Probabilistic Joint Recovery Method for CO$_2$ Plume Monitoring

Abhinav Prakash Gahlot; Felix J. Herrmann; Rafael Orozco; Zijun Deng

REVIEW 3 major objections 4 minor 9 references

Probabilistic Joint Recovery Method for CO$_2$ Plume Monitoring

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Probabilistic joint recovery reconstructs CO2 plumes with uncertainty maps, not just a single image.

desk verdict Adapts a shared-generator idea to time-lapse seismic with qualitatively better reconstructions, but the posterior claim is never derived or calibrated—the central UQ contribution is unsupported as it stands. read the letter →

arxiv 2501.18761 v1 pith:EMIU2B5F submitted 2025-01-30 cs.LG physics.ao-ph

Zijun Deng , Rafael Orozco , Abhinav Prakash Gahlot , Felix J. Herrmann This is my paper · ORCID

classification cs.LGphysics.ao-ph

keywords CO2plumemonitoringtime-lapseseismicimagingjointrecoverymethodposteriordistributionsharedgenerativemodeluncertaintyquantificationcarboncaptureandstoragedeepdecoder

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 claims that repeated seismic surveys of a CO2 storage site can be inverted together to produce not one time-lapse image but a posterior distribution of possible plume images, with explicit uncertainty. The method, called the Probabilistic Joint Recovery Method (pJRM), shares a single generative model across surveys so that common subsurface features are reconstructed jointly, while per-survey latent variables capture what changes. In synthetic post-stack experiments the joint reconstruction outperforms independent probabilistic inversion, and adding more surveys sharpens the posterior. If the claims hold, CCS operators would gain a practical way to turn monitoring data into risk estimates instead of a single best guess.

What carries the argument

The mechanism is a Shared Generative Model (SGM) combined with a weak formulation of the inversion. The SGM decodes shared features that appear in every survey, while each survey gets its own Gaussian mixture latent $q_{\phi_i}(z_i)$ to represent innovations such as the growing CO2 plume; minimizing the forward data misfit with all surveys tied through the same generator is what makes recovery joint. The weak formulation splits the optimization into an outer loop that evaluates the forward operator and its gradient roughly 200 times and an inner loop that performs roughly 100,000 network parameter updates, so expensive forward evaluations are not repeated at every network step.

What would settle it

On a synthetic model with known ground truth, compute posterior samples from pJRM, build pixel-wise 90% credible intervals for the time-lapse difference, and check the frequency with which the true difference falls inside them. If observed coverage is far below 90%, the uncertainty maps misrepresent monitoring risk and the central claim fails.

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Extended reading notes

Core claim

The paper's central claim is that probabilistic joint recovery computes a posterior distribution for every monitoring survey while leveraging shared structure across surveys through a shared generative model: each subsurface image is decoded as $x_i \sim G_\theta(q_{\phi_i}(z_i))$ with $z_i \sim N(0,I)$, and the generator and per-survey Gaussian-mixture latents are optimized to match all observed surveys. On synthetic CCS monitoring data, the paper finds that joint recovery significantly improves time-lapse signal reconstruction over independent recovery and lowers uncertainty, and that using six surveys instead of two further improves the result. The intended consequence is that pJRM supplies uncertainty information that deterministic JRM cannot, making the method a candidate tool for risk-aware decision making in CCS projects.

Load-bearing premise

The method's uncertainty maps are only valid if the optimized generative model's outputs actually represent the full range of plausible subsurface images that fit the data; the paper does not include a Bayesian sampling step or a calibration check to verify this.

Editorial extensions

If this is right

Posterior mean and standard-deviation maps become available for time-lapse CO2 monitoring, so plume boundaries can be reported with confidence intervals rather than as a single image.
Because additional surveys improve reconstruction and reduce uncertainty, monitoring campaigns can be planned adaptively by adding surveys until a target confidence level is reached.
The weak formulation cuts the number of forward-operator evaluations to a few hundred, which makes posterior-style inversion more tractable for expensive wave-based operators and, the paper argues, for future nonlinear full-waveform inversion.
The framework generalizes to any number of surveys, so legacy monitoring data can be folded into the joint reconstruction.
Beyond CCS, any time-lapse monitoring problem with a static background and localized changes could use the same shared-generator assumption.

Reading between the lines

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

A natural extension the paper leaves implicit is using the posterior maps for sequential decisions, such as locating an observation well or modifying injection rate based on the probability that the plume has crossed a boundary.
The shared-generator assumption could be stress-tested on out-of-distribution geology: if the baseline changes between surveys, the joint posterior may become overconfident, and a comparison against per-survey generators would show where the shared model breaks.
The paper validates only a linear post-stack operator; testing the same machinery on nonlinear full-waveform data would reveal whether the weak formulation's decoupling survives stronger nonconvexity.

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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

Quick take: the paper has a genuinely new construction, but its main claim—that pJRM computes posterior distributions—is unsupported. The algorithm minimizes data misfit; nothing in Section 2.1 or Algorithm 1 turns that minimizer into a posterior sample. I agree with the stress-test note: the generator is trained on the same data that defines the 'posterior,' so drawing z ~ N(0,I) after convergence gives you the model's internal variability, not calibrated monitoring risk. Without a derivation or a calibration check, Figures 3a-f don't measure what the abstract promises.

What's actually new: combining a shared generative model with per-survey GMM latents for time-lapse joint recovery is a reasonable adaptation of the black-hole shared-features framework [Leong et al.]. The weak formulation that decouples forward evaluations from network updates is a practical idea worth borrowing. The synthetic experiments show a clear qualitative improvement of pJRM over pIRM, and adding surveys appears to sharpen the reconstruction. Those are real, if modest, contributions.

The soft spots are exactly where the reader put them. The missing Bayesian foundation is load-bearing: the objective is a sum of data-misfit terms with no expectation over z, no explicit likelihood, no prior, no ELBO. The penalty term in Algorithm 1 is a coupling constraint, not a likelihood-prior product. Because θ is fit to the same y_i, the 'posterior' is a data-dependent implicit prior, which breaks any standard Bayesian reading. The paper also omits quantitative metrics: no reconstruction error numbers, no comparison with deterministic JRM, no code or data. The '(author?)' placeholders in Sections 2.3 and the experiment section are sloppy.

I would not reject out of hand. The idea deserves a serious referee who can push the authors to either formulate a proper variational posterior with an ELBO and a calibration experiment, or at minimum compare their uncertainty maps against a known posterior on a synthetic with ground-truth noise. As is, the UQ claim is not established, so the paper is not ready for publication. If the authors can close that gap, it could be a useful contribution for applied geophysics and CCS monitoring.

For a desk editor: send to peer review, but with a clear signal that the posterior derivation and evaluation are the make-or-break points. A good referee report could turn this into a solid workshop or conference paper.

Referee Report

3 major / 4 minor

Summary. The paper presents the Probabilistic Joint Recovery Method (pJRM) for time-lapse seismic monitoring of CO2 plumes. Building on the deterministic Joint Recovery Method (JRM), pJRM couples a shared generative model (a deep decoder) with per-survey Gaussian Mixture Model (GMM) latents, and uses a 'weak formulation' (Algorithm 1) that alternates gradient updates on reconstructed images x_i and on the generator/encoder parameters to minimize a sum of per-survey data-misfit terms. The authors claim that this procedure computes a posterior distribution over the subsurface at each monitoring survey: the difference of the posterior means yields the time-lapse image, and the standard deviation of sample differences yields an uncertainty map. Validation is performed on one synthetic 2D example: a CO2 plume generated with Jutul, observed through a post-stack convolutional operator implemented with Pylops, with two and six surveys. The paper concludes that joint recovery outperforms independent recovery (pIRM) and that additional surveys reduce uncertainty, making pJRM suitable for risk-averse CCS monitoring.

Significance. CCS monitoring is an important application, and extending the shared-feature idea of JRM to produce uncertainty information is a sensible research direction; if the posterior claim could be substantiated, the method would be a useful contribution to risk-based monitoring workflows. The authors deserve credit for assembling a realistic synthetic pipeline (Jutul flow simulation followed by a Pylops post-stack operator), for the computationally motivated weak formulation that decouples forward-operator evaluations from network updates, and for directly comparing joint versus independent recovery. However, as written the paper does not deliver on its central promise: there is no Bayesian derivation, no sampling step, no calibration check, and no quantitative evaluation. The contribution is currently a plausible but unvalidated claim supported by a single qualitative example; the strengths listed above are real but cannot carry the paper without the missing probabilistic and empirical support.

major comments (3)

[§2.1; Algorithm 1] The load-bearing claim that pJRM 'computes the posterior distribution at each monitoring survey' (abstract and §1) is neither derived nor implemented. The objective in §2.1 is a plain sum of data-misfit terms; no likelihood or noise model is stated, no prior over x_i is defined, and no normalization, expectation, or sampling step appears anywhere in the method. The notation x_i ∼ G_θ(q_{φ_i}(z_i)) is ill-posed: G_θ is a deterministic map whose input must be a realized latent vector, so applying it to the GMM itself is undefined. Algorithm 1 is a deterministic alternating optimization, with gradient descent on x_i in lines 3–4 and ADAM on (φ_j, θ) in lines 7–8; the term (1/2γ²)‖x_j − G_θ(q_{φ_j}(z_j))‖² is a quadratic penalty, not a likelihood–prior product, and the paper never describes how the 'posterior samples' used to produce Figure 3 are drawn. Moreover, because θ is optimized against the very data y_i that define the target distribution, the generator acts as a data-dependent implicit prior, which is incompatible with the Bayesian interpretation. The authors must either derive the objective from a proper likelihood–prior–posterior decomposition (e.g., a variational bound or an explicit sampler) and specify the sampling procedure, or explicitly reclassify the standard-deviation maps as a heuristic internal variability measure and validate that measure with calibration experiments.
[§3] The evaluation contains no quantitative results. The comparisons of pJRM versus pIRM (Figures 2b and 2c) and of two versus six surveys (Figures 2c and 2d) rest on visual inspection of a single synthetic example, with no error metrics such as relative L2 error or SNR, no error bars across repeated runs, and no tabulated numbers. The assertions that joint recovery 'significantly outperforms' independent recovery and that the juxtaposition of uncertainty with error in Figure 3 'validates the expected correlation between uncertainty and error' are therefore unsupported. For a paper whose entire contribution is uncertainty quantification for risk assessment, a calibration check (e.g., empirical coverage of nominal credible intervals, or prediction-interval coverage probability) is essential before the standard-deviation maps in Figures 3c and 3e can be interpreted as measuring monitoring risk.
[Algorithm 1; §3] The experimental configuration is underspecified to the point of non-reproducibility: the noise scale σ and the penalty weight γ appear in Algorithm 1 without definition or numerical values, and the latent dimension of z_i, the number of Gaussian-mixture components, maxiter1, maxiter2, the ADAM learning rate, and the number of samples used to estimate the standard deviations in Figure 3 are not reported. Since the empirical claims rest on a single unquantified run, these omissions are not cosmetic; they are needed to judge whether the reported behavior is robust or an artifact of particular hyperparameter choices.

minor comments (4)

[§2.3; §3] The placeholder '(author?)' appears in place of citations for the weak formulation in §2.3 and for the non-probabilistic JRM comparison in §3; these should be resolved before submission.
[References] Reference [6] (ASPIRE) appears in the bibliography but is not cited in the body text, and reference [3] is missing its arXiv identifier.
[§3] The two-survey versus six-survey comparison conflates the number of surveys with the total volume of observed data, so the 'decreasing trend in uncertainties' could be explained by the larger data budget rather than by the exploitation of shared structure; a controlled comparison with equal data volume per configuration would clarify this.
[§2.1] The time-lapse construction is described informally ('we take their difference', 'standard deviation of the differences between posterior samples'); the paper should define the sample-wise difference of posteriors and the estimator of its standard deviation explicitly, and should also fix the typographical '∥·∥²₂' notation.

Circularity Check

1 steps flagged · score 6.0 of 10

The claimed posterior distribution is defined as the generator output rather than derived, so the uncertainty-quantification claim is self-definitional; the reconstruction improvement itself is tested independently.

self definitional [Section 2.1, 'Probabilistic Joint Recovery Model', defining x_i ~ G_theta(q_phi_i(z_i)) and the objective; Algorithm 1]
"To incorporate uncertainty, we model the solutions probabilistically by using a SGM to decode latent distributions into the posterior estimates x1 and x2—i.e., x1 ∼ Gθ(qφ1(z1)) and x2 ∼ Gθ(qφ2(z2)), where zi ∼ N(0,I) is the standard Gaussian distribution ... Consequently, our objective function is defined as follows: min θ,φ1,φ2 ∥A1Gθ(qφ1(z1)) − y1∥2 2 + ∥A2Gθ(qφ2(z2)) − y2∥2 2."

The 'posterior estimates' are introduced as the outputs of the shared generative model, and no Bayesian derivation is provided: there is no likelihood model, prior, Bayes update, ELBO, or sampling step in Section 2.1 or Algorithm 1. The objective is only data misfit, and Algorithm 1 alternates gradient updates on x_i and on the parameters with penalty terms. Therefore, after convergence, the distribution of x_i is, by construction, the distribution of G_theta(q_phi_i(z_i)). Calling this distribution 'the posterior' restates the generative model output distribution rather than computing it from the data.

full rationale

The empirical reconstruction comparison is not circular: the CO2 plume is simulated with Jutul, the data are generated with a convolutional operator in Pylops, and pJRM is evaluated against ground truth using the errors in Figure 3. No constants are fitted to force the conclusion that joint recovery improves time-lapse reconstruction. However, the paper's central probabilistic contribution, stated in the abstract and Section 1 as computing 'the posterior distribution at each monitoring survey,' is never derived. In Section 2.1 the posterior estimates are defined outright as the outputs of the shared generative model, and the optimization objective contains only data-misfit terms. Algorithm 1 contains no expectation over latent variables, no KL or entropy regularization, and no posterior calibration. The distribution of reconstructed images is therefore, by construction, the distribution of G_theta(q_phi_i(z_i)), so the uncertainty-quantification claim reduces to a definition rather than a computation. This is partial circularity: the reconstruction uplift is independently demonstrated, but the risk-assessment uncertainty maps are not valid posterior uncertainties unless a Bayesian derivation or calibration is supplied. Section 2.3 also contains a missing-reference placeholder for the weak formulation ('introduced by (author?)'); this is an omission, not a circular step, and it does not affect the reconstruction comparison. Overall score 6.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central method rests on architectural assumptions (shared decoder, GMM latents), an unstated posterior interpretation, and a synthetic validation setup. No physical entities are invented; the model components are trainable network parameters rather than new natural kinds. The main burden is the unsupported step from an optimization objective to a calibrated posterior.

free parameters (5)

noise scale sigma
Used in Algorithm 1 line 3 for data misfit weighting; no value reported, and the posterior interpretation depends on it.
regularization weight gamma
Controls coupling between reconstructed image and shared generative model in Algorithm 1; value not reported and is load-bearing for the tradeoff between data fit and prior.
latent dimension of z_i
z_i is defined as standard Gaussian in Section 2.1 but its dimension is unspecified.
GMM component count
q_phi_i is described as a Gaussian mixture model, but the number of components is not stated.
optimization iteration counts and ADAM learning rate
Algorithm 1 specifies maxiter1 and maxiter2 symbolically but not their values or the learning rate; these are needed for reproducibility.

assumptions (4)

ad hoc to paper A single shared generative model G_theta can represent common features across surveys while per-survey GMM latents represent survey-specific innovations.
Section 2.1 constructs the model this way; if the shared decoder cannot separate common and innovation structure, joint recovery would not improve over independent recovery.
domain assumption The forward operator A_i is known and linear (post-stack convolution with spatial derivative).
Section 2.1 states the proof of concept uses a linear post-stack operator; full-waveform inversion is nonlinear and listed as future work, so the demonstrated behavior may not transfer.
ad hoc to paper Optimizing the objective in Algorithm 1 yields posterior distributions, not just point estimates.
No Bayesian derivation, likelihood, or sampling step is provided; the paper simply states that posterior distributions are computed.
domain assumption The Jutul CO2 flow simulation and Pylops forward modeling produce a realistic CCS monitoring scenario.
Section 3 describes the synthetic case; the validity of the conclusions is limited by the realism of the simulator and operator.

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Cite this review

Pith. "Pith review of Probabilistic Joint Recovery Method for CO$_2$ Plume Monitoring." pith.science (2026). https://pith.science/paper/EMIU2B5F

@misc{pith2026250118761,
  author       = {Pith},
  title        = {Pith review of: Probabilistic Joint Recovery Method for CO$_2$ Plume Monitoring},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EMIU2B5F}},
  note         = {Machine review of arXiv:2501.18761}
}

abstract

Reducing CO$_2$ emissions is crucial to mitigating climate change. Carbon Capture and Storage (CCS) is one of the few technologies capable of achieving net-negative CO$_2$ emissions. However, predicting fluid flow patterns in CCS remains challenging due to uncertainties in CO$_2$ plume dynamics and reservoir properties. Building on existing seismic imaging methods like the Joint Recovery Method (JRM), which lacks uncertainty quantification, we propose the Probabilistic Joint Recovery Method (pJRM). By estimating posterior distributions across surveys using a shared generative model, pJRM provides uncertainty information to improve risk assessment in CCS projects.

Figures

Figures reproduced from arXiv: 2501.18761 by the authors.

**Figure 1.** Synthetic case study. (a) Earth model with CCS injection site and unknown CO2 plume. (b) Simulated post-stack data at first survey. (c) Simulated post-stack data at second survey. To incorporate uncertainty, we model the solutions probabilistically by using a SGM to decode latent distributions into the posterior estimates x1 and x2—i.e., x1 ∼ Gθ (qφ1 (z1)) and x2 ∼ Gθ (qφ2 (z2)), where zi ∼ N (0,I) is the standard G… view at source ↗

**Figure 2.** Comparison of time-lapse Images. (a) Time-lapse image with Independent pIRM. (b) Ground truth time-lapse difference. (c) Time-lapse image with pJRM and 2 surveys. (d) Time-lapse image with pJRM and 6 surveys. 4 Conclusions We introduce a novel approach to CCS monitoring that leverages shared structures between surveys to improve reconstruction while providing uncertainty quantification. Additionally, we present a we… view at source ↗

**Figure 3.** Comparison of uncertainty and errors. (a) Uncertainty of time-lapse image w/ independent method pIRM and 2 surveys. (b) Error of time-lapse image w/ pIRM and 2 surveys. (c) Uncertainty of time-lapse image w/ joint method pJRM and 2 surveys. (d) Error of time-lapse image w/ pJRM and 2 surveys. (e) Uncertainty of time-lapse image w/ joint method pJRM and 6 surveys. (f) Error of time-lapse image w/ pJRM and 6 surveys. … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Herrmann

Abhinav Prakash Gahlot, Mathias Louboutin, and Felix J. Herrmann. Time-lapse seismic monitoring of geological carbon storage with the nonlinear joint recovery model. In International Meeting for Applied Geoscience and Energy, 08 2023. (IMAGE, Houston)

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Deep decoder: Concise image representations from untrained non- convolutional networks, 2019

Reinhard Heckel and Paul Hand. Deep decoder: Concise image representations from untrained non- convolutional networks, 2019

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Ill-posed image reconstruction without an image prior

Oscar Leong, Angela F Gao, He Sun, and Katherine L Bouman. Ill-posed image reconstruction without an image prior. CoRR, 2023

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Felix Oghenekohwo and Felix J. Herrmann. Compressive time-lapse seismic data processing using shared information. In CSEG Annual Conference Proceedings, 05 2015. (CSEG, Calgary)

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Herrmann

Rafael Orozco, Ali Siahkoohi, Mathias Louboutin, and Felix J. Herrmann. Aspire: Iterative amortized poste- rior inference for bayesian inverse problems, 2024

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Pylops—a linear-operator python library for scalable algebra and opti- mization

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Storage of Carbon Dioxide in Saline Aquifers: Building confidence by forecasting and monitoring

Philip Ringrose. Storage of Carbon Dioxide in Saline Aquifers: Building confidence by forecasting and monitoring. Society of Exploration Geophysicists, 2023. Page 4

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Compressive time-lapse seismic monitoring of carbon storage and sequestration with the joint recovery model

Ziyi Yin, Mathias Louboutin, and Felix J Herrmann. Compressive time-lapse seismic monitoring of carbon storage and sequestration with the joint recovery model. InSEG International Exposition and Annual Meeting, page D011S145R001. SEG, 2021. Page 5

2021

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Herrmann

Abhinav Prakash Gahlot, Mathias Louboutin, and Felix J. Herrmann. Time-lapse seismic monitoring of geological carbon storage with the nonlinear joint recovery model. In International Meeting for Applied Geoscience and Energy, 08 2023. (IMAGE, Houston)

work page 2023

[2] [2]

Deep decoder: Concise image representations from untrained non- convolutional networks, 2019

Reinhard Heckel and Paul Hand. Deep decoder: Concise image representations from untrained non- convolutional networks, 2019

work page 2019

[3] [3]

Ill-posed image reconstruction without an image prior

Oscar Leong, Angela F Gao, He Sun, and Katherine L Bouman. Ill-posed image reconstruction without an image prior. CoRR, 2023

work page 2023

[4] [4]

sintefmath/jutuldarcy.jl: v0.2.3, April 2023

Olav Møyner, Grant Bruer, and Ziyi Yin. sintefmath/jutuldarcy.jl: v0.2.3, April 2023

work page 2023

[5] [5]

Herrmann

Felix Oghenekohwo and Felix J. Herrmann. Compressive time-lapse seismic data processing using shared information. In CSEG Annual Conference Proceedings, 05 2015. (CSEG, Calgary)

work page 2015

[6] [6]

Herrmann

Rafael Orozco, Ali Siahkoohi, Mathias Louboutin, and Felix J. Herrmann. Aspire: Iterative amortized poste- rior inference for bayesian inverse problems, 2024

work page 2024

[7] [7]

Pylops—a linear-operator python library for scalable algebra and opti- mization

Matteo Ravasi and Ivan Vasconcelos. Pylops—a linear-operator python library for scalable algebra and opti- mization. SoftwareX, 11:100361, 2020

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[8] [8]

Storage of Carbon Dioxide in Saline Aquifers: Building confidence by forecasting and monitoring

Philip Ringrose. Storage of Carbon Dioxide in Saline Aquifers: Building confidence by forecasting and monitoring. Society of Exploration Geophysicists, 2023. Page 4

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