the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 23 Dec 2025
A State-Space Model for Monitoring Greenland Ice Sheet Surface Elevation Change from CryoSat-2
Abstract. We present a data-driven State-Space Model (SSM) for deriving monthly surface elevation changes of the Greenland Ice Sheet from CryoSat-2 radar altimetry (2011–2025). The model combines a Gaussian Markov Random Field for spatial dependence with an autoregressive process for temporal evolution, allowing seasonal cycles and long-term trends to emerge directly from the data.
The resulting 5 km gridded dataset captures both large-scale and local variations, showing widespread thinning along the margins and near-stable conditions in the interior. Validation against ICESat-2, Operation IceBridge, automatic weather stations, and laser-altimetry-based time series shows strong agreement and a 40–45% reduction in noise after smoothing.
This flexible and mission-independent approach provides monthly, uncertainty-quantified elevation change records that enhance understanding of Greenland Ice Sheet dynamics and support long-term, multi-sensor monitoring of its contribution to sea-level rise.
Competing interests: At least one of the (co-)authors is a member of the editorial board of The Cryosphere.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.- Preprint
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CC1: 'Comment on egusphere-2025-5015', Jichang Shen, 28 Dec 2025
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This preprint stands out for its innovative use of a GMRF in processing CryoSat-2 altimeter data, which effectively captures spatiotemporal dependencies to derive robust monthly SEC estimates, even in regions with sparse sampling. However, a notable gap is the lack of direct intercomparison with public datasets such as Zhang et al.’s (2022, ESSD) and Khan et al.’s (2025, ESSD) products (the corresponding references for these two datasets are both cited in this article). Such a comparison would validate trends, strengthen credibility, and highlight the 3D-ECM’s unique value, enhancing the work’s impact upon formal publication.ReplyCitation: https://doi.org/
10.5194/egusphere-2025-5015-CC1 -
AC1: 'Reply on CC1', Natalia Havelund, 16 Jan 2026
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We thank the commenter for the positive assessment and the constructive suggestion regarding additional intercomparisons.
The primary aim of this study is to present and validate the 3D-ECM framework for deriving monthly surface elevation change from CryoSat-2, with emphasis on spatiotemporal consistency, and performance under sparse sampling. The validation is therefore centered on independent airborne and in situ reference data (Operation IceBridge, AWS), complemented by a consistency comparison with ICESat-2 ATL15.
The datasets of Zhang et al. (2022) and Khan et al. (2025) are valuable community products, but represent fully processed altimetry-based SEC datasets with overlapping information content. As such, they are not fully independent validation sources as they contain the same data as used in this study. As such, a product-level intercomparison was not a central focus of this study.
A qualitative comparison of long-term trends against these products may nevertheless be explored in future work, particularly once an extended 3D-ECM time series is available. A full product-level intercomparison would benefit from matching temporal coverage and data density. Here, we aim to keep the primary validation focused on independent reference data.
Citation: https://doi.org/10.5194/egusphere-2025-5015-AC1
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AC1: 'Reply on CC1', Natalia Havelund, 16 Jan 2026
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RC1: 'Comment on egusphere-2025-5015', Anonymous Referee #1, 20 Jan 2026
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The manuscript "A State-Space Model for Monitoring Greenland Ice Sheet Surface Elevation Change from CryoSat-2" presents a method to derive surface elevation changes of the Greenland ice sheet at a monthly temporal resolution and at 5 km spatial resolution. For estimating the surface elevation changes a state-space filtering technique is applied to CryoSat-2 data (radar altimetry). A comparison is made with satellite and airborne laser altimetry data. The data is validated with data from automatic weather stations at a total of three locations. Further development of surface elevation change products is highly appreciated, as they could provide valuable insights into the ongoing processes related to the Greenland Ice Sheet. Advances in the processing of radar altimetry data to derive spatially and temporally coherent SEC are a very important part of this, which makes the topic of this study significant.
In its current form, the article essentially presents a data set and the underlying method, a state-space model.
However, much remains unclear to me. In particular, after reading the manuscript, I am left with the following questions:- Is this method as good as or better than other methods?
- Which limitations of existing methods does the presented method overcome, and what limitations does it have itself?
- Does the final SEC product agree better with independent data than existing products that have already been published?
As long as these questions remain unanswered, the relevance of the study remains unclear to me. The application of a method alone does not justify this study.
The article's technical focus also makes it difficult to identify the relevance of the study. The authors should state more clearly what the actual problem is that they are working on. If I understand correctly, the challenge lies in the fact that the retracked heights (POCAs) are highly irregular in both space and time. The task now is to generate regular monthly grids from these irregularly sampled observations. I consider the choice of a state space filter to be extremely suitable for this task. It would be helpful to provide a clearer justification for this choice and demonstrate to the reader what the spatial and temporal sampling of the initial data looks like. Additionally, a review of alternative methods for this task and their limitations is missing, which are apparently to be resolved here.Essentially, I can think of two ways of how to proceed with the manuscript.
- The development and analysis of the methodology could be more rigorous. It would be interesting to see a comparison of the state space model with other methods of creating temporally and spatially coherent grids from POCAs. If applicable, the manuscript might then be better suited to a remote sensing journal specializing in methodology or a journal focusing on geodata analysis methods. The authors may also consider to focus on publishing a dataset and aiming for an article in a data journal like ESSD. The authors should consider which readers they want to reach, bearing in mind that readers of The Cryosphere are potentially more interested in insights into cryospheric processes than in data analysis methods. The choice of title makes it already clear that this is a technical and methodological study, rather than one focusing on conclusions about the Greenland ice sheet.
- The study could be expanded by demonstrating that signals can be detected with a higher accuracy compared to other SEC products, or maybe even compared to other remote sensing methods. I agree with Jichang Shen's comment that a comparison with other SEC products should be included. While this would not be an independent validation, it would be a helpful comparison. Ideally, a validation dataset would be used to determine which SEC product performs best. I think it is problematic that the current validation is supposed to be meaningful for the entire SEC product across the entire GIS, based on only three validation points. Ideally, a validation is designed to allow statements to be made about the entire GIS as extensively as possible. Furthermore, I think it is necessary to expand the analysis of the signals contained in the determined SEC. The abstract promises that it should be possible to "detect seasonal cycles, long-term trends and abrupt changes". Accordingly, the following questions could be answered after some more analysis: What can we learn about the seasonal cycles of the Greenland ice sheet in space and time? How is summer ablation changing over time and in which regions? How significant is glacial thinning over the observation period? How does the long-term trend and its uncertainty compare to that obtained using other methods? The authors may also be aware of questions that could not be answered using previous processing strategies, but which can be answered using the developed "State Space Model".
General Comments
- The abstract is very technical and does not provide enough contextualization of the relevance of the study presented. What is the motivation for the article? Why are monthly surface elevation changes needed, and what is the problem with existing products? Why is the SSM needed? It would also be desirable to quantify the statement "... consistent elevation change records that capture both seasonal variability and long-term trends of the Greenland Ice Sheet". Can you state the "seasonal variability" and "long-term trends" with a value and an uncertainty?
- The introduction does not mention any research questions or hypothesis derived from the current state of knowledge. What are the research questions being investigated here? What are the hypotheses being tested here? Could it be said that the existing methods of altimetry processing listed in l37 are not capable of quantifying the processes mentioned in l32 with sufficient accuracy? Currently, it sounds as if the existing methods are very good ("mark a major step forward"). Why is the SSM needed then?
- The introduction does not clearly state which specific step of altimetry processing is being addressed here. This should be clarified in advance for readers who are not familiar with all the details of altimetry processing. State-space filtering is used here as a spatial-temporal interpolation method (or extrapolation method), is that correct? Is the step then similar to, e.g., IDW, collocation/kriging or similar?- The question that keeps coming to mind as I read, and where I make comments in several places: How compares your result to existing published SEC products (e.g. 10.5194/essd-14-973-2022, 10.5194/essd-2024-311, 10.5194/essd-17-3047-2025) in space and time? Can we conclude that this SEC product represents an improvement on existing ones? Or more general: Why is there no quantitative comparison with other studies that did similar things?
- Am I correct in understanding that validation of short-term signals only takes place at three individual AWS stations? The caption for Figure 1 does not state that these are shown there. This should be added. Section 2.4 contains a lot of general information about AWS measurements. Please clearly state at the outset which data is used, and justify the selection of data. Furthermore, I do not understand why it is justified not to use data from all the other AWS stations (e.g. 10.5194/essd-13-3819-2021). I'm wondering whether the authors only selected stations where the comparison with the SEC works well. In my opinion, a comparison with all available AWS stations is necessary in order to assess the validity of the study. How do you validate short-term signals from ice flow changes? Maybe firn thickness changes from publicly available firn models are useful to get further insights in some parts of the GIS. How do the seasonal signals from the SEC product align with results from regional climate modelling?
- If I understand correctly, one objective is also to determine SEC in glacial valleys where other data sets encounter difficulties. Is laser altimetry data even suitable for comparison or validation there, given that these glacial valleys are often cloud-covered? It would be also very useful to know what the spatial and temporal sampling of the laser altimetry data looks like in space and time. Are there regions in Greenland where you could first demonstrate how well CryoSat-2 and ICESat-2 agree before generating the SEC grids?
- A somewhat more detailed explanation of the motivation for the stochastic description of SEC in space and time would be desirable. Are there any statistical analyses of SEC that could be cited as a basis here? How do you justify to use the GMRF in space and the AR(1) process in time?
- It is not clear to me from the description of the method how a distinction can be made in the approach between uncorrelated neighbours (phi = 0) and measurement noise (eps_i). How are temporal and spatial error correlations taken into account, which can be particularly significant in radar altimetry? Is it justified to describe observational errors with a normal distribution (Eq. 1)? I think it is very necessary to clearly state and justify the assumptions made in the methodology.
- Assumptions made about stochastic processes are not discussed or critically evaluated within the discussion.
- The discussion lacks a critical assessment of the method presented. Why are the results initially so noisy and why is a post-processing step necessary? This suggests that the SSM method has limitations that need to be explored and discussed.
Specific Commentsl14: Please specify "high-resolution"
l14: What do you mean by "temporally consistent"?
l21: Please provide a number to contextualize "significant"
l33: Please provide references for the "annual SEC products traditionally derived ...".
l37: You may add Helm et al. (2024, 10.5194/tc-18-3933-2024) when you provide references for "new processing methods".
l39: What do you mean by "integration" in "better integration with regional climate models and in situ observation"?
l40: Please provide an introduction of state-space models and please argue which limitations you aim to resolve with them. In the paragraph before you mention "These efforts mark a major step forward ..." But where are the limitations? What justifies the need of a new method? (See also my general comments given above).
l40: "detect subtle patterns" That would indeed be interesting, but as far as I can see, it plays no role at all in the rest of the manuscript.
l55: You may mention that you use level-2 data at the beginning of Section 2.
l110: Please argue why the comparison with SERAC is useful as there is already a comparison with laser altimetry (Sect. 2.2 and 2.3).
l112: Why is the SERAC time series only created for the three AWS points? What are the arguments for not doing this for all points on the ice sheet?
l145: I don't understand what was inefficient about previous approaches. What is the problem?
l154: Please clarify up to what correlation lengths processes can be captured temporally and spatially with the GMRF and AR(1), or does l174 mean that only direct neighbours are considered?
l159: Some motivation would be desirable for the chosen spatial and temporal resolution.
l187: A more mathematical explanation would be desirable, detailing how the problem is solved and how it is implemented numerically. I do not see how one would be able to reproduce the results based on the description. The authors could also consider making the routines used publicly available alongside detailed documentation. Am I correct in understanding that no Kalman filter or smoother is used here to estimate the states?
l191: Please explain "generalized delta-method".
l196: Where does the unrealistic high-frequent variability comes from? Are these errors? In l46 you promised SSMs explicitly separate signal from noise.
l200: How do you estimate the seasonal component and the trend? As far I can see both are not separate parameters in your model.
l205: Is the uncertainty then representative? Why is it not propagated to the post-processed SEC?
l207: I don't quite understand that. Apparently, the estimated parameters still contain too much noise, so the estimated model is smoothed spatially and temporally in a post-processing step. Shouldn't this noise, which is filtered out in the post-processing step, also be included in the uncertainties? Having read section 3, I don't really get the impression that the methodology can sufficiently separate signal and noise. Ideally, the SSM would be designed so that the post-processing step would not be necessary. Or, to put it another way: How does this variant perform compared to other approaches for generating spatially and temporally consistent SEC grids from radar altimetry observations?
l220: But this does not account for spatial correlations across grid cells, is that right? This explanation should be included in the Methods, not the Results.
Figure 1: I suggest showing the Greenland map (Panel D) as a separate, much larger figure. It would be helpful to plot metres per year, as in Figure 3.
Figure 1: The uncertainties are constant over time, am I right? Is that realistic? Other products identify significant uncertainty of the trend (e.g. Fig. 16 in 10.5194/essd-17-3047-2025), and I would have expected time-correlated errors to be reflected in the SSM result. Are time-correlated errors taken into account?
Figure 2: It would be helpful to show an integrated volume time series here, so that interior + ablation = whole Greenland. Here, too, the total uncertainty does not seem to include any uncertainties in the trend, which I consider unrealistic.
Section 4.1.1: The name of the section is "Validation of trend". Essentially, this section deals with how the post-processing step minimizes noise in the differences between the result and ICESat-2 and OIB. How does the linear trend rate of ICESat-2 and the presented product compares over the period from 2019 to 2025? It would be helpful to see this as a spatial grid. Furthermore it would be interesting to a time series comparison between the SEC product and ICESat-2.
Section 4.1.2: I find it highly problematic to refer to "validation of seasonal amplitude" when a comparison is only carried out at three points, because the section title suggests this is representative for the whole GIS. Why is the phase not taken into account?
l298: The conclusion is not clear to me. "Reducing noise" is only achieved through the post-processing step.
Figure 6: How were the average seasonal cycles calculated?
l323: "Independent validation against ICESat-2 ATL15 and OIB confirms the 3D-ECM method’s effectiveness." Effectiveness sounds quite vague. What does it mean? Can it be quantified?
l343: Please show and quantify these patterns that were not revealed before.
l344: What do you mean by "great detail"? Which integration do you refer to? Please explain how the SEC product would be valuable for climate models?
l347: Please explain more extensively why "entirely data-driven" is a great advantage? What are the caveats of the other studies you are citing? Here it becomes clear once again that a comparison with these products would be very useful.
l350: The authors could consider expanding the study to include a comparison with swath-processed data. It would certainly be very interesting to see how these data sets compare in the outflow glacier regions where the sampling is sparse.
l356: Please clarify "ice mask". Do you mean the mask of the Greenland ice sheet without peripheral glaciers? Please provide a reference to the mask you are applied.
l364: Please explain what do you mean by scalability and statistical rigour.
l367: It's not entirely clear to me how the approach enhances the temporal resolution. Maybe you could use "unifies" instead of "enhances".
l368: You conclude here that the approach "supports improved assessments of ice-sheet mass balance and dynamics". On what findings is this conclusion based? The manuscript contains no analyses of mass balances or ice sheet dynamics.
Technical CorrectionsYou introduced SSM as an abbreviation for State Space Model. Please use throughout.
When referring to uncertainties, please indicate the sigma level.
Figures: Please increase labels to a sufficient font size as they are hard to read at some places (e.g. Figure 1, panel C Figure 5 etc.)
l56: Please introduce the abbreviation as usual: Surface Elevation Reconstruction And Change detection (SERAC)
l195: "Thorson and Kristensen, 2024" is not in the reference list.
Figure 5: Please provide time periods.
l337: "Taken together, these comparisons indicate that dSEC generally agrees well with both the AWS records and the altimetry-based SERAC product, underscoring the ability of the 3D-ECM framework to extract physically meaningful elevation change signals that are consistent with in situ observations and the altimetry/model-based SERAC." This is redundant and could be formulated more concisely.Citation: https://doi.org/10.5194/egusphere-2025-5015-RC1
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