the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 08 May 2026
Greenland mass change using range acceleration from GRACE-FO
Abstract. Observations of ice mass change with GRACE and GRACE-FO are limited in both temporal and spatial resolution. For the Greenland ice sheet, standard processing can resolve monthly mass change at the scale of large drainage basins, not at the level of individual glaciers or smaller regions. For GRACE-FO, a method using the range acceleration measurements to calculate instantaneous line-of-sight gravity has been used to study hydrological extremes such as floods and storm surges. In this paper, we look at the Greenland ice sheet and use the line-of-sight gravity signal to estimate 5-day mass change for four summers. We also include altimetry data to try to improve the spatial resolution of the observed mass change. The amplitude of the estimated mass change is within the range of monthly mass change estimates. The choice of regularisation clearly affects the spatial distribution, which in turn affects estimates at basin scales or even at smaller scales. Additionally, the noise level increases relative to the monthly signal, and the regularisation based on altimetry data increases the correlation among mass points compared to Tikhonov regularisation. Estimating mass change based on line-of-sight gravity is possible, although the spatial distribution remains sensitive to the choice of regularisation. Nonetheless, the approach shows promise for resolving short‑term mass‑change events and motivates further refinement of line-of-sight gravity-based inversion techniques for future satellite gravimetry missions.
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Status: final response (author comments only)
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RC1: 'Comment on egusphere-2026-2451', Paul Tregoning, 09 Jun 2026
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AC1: 'Reply on RC1', Barbara Jenny, 12 Aug 2026
Thank you for your comments. We will address the comments below one by one.
Major comments
- Since the objectives of the paper are to assess whether and how well the LGD approach can estimate high spatial resolution mass change, it seems strange that the size of the mascons used to parameterise the mass changes is simply set to 45 km. Why was this value chosen? It seems likely that the setting of this value had significant impact on the results, yet no other values seem to have been used to make comparisons, nor has any justification been given for why 45 km was the chosen value. Also, the choice of regularisation will trade off with the size of the mascons used - the impact of the arbitrariy chooice of this parameter pervade through all the results, interpretations and conclusions of the paper. This deserves greater assessment, probably including showing results using different sizes of mascons to quantify the sensitivity of the approach to the mascon size.
We did consider this but, in the end, did not have enough time to look at denser grids. We will conduct some tests with denser grids for one summer and decide on how we proceed afterwards. As increasing grid density significantly increases both computation time and memory requirements and affects regularisation especially as the number of observations stays the same, we expect a trade-off between spatial resolution and number of estimates per month. - It is not clear why Iceland and Ellesmere Island were included as point masses in the setup of the experiment. Mass changes there are never discussed, so why are they parameterised at all?
Including Iceland and Ellesmere Island serves two purposes. Ellesmere Island is very close to northern Greenland and previous studies have shown that including mascons on Ellesmere Island helps reducing leakage effects by allowing signal separation between the neighbouring ice caps. The inclusion of Iceland was done as once the objective was to test the approach for Iceland. As the validation of these results is not straight-forward, we decided to address this in a separate study but as Sørensen et al (2017) showed, leakage from Greenland is an important factor for mass change estimates in Iceland and the signal from Greenland would have to be removed in some way anyway. By estimating Ellesmere Island, Greenland and Iceland, we hope to reduce leakage from Ellesmere Island on Greenland for this study and to reduced leakage from East Greenland on Iceland for a future study. In a revised version of the paper, we will try to make this clearer and will consider including the mass change estimates for Ellesmere Island and Iceland in the appendix/supplementary material.
Minor comments
line 39: the LRI has not been running in diagnostic mode to reduce thruster activation and overall fuel usage. It has been running in diagnostic mode because no inter-satellite measurements are being made anymore due to reduced thruster activation.
RC2 also mentioned this and we will revise this part of the text.
line 42: change "to get" to something more formal
We will change this to “to obtain”.
line 100: could a sigma of 10 nm/s^2 be throwing out actual signals? Some of the large mass loss signals are now causing range acceleration observations over 100 nm/s^2 (or is that only over Antarctica where the mass loss is larger?)
We will investigate this but LGD from L2 from one month to the next only show changes of the order of 5-10nm/s^2 over fast flowing glaciers and we would expect mostly ice mass loss in June-August and only limited mass gains during these months.
line 238: have you calculated altimetry height changes using linear trends? Won't the inclusion of linear changes affect the ability of the LGD approach to identify correctly any short-term mass changes? What is the impact of imposing linear height changes through the inclusion of the altimetry observations?
To minimize the effect of the linear height changes used in the regularisation, we estimate mass change since 2010. The main signal is thus the accumulated mass change and which should show the same pattern as the linear trends. It will affect the possibility of detecting really small scale anomalies but events like glacial lake drainages or snowfalls are not expected to be stronger than the background signal of continuous mass changes since 2010.
line 168: what is the origin of the -231.48 GT/yr trend that was restored?
This is the trend mentioned in the paper that accompanies the GNET GMB data set.
Figure 5: please add the basin numbers (again) to the central figure. Also, why are all your time series chopped up into four individual boxes? Why can't they be shown as continuous time series?
We will add the basin numbers in a revised version of the paper. The time series is chopped up as we only estimated mass changes from LGD for the summer months and showing it as a continuous time series adds a lot of white space and makes it hard to see the details. We had many discussions on how to make the figures such that the important information can be seen and decided on splitting it up to only show the summers for now.
line 204: the term "mascon" first appears here, without definition.
We will revise the text accordingly.
Citation: https://doi.org/10.5194/egusphere-2026-2451-AC1 - Since the objectives of the paper are to assess whether and how well the LGD approach can estimate high spatial resolution mass change, it seems strange that the size of the mascons used to parameterise the mass changes is simply set to 45 km. Why was this value chosen? It seems likely that the setting of this value had significant impact on the results, yet no other values seem to have been used to make comparisons, nor has any justification been given for why 45 km was the chosen value. Also, the choice of regularisation will trade off with the size of the mascons used - the impact of the arbitrariy chooice of this parameter pervade through all the results, interpretations and conclusions of the paper. This deserves greater assessment, probably including showing results using different sizes of mascons to quantify the sensitivity of the approach to the mascon size.
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AC1: 'Reply on RC1', Barbara Jenny, 12 Aug 2026
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RC2: 'Comment on egusphere-2026-2451', Khosro Ghobadi-Far, 11 Jul 2026
The manuscript “Greenland mass change using range acceleration from GRACE-FO” computes LGD observations from GRACE-FO LRI and then inverts them to estimate 5-day mass change solutions over Greenland. The aim is to detect sub-monthly signals which are not captured by the standard GRACE-FO monthly solutions. The methodology is clearly presented, the results and discussion are well organized, and the conclusions are supported by the results to a large extent. I find this to be an interesting paper and believe it should be published after addressing the comments below. I have several minor to moderate comments that I believe would further improve the quality and presentation of the manuscript.
In addition to all the comments and recommendations provided below, I have one suggestion regarding the validation of the sub-monthly signals in the 5-day solutions. I recognize that rigorous validation of sub-monthly mass change estimates is inherently challenging. However, since the authors already use the daily GNET-GMT mass change solution, which shows good agreement with the 5-day LGD solutions (see Figure 4), I believe this dataset could be further exploited as a reference to evaluate the sub-monthly signal content of LGD solutions. Specifically, the authors could compute 5-day averages from the daily GNET-GMT solution and compare them with both the 5-day LGD solutions and the monthly GRACE-FO GravIS solutions using metrics such as correlation and RMS reduction. This analysis could be performed for the entire Greenland Ice Sheet (as in Figure 4) and, if GNET GMT product is available, for the 8 Greenland basins as well. This analysis would provide a more quantitative assessment of the ability of the 5-day LGD solutions to capture sub-monthly mass variations beyond those resolved by the standard monthly GRACE-FO products. Therefore, it would further strengthen the primary scientific contribution of this work.
------------------------- Detailed Comments -------------------------------
** * Title
The approach you are using is based on “line-of-sight gravity difference”, not range-acceleration. Unlike line-of-sight gravity difference, range-acceleration observations cannot be directly related to surface mass change and require numerical integration.
*** Abstract
“line-of-sight gravity” must be replaced with “line-of-sight gravity difference” in abstract, as well as throughout the text. The observable from GRACE/-FO is gravity difference from the two satellites, not gravity.
*** 1 Introduction
The introduction provides a good overview of the technical aspects of the proposed research. However, it lacks background on Greenland Ice Sheet mass change as measured by GRACE/GRACE-FO. The introduction would benefit from a discussion of GRACE/GRACE-FO results for the Greenland Ice Sheet and the limitations of existing approaches that the proposed methodology seeks to address.
*** 2 Materials and methods
L39: As far as I know, the LRI is still operating in its nominal mode. However, the precision pointing between the two satellites has been disabled to conserve propellant following the fuel leak (therefore, deciding not to use the thruster firings all the time), so the laser beams are no longer aligned with the companion spacecraft. You may need to revise this sentence.
Table 1 mentions KBR1B is the dataset used in the study, while LRI1B was actually used. Please revise.
L69: “fourier inversion” must be replaced with “Fourier tranform”
Transfer function equation in L70: (1) There is a typo in the analytical transfer function provided by Ghobadi-Far et al. (2022). The correct equation is 1.0+3.5*1E−4*f−1.04 (not 1.0+3.5^−4*f−1.04). (2) There is no upper limit for the validity of transfer function above 1 mHz. Therefore “f <= 0.25 Hz” must be removed.
L119-120: “To reduce cut-off error, we only calculate the loading correction in the spectral domain and define the full potential” => The meaning behind “To reduce cut-off error” is not clear to me. Please revise and clarify.
L140: The meaning of “max(h^{alt})” in the numerator of Eq. (18) is not clear to me. I thought the authors are using an icehseet height trend from altimetry for that purpose, which has only one value for each grid cell. Please clarify.
L156-157: “we chose to go with a slightly lower regularisation than the L-curve would suggest to include most of the signal.” => This kind of makes sense to me. In my own experience with regional gravity modelling, sometimes I have seen that the optimal “knee” of the L-curve does not provide the best/optimal results, in which case I had to manually play slightly with “\lambda” to improve the solution. With that being said, the meaning of “to include most of the signal” in that sentence is not clear to me. Please clarify.
*** 3 Results and discussion
Figure 4: Do you know why the CCI and GravIS solutions show a bias, resulting in lower and upper estimates compared to GMB and your LGD solutions? Particularly, I thought the GravIS solution, which is based on GRACE-FO (is that correct?), should not show a bias. Do you have any guess why that is?
L176: How did you derive the equation for covariance matrix of estimate mass change? Please elaborate.
Table 2: It is very interesting to see that STD of residuals decreases as we go from 2019 to 2022! I do not think this can be fully attributed to the improved transplant data. Based on Figure 6, the magnitude of the mass change decreases progressively from 2019 to 2022, and this pattern seems consistent with the behavior of the residual STDs reported in Table 2. Specifically, the larger LGD signals associated with the greater mass changes in the earlier years (e.g., 2019) would be expected to produce larger residuals and, consequently, higher STD values. This suggests that the observed decreasing STD values from 2019 to 2022 may mainly reflect changes in signal amplitude rather than other factors. I encourage the authors to revise the discussion and interpretation of the results in Table 2 based on the this observation.
L183-188: “Table 2 also shows …. . “The reason for these higher residuals in the point-mass inversions is not investigated”. Since the inversion seems to perform worse ….. each with 2 to 6 orbits crossing our observation area.” => The discussion here must be revised as it includes some statements which are not accurate such the point you mentioned about the “limited spatial distribution of the observations”.
Your results show that residuals from L2 solutions are lower than those from your regional solutions. This result is expected (if there is not too much sub-monthly mass variability!) because L2 solutions are unconstrained solutions, which means they are able to better capture LGD signal variability. The regional mascon solutions, on the other hand, are regularized, and as such, cannot capture observations as good as the global, unconstrained L2 solutions. You can design your regional solution such that it gives you a much better fit to LGD data, but the mass change solution you get out of that would not be good. This is a typical characteristic of mascon solutions. Please revise.
Citation: https://doi.org/10.5194/egusphere-2026-2451-RC2 -
AC2: 'Reply on RC2', Barbara Jenny, 12 Aug 2026
Thank you for your comments. We will address the comments below one by one.
In addition to all the comments and recommendations provided below, I have one suggestion regarding the validation of the sub-monthly signals in the 5-day solutions. I recognize that rigorous validation of sub-monthly mass change estimates is inherently challenging. However, since the authors already use the daily GNET-GMT mass change solution, which shows good agreement with the 5-day LGD solutions (see Figure 4), I believe this dataset could be further exploited as a reference to evaluate the sub-monthly signal content of LGD solutions. Specifically, the authors could compute 5-day averages from the daily GNET-GMT solution and compare them with both the 5-day LGD solutions and the monthly GRACE-FO GravIS solutions using metrics such as correlation and RMS reduction. This analysis could be performed for the entire Greenland Ice Sheet (as in Figure 4) and, if GNET GMT product is available, for the 8 Greenland basins as well. This analysis would provide a more quantitative assessment of the ability of the 5-day LGD solutions to capture sub-monthly mass variations beyond those resolved by the standard monthly GRACE-FO products. Therefore, it would further strengthen the primary scientific contribution of this work.
Thanks for the suggestion. We agree that this will improve the quality of the paper. We will investigate the availability of basin scale GNET GMT products and will conduct the quantitative comparison of all available time series as suggested.
------------------------- Detailed Comments -------------------------------
** * Title
The approach you are using is based on “line-of-sight gravity difference”, not range-acceleration. Unlike line-of-sight gravity difference, range-acceleration observations cannot be directly related to surface mass change and require numerical integration.We chose range-acceleration in title as it is the observation and LGD is a quantity derived from range acceleration through multiple processing steps. We will discuss the proposed change in wording.
*** Abstract
“line-of-sight gravity” must be replaced with “line-of-sight gravity difference” in abstract, as well as throughout the text. The observable from GRACE/-FO is gravity difference from the two satellites, not gravity.We agree and will change this for the whole manuscript.
*** 1 Introduction
The introduction provides a good overview of the technical aspects of the proposed research. However, it lacks background on Greenland Ice Sheet mass change as measured by GRACE/GRACE-FO. The introduction would benefit from a discussion of GRACE/GRACE-FO results for the Greenland Ice Sheet and the limitations of existing approaches that the proposed methodology seeks to address.We agree and will revise the introduction accordingly
*** 2 Materials and methods
L39: As far as I know, the LRI is still operating in its nominal mode. However, the precision pointing between the two satellites has been disabled to conserve propellant following the fuel leak (therefore, deciding not to use the thruster firings all the time), so the laser beams are no longer aligned with the companion spacecraft. You may need to revise this sentence.RC1 also mentioned this and we will revise this part of the text.
Table 1 mentions KBR1B is the dataset used in the study, while LRI1B was actually used. Please revise.
LRI1B was used as the observation for range acceleration but KBR1B is used in estimating the reference orbits. We will add a sentence to make this clearer for the reader.
L69: “fourier inversion” must be replaced with “Fourier tranform”
We will change this.
Transfer function equation in L70: (1) There is a typo in the analytical transfer function provided by Ghobadi-Far et al. (2022). The correct equation is 1.0+3.5*1E−4*f−1.04 (not 1.0+3.5^−4*f−1.04). (2) There is no upper limit for the validity of transfer function above 1 mHz. Therefore “f <= 0.25 Hz” must be removed.
We will correct the function. The upper limit is due to the observation frequency but this can of course be removed for the analytical description of the transfer function.
L119-120: “To reduce cut-off error, we only calculate the loading correction in the spectral domain and define the full potential” => The meaning behind “To reduce cut-off error” is not clear to me. Please revise and clarify.
Here, cut-off error relates to the error coming from cutting of the infinite sum at a maximum order. We will revise the text accordingly.
L140: The meaning of “max(h^{alt})” in the numerator of Eq. (18) is not clear to me. I thought the authors are using an icehseet height trend from altimetry for that purpose, which has only one value for each grid cell. Please clarify.
We normalise the gridded trends by the maximum of the whole grid to try to achieve a regularisation only for the location of the mass change and not for the actual amplitude. This also helped to make the inversion numerically more stable.
L156-157: “we chose to go with a slightly lower regularisation than the L-curve would suggest to include most of the signal.” => This kind of makes sense to me. In my own experience with regional gravity modelling, sometimes I have seen that the optimal “knee” of the L-curve does not provide the best/optimal results, in which case I had to manually play slightly with “\lambda” to improve the solution. With that being said, the meaning of “to include most of the signal” in that sentence is not clear to me. Please clarify.
This relates to figure 2 where (a) shows that the total mass converges at a slightly lower value of lambda than the “knee” of the L-curve. We will add one sentence to clarify what we mean.
*** 3 Results and discussion
Figure 4: Do you know why the CCI and GravIS solutions show a bias, resulting in lower and upper estimates compared to GMB and your LGD solutions? Particularly, I thought the GravIS solution, which is based on GRACE-FO (is that correct?), should not show a bias. Do you have any guess why that is?Both GravIS and CCI are based on GRACE and GRACE-FO. Since the LGD is with respect to the static GOCO06S, we shifted both CCI and GravIS timeseries such that they cross zero at January 2010. This might introduce a bias as it is not done using a fit but the actual value of each time series of that month.
L176: How did you derive the equation for covariance matrix of estimate mass change? Please elaborate.
This is based on the gaussian error propagation. We will add a sentence to clarify.
Table 2: It is very interesting to see that STD of residuals decreases as we go from 2019 to 2022! I do not think this can be fully attributed to the improved transplant data. Based on Figure 6, the magnitude of the mass change decreases progressively from 2019 to 2022, and this pattern seems consistent with the behavior of the residual STDs reported in Table 2. Specifically, the larger LGD signals associated with the greater mass changes in the earlier years (e.g., 2019) would be expected to produce larger residuals and, consequently, higher STD values. This suggests that the observed decreasing STD values from 2019 to 2022 may mainly reflect changes in signal amplitude rather than other factors. I encourage the authors to revise the discussion and interpretation of the results in Table 2 based on the this observation.
Interesting observation. We will look at both the range accelerations and the LGDs and have another discussion before we revise the section accordingly.
L183-188: “Table 2 also shows …. . “The reason for these higher residuals in the point-mass inversions is not investigated”. Since the inversion seems to perform worse ….. each with 2 to 6 orbits crossing our observation area.” => The discussion here must be revised as it includes some statements which are not accurate such the point you mentioned about the “limited spatial distribution of the observations”.
Your results show that residuals from L2 solutions are lower than those from your regional solutions. This result is expected (if there is not too much sub-monthly mass variability!) because L2 solutions are unconstrained solutions, which means they are able to better capture LGD signal variability. The regional mascon solutions, on the other hand, are regularized, and as such, cannot capture observations as good as the global, unconstrained L2 solutions. You can design your regional solution such that it gives you a much better fit to LGD data, but the mass change solution you get out of that would not be good. This is a typical characteristic of mascon solutions. Please revise.
Thank you for this comment. We will investigate that and revise the discussion.
Citation: https://doi.org/10.5194/egusphere-2026-2451-AC2
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AC2: 'Reply on RC2', Barbara Jenny, 12 Aug 2026
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- 1
This is an interesting and well-presented manuscript, using line-of-sight gravity differences to attempt to estimate very high spatial resolution changes in mass over Greenland. The authors use an established technique but apply it to a cryospheric setting rather than hydrological as has been done previously.
major comments
1. Since the objectives of the paper are to assess whether and how well the LGD approach can estimate high spatial resolution mass change, it seems strange that the size of the mascons used to parameterise the mass changes is simply set to 45 km. Why was this value chosen? It seems likely that the setting of this value had significant impact on the results, yet no other values seem to have been used to make comparisons, nor has any justification been given for why 45 km was the chosen value. Also, the choice of regularisation will trade off with the size of the mascons used - the impact of the arbitrariy chooice of this parameter pervade through all the results, interpretations and conclusions of the paper. This deserves greater assessment, probably including showing results using different sizes of mascons to quantify the sensitivity of the approach to the mascon size.
2. It is not clear why Iceland and Ellesmere Island were included as point masses in the setup of the experiment. Mass changes there are never discussed, so why are they parameterised at all?
Minor comments
line 39: the LRI has not been running in diagnostic mode to reduce thruster activation and overall fuel usage. It has been running in diagnostic mode because no inter-satellite measurements are being made anymore due to reduced thruster activation.
line 42: change "to get" to something more formal
line 100: could a sigma of 10 nm/s^2 be throwing out actual signals? Some of the large mass loss signals are now causing range acceleration observations over 100 nm/s^2 (or is that only over Antarctica where the mass loss is larger?)
line 238: have you calculated altimetry height changes using linear trends? Won't the inclusion of linear changes affect the ability of the LGD approach to identify correctly any short-term mass changes? What is the impact of imposing linear height changes through the inclusion of the altimetry observations?
line 168: what is the origin of the -231.48 GT/yr trend that was restored?
Figure 5: please add the basin numbers (again) to the central figure. Also, why are all your time series chopped up into four individual boxes? Why can't they be shown as continuous time series?
line 204: the term "mascon" first appears here, without definition.