the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 07 Nov 2025
Joint characterization of heterogeneous conductivity fields and pumping well attributes through iterative ensemble smoother with a reduced-order modeling strategy for solute transport
Abstract. We develop and test an efficient and accurate theoretical and computational framework to jointly estimate spatially variable hydraulic conductivity and identify unknown pumping well locations and rates in a two-dimensional confined aquifer. The approach (denoted as iES_ROM) integrates an iterative Ensemble Smoother (iES) with a Reduced-Order Model (ROM) for solute transport taking place across an otherwise steady-state groundwater flow field. This offers a computationally efficient alternative to the Full System Model (iES_FSM) upon addressing the high computational demands of ensemble-based data assimilation methods, which typically require large ensemble sizes to characterize uncertainties in (randomly) heterogeneous aquifers. Our iES_ROM is constructed through proper orthogonal decomposition. It is then evaluated across a collection of 28 test cases exploring variations in model dimension, ensemble size, measurement noise, monitoring network, and statistical properties of the (underlying randomly heterogeneous) conductivity field. Our results support the ability of iES_ROM to accurately estimate conductivity and identify pumping well attributes under diverse configurations, attaining a quality of performance similar to iES_FSM. When using moderate ROM dimensions (n = 25–30) and ensemble size (i.e., 500–1000), the accuracy of iES_ROM does not vary significantly while computational time is reduced by nearly an order of magnitude. Our approach thus provides a reliable and cost-effective tool for inverse modeling in groundwater systems with uncertain parameters.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Hydrology and Earth System Sciences.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.- Preprint
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Status: open (until 04 Jan 2026)
- RC1: 'Comment on egusphere-2025-5320', Anonymous Referee #1, 16 Nov 2025 reply
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RC2: 'Comment on egusphere-2025-5320', Anonymous Referee #2, 17 Dec 2025
reply
Based on the HESS principal criteria (scale 1 = Excellent, 2 = Good, 3 = Fair, 4 = Poor).
Scientific significance: 2 (Good)The manuscript provides a substantive contribution to ROM–DA coupling for groundwater systems by proposing iES_ROM for the joint estimation of hidden pumping-well attributes (rate and location), and by demonstrating it through a broad multi-factor sensitivity assessment (28 test cases). The contribution is clearly relevant to the scope of HESS; however, the manuscript should state more explicitly—both in the Introduction and Conclusions—what is novel relative to existing ROM+DA studies to firmly support an “Excellent” rating.
Scientific quality: 2 (Good)The overall scientific approach is sound: iES is clearly formulated, the POD/Galerkin transport ROM is properly derived, the method is benchmarked against iES_FSM, and the discussion acknowledges nonlinearity and occasional non-monotonic behavior (especially for well attributes). The literature coverage is generally appropriate, and the discussion is balanced. That said, traceability and reproducibility would be strengthened by a more concrete snapshot-generation protocol (how snapshots are selected, whether they are mean-centered, and a brief discussion of sensitivity to “prior mismatch”), which would further reinforce methodological robustness.
Presentation quality: 2 (Good)The presentation is clear and well-structured; the tables (e.g., Table 1, which serves as a roadmap for Groups A–E) and figures adequately support the conclusions. The English is generally correct and technically appropriate. Suggested improvements include condensing repetitive result descriptions, emphasizing cross-group patterns and non-intuitive behaviors (e.g., non-monotonic trends), and adding a small workflow schematic/flowchart for iES_ROM (since the POD basis is built once and reused).
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Does the paper address relevant scientific questions within the scope of HESS?
Yes. The manuscript tackles a core HESS-relevant problem: data assimilation and uncertainty quantification in groundwater systems, combining steady-state flow and transient solute transport to infer hydrologic/hydrogeologic properties and unobserved stresses. Specifically, it targets the joint identification of a heterogeneous conductivity field Y = ln K and hidden pumping-well attributes (rate and location) from head and concentration observations. -
Does the paper present novel concepts, ideas, tools, or data?
Yes, although the novelty should be stated more explicitly in the Introduction and Conclusions. The main novelties are: (i) joint estimation of heterogeneous Y=lnKY=\ln KY=lnK and hidden well attributes (rate qsq_sqs and coordinates), (ii) a hybrid strategy that reduces only the transport equation (POD-ROM) while keeping steady-state flow full order, and (iii) a systematic multi-factor evaluation with 28 test cases (Groups A–E) spanning ROM dimension, ensemble size, observation noise and network density, prior statistics, and snapshot size. -
Are substantial conclusions reached?
Yes. The paper concludes that iES_ROM can achieve accuracy close to the full-system benchmark iES_FSM within the tested setting, while substantially reducing CPU cost. It supports this with multi-group results and a concrete CPU-time comparison (e.g., the reported TC6 vs TC28 timing). -
Are the scientific methods and assumptions valid and clearly outlined?
Mostly yes. The iES formulation, the joint parameter vector (spatial field plus well parameters), and the POD/Galerkin ROM derivation for transport are clearly presented, and the comparison against iES_FSM is a strong methodological choice. The physical assumptions (2D confined aquifer, steady-state flow, non-reactive transport, observation noise) are stated. Remaining points that would benefit from clarification are whether snapshots are mean-centered and a more explicit discussion of the implications of not using a “mean + anomalies” representation. -
Are the results sufficient to support the interpretations and conclusions?
Yes. The manuscript provides structured evidence across Groups A–E, with summary tables, convergence/accuracy metrics, and distributional comparisons (PDFs and KLDs) that support the claim that ROM can closely match FSM when the ROM dimension and snapshot size are adequate. The discussion also acknowledges non-monotonic behaviors, which strengthens interpretive balance. -
Is the description of experiments and calculations sufficiently complete and precise to allow reproduction (traceability of results)?
Partly. Many key ingredients are specified (domain and boundary conditions, mesh size, time step, simulation horizon, test-case matrix, and the definition Nsn=mNtN_{sn}=mN_tNsn=mNt). However, full traceability would be improved by an explicit operational snapshot protocol: how mmm realizations are selected in practice (the text indicates “arbitrarily chosen”), whether snapshots come from prior draws or a reference field, whether snapshot-generation statistics match those used in DA, and a brief discussion of expected sensitivity to prior mismatch (snapshot prior vs assimilation prior). -
Do the authors give proper credit to related work and clearly indicate their own new/original contribution?
Generally, yes in terms of literature coverage and positioning. The manuscript cites relevant ROM/ROMC and DA foundations and explains how iES_ROM is constructed and benchmarked. Still, the paper would benefit from a concise “what is new compared to existing ROM+DA studies” paragraph with a few closely related citations, so the original contribution is unmistakable. -
Does the title clearly reflect the contents of the paper?
Yes. The title accurately reflects joint characterization, IES-based estimation, and a reduced-order strategy specifically for solute transport, as well as the inclusion of pumping-well attributes. - Does the abstract provide a concise and complete summary?
Yes. It describes the problem, the proposed iES_ROM method, and the iES_FSM benchmark, the scope of the test campaign, and the key takeaways on recommended ROM dimensions/ensemble sizes and computational savings. A minor improvement would be to state the validity domain (2D confined, steady-state flow, single non-reactive solute, single well) explicitly in one sentence. -
Is the overall presentation well structured and clear?
Yes. The manuscript follows a logical flow from methods to test design to results, organized by Groups A–E, and is supported by a functional “roadmap” table. The presentation could be further improved by more explicit signposting to Table 1 at the start of each group subsection, by condensing repetitive descriptions, and by highlighting cross-group patterns and non-intuitive behaviors. -
Is the language fluent and precise?
Yes overall. The English is technically appropriate and generally clear. Minor consistency edits (terminology and symbols) would further improve precision. -
Are mathematical formulae, symbols, abbreviations, and units correctly defined and used?
Mostly yes. Key quantities (Y=lnKY=\ln KY=lnK, nnn, NMCN_{MC}NMC, σobs\sigma_{obs}σobs, NmN_mNm, NsnN_{sn}Nsn) and the ROM/iES formulations are defined and used consistently in the core derivations, and the manuscript notes consistent units. Minor issues include standardizing notation (e.g., NMCN_{MC}NMC vs NmcNmcNmc) and clarifying whether “time levels” include the initial condition or only simulated steps, which affects NtN_tNt and thus NsnN_{sn}Nsn. -
Should any parts of the paper be clarified, reduced, combined, or eliminated?
Yes. The main clarifications are the snapshot protocol, mean-centering, and a short discussion of prior mismatch. In Results, some case-by-case narratives can be reduced/combined by summarizing each group with a compact set of key findings and then highlighting the few non-intuitive outcomes (e.g., small-ensemble inbreeding and non-monotonic trends for well attributes). Adding a small workflow schematic/flowchart of iES_ROM would also improve clarity without inflating text length. -
Are the number and quality of references appropriate?
Yes. The manuscript cites relevant ROM/POD/ROMC and DA literature and provides an adequate methodological context. Strengthening the novelty statement with a handful of the closest ROM+DA references would further improve the framing. - Is the amount and quality of supplementary material appropriate?
Yes. The supplement is used appropriately to support distributional comparisons (e.g., KLD-based agreement between iES_ROM and iES_FSM) and to provide additional supporting results that would otherwise distract from the main narrative. A slight improvement would be a brief sentence in the main text indicating what each supplementary table contributes and when the reader should consult it.
Technical and typographical suggestions (minor)
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Ensure consistent notation for ensemble size (e.g., N_{MC} vs N_{mc})
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Clearly define whether the number of time levels N_t includes the initial condition or only the simulated time steps.
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Where CPU time is used to support the efficiency claim, consider presenting it once in the Results in a compact way (e.g., a short table or a concise paragraph) and referencing it in the Conclusions.
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Double-check symbol definitions at first use (e.g., n, N_{sn}, N_m, σ_{obs}) and keep units explicit where relevant.
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If distributional comparisons (e.g., KLD in supplementary tables) are key to the argument, add one sentence in the main text explaining what the supplement contributes.
Citation: https://doi.org/10.5194/egusphere-2025-5320-RC2 -
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- 1
The paper presents a solid and carefully executed study on coupling a POD-based reduced order transport model with an iterative ensemble smoother for joint estimation of K and well attributes. Results are convincing, and the work is publishable after some focused improvements.
Improve the novelty statement in the Introduction and Conclusions; what is new relative to existing ROM+DA studies: (i) joint estimation of heterogeneous K and hidden pumping well attributes, (ii) reduction of only the transport equation while keeping flow full-order, and (iii) the systematic multi-factor analysis (ROM size, ensemble size, prior stats, noise, snapshot size).
Provide concrete numbers and protocol: how many realizations and time levels are used, whether snapshots come from prior draws or a single reference field, and whether their statistics match those used in the DA experiments. Add a short discussion of how ROM performance might change if the prior used for snapshot generation differs from that used in assimilation. Add also a short discussion (no need for new runs) on how sensitive the ROM is if the prior used for snapshot generation differs from the prior used for DA.
State whether snapshots are mean-centered, and discuss briefly how omitting a separate mean field affects accuracy. Add a short justification of why a “mean + anomalies” representation is less convenient in your iES implementation, and whether it might reduce ROM error.
Make clear which ensemble sizes are realistic for applied hydrogeological problems (e.g. a few hundred to 1000), and present N_MC = 10000 explicitly as a reference benchmark. Emphasize results and cost accuracy trade offs for the practically relevant range.
Use Tables 1–5 and possibly a small schematic/flowchart to clearly show what each group (A–E) varies and why. In the results, slightly condense repetitive descriptions and highlight cross-group patterns and any non-intuitive behaviors (e.g. non-monotonic trends).
In the Conclusions, clearly delimit the domain of validity: 2D confined aquifer, steady-state flow, single non-reactive solute, single well. Briefly comment on expected challenges and required modifications for transient flow, multiple wells, or reactive/density-dependent transport.
Overall, these changes are mostly clarifications and presentation refinements; the core methodology and results appear sound.