Physics-aware machine learning for glacier ice thickness estimation: a case study for Svalbard

Steidl, Viola; Bamber, Jonathan Louis; Zhu, Xiao Xiang

doi:https://doi.org/10.5194/tc-19-645-2025

Articles | Volume 19, issue 2

https://doi.org/10.5194/tc-19-645-2025

© Author(s) 2025. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/tc-19-645-2025

© Author(s) 2025. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 19, issue 2

Research article

|

07 Feb 2025

Research article |

| 07 Feb 2025

Physics-aware machine learning for glacier ice thickness estimation: a case study for Svalbard

Viola Steidl, Jonathan Louis Bamber, and Xiao Xiang Zhu

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Final revised paper (published on 07 Feb 2025)
Preprint (discussion started on 14 Jun 2024)

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-1732', Guillaume Jouvet, 09 Jul 2024

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-1732/egusphere-2024-1732-RC1-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2024-1732-RC1
- AC1: 'Reply on RC1', Viola Steidl, 28 Aug 2024
  
  Please see the attached file.
  
  Citation: https://doi.org/10.5194/egusphere-2024-1732-AC1
RC2:
'Comment on egusphere-2024-1732', Anonymous Referee #2, 24 Jul 2024

Please see attached review

Citation: https://doi.org/10.5194/egusphere-2024-1732-RC2
- AC2: 'Reply on RC2', Viola Steidl, 28 Aug 2024
  
  Please see the attached file.
  
  Citation: https://doi.org/10.5194/egusphere-2024-1732-AC2

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

ED: Reconsider after major revisions (further review by editor and referees) (02 Sep 2024) by Ben Marzeion

AR by Viola Steidl on behalf of the Authors (30 Sep 2024) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (22 Oct 2024) by Ben Marzeion

RR by Anonymous Referee #3 (05 Nov 2024)

Suggestions for revision or reasons for rejection

## General comments

The manuscript "Physics-aware Machine Learning for Glacier Ice Thickness Estimation: A Case Study for Svalbard" presents an interesting approach for reconstructing spatially-continuous glacier ice thickness from sparse in situ measurements with a neural network. The sparsity of the ice thickness measurements is mitigated by incorporating physical knowledge into the objective function, which acts as a regularizer on the neural network predictions. The authors demonstrate that the their method produces thickness estimates that are consistent with previous studies using physical models of ice flow.

After the first round of revisions, I believe that the authors have done a good job in addressing much of the concerns regarding the methodological details and factors that impact the accuracy of the thickness estimates. I particularly liked the inclusion of the SHAP analysis and the systematic analysis of the physics-aware loss functions.

However, based on the earlier reviews, it appears that a key remaining issue holding back this work is the lack of ground truth and the limited validation via comparison with previous ice thickness estimates. I would agree with the other reviewers that a synthetic experiment would be very beneficial in instilling confidence about the thickness estimates, especially since it appears that the estimates are systematically lower than the estimates from the previous approaches (Figure 5). While the authors state in their response that such an experiment would be a substantial effort, I believe that even simplified synthetics would go a long way towards clarifying the performance of the neural network.

Concretely, I would suggest that the authors generate 1D synthetics with a Blatter-Pattyn model with some random bed topography profiles. This can be done for a relatively small number of synthetic glaciers (~20) such that the number of velocity points is on the order of a few tens of thousands. A small fraction of ice thickness values can then be used as the auxiliary data. For the neural network features, since these are 1D, the features can be limited to the x-coordinate, slope, vx, and beta (and perhaps elevation if the modeled mass balance is elevation-dependent). These features (as well as the auxiliary mass balance) can be subjected to various amounts of noise. After the neural network is trained, then the thickness can be evaluated vs. the true thickness. Crucially, the thickness can also be evaluated against the reconstructed thickness using the approach of a model like Millan et al.'s. It may turn out that the previous approaches overestimate the thickness, which would be important to know. Overall, while I understand that such an experiment is not trivial by any means, it would really help increase the impact of this work and potentially highlight its technical advantages relative to previous approaches.

Other than the synthetics suggestion, I see no other major issues with this work. I've listed more specific comments below.

## Additional specific comments

- The Fourier dimension and scale adds two more hyperparameters to the neural network design. How do these affect the performance of the reconstructions, and how are they chosen in this work?

- In Reviewer 1's comments, it was suggested to move the surface velocity data to the network outputs. Then, the data can be used as additional "auxiliary data" for constraining the predictions of the surface velocity. I think this is a good idea as it would provide a way to smooth both the surface and depth-averaged velocities in a consistent manner. This approach could also be used to predict mass balance, constrained by more accurate in situ measurements. Thus, I think the authors should at least mention this as possible follow-up work.

- I'm still confused why there should be three different beta values for the different velocity components. It seems to me that beta should be calculated from the velocity magnitudes, and then the same beta is used for all three components. This would keep the partitioning consistent.

- Line 101: It's probably better to say "calculate the terms in the PDE" since there may be non-derivatives or products of derivatives depending on the PDE

- Figure 1: In the caption, please add a little bit more description on the inputs, outputs, and losses here. What do the colors in the loss boxes correspond to?

- Line 130: Why not just predict log(H)

- Line 153: I believe the use of the term "boundary condition" was changed to something like "internal condition" when referring to the ice thickness data, so this needs to be made consistent.

- Line 167: Does the varying grid resolution have an effect on generalization?

- Line 175: Please state the size of the filter used for Gaussian smoothing (perhaps in units of fraction of average ice thickness)

- Line 176: It's probably more accurate to say that $\beta$ is introduced to incorporate the effect of basal sliding on the measured surface velocity (instead of estimate)

- Line 189: Also here, please state the size of the filter

- Line 234: Since the in-sample and LOGO RMSD scores are significantly different, it would be useful to show a plot (probably in the Appendix) of the training performance (loss vs. training epoch) for the train and validation data. This way, we can get a better sense on the amount of over-fitting.

- Line 289: It seems like it would be good to show a comparison of velocity fields between ~2010 and 2017-18 (if such data are available).

- Line 298: Even though the loss weights are held fixed, you should briefly discuss how these weights affect the final thickness estimates (or perhaps show an L-curve for the most important weight).

- Figure 6: I would like to see one or two additional sentences in the caption that summarize the challenges or highlight the challenges that are most consequential for the proposed method.

- Line 355: This could be straightforwardly implemented by using thickness uncertainties as weights to the loss function

- Line 375-380: This paragraph is a bit too general and doesn't really add to the rest of the manuscript. Physics-aware machine learning has been around for quite some time now and has been applied to a wide variety of geophysical problems.

- Line 415: What about the batch size?

- Appendix D title: Please state up front what metric you are measuring importance against (validation accuracy?)

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RR by Anonymous Referee #2 (06 Nov 2024)

RR by Anonymous Referee #4 (08 Nov 2024)

Suggestions for revision or reasons for rejection

The manuscript presents the development of a new physics-informed neural network (PINN) method to infer glacial ice thickness, based on a case study of Svalbard. While the description of the method in the initial manuscript was somewhat rough, the revised manuscript shows significant improvement. The other reviewers have addressed most of my questions, and the authors provided detailed responses. Below, I list some additional questions specific to the PINN method. I recommend the publication of this paper if the authors can address these points.

Questions:

1. The authors employ Fourier feature encoding within the network, which requires setting a hyperparameter B. From Tancik et al. (2020), we know that an incorrect choice of B can lead to overfitting in the network’s predictions. Could the authors provide more details on how they determined the optimal value for B?

2. On one hand, the authors apply Fourier feature encoding to capture high-frequency features in the output; on the other hand, they include a smoothness loss term in the loss function to regulate high derivatives. However, high-frequency functions typically exhibit high local derivatives, which appears somewhat contradictory to the use of a smoothness loss. Could the authors provide an explanation to clarify and justify these settings?

3. In Appendix B (line 414), the authors state that they set the loss weights λiλi to ensure all loss terms in the loss function are of the same order of magnitude. However, if the smoothness loss term is kept at the same magnitude as others, could this result in an over-smoothed thickness prediction?

4. Based on Figure 2, it appears that surface velocity and β-values are used solely as input features. However, both β and surface velocity also appear in the velocity loss term L_{vel} (Equation 6). It may be helpful to update Figure 2 to indicate that β-value and surface velocity are part of the data required for the loss function.

5. As the authors mention, inferring glacial ice thickness is a highly under-constrained problem. Additionally, the sparsity of the measured thickness data complicates model validation. This may be beyond the scope of the current study, but it would be interesting if the authors could generate synthetic data to validate the robustness of the proposed PINN model.

6. Did the authors conduct experiments with different weight initializations or network structures to assess whether the PINN training converges to a unique thickness inference? If so, could they provide the standard deviation of these tests?

7. Given that the authors used various input features in the training, I suggest adding more detail on the input layers in Appendix B to help readers better understand the network structure used.

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ED: Publish subject to minor revisions (review by editor) (08 Nov 2024) by Ben Marzeion

AR by Viola Steidl on behalf of the Authors (18 Nov 2024) Author's response Author's tracked changes Manuscript

ED: Publish as is (11 Dec 2024) by Ben Marzeion

AR by Viola Steidl on behalf of the Authors (13 Dec 2024) Manuscript

Short summary

Glacier ice thickness is difficult to measure directly but is essential for glacier evolution modelling. In this work, we employ a novel approach combining physical knowledge and data-driven machine learning to estimate the ice thickness of multiple glaciers in Spitsbergen, Barentsøya, and Edgeøya in Svalbard. We identify challenges for the physics-aware machine learning model and opportunities for improving the accuracy and physical consistency that would also apply to other geophysical tasks.