrm{m}\times 263\,\mathrm{m}$ )	3.57%	30.49	0.7553
Marmousi2-P3 ( $1000\,\mathrm{m}\times 175\,\mathrm{m}$ )	2.81%	33.21	0.8975
Marmousi2-P4 ( $1500\,\mathrm{m}\times 175\,\mathrm{m}$ )	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

\ell_{2}

error

e_{\ell_{2}}

, PSNR, and SSIM for the reconstructed velocity model (arrows indicate the preferred direction).

Diffusion-Based Inverse Problems

Diffusion-based inverse problems use generative priors to recover images and physical fields from incomplete, noisy, or indirect measurements.

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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.

Abstract

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