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Generative modeling of seismic data using diffusion models and its application to multi-purpose posterior sampling for noisy inverse problems

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that a single unconditional diffusion model trained on seismic data with a power-function noise schedule can generate diverse posterior samples for several noisy linear inverse problems—denoising, interpolation, and…

desk verdict Useful engineering on a known backbone, but the power noise schedule is undefined as written and the quantitative evidence is missing. read the letter →

arxiv 2506.12897 v1 pith:SNVLM66C submitted 2025-06-15 physics.geo-ph

Chuangji Meng , Jinghuai Gao , Wenting Shang , Yajun Tian , Hongling Chen , Tieqiang Zhang , Zongben Xu This is my paper · ORCID
classification physics.geo-ph
keywords diffusionmodelseismicdatapowerfunctionnoisescheduleposteriorsamplingnon-Markovinverseproblemsuncertaintyquantification
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Seismic inverse problems are ill-posed, so a single deterministic answer hides uncertainty. This paper tries to establish that one unconditional diffusion model trained on seismic patches with a power-function noise schedule can act as a multi-purpose prior: by plugging in the forward operator and an estimate of observation noise, the same network generates diverse posterior samples for each task without retraining on paired data. The authors also try to show that non-Markov accelerated sampling reaches near full-trajectory quality in roughly 20 function evaluations, and that the resulting posterior samples beat score-based and classical baselines on out-of-distribution synthetic and field data. If correct, the practical payoff is that one training run replaces many task-specific conditional models, and users can interactively trade noise suppression against detail by changing the assumed noise level.

What carries the argument

The load-bearing mechanism is the trained noise-prediction network $\epsilon_\theta$ of a DDPM, used in two roles: generating unconditional seismic samples and, through Algorithm 1, driving the conditional reverse chain in the spectral domain of the forward operator. Three design choices carry the argument: the power-function noise schedule, which the paper says injects noise slowly enough for low-amplitude seismic signals; the non-Markov DDIM subsampling rule, which cuts function evaluations from $T$ to $\dim(\tau)$ while preserving quality; and the singular-value-decomposition-based conditional transition (Equations 8 and 9) that connects observation noise $\sigma_y$ to the diffusion noise levels, with pixelwise noise levels estimated by a separate variational model.

What would settle it

Inspect the released training configuration or code for the exact sequence $\beta_t$ used when $T=1000$ and $a=3$; if $\beta_T$ is not below $1$, the documented forward process is ill-defined. Alternatively, retrain with an explicitly normalized power schedule (for example $\beta_t = (t/T)^a$) and compare standard sample-quality metrics and posterior SNR/SSIM on the Sigmoid and field test cases; a material difference would show the reported results depend on a schedule the paper does not state.

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Extended reading notes

Core claim

The paper's central claim is that an unconditional denoising diffusion probabilistic model, trained with a power-function noise schedule ($\beta_t = t^a$ with $a=3$ or $4$), gives a single pretrained network sufficient for posterior sampling across distinct noisy linear inverse problems. The posterior sampler works in the singular-value basis of the forward operator $G=U\Sigma V^\top$, reusing the same noise-predictor $\epsilon_\theta$ from the unconditional model and the estimated observation noise level $\sigma_y$; it produces stochastic solutions whose mean approximates the posterior expectation and whose spread provides uncertainty information. With the DDIM-style non-Markov sampling trajectory shortened to $20$ steps, the authors report output comparable to the full $1000$-step chain and faster than score-based posterior samplers that need hundreds of evaluations.

Load-bearing premise

The load-bearing premise is that the power noise schedule is specified precisely enough to reproduce the trained model; as written, $\beta_t = t^a$ with $T=1000$ and $a=3$ gives $\beta_T = 10^9$, contradicting the paper's own bound $0<\beta_t<1$, so an unstated normalization is required for the schedule to exist.

Editorial extensions

If this is right

  • One trained unconditional diffusion model can serve denoising, interpolation, and compressed sensing by swapping only the forward operator $G$, eliminating task-specific conditional retraining.
  • Using a subsequence of 20 reverse steps reproduces the posterior quality of the full 1000-step chain, giving more than a 50× speed-up in unconditional sampling.
  • The assumed observation noise level $\sigma_y$ acts as a user dial: larger values suppress noise more strongly while preserving signal, enabling interactive posterior exploration.
  • The method's time complexity scales as $O(\dim(\tau)\cdot d)$ with $O(d)$ memory, below the score-based baselines used, supporting larger seismic datasets.
  • Because the method generates multiple samples, posterior mean and variance provide uncertainty quantification for each inverse problem, which deterministic discriminative methods cannot supply.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An untested extension is to other low-amplitude geophysical or medical signals where the power schedule's slow noise injection should behave similarly; the paper only demonstrates seismic data.
  • The noise-level interaction suggests a practical uncertainty-quantification protocol: sample at fixed $\sigma_y$ for diversity, then sweep $\sigma_y$ to map the trade-off between resolution and noise suppression; the paper shows the trade-off qualitatively but does not formalize such a protocol.
  • The Discussion's claim that the method extends to nonlinear inverse problems remains a sketch; a concrete test would replace $G$ with a linearization of a nonlinear operator and compare against task-specific nonlinear solvers.
  • If the power schedule is made explicit and normalized, the same framework could be benchmarked against later conditional diffusion models on identical compute budgets; this comparison is not in the paper.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes an unconditional denoising diffusion probabilistic model (DDPM) trained with a newly introduced "power function" noise schedule for seismic data, and combines it with the DDRM posterior-sampling framework to solve noisy linear inverse problems (denoising, interpolation, compressed sensing) without task-specific retraining. The authors claim that 20-step accelerated sampling is comparable to 1000-step sampling, that the method outperforms score-based and classical baselines, and that it generalizes out-of-distribution. The experimental section reports only qualitative visual results on synthetic and field data, with no tables of quantitative metrics. The power schedule is defined as beta_t = t^a, which conflicts with the stated 0 < beta_t < 1 requirement, leaving the actual training schedule unspecified and making the central construction irreproducible as written.

Significance. The idea of reusing a single unconditional diffusion prior for multiple seismic inverse problems is practically appealing, and the qualitative demonstrations on field data suggest potential. The proposed power schedule, if properly normalized and validated, could be a useful adaptation for low-amplitude seismic signals. However, the posterior sampling machinery is directly inherited from DDRM, and the paper provides no code or numerical results; the claimed advantages over baselines are therefore unverified. The paper does not present machine-checked proofs, reproducible code, parameter-free derivations, or falsifiable quantitative predictions, so its significance currently rests on a promising but incompletely specified empirical recipe.

major comments (3)
  1. [Section 2.1.1, Equations (1)-(3)] The power noise schedule beta_t = t^a is incompatible with the stated bound 0 < beta_1 < ... < beta_T < 1. For the reported settings a = 3, a = 4, and T = 1000, beta_1000 = 10^9 and beta_2 = 8 > 1, so alpha_t = 1 - beta_t becomes negative for t >= 2, and the cumulative product alpha_bar_t in Equation (2) no longer yields a valid variance schedule. Consequently, the closed-form noising q(x_t|x_0) and the training objective in Equation (3) are not well defined for the stated schedule. The paper must specify the actual normalized schedule (e.g., beta_t = (t/T)^a or a similar normalization) and verify that 0 < beta_t < 1 for all t; without this, the trained model and all downstream posterior-sampling results are not reproducible.
  2. [Section 3, Figures 9-12] The paper reports no quantitative metrics. The text claims 'higher SNR and SSIM values' (Section 3.4, Figure 11) and 'best results' for interpolation and compressed sensing, but no tables or numerical values are provided. This makes it impossible to assess the central claims of superiority over SGMPS-RNA, SGMPS-SDR, SNIPS, DIP, MSSA, and OMP, especially because baseline hyperparameters (e.g., DIP iterations, MSSA singular values) are described only loosely. The authors should add quantitative tables with SNR and SSIM values and runtime comparisons for all tasks and baseline methods.
  3. [Section 2.2.1, Equations (7)-(9) and Algorithm 1] The posterior sampling construction is imported from DDRM, but the presentation contains several internal inconsistencies that prevent independent implementation. The condition in Equation (9) uses sigma_t, whereas the surrounding text defines sigma_hat_t as the diffusion noise level and earlier (Equation (4)) sets sigma_t = beta_t; the variance term in the third case of Equation (9) is written without a square root and is dimensionally inconsistent with the first two cases; Equation (8) uses z without specifying its distribution; and Algorithm 1 lines 4 and 7 are inverse operations of each other, making the update redundant. Since these formulas are the core of the proposed method, the authors need to re-derive them carefully and align the notation with the cited DDRM reference.
minor comments (5)
  1. [Section 1] There is a typo 'denoisng' in the first sentence; please correct it.
  2. [Section 2.1.1] The sentence 'the larger a is, the slower the noise level curve rises' appears inconsistent with beta_t = t^a for t > 1, where a larger exponent actually makes beta_t rise faster in the later steps. Please clarify the intended behavior, possibly by plotting the normalized schedule.
  3. [Section 2.2.4] The phrase 'on feild data' should be 'on field data'.
  4. [Table 1] The complexity entries for SGMPS-RNA are unclear ('O(Ln · d), L n < T'); please format the table so each method has a single time-complexity expression and a single space-complexity expression.
  5. [References] Reference [16] is a conference version of the present work; the relation should be explicitly stated to avoid self-citation ambiguity. Additionally, the paper does not compare against DDRM or DPS, which are the most direct diffusion-based posterior sampling baselines; such a comparison would contextualize the claimed improvements.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the central construction is an application of the external DDRM/DDIM/DDPM framework plus an explicitly stated (if under-specified) power noise schedule; self-citations appear only as baselines or auxiliary estimators and do not force the claimed results.

full rationale

The paper's claimed derivation chain is: (1) train an unconditional DDPM with a proposed power-function noise schedule; (2) accelerate sampling with DDIM; (3) reuse the same network in a DDRM-style posterior sampler; (4) estimate the observation noise level with an external VI-non-IID model. None of these steps equates a predicted output to a fitted input or reduces the target result to a self-citation by construction. The power schedule β_t = t^a is introduced as an explicit input hyperparameter, not inferred from the data or from the evaluation metrics; this is a design choice, not a fitted parameter renamed as a prediction. The posterior sampling update (Eq. 9) is explicitly adopted from the external DDRM paper [36] and uses the unconditional denoiser as a prior; the paper does not claim to derive this update from its own assumptions. The unconditional model is trained on Marmousi/OpenSEGY and then evaluated on held-out and out-of-distribution data (Sigmoid, field data), so the posterior samples are not statistically forced by the training fit. Self-citations appear for the SGM baselines [15, 20, 24] and for the VI-non-IID noise-level estimator [39]; these are independently published artifacts used as comparison methods or auxiliary inputs, and the main result does not reduce to them. The paper does contain a serious internal inconsistency: as written, β_t = t^a with a = 3 or 4 and T = 1000 violates the stated requirement 0 < β_t < 1 of Eq. (1), implying an unstated normalization; however, that is a reproducibility/correctness flaw in the specification, not a circular reduction. Under the hard rule that circularity must be exhibited as an equation-level equivalence or a fitted-input-called-prediction, no such step can be identified. Therefore the appropriate finding is no significant circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The method imports its main machinery (DDPM, DDIM, DDRM) from prior work. The only newly proposed ingredient, the power schedule, has an unnormalized free parameter a and no stated fitting criterion. The observations are assumed Gaussian, and the VI-non-IID estimator is used as a black box. No invented entities are introduced.

free parameters (4)
  • a (power schedule exponent) = a = 3, 4 (examples given)
    Controls the shape of the beta_t = t^a noise schedule; the paper provides no fitting criterion and no normalization formula.
  • gamma and gamma_b (posterior sampling noise injection) = gamma = 0.9, gamma_b = 1
    Chosen by the authors after testing; stated to be insensitive, but fixed for all experiments.
  • sigma_y (observation noise level) = Estimated per image via VI-non-IID; defaults to min/median/max: 0.0204, 0.0658, 0.1739 on the field data
    Input to posterior sampling; users can interactively choose it, so it is effectively a free parameter of the method.
  • eta (DDIM stochasticity) = eta = 0.8 for unconditional sampling in Figure 8
    Controls determinism versus diversity; not systematically studied.
assumptions (4)
  • domain assumption The unconditional DDPM trained on Marmousi and OpenSEGY captures the seismic data distribution accurately enough for posterior sampling.
    The whole posterior sampling pipeline assumes the pre-trained model is a valid prior for the data, including out-of-distribution field data.
  • domain assumption The observation noise n is Gaussian with known or reliably estimated level sigma_y.
    Sections 2.2 and 3.3 use a Gaussian noise model; field data noise is non-IID and handled by a separate estimator [39] whose accuracy is not validated in this paper.
  • standard math DDIM acceleration and DDRM posterior sampling theory apply unchanged when the DDPM is trained with the new power schedule.
    The paper cites [34] and [36] and does not re-derive these results for the modified schedule.
  • domain assumption The forward operator G is linear and its SVD is computable.
    Equation (7) requires SVD; for large-scale 3D problems this may be prohibitive.

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Cite this review

Pith. "Pith review of Generative modeling of seismic data using diffusion models and its application to multi-purpose posterior sampling for noisy inverse problems." pith.science (2026). https://pith.science/paper/SNVLM66C

@misc{pith2026250612897,
  author       = {Pith},
  title        = {Pith review of: Generative modeling of seismic data using diffusion models and its application to multi-purpose posterior sampling for noisy inverse problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SNVLM66C}},
  note         = {Machine review of arXiv:2506.12897}
}
read the original abstract

Geophysical inverse problems are often ill-posed and admit multiple solutions. Conventional discriminative methods typically yield a single deterministic solution, which fails to model the posterior distribution, cannot generate diverse high-quality stochastic solutions, and limits uncertainty quantification. Addressing this gap, we propose an unsupervised posterior sampling method conditioned on the noisy observations and the inverse problem, eliminating the need to retrain a task-specific conditional diffusion model with paired data for each new application. Specifically, we first propose a diffusion model enhanced with a novel noise schedule for generative modeling of seismic data, and introduce the non-Markov sampling strategy to achieve fast and quality-controllable unconditional sampling. Building upon this, we further present a posterior sampling method for various noisy inverse problems using the trained unconditional diffusion model. Our method requires only a small number of function evaluations to achieve competitive performance, while enabling flexible posterior sampling that interacts adaptively with different noise levels.Experiments on unconditional generation and posterior sampling across different tasks show that our method not only efficiently models the seismic data distribution and posterior conditioned on observations and tasks but also achieves substantially faster sampling and superior out-of-distribution generalization.

Figures

Figures reproduced from arXiv: 2506.12897 by the authors.

Figure 1
Figure 1. Schematic diagram of generative modeling for seismic data using a diffusion model. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Comparison of linear [14], cosine [35], and power noise schedules (a=3). [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Results of unconditional sampling (sample generation) using a trained diffusion model with different noise [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Unconditional sampling (sample generation) with different [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: An example of multi-task posterior sampling results, taking denoising, interpolation, and compressed sensing [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: The impact of diffusion models trained with different noise schedules on posterior sampling results. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Field data and corresponding pixelwise noise level map. [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: Unconditional sampling results of different generative models. Marmousi represents samples randomly [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: Posterior sampling results of subsequence trajectories of different lengths [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: Posterior sampling interacting with predicted noise level. Take (a) denoising, (b) interpolation, and (c) [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: Posterior sampling results of synthetic data, taking tasks (a) denoising, (b) interpolation, and (c) compressed [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: Posterior sampling results of field data, taking tasks (a) denoising, (b) interpolation, and (c) compressed [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]
Figure 13
Figure 13. Figure 13: Schematic diagram of the posterior sampling trajectory of different models in Figure 12. [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]

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Forward citations

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.