General comments

This paper aims to provide a robust theoretical background and test for practicality and usefulness of gravity inversion for determining sub-ice shelf bathymetry. Such a paper is useful as it has the potential to guide and optimise future real-world data collection over Antarctic ice shelves. The paper uses a prism-based forward model, coupled with an iterative least-squares approach to provide bathymetric estimates. The test results point towards the importance of higher quality/resolution gravity data in areas of low amplitude background field (simple underlying geology), while direct observations (e.g. seismic or AUVs) become increasingly important where the underlying geology is complex.

Overall the paper is well written and the results appear reasonable. However, I have one specific comment associated with the treatment of gravity errors which I feel should be addressed and a few additional more technical points. This will likely not significantly change the outcome of the paper, but may change the suggested likely minimum achievable error in bathymetry from gravity data.

Specific comments

Around L390 to 395 the authors talk about simulating noise in the gravity data. My understanding of the paper is that the authors simulate noise by first adding random Gaussian noise to the baseline gravity disturbance. This pixel by pixel noise has an amplitude in-line with the errors reported for typical airborne surveys. The initially adulterated data is then re-filtered to achieve a best noise reduction with minimal loss of gravity signal, and the subsequent re-filtered data inverted for bathymetry. However, the data loss from noise and filtering Fig. 10c is consistently below +/-1 mGal, which seems small compared to what would be expected for a real survey.

The authors justify re-filtering the data after adding noise because filtering is a standard method of noise reduction in airborne gravity processing. However, the errors quoted for gravity surveys are after filtering. I therefore don’t think this is the best way to simulate noise in a synthetic gravity dataset. I would suggest that a better method would be to create a random Gaussian noise field, which when filtered with a 10 km wavelength filter (to simulate gravity processing) had a 1 mGal standard deviation (equivalent to the error in high quality gravity data). Adding this filtered error field (with likely local maximum amplitudes of +/- 4 mGal) to the baseline gravity disturbance would be more representative of the likely errors in real Antarctic airborne gravity data. Other ways to create realistic noise could be considered. Use of this error field would likely amplify the errors in the recovered bathymetry, giving a higher, but more realistic, estimation of the expected error due to noise in the gravity data.

Technical corrections

L35 and other places in the text (e.g. L277, L343) refer to “regional gravity field strength”. It is not 100% clear what is meant by this. My understanding of this in other contexts in the paper is that the authors mean the “amplitude of the variability in the regional field”. High field strength could be a uniform value of 200 mGal, but this would have no impact on the inversion quality.  I would suggest re-wording.

L163 – It is not clear why the sensitivity matrix is populated by the vertical derivative of the gravity. This should probably be justified in a little bit more detail. – I think high gradient areas might have shallower sources so be more sensitive, but this is a guess? This is covered in Appendix 1, which could be cited. However, in the appendix the example of varying density was given. As this is fixed in the inversion then the matrix can be filled just with the gravity gradient. However, the parameter which is varied is the topography, which isn’t fixed at each iteration. Therefore is the sensitivity matrix re-computed at each step as well (L191-193)?

L210 – constructing training datasets for Damping value cross-validation. This is done by creating two raster’s – training and testing, which are on meshes with cell size X, shifted by ½ X. In effect taking a mesh with cell size ½ X and considering alternating points.  A concern with this is that the mesh size X must leave some ambiguity. For example if you have 10 km wavelength gravity data and training/testing meshes of 100 m both will be in effect identical. Mesh size therefore matters in this case and is related to the wavelengths considered. The mesh size used for generating the observation and test data, or how it could be estimated, should be stated here.

229-237 – Uncertainty constraint. It is not clear if/how the uncertainty is quantified given the control points form part of the inversion, so should have zero offset. Were random control points left out?

This manuscript presents a new open-source geometry inversion tool (Invert4Geom) for recovering sub-ice-shelf bathymetry from gravity data, together with a comprehensive suite of synthetic tests and Antarctic ice-shelf survey analysis. The authors demonstrate the algorithm’s behavior under ideal and realistic conditions, investigate the influence of key parameters (data noise, survey spacing, regional field strength), and quantify uncertainty via Monte Carlo sampling. They conclude with practical recommendations for future airborne gravity surveys and bathymetric constraint collection.

The topic is timely and the open-source implementation will benefit the glaciological and geophysical communities. The manuscript is generally well structured, and the figures are clear. However, some areas require clarification or rephrasing to improve readability and scientific rigor. In particular, I list some minor comments to help strengthen the manuscript.

L7 & L 24 & L293 Definition of “real” vs “synthetic” bathymetric data

L313 The description of the four ensembles (especially the parameter ranges and sampling strategy) remains too general.

Figure 8 The thick grey line in the profile panels can be misinterpreted as an uncertainty envelope. Replace the thick grey line with a thin black line for the profile of the inverted bathymetry, and show the starting model with a dashed line.

Figure 12 Use a slightly darker color for the “true regional” field so it is distinguishable from the estimated field.

L524 You introduce “RMSE” and then immediately write it out (“root mean squared error”). This is redundant. 

Figure 16 The red and black colors in the ice-shelf names denote previous versus new inversions, but this could be repeated in the figure caption for clarity.

Section 3.9 presents results that the authors acknowledge are “expected. You could move the detailed maps and synthetic summaries of Section 3.9 to the Appendix, and condense the main text to a short paragraph highlighting only the key findings.

L590 Change to “resembles those of Ensemble 2”.

L675 Do you mean “dense constraints”?

L683  Change “Gravity inversions in Antarctica…” to “Gravity‐based bathymetry inversions...”

Recommendation:

Once revised, this work will be a valuable resource for the bathymetry‐ and ice-sheet modeling community.

General comments

This paper employs a rigorous method to test gravity-based bathymetric inversions for sub–ice shelf cavities using a new open-source algorithm. The algorithm calculates gravity reduction through a forward model that uses prism-based density discretization. It assesses gravity misfit and integrates bathymetric constraint points to minimize the influence of regional gravity effects on the inverted bathymetry.

Synthetic tests were conducted in front of the Ross Ice Shelf to estimate the influence of gravity noise, gravity measurement spacing, and long-wavelength gravity distribution on inversion uncertainties. These uncertainties are computed via Monte Carlo simulations. This work has the potential to significantly impact the scientific community, as gravity-based bathymetric inversion remains one of the few viable techniques for estimating bathymetry beneath ice shelves in the absence of direct measurements.

Although various techniques have been developed and applied, quantifying their associated uncertainties and determining how closely they reflect actual bathymetry remain challenges. The tools proposed here represent a valuable and timely contribution to addressing this gap.

The tests are well designed, and the figures are clear. The paper uses several methods for estimating uncertainties with optimized parameters, presenting a structured and coherent approach focused on the primary objective.

Specific Comments

You need to be careful with your choice of test region. You selected the area in front of the Ross Ice Shelf due to its dense bathymetric coverage. However, (1) it does not cover a sub–ice shelf cavity area, and (2) it is a region with almost no gravity measurements in the ANTGG2021 dataset (see ANTGG2021's Standard Deviation map in Supplement).

1. Since your aim is to estimate uncertainties for sub–ice shelf cavities, it would be more appropriate to perform your tests in a region that actually includes a sub–ice shelf cavity. While it is difficult to find ice shelves with dense bathymetric coverage, some areas do have a substantial number of gravity measurements. You could consider comparing estimated uncertainties from tests conducted over an ice shelf with those from open-ocean regions where both gravity and bathymetry are well constrained.
2. You should more clearly define the ANTGG2021 gravity grid, as it is a major source of uncertainty in gravity-based bathymetric inversions. The ANTGG gravity grid combines all available gravity measurements (which do not cover the entirety of Antarctica) and estimates gravity in data-sparse regions using satellite observations (GRACE and GOCE) and topographic data from Bedmap2 (see Hirt et al., 2016; Scheinert et al., 2016; Zingerle et al., 2019; Zingerle et al., 2021). The use of Bedmap2 topographic data to reconstruct gravity in areas lacking measurements is especially concerning, since you ran your tests in an area with very limited direct gravity data (see the figure below showing the standard deviation map of the ANTGG2021 grid). As a result, your inverted bathymetry is likely very similar to the measured bathymetry because the gravity signal was reconstructed from Bedmap2 data. This introduces bias into your results. However, if you were to relocate your tests to a region with dense gravity and bathymetry measurements, your results would likely improve significantly and avoid this bias. See attached file with ANTGG2021 grid standard deviation map.

Your goal is to enhance gravity-based bathymetric inversions for sub–ice shelf cavities. While many studies have previously addressed this topic using various methods and assumptions, it is unclear whether the uncertainties and test results in your work are applicable to existing gravity inversion techniques. Adding a section that clearly reviews the current state of the art—including the limitations of previous studies—would fill this gap. A map showing the inverted bathymetry derived from your method, along with the differences compared to prior models, would help identify where your approach is most effective and where existing inversions might need to be recalculated due to high uncertainty or outdated methods.

Technical comments

L14-16 “We analyzed Antarctic ice shelves and found that, if high-resolution gravity data were available, gravity inversion could improve bathymetry models for 94% of them compared to interpolated products like Bedmap2.”  Does that mean that we only have 6% of high-resolution gravity data covering Antarctica's ice shelves?

L18-20 To rephrase in a better logical order. (1) The shape and depth elevation of the continental shelf seafloor may allow warm water (name of warm water, usual depth) to reach the subglacial cavity. (2) The shape of the subglacial cavity may lead warm water to reach the grounding zone, where the ice is at its deepest, steepest, most pressurized, thus vulnerable to significant basal melting. (3) This significant basal melting may affect the stable state of ice shelves due to the long-term retreat of their grounding line.

L21 “echo sounders”. What about seismic data?

L21-22 “ are often impractical or expensive when applied to the vast ice shelves that fringe Antarctica’s ice sheets.” Why? Explain that this is because the instruments have to be sent under the ice using AUVs (Automated Underwater Vehicles).

L22 “acquiring”. Do you mean measuring or estimate? Acquiring is too vague.

L22 “gravity”. When you say "gravity" are you talking about free-air gravity anomalies?

L23-24 “difference in density between seawater and the seafloor.” What about the ice density?

L24 “a gravity inversion”. Using which gravity data/model? In which areas?

L24 “synthetic data”. To define, you are also using existing datasets to set up your tests, right? Not all data are synthetic.

L25-26 “We find that removing the portion of the gravity data that results from deep geologic structures is the largest source of error.”  Does that mean that we poorly remove the deep geologic gravity signal? Does this affect our free-air gravitational anomaly signal, which is therefore not correct? The deep geological gravitational signal occurs at long wavelengths and the surface topography gravitational signal occurs at shorter wavelengths. Are you talking about deep geology or bedrock geology (which can vary locally, implying changes in density)?

L30 “existing bathymetry models”. The existing bathymetry models are also gravity inversions. Is your inversion method better than existing ones?

L42-43 “These ice shelves play a key role in holding back the flow of inland ice by exerting a resistive force, buttressing, which comes from lateral drag and resistive stresses where the ice touches the sea floor at pinning points “. A lot of redundancies can be avoided in this sentence.

L45  “across the grounding zone”.  I am not sure if it is correct to say that. The grounding zone is changing over time and retreating due to the same processes (not independent of the reducing buttressing effect). You might have wanted to say: "toward the ocean" instead?

L49  “cold-water shelves”. Why are you talking about the cold-water shelves to highlight the importance of sub-glacial bathymetry? Cold-water shelves are in general in a steady state. It means that they are currently not affected by changes in the ocean currents due to global warming. You should explain instead the unsteady-state ice shelves that are affected by the intrusion of the warm Circumpolar Deep Water reaching the grounding line thanks to the sub-glacial bathymetry.

L49 “grounding zones”. To define before. It could be: The grounding line is the transition point where the ice goes from grounded to floating. This line migrates over the grounding zone due to short-term effects of the tides, and long-term effects of the basal melt or refreeze.

L51 “This water”. Which water are you talking about? "This water" is confusing because you're talking about cold and salty waters, and basal melting (thus fresh water) before, making it confuse.

L51 “such as”. If you say "such as" it means that the dense and cold water you're talking about can come from a different origin than the one formed from sea ice formation. Explain which one, it should be the water originating from the deep ocean.

L57-58 “(AUVs) are impractical for large ice shelves.”  They are not impractical for large ice shelves, because such data were measured on large ice shelves. I guess you wanted to say that it is impractical to cover all the surface of large ice shelves with such measurements because they are point data.

L58 “than water”. To add “and than ice”.

L59 “gravity”. Do you mean free-air gravity anomaly field?

L61 “Antarctica”. Cite Charrassin et al., 2024, bathymetric inversions have been calculated for the whole of Antarctica. You cite this article at the end of your paper, but it's important to refer to it first as a state of the art, as it is the most recent gravity-based inversion to have been carried out on such a large scale.

L65 “They’re”. To replace by “They are”.

L69 “that’s” to replace by “that is”

L70 “don’t do a good job of” to replace by “struggle to”?

L71 “geometry inversion”. Why did you choose to do a geometry inversion then? Is it better than density inversions? Why?

L71 “It’s” to replace by “It is”.

L72 “from”. Add “inverted from”.

L74 “regional gravity fields” to define.

L76-77 “Finally, we highlight which ice shelves are most likely to benefit from inversion and which ones probably won’t.” to rephrase maybe like “we highlight ice shelves for which bathymetry would be improved using a free-air gravity anomaly inversion, and ice shelves were it would degrade the bathymetry reliability".

L80 “forward model” to define.

L107-108  “These sources can then be used to predict the gravity anomaly at any desired point, such as each location on an even grid.” This is not clear. You use the “equivalent sources” technique, which calculates the expected gravity from the observed bathymetry and inserts it into your matrix as input, right? Will this improve the Jacobian solution to be more accurate? If you don't have bathymetry data over a large area, how do you know it really works?

L109  “)” parenthesis to remove.

L121 “densities(Figure 3b)”. Add space.

L134 “most of which” to cite.

L147-148 “it avoids the subjective parameter selection required by the other techniques.”. This sentence is too general. The use of constraint points is one of the only known ways of using actual bathymetry measured in inverted results. They are therefore essential for representing bathymetry accurately and avoiding false assumptions about bedrock density. This allows us to avoid using very precise geological assumptions, since they will be corrected thanks to the actual bathymetry data.

L166 “sector of an annulus”. How do you choose the diameter? Is it the initial horizontal length of the prism?

L175 “to remain close to known bathymetry measurements”. Do you keep the exact values of existing bathymetric measurements in the final inversion results?

L204 “Too high a value under-fits the data; too low”. To replace with “Too high,  […] too low, […]”.

L255 “it’s” to replace by “it is”.

L259 “it’s” to replace by “it is”.

L260 “doesn’t” to replace by “does not”.

L280-281 “included data from multibeam and single-beam sonar, over-ice seismic surveys, airborne and surface-based radar, and exposed rock outcrops.”. To cite.

L283 “AntGG-2021 gravity data”. You should know that this gravity grid partially uses inverted Bedmap bathymetry to reconstruct gravity when it is not measured. You need to use the error map they have created for your results to be correct.

L283 “Bedmap2”. I think they have added a lot more measurement data (outside the ice shelves) in the Bedmap-3 version. Perhaps it's better to use Bedmap-3 and place Bedmap-2 at the location of the ice shelves? Why don't you want to use the gravimetric inversions that already exist? Your final bathymetry will be different and improved anyway, won't it? Bedmap is just the initial bathymetry to be improved.

L290 “a range of published sub-ice shelf bathymetry studies”. To cite.

L292 “showcase” to replace by “introduce”?

L339-340 “he standard deviation of the topography-corrected

L340 “gravity disturbance and the standard deviation of the regional gravity misfit.”. Do they consider the standard deviation of the initial ANTGG grid?

L346 “we reviewed published studies”. To cite.

L371 “true value”. Cite. If you say it is a true value, it sounds like it has been measured? The word “real” may be a bit strong, perhaps you meant “most realistic”?

L412 “The synthetic airborne survey follows a typical Antarctic design”. To rephrase, what is a “typical Antarctic design”?

L437 “regional component”. Maybe you could define again what is it?

L438 What is “crystalline basement topography”?

L438-439 “Brancolini et al., 1995) ”. Is this citation from the ANTOSTRAT seismic compilation? If so, does this mean that it contains only seismic measurements taken before 1995? Why did you use it when it is not updated?

L608 “this means adding gravity data when there is very little”. To rephrase.

L629 “skewness” Are you talking about the asymmetry of the distribution of the constraint points?

L643 “won’t” to replace by “will not”.

L721 “Gravity inversion” to replace by “Free-air gravity-based bathymetry inversions”?

L722-723 “Using synthetic tests based on real bathymetry data from the Ross Sea” and using Antgg2021, right? To be added in the sentence. See comment on test location in specific comments.

Figure 1.

• Above ellipsoid: “ρ earth-ρ air”: Maybe I didn't understand, but for the bedrock above the ellipsoid and below the ice, isn't the equation supposed to be something like “ρ earth + ρ ice – ρ air”?
• Put a full black contour around your color legends.

Figure 2: What is “Normal gravity” in your input data?

Figure 3: “One prism layer either above (c) or below (b)”. To replace “or below (b)” by “or below (d)”.

Figure 7.

• Some ice shelf names are missing on a) and b) like Shackleton, West, Ronne.
• “24 previously inverted shelves are in red”. Read Charrassin et al., 2024. All ice shelves were already inverted (even the ones in black).

Figure 12.

• “e)”. Describe before d), as d) is the difference between c) and e). We need to know what is e) before knowing what is d).
• “true regional component of the misfit”. How can you compute that?

Figure 16: Where does come from the RMSE for each ice shelf? Why do you have two values per ice shelf? Is it the min/max?

Figure 18. The captions of the plots are illegible.