Three-Dimensional Time-Lapse Joint Inversion of Resistivity and Time-Domain Induced Polarization Methods

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper develops a 3D time-lapse joint inversion that couples ERT and TDIP using L2 temporal smoothing and a cross-gradient structural term. The formulation is clear and the problem is relevant to 4D monitoring. The idea is sound, but the manuscript needs substantial clarification of choices, notation, and evaluation before it’s ready for publication.

Major points to clarify:

1. Regularization weights. Explain how you chose λm, λt, λcg for each property and epoch (L-curve, grid search, continuation). Include a compact trade-off figure or a sensitivity table; current conclusions may depend on tuning.
2. Cross-gradient scaling. Because conductivity and polarizability differ in scale and roughness, say whether you use a normalized/weighted cross-gradient. If not, discuss the effect and show a small sensitivity test.
3. Parameter bounds. State exactly how you enforce σ>0 and 0≤η<10 (log/logit transform, barriers, clipping).
4. Noise and data covariance. Specify the covariance model (percent + floor), list the values for ERT and TDIP, and report per-epoch RMS plus convergence plots for each inversion type.
5. Forward/mesh setup. Provide mesh size, domain extents, boundary conditions (air layer, sides, bottom), electrode geometry/arrays, and any topography treatment—these details materially affect 3D responses.
6. TDIP acquisition. List time windows/gates and how apparent polarizability is computed. Clarify whether η is treated as frequency-independent or as a DC-IP proxy.
7. Algorithmic details. Report the linear solver tolerances, average iterations, memory footprint, and the stopping criteria for the outer loop (RMS threshold, objective decrease, iteration cap).
8. Structural mismatch test. Add a case where the ERT and TDIP targets are intentionally offset or have different sizes. Vary λcg (and any normalization) to show leakage/over-coupling behavior.
9. Quantitative evaluation. Beyond RMS and cross-gradient magnitude, include volumetric error, centroid offset, IoU with the truth, and one or two diagnostic 1D profiles through the anomaly centers.
10. Practical validation. A short field example would strengthen the paper. If that’s not possible, add a mismatch test (different forward vs. inverse mesh, electrode jitter, non-Gaussian noise).

Technical and clarity fixes:

• Keep notation consistent for the two objectives (σ and η) and define all symbols on first use.
• Write out the temporal difference operator once with its block sizes (… I, −I …).
• Use common color scales across epochs/methods; label units (Ω·m or S/m; η as dimensionless but commonly shown in mV/V).
• If you include a workflow figure, add the three λ weights, the data-covariance block, and the explicit stopping criteria.
• Proofread subscripts/superscripts (Mσ, Mη; Vσ, Vη) and apply bold consistently to vectors/matrices.
• Provide a minimal bundle: synthetic meshes, electrode layout, injection patterns, noise generator, and inversion scripts with the actual λ values and tolerances.
• If full code can’t be shared, add clear pseudocode and a table of all hyperparameters per experiment.
• List all truth-model parameters (ρ/σ and η, anomaly positions/sizes, epoch timing, noise seeds).
• Define ERT and TDIP at first mention (Abstract and Introduction).
• Prefer “dimensionless polarizability η” to “nondimensional.”
• Shorten long sentences, avoid repeating “objective function”.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript proposes and implements a three-dimensional time-lapse inversion algorithm that integrates the time-domain resistivity and induced polarization (TDIP) methods using intergradient constraints and temporal continuity functions. Synthetic modeling is used to evaluate the algorithm under varying noise conditions, and the results are compared with separate inversion schemes and conventional time-lapse inversion schemes. The study demonstrates that the proposed approach improves temporal consistency, reduces structural discrepancies between resistivity and polarizability models, and increases robustness to noise contamination. The work is technically robust, relevant to the field of geophysical monitoring, and contributes a novel algorithmic framework to time-lapse inversion research.

However, a few issues and suggestions are outlined below to enhance the manuscript’s clarity and scientific rigour:

1. Although the manuscript demonstrates significant technical progress, the novelty compared to existing combined inversion methods could have been expressed in more detail. The authors should emphasize the fundamental novelty of their approach, beyond the combination of intertemporal gradient theory and time-lapse constraints.
2. In the summary, it can be briefly mentioned that the validation is based solely on synthetic tests to avoid misleading readers about real-world applications.
3. The study relies solely on synthetic examples. This is, of course, acceptable in the context of methodological development, but perhaps a discussion of potential field applications would enhance the validity of the solutions used (I leave this to the authors' consideration). Ideally, preliminary field validation would be added in future work.
4. The inversion results depend on several regularization weights. Could the authors perform a sensitivity analysis to show how varying these parameters affects the inversion results? Without this, reproducibility and general applicability are limited?
5. The study provides RMS values, gradient-to-gradient values, and model misfit indices, which are commendable. However, it lacks statistical stability (i.e., repeatable realizations under different noise distributions). This would strengthen the claims about noise robustness.

This manuscript is a technically sound and valuable contribution. However, before acceptance, I recommend minor revisions to address the issues mentioned above—in particular, clarifying the novelty issue and discussing parameter sensitivity.

Comments for author File: Comments.pdf

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The comments have been incorporated, and the paper is suitable for publication in its present form.

Reviewer 2 Report

Comments and Suggestions for Authors

Thank you for sending me a revised version of the manuscript.
The authors have adequately addressed the issues raised in the first round of review. The novelty of the proposed approach has been more clearly emphasised, particularly in the context of combining frame-by-frame and cross-gradient constraints. The constraints on synthetic validation are explained in the abstract and discussion, and the authors have outlined their plans for future field validation. Details regarding the choice of regularisation parameters have been added, supported by L-curve analysis. The handling of noise robustness has been clarified by incorporating Gaussian noise into the synthetic examples, which mimic realistic conditions.

Overall, the revised version of the manuscript has been significantly improved.