Physical-State-Guided Diffusion Sampling for Full-Waveform Inversion

1 Introduction

• FWI’s nonlinear, ill-posed objective and limited observations create spurious basins, making reconstruction strongly dependent on the initial velocity model.Cycle skipping can occur when arrival-time discrepancies exceed approximately half a dominant period.Source: §1 · p. 2
• Classical regularizers and alternative misfits address instability, but handcrafted regularizers encode limited information about plausible subsurface structures.Tikhonov regularization can blur boundaries, while total variation can introduce staircase artifacts.Source: §1 · p. 2
• Learned generative priors can be combined with observation-specific wave operators without paired retraining, and prior FWI methods use varied routes to couple them.The introduction identifies coupling among the noisy diffusion state, prior-represented velocity, and nonlinear wave operator as a central algorithmic choice.Source: §1 · pp. 2–3
• Direct DPS-based FWI evaluates the wave equation at a denoised conditional mean that can blur geological modes, while recomputing estimates at each step without persistent physical state.This couples nonlinear physical gradients with denoiser sensitivity and does not preserve conventional FWI initialization or accumulated optimization history.Source: §1 · p. 3
• PSG couples a persistent physical velocity to a full conditional reverse-diffusion trajectory through Gaussian augmentation and a conditional-score message.Waveform fitting refines the physical state toward the denoiser’s conditional mean; the updated state guides reverse diffusion while separating the wave-equation adjoint from the denoiser Jacobian.Source: §1 · p. 3
• The paper evaluates PSG on OpenFWI families under clean, noisy, and missing-receiver conditions, studies stochastic trajectories, and tests transfer to larger models.The transfer experiments use a frozen OpenFWI prior on Marmousi, Overthrust, and BP2004 Salt.Source: §1 · p. 3

2.1 Full Waveform Inversion

• Incomplete surface illumination makes FWI nonlinear and ill-posed, with cycle skipping arising when phase errors exceed about half a dominant period.Contaminated or incomplete observations further weaken the information distinguishing admissible velocity models.Source: §2.1 · p. 6

2.2 Bayesian Formulation of FWI

• Bayesian FWI balances waveform likelihood with a velocity prior, while classical Tikhonov and TV regularizers can blur interfaces or create staircase artifacts.The posterior is the computational target for diffusion-based inversion, not a basis for calibrated posterior uncertainty quantification.Source: §2.2 · pp. 6–7

2.3 Diffusion Generative Models as Data Priors

• Diffusion models learn a geological prior by progressively corrupting velocity samples and training a denoiser that estimates clean models and supplies the marginal score.Embedding this pretrained prior in inversion requires specifying how its signal interacts with physics-based data fidelity.Source: §2.3 · pp. 7–8

2.4 Diffusion Posterior Sampling

• DPS can produce unreliable FWI guidance because its denoised mean may be nongeological, yielding misleading nonlinear wave-equation gradients while discarding initialization and optimization history.The paper identifies cycle skipping and spatially unbalanced waveform guidance as additional challenges for diffusion posterior sampling.Source: §2.4 · p. 9

2.5 Physical-State-Guided Diffusion Sampling

• PSG maintains a conventional-FWI-initialized physical velocity and couples it to the diffusion variable with a Gaussian bridge, regularizing physical refinement and guiding reverse diffusion.The nonlinear wave equation is evaluated only on the persistent physical state; the bridge's conditional-mean evaluation supplies a consistency message to the diffusion variable.Source: §2.4 · p. 10, §2.5 · pp. 10–11
• The surrogate-bound gap generally depends on the physical state, so reducing the surrogate need not lower the exact conditional energy or recover its MAP solution.The coupled trajectory is also not established to sample the augmented posterior exactly.Source: §2.5 · pp. 13–14
• The diffusion score is guided through a linear identity bridge comparing the denoised velocity with the physical state, while waveform fitting uses the nonlinear wave-propagation operator.The waveform likelihood disappears from the conditional-likelihood score when the physical state is fixed.Source: §2.5 · p. 13
• At each noise level, PSG uses the denoised velocity to regularize waveform-fitting updates to a persistent physical state, which then guides the next reverse-diffusion step.The physical branch carries its state across steps, starting from the conventional physical initialization.Source: §2.5 · pp. 12–14
• PSG returns denoised and physical estimates, using the denoised output for OpenFWI and the physical output for out-of-distribution inversion.The denoised estimate better preserves prior-encoded geological structures; the physical output can recover structures beyond those represented by the learned prior.Source: §2.5 · p. 15

3.1 Evaluation Metrics

• RMSE and MAE measure velocity-amplitude errors, while SSIM measures structural agreement; lower errors and higher SSIM indicate better results.Errors are reported in physical units after scaling by 1500 m/s.Source: §3.1 · p. 16

3.2 Validation on OpenFWI Datasets

• The evaluation uses four OpenFWI families spanning layered, faulted, curved, and combined curved-and-faulted geology, with shared pretrained priors among diffusion-based methods.Each family has its own EDM model; the test-set and generated samples are compared in Figure 3.Source: §3.2 · p. 16, Figure 3 · p. 17

3.2.1 Clean Seismic Data

• Classical regularization reduces artifacts, but Tikhonov blurs interfaces and TV preserves sharper contrasts while leaving fine structures unresolved.Unregularized FWI shows oscillatory artifacts and inaccurate amplitudes, particularly in deeper regions.Source: §3.2.1 · p. 17
• On clean OpenFWI data, PSG achieves the lowest mean MAE and RMSE and highest mean SSIM across all four families.Table 1 averages results over 100 test samples per family.Source: Table 1 · p. 18, §3.2.1 · p. 18

3.2.2 Noisy Seismic Data

• Under Gaussian noise, PSG leads all four families on MAE, RMSE, and SSIM at σ = 0.1; at higher noise, its metric advantages vary by level.At σ = 0.3, PSG has the lowest mean MAE and highest mean SSIM on all families, while RED-DiffEq has lower RMSE on three; at σ = 0.5, RED-DiffEq has lower RMSE on all four.Source: §3.2.2 · p. 19, Table 2 · p. 20

3.2.3 Seismic Data with Missing Traces

• With 10, 35, or 60 of 70 receiver traces removed, PSG has the lowest mean MAE and RMSE and highest mean SSIM on all four families at every level.Reconstructions generally deteriorate as traces are removed, with the strongest degradation when only 10 receivers remain.Source: §3.2.3 · p. 21, Table 3 · p. 22

3.3 Stochastic-Trajectory Sensitivity and Ensemble Uncertainty

• Across 20 runs per model, ensemble means remained close to ground truth, preserving bulk velocity contrasts, layer geometry, and fault patterns.Variability was concentrated mainly near faults and geological interfaces, with some additional dispersion at depth.Source: §3.3 · p. 23, Figure 7 · p. 23
• Mean Spearman and Pearson correlations between ensemble standard deviation and absolute error were 0.7063 and 0.6457, positive in 36 and 37 of 40 models.Both means were significantly positive (p < 10^-16, two-sided one-sample t-tests against zero).Source: §3.3 · p. 24
• Across all four families, higher-uncertainty deciles generally had larger local errors, although individual-model curves did not increase at every decile.Figure 8 groups pixels by ensemble standard deviation within each model; family medians summarize ten models and shading spans the 25th–75th percentiles.Source: §3.3 · p. 24, Figure 8 · p. 24
• Across 40 test models, mean within-model standard deviations were 40.9 m/s for RMSE, 27.5 m/s for MAE, and 0.0266 for SSIM.Different seeds generally preserved the main layer and fault patterns, while local interface positions and velocity values varied.Source: §3.3 · p. 25

3.4 Validation on Out-of-Distribution Models

• The evaluation tested a frozen CurveFault-B EDM on resized, rescaled 70 × 190 targets, using three overlapping 70 × 70 patches; Salt used two physical updates per diffusion step.The selected Salt solver configuration followed preliminary parameter and iteration-budget evaluation.Source: §3.4 · p. 25

3.4.1 Reconstruction of Out-of-Distribution Velocity Models

• On Marmousi and Overthrust, PSG recovered dominant curved layers, fault-related discontinuities, displaced layers, and broad deep high-velocity structure.RMSE values were 331.6 and 216.2 m/s, and SSIM values were 0.745 and 0.825, respectively; thin deep layers and local contrasts remained imperfectly resolved on Marmousi.Source: §3.4.1 · p. 25, Figure 9 · p. 26
• Salt reconstruction recovered the major salt body and irregular upper boundaries; its MAE, RMSE, and SSIM were 43.9 m/s, 123.1 m/s, and 0.956.Figure 10 shows the clean-data ground truth, smoothed initialization, and compared-method reconstructions at σinit = 20.Source: §3.4.1 · pp. 25, 27, Figure 10 · p. 27

3.4.2 Sensitivity to Initialization and Observation Noise

• From σinit = 20 to 32, Salt RMSE rose from 123.1 to 404.7 m/s and SSIM fell from 0.956 to 0.770, while Marmousi and Overthrust were less sensitive.Across the sweep, Marmousi RMSE decreased from 331.6 to 299.6 m/s; Overthrust RMSE remained between 215.5 and 217.4 m/s.Source: §3.4.2 · p. 27, Figure 11 · p. 28, Figure 12 · p. 29
• Increasing observation noise generally raised velocity errors and reduced structural agreement; at σ = 0.5, SSIM was 0.552 for Marmousi and 0.582 for Overthrust.On Salt, SSIM was 0.876 at σ = 0.1 and 0.782 at σ = 0.3; strongest noise caused substantial boundary and background distortion.Source: §3.4.2 · p. 27
• PSG recovered complex, irregular velocity structures beyond OpenFWI and showed robustness to moderate observation noise and limited sensitivity to initialization.The experiments reused a frozen OpenFWI-trained prior without retraining.Source: §3.4.2 · p. 28

4 Conclusion

• PSG integrates a pretrained diffusion prior into full-waveform inversion through a persistent physical velocity state.Waveform-consistency-driven physical updates couple to diffusion sampling while separating the nonlinear forward operator from the denoiser in gradient computation.Source: §4 · p. 30
• On four OpenFWI families, terminal denoised estimates recovered sharp geological structures and achieved the best mean reconstruction metrics under clean and missing-trace acquisitions.They also maintained strong structural recovery under measurement noise.Source: §4 · p. 30
• Stochastic runs retained dominant structures, with variability near geological interfaces positively associated with local inversion error; the frozen OpenFWI-trained prior also supported larger, more complex models without retraining.The additional models were Marmousi, Overthrust, and BP2004 Salt.Source: §4 · p. 30
• Future work will focus on more realistic geological priors and acquisition settings, and extending PSG to other PDE-governed inverse problems.Source: §4 · p. 30

A Diffusion Model Preliminaries

• The noisy family forms xt by scaling x0 and adding Gaussian noise, with at = √¯αt and bt = √(1 − ¯αt).The clean state follows p0 and the added noise follows N(0, I).Source: A Diffusion Model Preliminaries · p. 36
• The derivation transforms density evolution between Fourier and original coordinates, where a divergence term accounts for changing signal scale in the VP drift–diffusion equation.Source: A Diffusion Model Preliminaries · p. 36
• EDM uses zσ = x0 + σϵ, keeping signal scale fixed as noise variance increases; at fixed VP noise level, conditional means agree under invertible scaling.An EDM denoiser can be used for VP sampling by rescaling its input and matching the noise level.Source: A Diffusion Model Preliminaries · pp. 36–37
• The optimal denoiser supplies the conditional mean and yields the score (D⋆(z, σ) − z)/σ².In practice, a neural denoiser minimizes a noise-level-weighted denoising objective.Source: A Diffusion Model Preliminaries · p. 37
• Diffusion sampling uses score-driven reverse SDEs or a probability-flow ODE to transform initial Gaussian noise into approximate samples from the data distribution.The noisy family prescribes density evolution, and matching its Fokker–Planck equation yields the reverse dynamics.Source: A Diffusion Model Preliminaries · pp. 36–37

• The appendix covers prior pretraining, the acoustic solver, method implementation protocols, hyperparameters, and computational cost.Source: B Implementation Details · p. 37

Diffusion Model Pretraining

• 24,000 fields train each velocity-family prior and 48,000 fields train each fault-family prior, with a separate 6,000-field monitoring split excluded from gradient training.The monitoring split includes inversion test files; velocities are normalized as x = (v −3000)/1500.Source: §B.1 · p. 37
• The denoiser is a 25.4-million-parameter Dhariwal U-Net with four channel widths, two residual blocks per level, and attention at resolutions 36, 18, and 9.Native 70 × 70 fields are boundary-replicated to 72 × 72 and cropped after denoising.Source: §B.1 · p. 38
• Training uses Adam at learning rate 10−4 and batch size 32 for 400,000 updates, with a 500-step warm-up and FP16 mixed precision.The final EMA weights are used without monitoring-loss selection; CurveFault-B loss oscillates around a slowly decreasing plateau after roughly 3 × 10^5 steps, with sampling variability near convergence.Source: §B.1 · p. 38, Figure 13 · p. 39
• Diffusion inverse-solver baselines use a discrete linear DDPM schedule, while DPS and PSG use the continuous VP schedule of the sampling implementation.The discrete schedule uses T = 1000, β1 = 10−4, and βT = 2 × 10−2; the continuous rate is β(t) = 0.1 + 19.9t, with 1000 evaluation times from 1 to 10−3.Source: §B.1 · p. 38

Acoustic Wave-Equation Forward Solver

• The solver uses fourth-order spatial finite differences, second-order time stepping, a Ricker source wavelet, and a quadratically damped sponge boundary layer.Sources are approximately uniformly spaced horizontally, and receivers occupy every horizontal grid point; both are at fixed shallow depth.Source: §B.2 · p. 39
• Velocity gradients come from the discrete adjoint, which back-propagates waveform-residual derivatives through time stepping and combines them with forward wavefields.Source: §B.2 · p. 39
• OpenFWI and OOD targets share the numerical scheme, source wavelet, and boundary treatment, while wider targets use increased horizontal grid sizes and receiver counts.Table 4 identifies Marmousi, Overthrust, and Salt as OOD targets and reports grid footprints as depth by width.Source: §B.2 · p. 39, Table 4 · p. 40

Method Implementation Protocols

• All methods use the same acoustic operator and observations, with normalized velocity fields and shared family-specific EDM priors for diffusion methods.VP denoiser calls use the conversion specified in Eq. (50).Source: §B.3 · p. 40
• Unregularized FWI fits waveform loss, Tikhonov-FWI penalizes neighboring-velocity differences, and TV-FWI favors piecewise-constant fields while preserving sharp contrasts.Tikhonov regularization can smooth sharp interfaces and fine-scale structure, whereas TV can create staircase patterns when true velocities vary smoothly.Source: §B.3 · p. 40
• DPS differentiates waveform loss at the denoised velocity through both the wave solver and denoiser, while RED-DiffEq adds a predicted-versus-injected-noise residual to the waveform gradient.The implementation applies Adam to the predicted-noise direction while holding predicted noise fixed in the regularization term, avoiding denoiser-Jacobian evaluation.Source: §B.3 · p. 41
• DiffusionFWI alternates DDPM prior updates with physical velocity-update blocks, while SGDS-FWI alternates likelihood-driven physical refinement with diffusion denoising that encourages geological structure, after FWI–TV and SDEdit initialization.Source: §B.3 · p. 41
• PSG keeps a persistent Adam physical state, uses one physical update and one denoiser evaluation per reverse step, and reuses the denoised prediction for regularization and diffusion guidance.It clips normalized velocities to [−1, 1], retains optimizer state, and clips the terminal denoised estimate to the same bounds.Source: §B.3 · p. 41
• For 70×190 OOD fields, learned methods apply priors to three overlapping 70×70 patches and average predictions equally across each ten-column overlap.The patch ranges are [0, 70), [60, 130), and [120, 190); SGDS-FWI maintains independent patch states and noise draws through each complete SDEdit or Tweedie call.Source: §B.3 · pp. 41–42

B.4 Hyperparameter Settings

• PSG and the baseline methods use mean ℓ1 waveform loss, while DPS and SGDS-FWI use squared residuals.Hyperparameters are fixed across the four OpenFWI families and all observation conditions.Source: §B.4 · p. 42
• Classical methods and RED-DiffEq use 300 Adam updates, while Tikhonov-FWI and TV-FWI use regularization weights of 0.01 and RED-DiffEq uses 0.75.The spatial penalties use directional edge averages, with a smoothed absolute value for TV.Source: §B.4 · p. 42
• DiffusionFWI uses 10 physical Adam updates between denoising steps; SGDS-FWI starts with 300 FWI–TV updates, then uses five blocks of 150 updates.DiffusionFWI starts at t = 100 on a 1000-step schedule; SGDS-FWI uses a sigmoid diffusion schedule starting at level 250.Source: §B.4 · p. 42
• DPS and PSG use 1000 stochastic steps with normalized guidance weights of 0.01 and 0.3, respectively; PSG performs one physical Adam update per step.PSG uses a 0.5 coupling weight on the pixelwise mean squared velocity residual.Source: §B.4 · p. 42
• For Salt, learned-method parameters are selected using clean-target SSIM and then fixed across initialization and noise conditions; PSG uses two physical updates per diffusion step.OOD inversion reuses the CurveFault-B EDM through RED-DiffEq domain decomposition; other listed parameters retain OpenFWI settings.Source: §B.4 · p. 42, Table 5 · p. 42

B.5 Computational Cost

• A PSG step costs approximately 164% of a standard FWI iteration and 121% of a RED-DiffEq iteration; physical forward and backward operations account for 61.8% of PSG runtime.Table 6 reports the per-step breakdown for a 70 × 70 OpenFWI model with five sources, averaged over 1000 reverse steps.Source: §B.5 · p. 43, Table 6 · p. 43
• 547% of FWI’s and 405% of RED-DiffEq’s total core iteration time is required by PSG under the profiling setup.PSG and DPS use 1000 reverse steps, compared with 300 iterations for FWI and RED-DiffEq.Source: §B.5 · p. 43
• PSG’s 1000-step trajectory is approximately three times the 300 iterations used by FWI and RED-DiffEq; fewer-step sampling remains a direction for evaluating lower-cost inversion.The stated future investigation is whether inversion benefits can be retained with fewer reverse steps.Source: §B.5 · p. 43