| 25 Aug 2026
Improving Spatial Resolution of Ice Sheet Mass Change using a Data Combination of Satellite Gravimetry and Altimetry
Abstract. We present two decades of monthly Antarctic mass change at a spatial resolution of (100 km)2, representing an order of magnitude improvement over the current state-of-the-art for gravity-derived estimates. This advancement is achieved by leveraging the accuracy of satellite gravimetry over larger spatial scales and the spatial resolution of altimetry at finer spatial scales. We do this through a formal data combination of satellite gravimetry and altimetry observations at the level of the gravimetry normal equations, estimating 1° circular disk mascon elements. The data combination yields independent mascon estimates, yielding effective spatial resolution of 1°. Gravimetry observations dominate the solution at long wavelengths, while altimetry observations are the dominant contributor at the spatial scale of an individual mascon. This yields a mass change solution that is closely aligned with gravimetry-only solutions from GRACE and GRACE-FO over larger spatial scales, yet spatially resolves mass change variability at finer scales. Additionally, we provide a set of gain factors to further downscale the mass within each mascon, providing monthly mass change estimates at a 1.92 km spatial scale. Uncertainties from measurement systems and model corrections (glacial isostatic adjustment, firn air content) are propagated through the data combination and presented. The methodology presented is extensible to other regions around the globe.
Summary
This manuscript extends a GRACE/GRACE-FO Level-1-based mascon approach by incorporating an additional observation type (satellite altimetry over the Antarctic Ice Sheet) in order to improve estimates of ice sheet mass change.
The approach unites two important advantages: the high spatial resolution of satellite altimetry and the complete mass information provided by satellite gravimetry (GRACE/GRACE-FO).
The resulting product will be of great value for the community in investigating and understanding cryospheric processes. The methodology is presented clearly and in a comprehensible manner, and the use of flat mascon disks for this application is both interesting and promising. I have two general comments relating to the study design and the focus of the manuscript. My specific comments mainly concern points that remained unclear to me during reading.
General comments
1. Inconsistency in the processing levels of the two observation types
In my view, there is a certain inconsistency in the use of the two observation types, as they originate from very different processing levels. For GRACE/-FO, Level-1B data are used. To the best of my knowledge, this is the first time that L1B data have been used in a combination study of this kind as the authors write. An essential argument for doing so is that it avoids the use of more extensively post-processed observations, even though the methodological effort involved is considerably higher. At the same time, however, a highly processed altimetry product is used. Deriving surface elevation changes from the various satellite altimetry missions requires a large number of post-processing steps (retracking, calibration, interpolation, filtering, etc.). The same line of reasoning that motivates the use of L1B data for GRACE could equally be applied to the altimetry product, e.g. by starting from retracked heights rather than interpolated grids.
Furthermore, it would be interesting to demonstrate how the final result depends on the choice of altimetry product, or on processing decisions made during the altimetry data processing. Are there significant differences at small spatial scales when a different altimetry product is used? The authors show very nicely in Sect. 3.3 and Fig. A5 that the contribution of altimetry at short spatial scales is considerable. Moreover, the authors demonstrate how the additional information from altimetry improves the estimation by showing that the correlation between mascons is removed. In the polar gap of the altimetry data, however, this additional information comes solely from an interpolation, which may introduce systematic errors, e.g. by inter-mission calibration. In the combination result (Figure 9a) and in the gain-factor downscaled result (Figure 9b), the mission-specific polar gap transition becomes visible. This may be particularly relevant in Envisat-only months, when the polar gap starts already at a latitude of ~81.5°. This further suggests that the solution adopts systematic altimetry errors. This is neither discussed nor investigated further in the manuscript.
2. Conclusions about ice sheet dynamics
The authors conclude that the product "improves knowledge of ice sheet dynamics". I fully agree that the data combination product presented here makes it possible to investigate the dynamics of the Antarctic Ice Sheet in greater detail. However, this is not the subject of the present manuscript. The phenomena driving the mass changes of the ice-sheet system are neither investigated nor discussed further. The manuscript primarily presents a method for combining two satellite datasets, along with the resulting product, and in this respect it is certainly an important and very useful contribution.
Nevertheless, I have some reservations as to whether the manuscript in its present form fits the scope of The Cryosphere. The manuscript essentially reads as a methods or data description paper presenting a data product that meets certain community baselines. Given this, the statement in the conclusions that the product "improves knowledge of ice sheet dynamics" raises the question of what this claim is based on. What have we learned about ice-sheet dynamics from the data combination that we could not have learned from GRACE-only and altimetry-only solutions? The authors may wish to consider whether the manuscript in its present form, with revisions, might be better suited to a data journal such as ESSD, or to a journal focusing on processing methods in satellite geodesy. Alternatively, the manuscript could be extended with analyses demonstrating what the improved spatial resolution allows us to learn about ice-sheet processes that was not previously possible. I appreciate that this is ultimately an editorial decision and this is not a criticism of the scientific quality of the presented work, which I consider high.
Specific comments
l. 20: Please explain why the spatial resolution and accuracy currently provided by GRACE/GRACE-FO are insufficient for capturing cryospheric processes.
l. 40: It would be appropriate to also cite our study (doi.org/10.1093/gji/ggag059) at this point, in which, very similarly to the study presented here, we introduced satellite altimetry over Antarctica into a joint global inversion framework that uses GRACE and GRACE-FO alongside further datasets in order to increase the spatial resolution of the derived Antarctic mass changes to up to 50 km.
l. 46: The direct processing of Level-1 data within such a combination approach is indeed novel. At this point in the introduction, however, I would appreciate a justification for the necessity of taking this methodologically much more demanding way. What are the limitations of previous combination methods that use Level-2 data and the full covariance information? What can be overcome by using Level-1 data?
l. 51: It would be helpful to provide the reader with somewhat more background on the community goals and recommendations. Which scientific questions lie behind them? Can the data combination presented here answer these questions?
Eq. 1: An explanation of why the normal equation system was set up in this particular form would be desirable. This only became clear to me after reading further, or after consulting the cited literature. Without prior knowledge of mascon approaches, it would arguably be more intuitive to first set up a normal equation system of the form
HT W H x = HT W y
where W is then the inverse of the observational covariance matrix. Through the addition of P-1, the representation given in the manuscript corresponds to a regularized least-squares problem, for which additional information is required. This only becomes clear later (l. 267) and is due to the chosen mascon parameterization, since linear dependencies would otherwise arise, is that correct? For the altimetry, however, the system of equations reduces to an unregularized problem (l. 255). I found it somewhat confusing that the mathematical description initially introduces the system differently from the way it is subsequently implemented in practice.
l. 73: I can follow the reasoning behind the parameterization with 1° flat mascon disks in the region where ice altimetry observations are also available, since their determination is constrained by the additional observations. Outside this region, however, where no additional information is available, a 1° parameterization constitutes an over-parameterization and necessarily requires regularization in order to obtain a stable solution. A parameterization oriented towards the GRACE-only resolution would seem more appropriate there unless the authors see specific advantages in retaining a globally uniform parameterization.
l. 88: It states that each mascon disk is centred at body-fixed spherical coordinates, while l. 74 states that the disks are distributed on the surface of an ellipsoidal Earth. Other studies have shown that neglecting the oblateness of the Earth leads to significant differences, particularly for mass changes in the polar regions (doi.org/10.1007/s00190-018-1128-0). Or is this not critical here, because the actual distance between the satellites and the mascon is considered in the determination of the mascons?
Eq. 2: A suggestion: a simple sketch illustrating the flat disk mascon and its parameters could be helpful here.
l. 118: As a reader, one has to take this statement on trust. Is there a reference demonstrating this numerical benchmark, or could it be shown in a supplement?
l. 196: An extended explanation of the relationship between depth-averaged density and FAC would be helpful for all readers who are not familiar with firn processes.
l. 205: Does the FAC uncertainty depend solely on the magnitude, or would it also be sensible to introduce some form of noise level?
l. 214: I appreciate the practical rationale for a percentage-based uncertainty estimate, and Fig. A1 convincingly demonstrates that the solution is largely insensitive to this choice. It might nevertheless strengthen the manuscript to complement the operational argument with a brief physical justification for the 30 % value specifically.
l. 250: How many parameters are estimated, and how many observations are used per month?
l. 258: The necessity of the stepwise procedure surprises me, given that satellite altimetry is also used. I would have expected that the information from satellite altimetry leads to a system of equations which is overdetermined, i.e. that the spatial allocation over the ice-covered regions is already constrained and that regularization is therefore not necessary. The authors also show very nicely in Sect. 3.4 that the spatial correlation between the mascons is considerably reduced (effectively removed) by the addition of altimetry. The authors may wish to provide somewhat more insight into the system of equations, the degrees of freedom, the resolution matrix, etc.
More general: The authors conclude on an effective spatial resolution of 1° from the reduction of inter-mascon correlation to 0.02. Decorrelation of estimated parameters and spatial resolving capability are not necessarily the same thing, particularly when a regularization term is present. An analysis of the resolution matrix, or the response to a synthetic point-mass input, would provide a more direct demonstration of the claimed resolution.
l. 275: The 13-month running mean acts as a temporal low-pass filter, and the authors admit that the formal uncertainties are artificially low (l. 305). This suggests that this step adds information that is not strictly observational. It would be helpful if the authors could discuss what this implies for the interpretation of sub-annual variability in the final product, and whether the reported effective spatial resolution is partly a consequence of this constraint rather than of the altimetry information alone.
Sect. 2.7: Temporally correlated errors are neither accounted for nor discussed. The uncertainty determined in Eq. 15 therefore describes only part of the total uncertainty, as the authors themselves note in l. 309 and following. Satellite altimetry, for example, has temporally correlated errors, which presumably propagate into the combination result. Furthermore, both GIA and FAC have a trend error, i.e. an uncertainty that increases with temporal distance from a chosen reference epoch. Other GRACE-based ice-mass balances provide a more comprehensive uncertainty characterisation (see, e.g., Table S1 in https://doi.org/10.3390/rs13091736). An estimate of the trend errors, and their inclusion in Table 1, would be desirable in order to allow the values presented here to be assessed and compared. In addition, neither the interpolation across the satellite altimetry polar gap nor the hypsometric interpolation of low quality data (Sect. 2.3) appears to contribute separately to the overall uncertainty (Eq. 15) and are difficult to assess as a user of the data set.
Sect. 2.8: Are the results presented here consistent, within the stated uncertainties, with https://doi.org/10.1038/s41597-026-08088-0?
Sect. 2.8: The validation promised in the section heading is, to be honest, rather a comparison, is it not? In particular, expectation 2 in Sect. 2.8 states that mass change at scales <300 km should more closely follow altimetry observations. Since altimetry is itself an input to the combination, this criterion appears to be satisfied by construction rather than independently verified. The same may hold for the first expectation, although, admittedly, there are no genuinely independent observations available to validate it.
However, a validation against independent datasets would indeed be interesting. Given the high spatial resolution that the authors achieve with their product, a comparison with GNSS-measured elastic loading effects, for example, would be very interesting. Admittedly to the applied GIA correction, no agreement in the trend is to be expected, but possibly on shorter time scales (e.g. seasonal signals). A comparison with independent GNSS data could validate/falsify the data combination to a certain extent.
Fig. 2 + Fig. 3 + Fig. 5: Why is the altimetry-only uncertainty larger than the assessed uncertainty of the data combination product?
Fig. 7: The figure indicates that beyond approximately degree 25, the a priori information is the largest contributor to the global solution, acting comparably to a 330 km Gaussian smoothing operator, while altimetry contributes only a few percent globally. Fig. 6, in contrast, shows altimetry to be dominant at the mascon scale over Antarctica. These two representations are not easily reconciled by the reader. A brief discussion of how the global spectral picture relates to the regional contribution maps would help to clarify to what extent the reported 100 km resolution over Antarctica is supported by observations rather than by the a priori constraint.
Sect. 3.5 + Fig. 9: A difference map between Fig. 9b and Fig. 9c would be very interesting, together with a brief discussion of the differences. And one further question for my understanding: would it also be possible to arrive at the downscaled result directly from a classical mascon solution? It remains unclear to me why it is preferable to first determine mascons jointly from GRACE and altimetry, which are then downscaled using the gain-factor approach, rather than determining mass changes directly at the high resolution from a GRACE-only mascon solution.
Sect. 4: A critical discussion of the limitations of the presented dataset is missing. At the very least, the considerable uncertainty that propagates from the GIA correction into the derived mass estimates should be discussed here (cf. Table S1 in https://doi.org/10.3390/rs13091736). In addition, further systematic error sources arising from the altimetry processing and the post-processing (l. 279ff) should be discussed.
Sect. 5: The conclusions are very methodological. Are there any conclusions about the estimated mass changes in space and time that the authors have found?
Technical comments:
Eq. 5: Please introduce kappa.
l. 330: "Independent" may be misleading here, as as both the gravimetry-only and the altimetry-only solutions are derived from the same observations that enter the data combination.
Fig. 2 + Fig. 3: The diagrams are difficult to read. A larger presentation, as well as grid lines, would be helpful. In addition, the use of red and green within the same figure may be a barrier for some readers.
Table 1: Please specify the time period over which the trends were estimated. Given the uncertainties of Antarctic mass balance estimates, reporting two decimal places does not seem appropriate.
Fig. 4 and similar figures: Highlighting the mascons shown in the figure would be helpful.