Enhancing Robustness Of Digital Shadow For CO2 Storage Monitoring With Augmented Rock Physics Modeling

Abhinav Prakash Gahlot; Felix J. Herrmann

REVIEW 3 major objections 5 minor 18 references

Enhancing Robustness Of Digital Shadow For CO2 Storage Monitoring With Augmented Rock Physics Modeling

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

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Pith's one-line read When a CO2 plume monitor is trained on ten rock physics models instead of one, its forecasts stay accurate even when the true saturation-mixing law is unknown, within the trained family.

desk verdict A plausible but under-evidenced augmentation trick for Digital Shadow CO2 monitoring; the 'out-of-distribution' test is actually in-distribution for the augmented model. read the letter →

arxiv 2502.07171 v1 pith:MJOMZ4HJ submitted 2025-02-11 physics.comp-ph cs.LGphysics.geo-ph

Abhinav Prakash Gahlot , Felix J. Herrmann This is my paper · ORCID

classification physics.comp-phcs.LGphysics.geo-ph

keywords CO2storagemonitoringDigitalShadowrockphysicsuncertaintyBriesaturationmodelconditionalnormalizingflowstime-lapseseismicensembleaugmentationBayesiandataassimilation

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

This paper argues that a CO2 plume monitoring system built on machine-learning data assimilation can be made insensitive to a key modeling guess: the rock physics law that converts reservoir fluid saturation into measurable seismic properties. The authors train a normalizing-flow digital shadow on a forecast ensemble in which each simulated plume is paired with seismic images generated under ten different Brie saturation models, with the mixing exponent drawn uniformly from 1 to 10. They report that when the true rock physics model used to create a test seismic image is unknown, this augmented training keeps the posterior mean close to the true plume and lowers both error and uncertainty, whereas a shadow trained on one rock physics model degrades. The practical stake is that geological CO2 storage sites have poorly constrained rock physics, and safe monitoring needs forecasts that do not silently fail when the assumed model is wrong.

What carries the argument

The load-bearing object is the conditional normalizing flow as a neural posterior density estimator, trained on simulation pairs of plume state and seismic observation. The augmentation mechanism is the Brie saturation model, a rock physics mixing law whose exponent controls how CO2 saturation maps to the effective bulk modulus of the pore fluid, interpolating between uniform and patchy fluid distributions; drawing the exponent uniformly from 1 to 10 and generating one seismic image per exponent multiplies the forecast ensemble tenfold. This teaches the network a family of observation operators instead of a single assumed model, so the learned posterior marginalizes over rock physics uncertainty. The seismic observations are produced by simulating wave propagation and imaging with colored Gaussian noise added before migration.

What would settle it

Run the same augmented digital shadow on a test seismic image generated with a rock physics law that is deliberately outside the Brie family, for example a different effective-medium mixing model, or with a Brie exponent above 10. If the augmented shadow's posterior mean error and uncertainty are no better than the non-augmented baseline, the paper's claim of robustness to unknown rock physics models is refuted.

Watch

Extended reading notes

Core claim

The central claim is that data augmentation over rock physics models makes the Digital Shadow robust to misspecification of the saturation-mixing law. In the paper's formulation, the observation operator draws a rock physics model from the Brie family with exponent uniformly sampled from [1,10], and each CO2 flow simulation is converted into seismic images for ten such exponents, producing 1280 training pairs from 128 flow simulations. A conditional normalizing flow is then trained to map seismic observations to posterior plume states. At inference, when the seismic observation is generated with an unknown (but within-family) exponent, the augmented shadow yields a conditional mean closer to the ground-truth plume and reduced uncertainty compared with the non-augmented shadow trained on a single rock physics model. The authors frame this as mitigating the negative effects of incorrect rock physics assumptions and improving the reliability of CO2 storage monitoring.

Load-bearing premise

The whole robustness result rests on the true rock physics behavior being representable by a Brie saturation model with exponent between 1 and 10; if the real mixing law falls outside that family or range, the augmented training gives no guarantee.

Editorial extensions

If this is right

An augmented digital shadow remains accurate when the true rock physics exponent is unknown but within the trained range [1,10], where the non-augmented shadow's mean deviates from the true plume.
Posterior uncertainty is lower in the unknown-exponent case, giving operators narrower and more truthful confidence bands.
The augmentation multiplies the effective training set tenfold from the same 128 flow simulations, so the additional cost is seismic simulations rather than new flow simulations.
The approach inherits the amortized character of the shadow: after training, inference on a new seismic survey is fast, which matters for repeated monitoring.

Reading between the lines

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

Because the training distribution explicitly randomizes the Brie exponent, the learned posterior is effectively a mixture over rock physics models; this makes the observed uncertainty reduction a form of Bayesian model averaging, an interpretation the paper does not spell out.
The paper's out-of-distribution test is still in-distribution with respect to the augmented training family, so the result is interpolation; a stronger test would use a mixing law absent from the Brie family.
The same augmentation idea should carry over to other uncertain components of the observation operator, such as wavelets, noise statistics, or anisotropy parameters; the paper only randomizes the saturation exponent.
One could extend the flow to output a posterior over the rock physics exponent itself, turning an assumed nuisance parameter into a monitored quantity that flags when the true model leaves the trained family.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The genuinely new thing here is narrow but real: the authors show, on one synthetic case, that training their CNF-based Digital Shadow on seismic images generated with several Brie saturation exponents (e~U(1,10)) produces better looking posterior means and lower qualitative uncertainty than a shadow trained on a single exponent, when the test image comes from a different exponent in that range. That is a sensible extension of their earlier Digital Shadow work, and the augmentation cost is cheap because the expensive flow simulations are reused. The paper is honest in its conclusion that the robustness is only demonstrated within the Brie family, and it uses open-source tools throughout, which helps reproducibility. The soft spots are exactly where the stress-test lands. The Figure 1b/c setup is labeled 'unknown rock physics model,' but for the augmented model the test exponent is drawn from the same U(1,10) range used in training. That is in-support Bayesian model averaging over e, not out-of-distribution generalization to a misspecified forward model. Calling it 'out-of-distribution' oversells the result. The second problem is the evidence itself: no quantitative error or uncertainty metrics, no error bars, no repeated experiments, and the non-augmented baseline is not described beyond 'the correct rock physics model.' So the central claim is plausible but under-supported. The authors should report numbers (e.g., NRMSE or log-likelihood for the posterior mean/samples), show what happens for e outside [1,10] or for a different mixing model like Voigt/Reuss bounds, and clarify that the augmented model is marginalizing over the training distribution of e, not magically robust to anything outside it. This is a short preprint that reads like an extended abstract. The direction matters for GCS monitoring, and the result, if confirmed with quantitative evidence, would be a useful incremental contribution. As it stands, the paper is not publishable as-is, but it deserves a serious referee rather than a desk reject, because the idea is sensible and the missing evidence is obtainable with modest additional work. Tell the authors to add metrics and a true OOD test; then it becomes a solid workshop or short-conference paper.

Referee Report

3 major / 5 minor

Summary. The paper proposes making a Digital Shadow data-assimilation framework for CO2 storage monitoring more robust to rock-physics model uncertainty by augmenting the training set of a conditional normalizing flow with multiple Brie saturation models, with exponent e drawn from U(1,10). The authors train the augmented and non-augmented CNFs on synthetic forecast/observation pairs from a 2D Compass-based model, then compare their posterior plume estimates on a synthetic test in which the seismic observation is generated with an 'unknown' rock physics model. They report qualitatively that augmentation brings the conditional mean closer to ground truth and reduces error and uncertainty.

Significance. If the effect is real, ensemble augmentation over rock-physics models is a simple, practical way to hedge against misspecified saturation mixing models in learned seismic monitoring, and the paper builds on a reproducible pipeline of open-source tools (JutulDarcy.jl, JUDI.jl, InvertibleNetworks.jl). The authors are explicit that the range of protection is the Brie family with unspecified exponent. However, the current evidence is only qualitative, and the 'out-of-distribution' test is not outside the training support of the augmented model, so the central robustness claim is not yet established.

major comments (3)

[Section 3.2 and Section 4, Figure 1] The test labeled 'unknown rock physics model' in Figure 1b/c is generated from the same Brie family and the same U(1,10) exponent range used to augment the training data in Section 3.2. For the augmented model, this is an in-distribution sample from the marginal p(y)=∫p(y|e)p(e)de, so Figure 1c demonstrates Bayesian model averaging over the training prior on e, not robustness to an out-of-distribution or misspecified rock physics model. To support the OOD claim, the authors should either remove the 'out-of-distribution' label or add truly OOD tests, such as e values outside [1,10] or a different mixing model (e.g., Voigt/Reuss bounds or patchy versus uniform saturation).
[Section 4, Figure 1] No quantitative error, uncertainty, or statistical coverage metrics are reported; 'closer to the GT' and 'reduced uncertainty' are supported only by visual inspection of three panels. The authors should report numerical measures such as RMSE, mean absolute error, energy score, or interval coverage, and repeat the experiment over multiple flow realizations, noise draws, and test exponents to show that the improvement is consistent rather than a single cherry-picked realization.
[Section 3.2] The paper states that the Brie exponent is uniformly sampled from e~U(1,10), but then says 10 different exponents are used to augment the data tenfold. This is a discrete set, not a continuous uniform sample, and the number of distinct rock-physics variants is an important design choice. A sensitivity analysis varying the number of exponent values (e.g., 3, 10, 30) would clarify whether the observed robustness gain is stable or depends heavily on this choice.

minor comments (5)

[Equation (3)] The notation is inconsistent: the text defines the network weights as phi and the objective uses f_phi, while the surrounding sentence refers to the Jacobian of f_theta; please unify the symbol.
[Figure 1] The caption says 'at k=1' without explaining whether results at other timesteps are similar; please state whether this is representative or add the corresponding panels for other timesteps.
[Section 3.1] The injection rate is written as '0.0500 m3/s' and other quantities use plain text; use proper superscript formatting for units throughout.
[Introduction/References] The reference 'Panel on Climate Change) 2018' has a formatting error and should be corrected to the standard IPCC citation.
[Section 5, Conclusions] The conclusion mentions augmentation of 'fluid-flow properties', but the augmentation described in Section 3.2 only varies the rock physics model, not the flow simulation; please align the wording with the actual procedure.

Circularity Check

1 steps flagged · score 4.0 of 10

The 'out-of-distribution' rock-physics test is drawn from the same U(1,10) Brie-family distribution used to augment the training ensemble, so the claimed robustness is demonstrated in-distribution rather than against a truly unknown model.

self definitional [Sec. 3.2 (training augmentation), Sec. 4 / Fig. 1 (evaluation), Sec. 5 (conclusion)]
"The symbol Rk denotes the rock physics model, randomly drawn from the family of Brie Saturation models (Avseth, Mukerji, and Mavko 2010), with the exponent e uniformly sampled from the range e∼U (1, 10). ... Augmenting the ensemble improves generalization when seismic observations are based on an unknown rock physics model, as demonstrated in figure 1c. ... at least within the family of Brie saturation models where the exponent e is not specified."

The 'unknown rock physics model' used in the test is generated from the same Brie saturation family and the same e~U(1,10) range that defines the augmented training distribution. For the augmented model, the test observation is therefore a sample from the training marginal p(y)=∫p(y|e)p(e)de, not an out-of-distribution observation. The reported improvement is Bayesian model averaging / interpolation over the exponent e, not evidence of robustness to rock physics assumptions outside the assumed family or range. The conclusion restricts the claim to 'the family of Brie saturation models where the exponent e is not specified,' but the Section 4 wording 'unknown rock physics model' and Figure 1's 'out-of-distribution y' overstate the generalization.

full rationale

The numerical derivation chain--multi-phase flow simulation, seismic observation modeling with the Brie saturation model, and conditional normalizing flow training--is self-contained and does not reduce to fitting parameters to the test plume. The central issue is evaluative: the robustness claim is tested with an 'unknown' rock physics model that is in fact drawn from the same U(1,10) Brie-family distribution used to construct the augmented training set. Thus the augmented model sees the test as an in-distribution sample, and the demonstrated gain is interpolation within the assumed model family, not generalization to a misspecified rock physics law outside that family. The conclusion's parenthetical acknowledgment, 'at least within the family of Brie saturation models where the exponent e is not specified,' confirms this limitation. Self-citations to prior Digital Shadow and InvertibleNetworks work are methodological and are supported by open-source code and external solvers, so they are not load-bearing circularity. Score 4 reflects a partial, evaluation-level circularity in the 'unknown/out-of-distribution' framing while the central learning task retains independent empirical content.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No new physical entities, forces, or conserved quantities are introduced; the paper only varies parameters of an existing rock physics model. The free parameters are hand-chosen experimental design choices. The axioms are the standard domain assumptions of the simulation and ML pipeline, none of which are machine-checked.

free parameters (2)

Brie exponent range = e ~ U(1,10)
Chosen by hand, not estimated from data. Defines the augmented training distribution and the claimed robustness envelope.
Number of rock physics variants per flow simulation = 10
Selected to give 1280 training pairs from 128 flow runs; no sensitivity analysis is shown for this choice.

assumptions (4)

domain assumption Brie saturation model family spans uniform-to-patchy mixing and is a valid rock physics description for CO2-saturated sandstones.
Assumed in Section 2 (Eq. 2) and Section 3.2 when generating augmented seismic data; the robustness claim is only evaluated within this family.
domain assumption JutulDarcy multi-phase flow simulations correctly represent CO2 plume evolution in the Compass-based synthetic reservoir.
All ground-truth states x_k are produced by this simulator (Section 3.1), so the observed 'truth' is an artifact of the simulator.
domain assumption The conditional normalizing flow trained on 1280 pairs approximates the posterior p(x|y) sufficiently well for the comparison to be meaningful.
Section 3.3 specifies the loss but provides no calibration, convergence diagnostic, or architecture details.
domain assumption The permeability prior obtained from probabilistic full-waveform inversion and the empirical relationship of Gahlot, Orozco, et al. (2024) is representative.
Used to initialize the ensemble (Section 3); inherits all assumptions of that prior work.

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Cite this review

Pith. "Pith review of Enhancing Robustness Of Digital Shadow For CO2 Storage Monitoring With Augmented Rock Physics Modeling." pith.science (2026). https://pith.science/paper/MJOMZ4HJ

@misc{pith2026250207171,
  author       = {Pith},
  title        = {Pith review of: Enhancing Robustness Of Digital Shadow For CO2 Storage Monitoring With Augmented Rock Physics Modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MJOMZ4HJ}},
  note         = {Machine review of arXiv:2502.07171}
}

read the original abstract

To meet climate targets, the IPCC underscores the necessity of technologies capable of removing gigatonnes of CO2 annually, with Geological Carbon Storage (GCS) playing a central role. GCS involves capturing CO2 and injecting it into deep geological formations for long-term storage, requiring precise monitoring to ensure containment and prevent leakage. Time-lapse seismic imaging is essential for tracking CO2 migration but often struggles to capture the complexities of multi-phase subsurface flow. Digital Shadows (DS), leveraging machine learning-driven data assimilation techniques such as nonlinear Bayesian filtering and generative AI, provide a more detailed, uncertainty-aware monitoring approach. By incorporating uncertainties in reservoir properties, DS frameworks improve CO2 migration forecasts, reducing risks in GCS operations. However, data assimilation depends on assumptions regarding reservoir properties, rock physics models, and initial conditions, which, if inaccurate, can compromise prediction reliability. This study demonstrates that augmenting forecast ensembles with diverse rock physics models mitigates the impact of incorrect assumptions and improves predictive accuracy, particularly in differentiating uniform versus patchy saturation models.

Figures

Figures reproduced from arXiv: 2502.07171 by the authors.

**Figure 1.** Inference results of digital shadow at k = 1 (a) without data augmentation and known rock physics model for seismic observation (b) without data augmentation and unknown rock physics model for seismic observation (c) with data augmentation and unknown rock physics model for seismic observation. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

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Reference graph

Works this paper leans on

18 extracted references · 4 canonical work pages

[1]

Avseth, Per, Tapan Mukerji, and Gary Mavko. 2010. Quantitative Seismic Interpretation: Applying Rock Physics Tools to Reduce Interpretation Risk. Cambridge university press

work page 2010
[2]

Jones, C., J

E. Jones, C., J. A. Edgar, J. I. Selvage, and H. Crook. 2012. ``Building Complex Synthetic Models to Evaluate Acquisition Geometries and Velocity Inversion Technologies.'' In 74th EAGE Conference and Exhibition Incorporating EUROPEC 2012, cp--293. https://doi.org/https://doi.org/10.3997/2214-4609.20148575

arXiv 2012
[3]

Inference of CO2 flow patterns -- a feasibility study

Gahlot, Abhinav Prakash, Huseyin Tuna Erdinc, Rafael Orozco, Ziyi Yin, and Felix J. Herrmann. 2023. ``Inference of CO2 Flow Patterns a Feasibility Study.'' https://doi.org/10.48550/arXiv.2311.00290

work page Pith review arXiv doi:10.48550/arxiv.2311.00290 2023
[4]

Gahlot, Abhinav Prakash, Haoyun Li, Ziyi Yin, Rafael Orozco, and Felix J Herrmann. 2024. ``A Digital Twin for Geological Carbon Storage with Controlled Injectivity.'' arXiv Preprint arXiv:2403.19819

arXiv 2024
[5]

Herrmann

Gahlot, Abhinav Prakash, Rafael Orozco, Ziyi Yin, and Felix J. Herrmann. 2024. ``An Uncertainty-Aware Digital Shadow for Underground Multimodal CO2 Storage Monitoring.'' https://doi.org/10.48550/arXiv.2410.01218

arXiv doi:10.48550/arxiv.2410.01218 2024
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Herrmann, Felix J. 2023. ``President's Page: Digital Twins in the Era of Generative AI.'' The Leading Edge 42 (11): 730--32

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Kingma, Diederik P., and Jimmy Ba. 2014. ``Adam: A Method for Stochastic Optimization.'' CoRR abs/1412.6980. https://api.semanticscholar.org/CorpusID:6628106

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Louboutin, Mathias, Philipp Witte, Ziyi Yin, Henryk Modzelewski, Kerim, Carlos da Costa, and Peterson Nogueira. 2023. ``Slimgroup/JUDI.jl: V3.2.3.'' Zenodo. https://doi.org/10.5281/zenodo.7785440

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Orozco, Rafael, Philipp Witte, Mathias Louboutin, Ali Siahkoohi, Gabrio Rizzuti, Bas Peters, and Felix J. Herrmann. 2024. ``InvertibleNetworks.jl: A Julia Package for Scalable Normalizing Flows.'' Journal of Open Source Software 9 (99): 6554. https://doi.org/10.21105/joss.06554

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Witte, Philipp A., Mathias Louboutin, Navjot Kukreja, Fabio Luporini, Michael Lange, Gerard J. Gorman, and Felix J. Herrmann. 2019. ``A Large-Scale Framework for Symbolic Implementations of Seismic Inversion Algorithms in Julia.'' Geophysics 84 (3): F57--71. https://doi.org/10...

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Avseth, Per, Tapan Mukerji, and Gary Mavko. 2010. Quantitative Seismic Interpretation: Applying Rock Physics Tools to Reduce Interpretation Risk. Cambridge university press

work page 2010

[2] [2]

Jones, C., J

E. Jones, C., J. A. Edgar, J. I. Selvage, and H. Crook. 2012. ``Building Complex Synthetic Models to Evaluate Acquisition Geometries and Velocity Inversion Technologies.'' In 74th EAGE Conference and Exhibition Incorporating EUROPEC 2012, cp--293. https://doi.org/https://doi.org/10.3997/2214-4609.20148575

arXiv 2012

[3] [3]

Inference of CO2 flow patterns -- a feasibility study

Gahlot, Abhinav Prakash, Huseyin Tuna Erdinc, Rafael Orozco, Ziyi Yin, and Felix J. Herrmann. 2023. ``Inference of CO2 Flow Patterns a Feasibility Study.'' https://doi.org/10.48550/arXiv.2311.00290

work page Pith review arXiv doi:10.48550/arxiv.2311.00290 2023

[4] [4]

Gahlot, Abhinav Prakash, Haoyun Li, Ziyi Yin, Rafael Orozco, and Felix J Herrmann. 2024. ``A Digital Twin for Geological Carbon Storage with Controlled Injectivity.'' arXiv Preprint arXiv:2403.19819

arXiv 2024

[5] [5]

Herrmann

Gahlot, Abhinav Prakash, Rafael Orozco, Ziyi Yin, and Felix J. Herrmann. 2024. ``An Uncertainty-Aware Digital Shadow for Underground Multimodal CO2 Storage Monitoring.'' https://doi.org/10.48550/arXiv.2410.01218

arXiv doi:10.48550/arxiv.2410.01218 2024

[6] [6]

Herrmann, Felix J. 2023. ``President's Page: Digital Twins in the Era of Generative AI.'' The Leading Edge 42 (11): 730--32

work page 2023

[7] [7]

Kingma, Diederik P., and Jimmy Ba. 2014. ``Adam: A Method for Stochastic Optimization.'' CoRR abs/1412.6980. https://api.semanticscholar.org/CorpusID:6628106

arXiv 2014

[8] [8]

Louboutin, Mathias, Philipp Witte, Ziyi Yin, Henryk Modzelewski, Kerim, Carlos da Costa, and Peterson Nogueira. 2023. ``Slimgroup/JUDI.jl: V3.2.3.'' Zenodo. https://doi.org/10.5281/zenodo.7785440

doi:10.5281/zenodo.7785440 2023