After carefully reading and evaluating “Integration of automatic implicit geological modelling in deterministic geophysical inversion”, I am impressed with the thoroughness and meticulousness of the research. Good work! The authors have made a substantial contribution to both the geophysics and geology communities with their framework, which can be readily applied to a range of geoscientific problems using gravity data. This includes exploring critical minerals and mapping volcano structures. Overall, I believe this manuscript is of significant importance and quality, and I am pleased to recommend it for publication in Solid Earth, pending a moderate revision.

Below please find my major concerns followed by specific comments.

Major concerns:

1. The quality of the geological model is a critical aspect of the framework presented in the manuscript. It is noted that constructing a detailed prior geological model is not commonly practiced, as it requires a significant amount of field geological data and expert knowledge. A reference model that contains incorrect information may guide inversion in the wrong direction, even though the uncertainty may be reduced. The question arises as to whether geophysical data can correct any errors in the reference model. The authors mention that “their approach allows for the inversion to explore a part of the model space that remains within the neighborhood”. However, it is unclear how robust this method is. Therefore, further clarification on the efficacy of this approach is warranted, especially in dealing with errors in the reference model.
2. I recommend that the authors provide further elaboration on the term "rock unit" utilized in the manuscript. Although it is mentioned that each unit is associated with a unique range of physical properties and retains constant physical property values during inversion, it is unclear whether a unit corresponds to a single lithology or a mixture of multiple lithologies. Additionally, it is possible for a single lithology to be present in zones that exhibit varying degrees of alteration, mineralization, or mineral assemblages, leading to different physical properties. As a result, further clarifications regarding the definition of rock unit are required to enhance the understanding of the framework presented in the manuscript.
3. Perturbing dipping angle, azimuth, and strike for one or multiple units is a simple way to quantify the uncertainty of the deterministic inversions. Have you done any experimental results regarding uncertainty quantification? I would suggest mentioning UQ in Discussion.
4. The assigned density values utilized as a priori information can have a significant impact on the resulting model recovery. Small density values may lead to an overestimation of the rock unit's volume, resulting in a large and less dense body, while large density values may result in the underestimation of the volume corresponding to a small but dense body. To provide a more comprehensive understanding of how prior physical property values affect model reconstruction, it would be valuable to include numerical results reflecting the impact of the assigned density values. This information would enable readers to appreciate the extent of the influence of the a priori information on the inversion outcomes and aid in the interpretation of the model results.

Specific comments:

The manuscript is in a good shape, I just have a few minor comments.

Page 2L40: It is recommended that the authors consider citing the work of Wei and Sun (2022) in the manuscript. Wei and Sun quantified the uncertainty of rock units derived from geophysical joint inversion. Including a citation to this work would provide readers with additional context and insights into the understanding of rock units.

Wei, X. and Sun, J., 2022. 3D probabilistic geology differentiation based on airborne geophysics, mixed L p norm joint inversion, and physical property measurements. Geophysics, 87(4), pp.K19-K33.

Page 19 Figure 11: I would like to see the observed gravity data in Fig 11 as well, which can tell the mismatch between observed data and simulated data using reference and starting model.

Page 22 Figure 15: It is not clear from the manuscript why Cases 2, 3, and 4 do not converge.

• The authors demonstrate their method on a number of synthetic case studies. In all of these, the starting models is relatively close to the true reference model used to create the synthetic data. A more detailed analysis on how the proposed method depends on the starting model would be insightful.
• Inversion models should not under- or overfit the measured data. The very important topic of measurement error and data weights during the inversion should deserve more attention in this manuscript. Was the synthetic data noisified before inversion? If yes, which type of noise was used (e.g., Gaussian White Noise)? How were measurement errors treated in the inversion? Currently, this seems a bit arbitrary (e.g., no data weights in the formulas, inversion stops at 0.5 mGal.).
• Discussion of existing literature with similiar motivations: While it is true that the majority of literature either attempts to invert geophysical data using smoothness-constraints or simple geometrical definitions such layer-based parameterizations, there are a few recent examples where a geological modeling engine was used in the inversion of geophysical data. Putting these into context is in my view much more valuable for the readers (and the impact of the presented method) than pointing out differences to conventional approaches such as Thikonov regularization.
  • Güdük, N., de la Varga, M., Kaukolinna, J., & Wellmann, F. (2021). Model-Based Probabilistic Inversion Using Magnetic Data: A Case Study on the Kevitsa Deposit. In Geosciences (Vol. 11, Issue 4, p. 150). MDPI AG. https://doi.org/10.3390/geosciences11040150
  • Liang, Z., Wellmann, F., & Ghattas, O. (2022). Uncertainty quantification of geologic model parameters in 3D gravity inversion by Hessian-informed Markov chain Monte Carlo. In GEOPHYSICS (Vol. 88, Issue 1, pp. G1–G18). Society of Exploration Geophysicists. https://doi.org/10.1190/geo2021-0728.1

Thank you for the opportunity to review this exciting work!

• L52: In near-surface geophysics, "petrophysical (joint) inversion" is a widespread term implying the use of petrophysical relations for parameter transformation (e.g., electrical conductivity to water content) within the parameter estimation. 
• L119 / L120: "regularization" vs. "regularisation". Check the entire mansucript for the consistent user of either British or American English.
• L132: Is the opening bracket before "see" intended? If yes, make sure it closes.
• Eq. 6: This seems to be redundant. Maybe psi^prior could be already introduced in equation 4?
• L181: Remove the comma after data
• L199: Mentioning a few alternative geological modeling engines which could be used seems useful here.
• Eq. 16: Measurement errors seem to be neglected here. Was noise use to make the synthetic experiments more realistic? If yes, a data error estimate should also be used in the inversion. This needs to be clarified.
• L382: "... the data misfit visible in Fig. 5a and Fig. 5b" is misleading here, since no data misfits are shown. I recommend to rephrase to "... the corresponding data misfit of the inversion results shown in Fig. 5a and Fig. 5b".
• L459: Why 0.5 mGal? Was the data noisified and would this value mean the data was descirbed within its error bounds? The very important topic of data error/weights should deserve a bit more attention in this manuscript.
• L660: Is the code related to the method presented in this paper part of LoopStructural or is it made available with this manuscript in a separate repository?
• L738: The link to the pdf refers to a local filesystem.
• L797: The editor's last name is Schmelzbach (last letter h not k), the publisher is Elsevier, and the name of the book series ("Advances in Geophysics", https://www.sciencedirect.com/bookseries/advances-in-geophysics) may be added.

After reviewing the manuscript titled “Integration of automatic implicit geological modelling in deterministic geophysical inversion,” I was thoroughly pleased to see the development of new methodologies that address the problem of non-uniqueness in geophysical inverse problems. The manuscript is well-written and well-illustrated and has clear value that addresses the integration of geological and geophysical data. I look forward to seeing a field application of the proposed methods. Having said that, I do feel a few revisions would improve the quality of the manuscript.

First, equations (2) and (5) both present ways to map from signed-distances to physical properties, so which one is used in which stage of the inversion? The text (L 128 – 130) states that equation (5) is used to update the model, so when is equation (2) used then? Giraud et al. (2021a) suggest instead that equation (2) is used for the model update. Please clarify this, as equations (2) and (5) seem to be in conflict.

Second, I do feel that while the manuscript does make an effort to explain in detail the formulation of the inverse problems, it falls short in explaining how some of the parameters are determined. For example, the parameter “alpha” in equation (12) for the geological correction seems to be crucial in guiding the inversion, yet there is little mention on how “alpha” was determined. Results are shown for alpha = 0 and 0.5, but there is no explanation of how these values were determined. If “alpha” is a hyperparameter, then a sensitivity analysis that tests various values for “alpha” ranging between 0 and 1 would be appropriate.

Third, there is little mention of the robustness of the inverted models. Granted, this is a synthetic study, but how robust are some of the features that are consistently seen in the inverted models? Would changing the values of some parameters in the inversion result in different models, or would the features appear again (indicating some degree of robustness in the model)? This is also related to the fact that the method seems to be dependent on the starting model. How do the results change if the starting model changes? Ideally the proposed methods would give consistent results even when the parameters and starting model are changed. I would like to see some form of robustness test of the proposed methodologies.

Fourth, I would like to see the simulated synthetic data (i.e. gravity anomaly maps) and the predicted data from the inversions, as well as a data residual map. The data residual map would ideally show random patterns. A brief explanation on how the data was simulated would also be appropriate. Was the gravity data treated with random noise, or was it simply forward computed from the models? If the gravity data was treated with noise, how robust are the proposed methods to increasing levels of noise? In a field study, the observables would be gravity data maps and any additional geological information that may constrain the models. Yet, the manuscript does not address sufficiently in depth the data itself.

Fifth, this is somewhat addressed in the last case (5) of the second synthetic study, but what happens if a geological correction is applied to the inverted model from case (1)? This is also related to what is mentioned in L 57 – 60, where an a posteriori ad hoc process could also ensure geological plausibility of an inverted model. Could it be possible that applying a geological correction after geophysical inversion (not every iteration) gives similar results? Then, an argument could be made that the cumulative geological corrections applied at each iteration could be summed into one single geological correction applied at the end of the geophysical inversion. Or perhaps a geological correction is not necessary at every iteration. A plot showing the evolution of the geological correction with every iteration would be appropriate here.

Following are a few specific/technical comments.

L 38: Presumably “units” refer to rock units. Either add “rock” in front of “units”, or define what “units” are in this context

L 45: When the parenthetical phrase is removed, the sentence reads “representing equal to zero.” I suggest changing “equal” to “equality”

L 46: Change “inverts” to “invert”

L 51 – 53: The sentence “In comparison … potential field data.” makes it sound like petrophysical inversions are not geologically meaningful. Suggest rephrasing or clarify what is meant by geologically meaningful

L 103: It is unclear what “is” refers to. Presumably it’s the “tau” parameter

L 149: Add “a” between “as” and “prior”

L 220: I question the value of including Figure 1. Adding illustrations would add value to Figure 1, and would also elucidate the proposed workflow. I would even suggest a possibility of combining with Figure 2. But as it stands, Figure 1 only contains text and adds little value to the manuscript

L 316: Is it “qualitative” or “quantitative”?

L 362: The sentence “Figure 5a and Figure 5b … starting data misfit” is awkward. Is Figure 5 meant as a reference to the sentence? What is meant by a “strong” data misfit?

L 370: In Figure 5, the gravity anomaly maps on the top make it difficult to see the reference and starting models. I would suggest presenting the gravity anomaly maps as a separate figure

L 412, 515, 655: Either define ‘geologify’ or rephrase to something along the lines of “ensuring geological realism”

L 426: Similar to above comment, define “younging” or rephrase to something like “direction towards younger strata”

L 431: Add “are” between “area” and “shown”

L 509: What is the sentence “This indicate.” referring to?

L 511: Why is ‘improves’ in quotations?

L 534: Is reference to Figure 3 missing, or what is flow (1) referring to?

L 539: Did you mean “increase” in OC?

L 562 – 563: Sentence “As geophysical inversion … solution space.” is not clear. Are you saying that geophysical inversions can be used to explore different regions of the solution space?

L 603: Change “scenarii” to “scenarios”

L 627: Add “of” after “birth”

L 641: Remove “to” between “remediate” and “some”