{"id":"8945d904-93cc-4949-8863-b66a3e42f145","arxiv_id":"2506.10141","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Using a pretrained DDPM denoiser as a score-rematching regularizer in FWI improves synthetic inversion stability and accuracy without running reverse diffusion sampling.","lead":"This paper adds a pretrained diffusion model as a direct regularization term in Full Waveform Inversion, keeping the inversion in clean image space instead of sampling noisy diffusion states. If it works, it gives seismologists a practical way to inject learned geological priors into existing FWI codes with minimal changes.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 1 as printed has the FWI data-misfit gradient with the wrong sign, so the published update ascends on the misfit and cannot produce the reported convergence in Fig. 9.","rationale":"The central claim is that a pretrained DDPM can be inserted as a simple regularization term into the FWI update, and that this improves convergence and inversion quality. For that claim to hold, the published update rule must at minimum be a valid descent direction for the stated objective. Algorithm 1 as printed violates this: the minus sign before w_t ||F(m)−d_obs||² turns the FWI data-misfit gradient into an ascent direction, and the weighting is placed on the wrong term relative to the text and Eq. 23. This is load-bearing because the empirical evidence in Figs. 7–9 is generated by some update rule, but the manuscript does not give a rule that would produce those results. The reader's weakest_assumption concerned score approximation and transfer of the time/weight schedules; that is a genuine secondary limitation, and the paper's own Section 7.3 concedes the out-of-distribution risk. However, the update-rule inconsistency is more immediate: correcting the sign and supplying code would resolve it, but without that the claimed mechanism cannot be evaluated. I therefore recommend moving from CONDITIONAL to UNVERDICTED in the current form, rather than accepting or rejecting the method on its merits, since the manuscript as written does not specify a reproducible algorithm.","tokens_in":10623,"tokens_out":15082,"duration_ms":178196,"concrete_test":"Run Algorithm 1 verbatim on the Fig. 5 setup (or on a simple linear inverse problem with the same update rule) for 100 iterations and log ||F(m)−d_obs||². If the misfit does not monotonically decrease, the printed algorithm is not the one that produced Fig. 9. Then rerun the same experiment with the misfit sign corrected to m ← m − λ_t ∇(prior) − λ_t w_t ∇||F(m)−d_obs||² and check whether the reported convergence curves are reproduced; if they are, the manuscript must be revised to state the corrected update and release code.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"Algorithm 1, line 6, specifies m ← m − λ_t ∇_m [ E_ϵ ||ϵ − ϵθ(√ᾱ_t m + √(1−ᾱ_t)ϵ, t)||² − w_t ||F(m)−d_obs||² ]. With the minus before the data-misfit term, the update is m ← m − λ_t ∇(prior) + λ_t w_t ∇||F(m)−d_obs||², which is an ascent step on the FWI misfit. The FWI gradient is applied in the wrong direction, so the published algorithm cannot minimize the data misfit and cannot reproduce the monotonically decreasing misfit history in Fig. 9. The same line also attaches the hand-tuned weight w_t to the misfit term, whereas Section 5 and Eq. 23 describe the weight as modulating the diffusion-prior term. This is an internal inconsistency, independent of score-approximation quality or out-of-distribution transfer. Even if the sign is assumed to be a typographical error, the manuscript provides no code or data that would let a reader verify which objective was actually optimized, so the central claim that the method is a minimal modification of the standard FWI update is not testable from the paper as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes to regularize full waveform inversion by adding a score-rematching term based on a pretrained DDPM denoiser to the standard FWI update. The method avoids running the full reverse diffusion process and operates directly on clean velocity images. The authors present a variational derivation leading to an objective that combines a denoising-score-matching regularizer with the FWI data misfit, give pseudocode for training the DDPM and for the regularized FWI loop, and report synthetic acoustic experiments on Overthrust-derived and Marmousi2 models, claiming faster convergence and improved inversion quality relative to conventional FWI.","tokens_in":10920,"tokens_out":8235,"duration_ms":90573,"significance":"If the central algorithm is corrected, the proposal is a potentially useful and computationally light way to inject learned geological priors into FWI, and the choice to avoid noisy intermediate states is sensible for wave-propagation stability. The manuscript usefully provides training pseudocode, explicit schedules, and evidence on synthetic data. However, as printed, the FWI algorithm line has a sign inconsistency with the stated objective, the variational derivation contains algebraic errors, the main benchmark is in-distribution with respect to the prior training set, and the claimed GAN comparison is not present in the experiments. These issues are fixable but currently block acceptance.","major_comments":[{"comment":"In Algorithm 1, line 6, the update is written as m <- m - lambda_t grad_m[ E_epsilon ||epsilon - epsilon_theta(sqrt(alpha_bar_t) m + sqrt(1 - alpha_bar_t) epsilon, t)||^2 - w_t ||F(m)-d_obs||^2 ]. Because the data-misfit term enters with a minus sign, the update ascends on the data misfit; this contradicts Eq. (23), where the data-misfit term is added, and it cannot produce the monotonically decreasing misfit in Fig. 9. The paper does not provide code or data to verify which objective was actually optimized, so the central claim that the method is a minimal modification of standard FWI is not testable as printed. Please correct the pseudocode, clarify that w_t modulates the prior term, and make the exact objective explicit.","section":"Algorithm 1, line 6; Eq. (23)"},{"comment":"The variational derivation is algebraically incorrect as written: in Eq. (15), the equality E_q[log q - log p(m|d)] = E_q[log q - log p(m) - log p(m|d)] + c does not follow from Bayes' rule, since log p(m|d) is proportional to log p(d|m) + log p(m), not to log p(m) + log p(m|d). Eq. (16) also uses p(x|d) where the likelihood p(d|x) is intended and mixes the Dirac variational distribution with the DDPM forward-chain notation. The final objective in Eq. (23) is plausible, but the derivation does not support it in its current form; please rewrite this section with a correct Bayesian expansion.","section":"Section 3.1, Eqs. (15)-(16)"},{"comment":"The main numerical benchmark is in-distribution: the DDPM is trained on 40,000 Overthrust-derived 64x64 images (Section 4), and the target model used in Section 5 is a crop of the Overthrust model. The observed improvement therefore partly reflects the prior having already encoded the target geological style. Section 7.3 itself acknowledges that an unrepresentative training set can bias the inversion toward unrealistic solutions. To support the claimed robustness, the qualitative Marmousi2 experiment in Section 6 should be supplemented with acquisition details, quantitative misfit or similarity metrics, and ideally a genuinely out-of-distribution test, or the claims should be restricted to the in-distribution setting.","section":"Sections 4 and 5"},{"comment":"The abstract and conclusion state that the proposed method offers enhanced fidelity and robustness 'compared to ... GAN-based FWI approaches,' but no GAN-based FWI experiment is reported in the paper; Section 7.2 discusses GAN priors only in qualitative terms. Please either add the promised comparison or remove the claim from the abstract and conclusion.","section":"Abstract and Conclusions; Section 7.2"}],"minor_comments":[{"comment":"The training description first says the model was trained for 80 epochs and then says it was trained for a total of 120 epochs; please clarify the correct number.","section":"Section 4"},{"comment":"The grid spacing is reported twice with different values (9.5 m and 8.71 m for dz and dx); please reconcile or explain the discrepancy.","section":"Section 5"},{"comment":"The initial velocity model shown in Fig. 5c is not described; please state how the starting model was generated (for example, by smoothing the true model).","section":"Section 5"},{"comment":"The phrase 'w(βt) is a weighting function that can be negated' is unclear; if 'neglected' is intended, the role and eventual dropping of this weighting factor should be stated explicitly.","section":"Section 3.2, Eq. (22)"},{"comment":"The Marmousi2 experiment reports no quantitative result; please include the acquisition setup and at least one metric (e.g., final data misfit or structural similarity) so that the generalization claim can be assessed.","section":"Section 6"},{"comment":"Several references are incomplete as given: the Marmousi2 citation (Martin) lacks a publication venue, and the Aminzadeh et al. entry would benefit from complete pagination.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"Dear Editor, the sign inconsistency in Algorithm 1 is the most serious issue; if the authors can provide code or a corrected algorithm, the paper becomes viable. I would also ask them to substantiate the GAN comparison and move the main benchmark to a held-out model family. The contribution appears incremental relative to Graikos et al., so the authors should position it as an application study rather than a new diffusion-prior framework."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea is a natural transfer of Graikos-style score rematching into FWI, and the paper earns credit for naming a real practical issue: reverse-diffusion sampling injects noise into velocity models, which can cause nonphysical scattering in wave solvers. Avoiding that by evaluating the score-rematching loss on clean images at an annealed time step is a reasonable adaptation. I also appreciate the honest discussion in Section 7.3 about prior-data mismatch, and the attempt at a domain-shift test with a retrained GP prior on Marmousi2 gives the paper more substance than a single in-distribution demo.\n\nThe soft spots are real and one is load-bearing. Algorithm 1, line 6, as printed, is m ← m − λ∇[prior − w_t·misfit]. That makes the update ascend on the data misfit, so the published algorithm cannot produce the monotonically decreasing misfit history in Figure 9. Even if that is a typo, the line also attaches the hand-tuned weight w_t to the misfit term, whereas Section 5 and Eq. 23 describe the weight as modulating the prior term. So the actual optimized objective is ambiguous. No code or data is provided, so a reader cannot check which variant was run. This is not a minor style issue; it is a correctness problem in the central artifact.\n\nThe derivation is also sloppy: Eqs. 15–16 mix up logarithms and distributions, and the jump to the final score-rematching objective is not rigorous. The abstract and conclusion promise a comparison to GAN-based FWI, but I could not find any GAN results in Section 5. The main benchmark uses a model cropped from the Overthrust family on which the prior was trained, so the headline improvement is partly in-distribution; the Marmousi2 test with a retrained prior is more convincing but comes late and is under-analyzed.\n\nWho is this for? FWI practitioners who want a low-friction way to plug a pretrained denoiser into an existing pipeline. The idea is worth discussing, but the paper as written is not reproducible. I would send it to peer review because the approach is plausible and the experiments, if verified, would be useful. But the verdict should be major revision: fix the sign and weight placement in Algorithm 1, add or retract the GAN comparison, release code and data, and include a properly out-of-distribution test. As is, I would not cite the algorithm, though I might cite the discussion of why noisy intermediate states are problematic for FWI.","headline":"A sensible plug-and-play diffusion-prior recipe for FWI, but Algorithm 1 as printed has a sign error that makes the reported misfit curves impossible, and the paper lacks code or data to resolve it.","tokens_in":11432,"tokens_out":2248,"would_cite":false,"duration_ms":29366,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["86A22"],"pacs":[],"model":"deepseek-v4-flash","headline":"A pretrained diffusion denoiser can be added to FWI as a direct regularization term.","keywords":["full waveform inversion","diffusion prior","score rematching","denoising diffusion probabilistic model","generative regularization","seismic imaging","inverse problems"],"falsifier":"Run the same 100-iteration FWI benchmark with the score-rematching term replaced by a fixed random gradient of equal magnitude; if convergence and final misfit stay the same, the learned prior is not doing the work. Alternatively, invert a target whose geological features are absent from the 40,000-image training set and check whether the regularizer lowers or raises the final data misfit relative to conventional FWI.","tokens_in":10429,"feed_emoji":"🌊","tokens_out":5233,"duration_ms":54184,"temperature":0.7,"pith_summary":"The paper tries to show that a pretrained denoising diffusion probabilistic model can serve as a plug-in regularization term for full waveform inversion, without running the reverse diffusion process and without injecting noise into velocity models. The prior is derived from a score-rematching objective, and its gradient is inserted directly into the standard FWI update, balancing data misfit and learned geological structure through a single weight. If this works, seismic inversion would gain a low-cost way to inject learned structural knowledge into existing FWI codes, and the same recipe would extend to other inverse problems governed by differentiable forward operators.","feed_headline":"Diffusion prior regularizes seismic inversion without reverse sampling","feed_subtitle":"A pretrained DDPM denoiser enters the standard FWI update as a score-matching term, improving convergence and fidelity.","key_machinery":"The central object is the score-rematching prior built from a pretrained DDPM denoiser. Starting from a point-mass variational approximation $q(m)=\\delta(m-\\hat m)$, the prior term in the variational bound reduces to matching the noise $\\epsilon$ added to the current model with the denoiser's prediction $\\epsilon_\\theta(\\sqrt{\\bar\\alpha_t}m+\\sqrt{1-\\bar\\alpha_t}\\epsilon,t)$, and differentiating that mismatch updates the clean model toward the learned manifold. The U-Net denoiser, trained on 40,000 Overthrust-derived 64-by-64 images, supplies the learned score, while a prescribed annealing schedule for $t$ and a linear weight schedule control how strongly the prior acts as the inversion proceeds.","core_discovery":"The paper's central claim is that the pretrained DDPM denoiser $\\epsilon_\\theta$ defines a usable prior through the score-rematching loss $\\mathbb{E}_\\epsilon\\|\\epsilon-\\epsilon_\\theta(\\sqrt{\\bar\\alpha_t}m+\\sqrt{1-\\bar\\alpha_t}\\epsilon, t)\\|^2$, whose gradient with respect to the clean model $m$ acts as a direct regularizer for FWI. The inversion runs entirely in clean velocity-image space, with the diffusion time step $t$ annealed from large to small over iterations. On synthetic Overthrust-based benchmarks the authors report that this regularization converges faster, reaches a lower final data misfit, and produces more geologically plausible velocity models than conventional FWI, while a prior retrained on diverse synthetic images also improves recovery on the Marmousi2 dataset.","pith_inferences":["Because the prior term only needs a differentiable denoiser, the same score-rematching update could be attached to other grid-based inversions, such as electromagnetic or medical imaging, without altering the underlying forward solver.","An adaptive schedule that sets the diffusion time step and prior weight from the current data misfit could replace the hand-tuned cosine-modulated linear schedule and might make the method transfer more reliably to out-of-distribution targets.","The score-rematching gradient points along the denoiser's learned score, so the update behaves like an annealed score ascent; this suggests the regularizer may implicitly encode local uncertainty information even though the paper does not sample the posterior."],"forward_implications":["Existing FWI pipelines can incorporate a learned prior by adding one gradient term, with no reverse diffusion sampling and no noisy intermediate velocity states.","Because wave propagation always sees clean velocity fields, the method avoids non-physical scattering and numerical dispersion that can arise when noise is injected into the model.","The soft weighting of the diffusion prior provides a continuum between purely data-driven updates and strongly regularized solutions, which is more flexible than a hard GAN-manifold constraint.","The same score-rematching regularization applies to any differentiable forward operator, not only the acoustic wave equation used in the experiments.","Retraining the DDPM on a more diverse set of synthetic velocity models improves transfer to unseen geological settings such as Marmousi2."],"supporting_citations":[{"why":"Defines the DDPM training objective and noise-prediction reparameterization that the pretrained denoiser is trained with.","marker":"[Ho et al., 2020]"},{"why":"Supplies the plug-and-play diffusion-prior idea and the annealing-schedule inspiration used to set the diffusion time steps across FWI iterations.","marker":"[Graikos et al., 2022]"},{"why":"Provides the modified Overthrust-derived training dataset of 40,000 images and prior FWI work with deep generative priors.","marker":"[Xie et al., 2024]"},{"why":"Gives the adjoint-state method used to compute the data misfit gradient in the conventional FWI update.","marker":"[Plessix, 2006]"},{"why":"Supplies the full-waveform-inversion framework and wavefield modeling context that the experiments build on.","marker":"[Virieux and Operto, 2009]"},{"why":"Represents the reverse-diffusion-based FWI approach that the paper contrasts with its direct score-rematching regularization.","marker":"[Wang et al., 2023]"}],"fun_headline_variants":["FWI gains a direct diffusion-regularizer","Invert clean, not noisy: diffusion prior for FWI","Diffusion prior speeds up FWI convergence","Practical diffusion prior for FWI, no reverse sampling","Score-rematch diffusion prior improves FWI"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The pretrained denoiser must approximate the score of the target velocity-model distribution well enough, and the hand-tuned annealing and weighting schedules used in the synthetic tests must keep working for other targets; the main benchmark stays inside the training distribution.","fun_headline_variants_meta":{"raw":{"variants":["FWI gains a direct diffusion-regularizer","Invert clean, not noisy: diffusion prior for FWI","Diffusion prior speeds up FWI convergence","Practical diffusion prior for FWI, no reverse sampling","Score-rematch diffusion prior improves FWI"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000942,"raw_usage":{"total_tokens":4021,"prompt_tokens":937,"completion_tokens":3084,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":553,"completion_tokens_details":{"reasoning_tokens":3010}},"tokens_in":553,"tokens_out":3084,"duration_ms":22812,"temperature":1.0,"reasoning_tokens":3010,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:34:38.491845+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same 100-iteration FWI benchmark with the score-rematching term replaced by a fixed random gradient of equal magnitude; if convergence and final misfit stay the same, the learned prior is not doing the work. Alternatively, invert a target whose geological features are absent from the 40,000-image training set and check whether the regularizer lowers or raises the final data misfit relative to conventional FWI.","supporting_citations":[],"review_version":1}