the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 01 Sep 2026
Emulating high-resolution distributions of snow depth and density on Arctic sea ice
Abstract. Snow on Arctic sea ice significantly influences regional and global climate processes, yet accurately modelling its spatial and temporal variability remains challenging. We introduce a physics-guided Long Short-Term Memory (LSTM) emulator developed to efficiently simulate pan-Arctic snow density and depth distributions. The emulator is trained on high-resolution SnowModel-3D simulations that resolve wind-driven snow redistribution over sea-ice surface topography, and is applied over the Lagrangian parcel framework of SnowModel-LG for pan-Arctic inference. The topographic input required for inference comes from CryoSat-2 surface roughness estimates at a pan-Arctic scale. We assessed the emulator’s predictive accuracy across multiple years (2014–2021) and various seasonal conditions. Compared with airborne snow observation campaigns, the emulated snow depth has an RMSE of 0.08–0.10 m. By representing the influence of sea-ice topography on snow redistribution, which is not accounted for in SnowModel-LG, the emulator adds a process that brings the simulated snow depth closer to these airborne observations. While seasonal and interannual trends in snow density and depth were effectively captured, emulator accuracy varied during complex melt periods. Our emulator provides an efficient tool for enhancing pan-Arctic snow modelling capabilities in climate studies.
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Status: open (until 13 Oct 2026)
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RC1: 'Comment on egusphere-2026-4727', Anonymous Referee #1, 01 Oct 2026
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AC1: 'Reply on RC1', Ayush Prasad, 05 Oct 2026
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We thank Referee #1 for the careful and constructive review. Here we briefly address the main comments that determine the additional analysis we are now carrying out.
1. Isolating the effect of sea ice topography
We agree that the manuscript does not yet isolate the effect of topography and that a baseline without topography is needed to support this claim. However, this baseline cannot be obtained by running the existing emulator with the topography input removed or set to a fixed value, for two reasons. First, topography enters the emulator in two places. It is one of the inputs to the emulator, and it was also used to run the SnowModel-3D which produced the snow fields that the emulator was trained to reproduce. Changing the input at inference leaves these training targets unchanged, so the test measures how strongly the trained network relies on that input and not how much topography contributes to the snow cover. Second, the learned model weights depend jointly on all inputs. Setting one input to zero or to a fixed value at inference time can move the network outside the input distribution it was trained on, and in an autoregressive LSTM the resulting error is fed back and accumulates over the simulation (Bengio et al., 2015). The change in skill would then mix the effect of topography with the effect of this distribution shift and could not be read as the contribution of topography alone. This is a known problem with perturbation-based estimates of feature importance (Hooker et al., 2021; Molnar et al., 2022).
The standard remedy is to retrain the model without the feature in question (Hooker et al., 2019; Hooker et al., 2021). We could therefore build a proper no-topography baseline by running SnowModel-3D over flat, ridge-free domains and training the emulator on these runs with the same setup, without the topography input. At this stage, however, we do not consider the training of an additional emulator to be the most scientifically informative part of this experiment, since the physical contribution of topography can be isolated more directly by comparing otherwise identical SnowModel-3D simulations with and without surface topography.
2. Physics-guided loss and the compaction equation
Regarding the compaction equation, the form printed in Prasad et al. (2024) contains a typographical error. The factor exp(-A_2 rho_s) appears in the denominator, whereas it should multiply the right-hand side, as in Liston and Hall (1995) and in the submitted manuscript. The error is confined to the printed text. The actual implementation followed the SnowPack formulation used in SnowModel (Liston and Hall, 1995; Liston and Elder, 2006). In Prasad et al. (2024), the physics-guided LSTM lowered the density RMSE by about 1 kg m^-3 in four of the five regions and was slightly worse in the fifth. We see the physics-guided loss as a first step towards physically consistent output, and not as the main source of the emulator's skill.
3. Reformulation of the emulator architecture
The above point is also related to a change we have made since submission. As noted in Prasad et al. (2024), the physics-guided loss is a soft constraint, so there is no guarantee that the network satisfies the compaction equation at inference. We have now reformulated the emulator so that the LSTM predicts the daily change in snow water equivalent due to precipitation, the daily mass losses, and the compaction rate. The snow mass budget and the compaction equation are then applied to these predictions to compute snow density and snow depth. Density, depth, and snow water equivalent are therefore physically consistent through the model architecture, at inference as well as in training. We have also restructured the inference code, which makes the emulator significantly faster than SnowModel-LG.
We will address these points, together with the referee's remaining comments, in our full response and in the revised manuscript or a resubmission.
References
Bengio, S., Vinyals, O., Jaitly, N., and Shazeer, N.: Scheduled sampling for sequence prediction with recurrent neural networks, Advances in Neural Information Processing Systems, 28, 2015.
Hooker, G., Mentch, L., and Zhou, S.: Unrestricted permutation forces extrapolation: variable importance requires at least one more model, or there is no free variable importance, Statistics and Computing, 31, 82, https://doi.org/10.1007/s11222-021-10057-z, 2021.
Hooker, S., Erhan, D., Kindermans, P.-J., and Kim, B.: A benchmark for interpretability methods in deep neural networks, in: Proceedings of the 33rd International Conference on Neural Information Processing Systems, Curran Associates Inc., Red Hook, NY, USA, article 873, 12 pp., 2019.
Liston, G. E. and Elder, K.: A distributed snow-evolution modeling system (SnowModel), Journal of Hydrometeorology, 7, 1259-1276, https://doi.org/10.1175/JHM548.1, 2006.
Liston, G. E. and Hall, D. K.: An energy-balance model of lake-ice evolution, Journal of Glaciology, 41, 373-382, https://doi.org/10.3189/S0022143000016245, 1995.
Molnar, C., König, G., Herbinger, J., Freiesleben, T., Dandl, S., Scholbeck, C. A., Casalicchio, G., Grosse-Wentrup, M., and Bischl, B.: General pitfalls of model-agnostic interpretation methods for machine learning models, in: xxAI - Beyond Explainable AI, Lecture Notes in Computer Science, Springer International Publishing, Cham, 39-68, https://doi.org/10.1007/978-3-031-04083-2_4, 2022.
Prasad, A., Merkouriadi, I., and Nummelin, A.: Modeling snow on sea ice using physics-guided machine learning, Environmental Data Science, 3, e28, https://doi.org/10.1017/eds.2024.40, 2024.
Citation: https://doi.org/10.5194/egusphere-2026-4727-AC1
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AC1: 'Reply on RC1', Ayush Prasad, 05 Oct 2026
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- 1
Review for “Emulating high-resolution distributions of snow depth and density on Arctic sea ice”
This manuscript presents a physics-guided Long Short-Term Memory (LSTM) emulator developed to simulate pan-Arctic snow density and depth with the inclusion of wind-driven snow redistribution over sea ice topography and a Lagrangian sea ice motion framework. The emulator is trained on data from SnowModel-3D run over 5 localized Arctic domains, with sea ice topography information derived from CryoSat-2. The manuscript builds on previous work (Prasad et al., 2024), extending the emulator to a pan-Arctic domain. The emulator performs well in validation against observational data, though with limited improvement compared to the performance of SnowModel-LG (which does not account for sea ice topography). However, the emulator is more computationally expensive than SnowModel-LG, although less so than running SnowModel-3D (which does account for sea ice topography) over the entire Arctic domain.
The description of the LSTM emulator in the manuscript is clear, and the figures and diagrams are of good quality. However, although the emulator demonstrates some utility in its production of snow-on-sea-ice depth and density, in the manuscript’s current state, I am not convinced of the utility of the emulator in comparison with existing products such as SnowModel-LG. I have major concerns about some of the methods and scientific results of the manuscript, and I do not think the analysis in the manuscript in its current state is sufficient to claim that the emulator shows improvement in its representation of snow on sea ice relative to SnowModel-LG due to the inclusion of sea ice topography. There is also some redundancy with previously-published work. As such, unfortunately, my suggestion is rejection of the manuscript in its present state, though I will provide below some comments and suggestions which I hope will be constructive. I regret that I cannot be more positive in my assessment, since the idea of using an LSTM emulator in this way is conceptually interesting, but the results are not sufficiently motivated in my view, and I think significant additional analysis and revision would be necessary for the manuscript to be of sufficient significance for consideration for publication in The Cryosphere.
General comments:
As written, in the manuscript it is difficult to see where the novel contributions are of this study, and it is difficult to identify a clear problem or hypothesis that is being tested. To my understanding, it appears that the primary novel contribution is the introduction of the satellite-derived sea ice topography and the ice motion, which extends the emulator from producing regional snow depth/density to gridded pan-Arctic estimates of snow depth and density. The manuscript appears to claim that the inclusion of this topography is responsible for improved representation of snow on Arctic sea ice from the emulator, but this is not sufficiently substantiated by the analysis in the manuscript in my view. Parts of the manuscript read to me as if the physics-guided loss function is a novel contribution when that is not the case; Prasad et al. (2024) is insufficiently cited in e.g. Section 2.5. Although the authors state at the beginning of the manuscript that it is based on previous work, I think more care needs to be taken throughout the manuscript to indicate which parts are reproduced from previous work.
If one of the key points the authors wish to make in the manuscript is that the inclusion of sea ice topography improves the representation of snow on sea ice, then what is missing to me is a baseline result from the emulator when sea ice topography is not considered. In the manuscript’s current state, the emulator is shown to perform somewhat better by some validation metrics than SnowModel-LG, but SnowModel-LG and the emulator are distinct enough that I am not convinced that the inclusion of surface topography is the primary driver of differences between SnowModel-LG and the emulator. To better demonstrate this, I would suggest, for example, running the emulator in the same configuration as is presented in the manuscript, but without the CS2 sea ice topography input (or e.g. using synthetic ridge topography), and comparing how the resulting emulator output performs with the performance of the emulator when the satellite sea ice topography is included, with validation comparisons against SnowModel-LG and against observations as is done in this study. Particularly also given the computational expense of the emulator relative to SnowModel-LG, I think more needs to be done to demonstrate that there are improvements specifically from surface topography to motivate the inclusion of this topography. As-is, the evidence presented is unfortunately not sufficient for me to consider the use of the emulator over SnowModel-LG to be justifiable.
There is also some background in the article introduction missing to this point. The authors assert that the emulator can represent additional snow processes not captured by SnowModel-LG via being trained on SnowModel-3D, and by including sea ice topography data as an input dataset. I agree that at fine spatial scales (~1 m) such as those represented by SnowModel-3D, surface topography certainly has an impact for snow model fidelity. However, I am not aware of literature that asserts accounting for surface topography as being a necessity on the ~25-km spatial scale modelled by the emulator, and the authors do not appear to cite any literature to this point, either. I think the manuscript would benefit from further scientific background on this point to better motivate the need for including sea ice topography on these scales, since this motivation is currently missing.
Regarding the loss function, one concern I have is that the loss function equations in Section 2.5 differ from Prasad et al (2024) by being multiplied by a factor that was a denominator in the previous work; the factor of exp(-A_2rho_s). The expression in the current manuscript appears to be correct (it is consistent with the model formulation per Liston and Hall, 1995), but given that the use of the physics-based LSTM approach is motivated based on Prasad et al (2024), this raises some questions for me about the necessity of this physical constraint. How do you justify the necessity of a physics-guided loss function if an emulator with an unphysical loss function still performed successfully in Prasad et al. (2024)? (Assuming that there was not simply a typo in the equations in Prasad et al. (2024)). To be clear, I do not question that the emulator performs successfully in its current configuration, but this raises the question for me about how much of its good performance can be attributed to the physical constraint in the loss function. I think the authors would need to clarify if there was a typo in the previous manuscript (i.e. the physical constraint was implemented correctly in the emulator code), or if the constraint was incorrectly implemented in previous work, they would need to provide at the very least some justification for the necessity of a physical constraint that does not rely on the conclusions of Prasad et al. (2024).
The comparison against SnowModel-LG is also under-discussed in my view. The authors state that the topography-driven effects “should be responsible for some of the differences” between the model and the emulator, but this discussion would benefit from expansion. A lot of discussion of physical processes in this manuscript is lacking in detail. What sorts of differences would be expected when accounting for surface topography? Are the results from the SnowModel-LG comparison consistent with these expected differences? Identifying these differences would better motivate the emulator approach. Figure 6 shows a nice comparison of mean depth and density differences, but is hardly discussed. E.g. snow depth east of Greenland differs considerably between the emulator and SnowModel-LG, why is that? Also, when looking at Figure 6, SnowModel-LG appears to represent more fine-scale variability than the emulator does; I would presume more fine-scale features would emerge when accounting for ice topography (e.g. higher accumulation in areas with more ridging), so I would appreciate if the authors could explain this point. (I am aware that machine-learning approaches such as LSTM can have a tendency to produce “smooth” outputs, so I cannot help but wonder if that is not also a contributing factor.) Overall, I think much more expansion is needed on the physical implications of the topographical input and how the output data reflects this input. Again, as mentioned above, testing a configuration that excludes this topographical input could be a way to bolster this discussion.
The same point applies to the comparisons with airborne and in situ data; the authors assert that improvement relative to SnowModel-LG is due to the small-scale topography (line 333), but again, the evidence is insufficient to me since topography is not the only factor contributing to differences between the model and the emulator. There is less of a spread in the scatter plots in Fig 7 for the emulator, but is this a consequence of improved processes or simply a byproduct of the emulator including less fine-scale variability, which would tend to exclude more extreme values?
Finally, the emulator code and output is currently not available. I think at the very least output from the emulator should be provided prior to publication, though having the emulator code itself available (particularly the implementation of the loss function) would also be helpful for verification and reproducibility.
Line/figure-specific comments:
170 / Eq 4: As mentioned above, this section/these equations are closely reproduced from Prasad et al. (2024), please cite it here and rephrase to make clear that the physics-based LSTM constraint is based on previous work (e.g. “In Prasad et al. (2024), we developed a loss function [...] model, which we summarize here.”) and identify specific points in the text where novel changes are made relative to previous work. (Some clarification of the difference in multiplying/dividing the exp(-A_2rho_s) factor would be helpful as well, as discussed above.)
205: How are the initial values of snow depth and density determined for this initial state file? (E.g. is initial snow depth zero, or is it pre-initialized based on estimated values? Or if it is initialized based on the previous year final-day value, what are the implications for errors, as per line 217?)
266-267: Why the difference in grid cell assignment here? What is done if parcels overlap within a grid cell (does that happen?)
322: Can you comment on whether the spatial differences in snow depth are consistent with what you would expect given the inclusion of surface topography in the emulated product? Or are other factors more likely to drive differences in some regions? Can you explain why the depth difference is so large east of Greenland?
345: Do you have any sense of why there is such a large decrease in snow density in July in Figure 9 for SnowModel-LG and the emulator (e.g. can this be attributed to an observational sampling bias?)
387-388: As mentioned above, could you clarify why accounting for topography would yield a smoother output; could this not just be a machine-learning artefact?
Figure 7: Why are there more points in the SnowModel-LG scatter plots than the emulator scatter plots, given that they are on similar-resolution spatial grids? Are the projections different? (The caption mentions EASE grids.)
Figure 8: I suggest adding a panel showing the emulator Oct 2019-Sep 2020 mean to further illustrate the differences between SnowModel-LG and the emulator output.
Figure 9: The text at the bottom (n=2, etc) is too small to be easily legible; I suggest increasing the text size.
424: Re: including emulators like this in altimetry-based thickness retrievals: this seems somewhat circular since some altimetry information is being used to generate the dataset. I suggest rephrasing/clarifying, or omitting this point.
References:
Liston, G. E., and Hall D. K. , 1995: An energy balance model of lake ice evolution. J. Glaciol., 41 , 373–382. https://doi.org/10.3189/S0022143000016245
Prasad A, Merkouriadi I, Nummelin A. Modeling snow on sea ice using physics-guided machine learning. Environmental Data Science. 2024;3:e28. doi:10.1017/eds.2024.40