the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 21 Apr 2026
LFD (v1.0): Latent-Compression-Free Generative Diffusion with Geological Priors and Geophysical Regularization for Implicit Structural Modeling
Abstract. Diffusion models provide a promising way to model the distribution of implicit structural models, potentially improving generalization across surveys. However, existing diffusion Transformer pipelines scale poorly to high-dimensional geophysical data because noise- or velocity-prediction objectives are often unstable at large patch sizes, forcing the use of small patches that lead to long token sequences and high computational cost. To reduce computation, most approaches rely on Variational autoencoders (VAEs) and latent diffusion, but robust pretrained VAEs are scarce in geophysics, and enforcing geological priors in latent space is difficult. To address these scalability bottlenecks and the difficulty of enforcing geological priors in latent space, we propose Latent-Compression-Free Generative Diffusion (LFD) with Geological Priors and Geophysical Regularization for implicit structural modeling. Built on flow matching, LFD generates implicit structural models directly in the data space, enabling efficient large-patch Vision Transformer (ViT) inference and allowing fault/horizon constraints and geophysical regularization to be applied explicitly during generation. To strengthen structural conditioning, we design a structure-enhanced Transformer that injects horizon and fault embeddings at multiple layers. We further introduce two prior-guided losses: a horizon loss to match the generated models to the input horizons, and a fault-aware bending-energy term that regularizes smoothness while ignoring stencils across faults. By enforcing these priors directly in the data space, the model is effectively constrained to generate geologically reasonable structures. Experiments on both synthetic data and real surveys validate the effectiveness of LFD for prior-guided implicit structural modeling. Benefiting from large-patch inference, LFD generates a 512x512 model in 1.56 s on an NVIDIA H20 GPU. With relative positional encoding, LFD can be extended to higher resolutions via simple adaptation without retraining. Overall, LFD offers new insights into deploying diffusion models for high-dimensional geophysical data, enabling efficient generation with interpretable, prior-guided constraints.
- Preprint
(4601 KB) - Metadata XML
- BibTeX
- EndNote
Status: final response (author comments only)
-
RC1: 'Comment on egusphere-2026-1087', Samuel Thiele, 28 May 2026
- AC1: 'Reply on RC1', Xinming Wu, 03 Jun 2026
-
CEC1: 'Comment on egusphere-2026-1087 - No compliance with the policy of the journal', Juan Antonio Añel, 02 Jun 2026
Dear authors,
Unfortunately, after checking your manuscript, it has come to our attention that it does not comply with our "Code and Data Policy".
https://www.geoscientific-model-development.net/policies/code_and_data_policy.html
As already a reviewer has commented, the replicability of your manuscript can not be achieved with the information that you have provided. Given your study, you need to include in the repository with the code all the training data and pretrained weights used for your work.
The GMD review and publication process depends on reviewers and community commentators being able to access, during the discussion phase, the code and data on which a manuscript depends, and on ensuring the provenance of replicability of the published papers for years after their publication. Please, therefore, publish the requested information in one of the appropriate repositories and reply to this comment with the link and a permanent identifier for it (e.g. DOI) as soon as possible. We cannot have manuscripts under discussion that do not comply with our policy.
Later, if the Topical Editor decides to continue with the review or publication process of your manuscript and you are requested to upload a new version of it, then The 'Code and Data Availability’ section of your manuscript must also be modified to cite the new repository locations, and corresponding references added to the bibliography.
I must note that if you do not fix this problem, we cannot continue with the peer-review process or accept your manuscript for publication in GMD.
Juan A. Añel
Geosci. Model Dev. Executive EditorCitation: https://doi.org/10.5194/egusphere-2026-1087-CEC1 -
AC2: 'Reply on CEC1', Xinming Wu, 03 Jun 2026
Dear Prof. Añel,
Thank you for bringing this to our attention. We apologize for the initial non-compliance with the journal's Code and Data Policy.
We have now updated our repository to include the training data, pretrained weights, and all code necessary to reproduce the results presented in the manuscript. The repository is publicly available at the following locations:
1. Zenodo (permanent DOI): https://doi.org/10.5281/zenodo.20508635
2. GitHub: https://github.com/ProgrammerZXG/LFD
We will also update the 'Code and Data Availability' section in the revised manuscript to cite the new repository locations accordingly.
We hope this satisfies the journal's requirements and that the review process can continue.
Kind regards,
Zhixiang Guo
Citation: https://doi.org/10.5194/egusphere-2026-1087-AC2 -
CEC2: 'Reply on AC2', Juan Antonio Añel, 03 Jun 2026
Dear authors,
Thanks for addressing this issue so quickly. I have checked the repositories and we can consider now the current version of your manuscript in compliance with the code policy of the journal. Please, note that the GitHub site does not serve the purpose of the policy, and that GitHub sites are not valid repositories, so if you want you can remove its link from the Code and Data Availability statement.
Juan A. Añel
Geosci. Model Dev. Executive Editor
Citation: https://doi.org/10.5194/egusphere-2026-1087-CEC2
-
CEC2: 'Reply on AC2', Juan Antonio Añel, 03 Jun 2026
-
AC2: 'Reply on CEC1', Xinming Wu, 03 Jun 2026
-
RC2: 'Comment on egusphere-2026-1087', Pouria Behnoudfar, 05 Jul 2026
Summary
This manuscript proposes Latent-Compression-Free Generative Diffusion (LFD), a flow-matching-based generative model that operates directly in data space rather than in a VAE latent space, avoiding the scalability and prior-enforcement limitations of latent diffusion pipelines for implicit structural modeling. The authors introduce a structure-enhanced Vision Transformer with multi-layer horizon/fault conditioning, together with two prior-guided losses (a horizon loss and a fault-aware bending-energy loss) that are applied directly in data space. Experiments on synthetic and real data are used to demonstrate efficient, prior-constrained generation. The problem is relevant, and the data-space approach is a reasonable response to the scarcity of robust geophysical VAEs. However, I have several concerns, detailed below, regarding uncertainty quantification, reproducibility, and the formulation and justification of the fault-aware bending-energy loss (Eq. 14), which I believe should be addressed before the manuscript is suitable for publication.
General Comments
- Uncertainty quantification. The manuscript frames LFD as a generative model for the distribution of implicit structural models, which implies that the method should be capable of representing and communicating uncertainty across plausible structural realizations. This is, after all, one of the principal motivations for using a generative rather than a deterministic model in this setting. However, the manuscript does not clearly characterize the uncertainty produced by the method: it is not clear whether multiple samples from LFD given the same conditioning (horizons/faults) are shown to produce meaningfully diverse, geologically plausible realizations, how the spread of these realizations compares to the known or expected geological uncertainty in the synthetic/real test cases, and whether the prior-guided losses (horizon loss, bending-energy loss), by construction, narrow the ensemble in ways that could underrepresent genuine structural uncertainty. Given that uncertainty representation is central to the value proposition of a diffusion/flow-matching approach over deterministic implicit modeling, I recommend that the authors add an explicit quantitative and visual analysis of ensemble spread/variability (e.g., pixel-wise standard deviation maps or probability-of-occurrence maps across multiple samples) and discuss how the imposed priors affect this spread.
- Reproducibility of the code. I attempted to run the code accompanying the manuscript and was unable to get it to execute. As the manuscript's central claims rely on specific implementation choices (e.g., the structure-enhanced Transformer conditioning, the fault-aware finite-difference stencils in Eq. 14, and the reported inference time of 1.56 s for a 512×512 model on an NVIDIA H20 GPU), I am not currently able to independently verify these claims. I would ask the authors to (a) verify that the uploaded code corresponds to the exact version used to produce the results in the manuscript, (b) provide a minimal working example with pinned dependency versions and, if possible, a small pretrained checkpoint or synthetic toy dataset that reproduces at least one figure/table end to end. Given GMD's emphasis on reproducibility of model development, this should be resolved before publication.
- Uniform-grid assumption underlying Eq. 14. The fault-aware bending-energy term in Eq. 14 is written in terms of pixel-index derivatives (∂²ˆx/∂i², ∂²ˆx/∂i∂j, ∂²ˆx/∂j²). This formulation implicitly assumes that the grid spacing is uniform and identical in both the i- and j-directions (i.e., an isotropic, regular sampling grid), since otherwise the index-space second differences do not correspond to a consistent physical curvature. This assumption should be stated explicitly, since implicit structural models are frequently defined on grids with unequal vertical/horizontal spacing, or on locally warped/flattened grids. I would ask the authors to: (a) explicitly state whether the ViT patch grid used by LFD is always uniform and isotropic in physical units; (b) if not, clarify whether the derivatives in Eq. 14 are normalized by the local grid spacing (e.g., divided by Δi², ΔiΔj, Δj², respectively); and (c) provide at least one experiment or discussion addressing whether/how the method would behave on non-uniform grids, since this materially affects the physical meaning of the bending-energy penalty and, by extension, the geological plausibility of the generated structures.
- Sensitivity to noise level, number of sampling steps, and initial state. The manuscript reports a single operating point for the flow-matching/diffusion sampler (e.g., a fixed number of integration steps and a fixed initialization) without a systematic sensitivity analysis. Since LFD is positioned as a generative model whose outputs should be both efficient and geologically reliable, I believe such an analysis is necessary. Specifically: (a) Noise level/integration step size: flow-matching and diffusion samplers are known to be sensitive to the discretization of the noise/time schedule, and coarser schedules can introduce integration error that is not obviously distinguishable from genuine multi-modal geological variability. The authors should report how the quality of the generated structural models (fidelity to horizons/faults, bending-energy statistics, and any other structural metrics used) varies as the step size/noise schedule is coarsened or refined. (b) Number of sampling steps: relatedly, the manuscript should report how output quality and inference time trade off as the number of steps is varied, since the headline efficiency result (1.56 s for a 512×512 model) is only meaningful in the context of a stated accuracy level, and readers need to know whether fewer steps could be used without materially degrading geological plausibility, or whether more steps are required to reach convergence. (c) Initial state: because LFD generates directly in data space rather than in a compact latent space, the choice of initial noise/state may have a more direct and visible effect on the final structural model than it would in latent diffusion. The authors should clarify whether the initial state is drawn from a fixed prior (e.g., standard Gaussian noise) for every sample, and, if so, present an ablation showing how the diversity and quality of generated structural models depend on the initial state. Without this sensitivity analysis, it is difficult to assess how robust the reported results are to these sampler-level choices, and whether the reported timing/quality figures represent a carefully chosen operating point or a somewhat arbitrary one.
Specific Comment: The Second-Order Finite-Difference Bending-Energy Loss (Eq. 14)
I have a few concerns about using this discrete, second-order finite-difference bending energy as a training loss, for the following reasons, which compound the grid-uniformity issue raised above.
- Grid metric is not accounted for. As noted above, Eq. 14 is written entirely in index space. Even setting aside non-uniform grids, vertical and horizontal sampling intervals in seismic/structural volumes are typically different by construction, so the mixed term implicitly assumes an isotropic metric that generally does not hold. The authors should clarify whether/how the loss accounts for anisotropic or spatially varying grid spacing, and, if it does not, discuss the resulting bias.
- The "clean" prediction is still an uncertain estimate at high noise levels, and this is where curvature penalties are most fragile. The manuscript states that "as the model directly predicts the clean data, geological observations and prior constraints can be imposed in a straightforward and differentiable manner", and applies this prediction. I would like to point out that even though is nominally the "clean" target, at early sampling steps / high noise levels the network's prediction of given a heavily corrupted input is fundamentally underdetermined: many plausible clean structural models are consistent with the same noisy observation, and the network's point estimate at this stage is typically a blurred, low-confidence average over these possibilities rather than a single coherent geological structure. Its discrete second-order curvature at this stage therefore reflects the shape of the network's predictive uncertainty, not a genuine geological feature (or lack thereof). Penalizing that curvature is a much less well-posed operation than penalizing the curvature of a final, high-confidence output, and could bias the network toward predicting overly smooth estimates at high noise levels purely to reduce this loss term, independent of whether the final sample should be smooth. I would ask the authors to clarify: (a) at which noise levels/timesteps it is applied during training (all, or only near the clean end of the schedule), and (b) whether the loss weighting is scheduled/annealed as a function of noise level to account for the fact that it is far less reliable at high noise levels. If the loss is applied uniformly across all noise levels, I would ask for evidence (e.g., a comparison of results with vs. without this weighting) that it is not simply driving the network toward generically smoother predictions rather than toward geologically faithful ones.
- Inconsistent stencil order near fault-mask boundaries. "Fault-aware stencils that do not cross fault pixels" implies one-sided or truncated finite differences adjacent to. Standard central differences are second-order accurate, whereas one-sided or truncated stencils are typically only first-order accurate or have a different truncation-error constant. This means the effective smoothness penalty is not spatially homogeneous: pixels near a fault are regularized with a different bias/strength than interior pixels, purely as an artifact of discretization rather than of the underlying geology. Please specify the exact one-sided/modified stencils used near fault boundaries and confirm that they remain numerically consistent (i.e., that they converge to the correct derivative as ), rather than being a central stencil with missing terms simply set to zero, which would not be consistent.
- Risk of over-smoothing genuine, non-fault geological curvature. A quadratic penalty on bending energy favors globally low curvature (thin-plate-like behavior). However, fold hinges, drag folds, and tight synclines/anticlines away from faults are geologically real, legitimate high-curvature features that are not associated with a discontinuity. A blanket quadratic penalty risks over-smoothing these features unless the loss weight is very carefully tuned. I would ask for an ablation showing the sensitivity of the results to the weight of , and specifically its effect on fold-heavy synthetic or real test cases, together with a discussion of whether a more robust penalty (e.g., an L1 curvature term or edge-preserving weighting) was considered and, if so, why it was not adopted.
In summary, I would ask the authors to reframe and justify Eq. 14 more carefully: to state explicitly whether/how it accounts for grid anisotropy and non-uniformity, to clarify at which noise levels the loss is applied and how its reliability is affected by prediction uncertainty at high noise, to give the exact form of the fault-aware stencils and confirm their consistency, and to demonstrate through ablation that the loss does not over-smooth genuine, non-fault geological curvature.
The core idea of enforcing geological priors directly in data space is promising and well motivated, but the concerns above regarding uncertainty quantification, code reproducibility, and the formulation of the fault-aware bending-energy loss (including its implicit grid-uniformity assumption) should be addressed before the manuscript can be considered for publication in GMD.
Citation: https://doi.org/10.5194/egusphere-2026-1087-RC2 - AC3: 'Reply on RC2', Xinming Wu, 14 Jul 2026
Viewed
| HTML | XML | Total | BibTeX | EndNote | |
|---|---|---|---|---|---|
| 304 | 138 | 35 | 477 | 29 | 19 |
- HTML: 304
- PDF: 138
- XML: 35
- Total: 477
- BibTeX: 29
- EndNote: 19
Viewed (geographical distribution)
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
Dear Editor, Authors,
This manuscript introduces an interesting and novel framework for applying diffusion architectures to implicit structural modeling. It is an innovative and thought provoking approach, however I have several major comments that should be addressed before the manuscript is accepted for publication.
1) Geophysical vs. Geological Data: The manuscript frequently refers to "geophysical data," yet the model constraints and inputs appear to consist exclusively of geological interpretations (e.g., horizons and faults). While these are presumably derived from seismic data, they are geological interpretations rather than geophysical data. Please either clarify what is meant by "geophysical" or modify the text to remove references to "geophysical data" .
2) Ablation study and uncertainty: Please extend the results or discussion section to include an ablation study showing how the model's predictions change as the horizon constraints become increasingly sparse. A multi-realisation comparison (using metrics such as information entropy) to demonstrate how well the generated realisations cover the plausible, data-consistent model space is also needed. Ideally this uncertainty analysis should be explicitly linked to the sparse horizon ablation study, to illustrate how realisation variance increases as the amount of conditioning data is reduced.
3) Expanded Discussion: The current discussion is quite short. I suggest expanding it significantly, including to address the models generalisation capacity. How close (geologically speaking) do applications need to be to the synthetic training data for the model to perform reliably? How much variability can the model produce, especially when data is quite sparse? Does it / can it produce geologically impossible results (e.g., "bubbles")?
4) Reproducibility: I attempted to run the code provided in the Zenodo repository but was unable to do so. To ensure the presented method is FAIR and usable, please update it to include a comprehensive README.md file explaining the code structure and how to get started. Specifically, the documentation should explain how to setup the required data, dependencies, and paths. Currently, it is unclear where to download the required ImageNet model, how to set the IMAGENET_PATH, and where to download the pretraining checkpoint (PRETRAIN_CKPT). Where to download the training and testing datasets (and so set the associated paths) is also unclear.
5) Figure Placement: Please check that all figures appear after their first mention in the text. Currently, several figures seem to appear very early in the manuscript and are not discussed until much later. Additionally, please clarify in Fig. 1 that the implicit field is predicted as a continuous variable. It would be useful to plot this continuous field alongside the discrete lithology/color visualisations (in at least one figure), as this gives a clearer representation of the models output.
Minor Points
Abstract (Line 1): The phrase "diffusion models provide a promising way to model the distribution of implicit structural models" is awkward (do the models really model the distribution of implicit structural models? Which implicit structural models?). Consider rewording for clarity.
Abstract (Line 5): Change "Variational" to lowercase.
Abstract (Line 11): Change "Transformer" to lowercase.
Line 116: Please explicitly clarify if the model predicts continuous implicit field values or discrete lithology classes. I assume the former is true, but the text and figure captions should be updated to make this unambiguous.
Line 125: Please define what is meant by "geophysical data" in the context of this study, or remove the term (as the manuscript does not currently feature integration with geophysical datasets).
Figure 3: Please discuss what happens to the framework's performance when the fault mask is also sparse?
Figure 3: There appears to be an absence of unconformities in the demonstration data (and potentially the training set?). Please address this limitation or design choice in the discussion. Was the model able to accurately reproduce unconformitites? How were these parameterised in the implicit framework (as the implicit value above and below an unconformity is not comparable).
Line 210: When creating the structural data, was the timing of the structural events allowed to vary (e.g., scenarios involving faults that are truncated by an unconformity)? Please expand a little on the synthetic modeling logic.
Line 216: Please expand upon the explanation of the random subsampling of horizons. What percentage of the horizons were retained, and how were the removed horizons selected? This more detailed description should ideally feed directly into the major ablation study requested above.
I look forward to seeing an updated version of this interesting work.
Kind regards,
Sam Thiele