Numerical stabilization methods for level-set-based ice front migration

Cheng, Gong; Morlighem, Mathieu; Gudmundsson, G. Hilmar

doi:https://doi.org/10.5194/gmd-17-6227-2024

Articles | Volume 17, issue 16

https://doi.org/10.5194/gmd-17-6227-2024

© Author(s) 2024. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/gmd-17-6227-2024

© Author(s) 2024. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 17, issue 16

Development and technical paper

|

22 Aug 2024

Development and technical paper |

| 22 Aug 2024

Numerical stabilization methods for level-set-based ice front migration

Gong Cheng, Mathieu Morlighem, and G. Hilmar Gudmundsson

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Final revised paper (published on 22 Aug 2024)
Preprint (discussion started on 10 Oct 2023)

Interactive discussion

Status: closed

CC1:
'Comment on gmd-2023-194', Michael Wolovick, 13 Nov 2023

Thanks for this interesting study! I have a question that I might want your input on.
During my own studies using ISSM to simulate the Filchner-Ronne sector of Antarctica, I found that small values of the reinitialization interval tended to produce a bias towards front retreat. In your analysis of error magnitude (Figures 4 and 5) you show the absolute misfit area, but not the sign of the error. In Figure 2, the maps of ice front error seem to show a bias towards retreat with n_R=1. Am I interpreting that figure correctly? While you are analyzing periodic forcing in this paper, a systematic bias towards either advance or retreat is obviously very important for projecting the response of the real ice sheet to a changing climate. Would you say that your results also support the conclusion that very small values of n_R bias the model towards retreat? Did you see any systematic trend in the sign of the error in addition to the differences in overall error magnitude presented in your results?
Thanks in advance for your answer,
Mike Wolovick

Citation: https://doi.org/10.5194/gmd-2023-194-CC1
- AC1:
  'Reply on CC1', Gong Cheng, 13 Nov 2023
  
  Hi Mike,
  Thank you for your insightful question. Following the convention in ISSM, the sign distance function indicates negativity for the ice-covered region. Thus, the negative segment of d(\phi1, \phi2) in Figure 2 implies that \phi1 (numerical solution) is ostensibly more 'advanced' than \phi2 (true solution). This discrepancy arises from the strong diffusion introduced by n_R=1 to the 0-level set contour. Depending on the ice front's shape (convex or concave) and the velocity profile, this effect can introduce a bias towards either advance or retreat.
  However, it's crucial to emphasize that one of the primary focus of this paper is to underscore the inadequacy of choosing n_R=1. This choice introduces a significantly higher level of numerical errors compared to other sources. Consequently, we strongly advise to avoid using n_R=1 in any case.
  The bias associated with other choices of n_R is generally contained within the scale of the mesh, rendering it negligible when compared to errors originating from other sources, such as model inaccuracies and data discrepancies.
  Cheng Gong
  
  Citation: https://doi.org/10.5194/gmd-2023-194-AC1
  - CC2: 'Reply on AC1', Michael Wolovick, 14 Nov 2023
    
    Thanks for your response!
    I see I misinterpreted the color scale in Figure 2. I had already concluded that using n_R=1 was a bad idea, so there's no need to worry about that issue. But it's interesting to know that the bias towards retreat that I saw in my domain was a consequence of the particular details of front geometry and velocity profile, not anything fundamental to the algorithm.
    In the case of spatially refined meshes, what do you expect to be the effect of gradients in mesh size near the front? If the numerical errors depend on mesh size, how do you think that will affect front geometry and migration rate, if the front is migrating through an area where mesh size is changing?
    
    Citation: https://doi.org/10.5194/gmd-2023-194-CC2
    
    AC2: 'Reply on CC2', Gong Cheng, 14 Nov 2023
    
    Thank you for bringing attention to this aspect. We systematically examined diverse mesh resolutions, and the outcomes align closely with the principles of finite elements and corresponding stabilization techniques. Consequently, we opted to present results solely at a resolution of 100m, the typical finest mesh resolution in ice modeling.
    In the context of spatially refined meshes, we do expect the mesh size will influence the level-set function. However, our insight is that this influence, while present, is overshadowed by the more dominant effect from the velocity solution. The ice velocity has a direct influence on the frontal migration rate, therefore, the front geometry as well.
    The primary message of our paper centers on the pivotal role of thoughtful choices in stabilization and reinitialization. Our findings affirm that, with careful selection of these parameters, we will not introduce higher numerical errors beyond those inherent to the system.
    
    Citation: https://doi.org/10.5194/gmd-2023-194-AC2
RC1:
'Comment on gmd-2023-194', Matt Trevers, 17 Nov 2023

Thank you for the opportunity to review this manuscript. Please find my full referee comment in the attached document.

Citation: https://doi.org/10.5194/gmd-2023-194-RC1
- AC3: 'Reply on RC1', Gong Cheng, 28 Nov 2023
  
  The comment was uploaded in the form of a supplement: https://gmd.copernicus.org/preprints/gmd-2023-194/gmd-2023-194-AC3-supplement.pdf
  
  Citation: https://doi.org/10.5194/gmd-2023-194-AC3
RC2:
'Comment on gmd-2023-194', Anonymous Referee #2, 20 Nov 2023
The paper explores different stabilization methods for level-set equations and the impact of reinitialization on the accuracy of the solution.
The methods are demonstrated on an idealized geometry (union of a rectangle and a semidisk) to mimic a fjord with a semi-circular ice front. The front velocity is prescribed. The authors nicely present how the different stabilization approaches and the frequency of the reinitialization affect the accuracy of the position of the level set.
The paper addresses a very important topic in ice-sheet modeling and it is easy to read. However, I have major concerns which prevent me from recommending the paper for publication in its present form.
My main concern is that despite the title and the presentation of the work, there is little about ice front migration in this manuscript. In fact, the geometry and the prescribed velocity are too simplified to be representative of an ice front migration problem. In addition to the very simplified description of the fjord, the prescribed front velocity is aligned with the fjord axis, which is at odds with the fact that the calving component of the front velocity is typically assumed to be orthogonal to the ice front.

The standalone advection level-set equations have been extensively studied in the literature, and this paper adds little to what is already available. On the contrary, I would have found the paper very valuable if the authors targeted a more realistic ice sheet problem as well, where the level-set velocity was computed using ice flow equations (e.g., the Shallow shelf Approximation) for the ice velocity and at least one of the calving laws typically used in the literature.

Another concern I have is that the authors do not explain what reinitialization method they are using, despite the fact that the effect of reinitialization is one of the main topics of the paper. When they introduce the reinitialization they reference two papers they co-authored but I could not find any detail there either. Further, plots in figure 2 show a loss of symmetry, which is likely due to the reinitialization procedure, but the authors do not offer any explanation of why that is happening. I worry that there might be an issue with the reinitialization procedure which would affect the results and possibly the paper conclusions.

Finally, the forward and backward diffusion stabilization considered in this paper aims at keeping the level-set function close to the distance function, so that no reinitialization is needed. This is qualitatively confirmed by their results. However, the authors miss this point in the discussion of the results. Also the authors do not provide any reference for this stabilization method.
Citation: https://doi.org/10.5194/gmd-2023-194-RC2
- AC4: 'Reply on RC2', Gong Cheng, 28 Nov 2023
  
  The comment was uploaded in the form of a supplement: https://gmd.copernicus.org/preprints/gmd-2023-194/gmd-2023-194-AC4-supplement.pdf
  
  Citation: https://doi.org/10.5194/gmd-2023-194-AC4

Peer review completion

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

AR by Gong Cheng on behalf of the Authors (03 Jan 2024) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (08 Jan 2024) by Ludovic Räss

RR by Anonymous Referee #2 (23 Jan 2024)

Suggestions for revision or reasons for rejection

The revised version of the paper improves on the original version. In particular the authors added numerical experiments to show the impact of time step and mesh resolution on the solution. However, several of my concerns were not adequately addressed.

The main concern I had about the study being too simplified to represent an ice front migration problem has not been addressed. To be clear, I strongly support studies that consider simplified settings to analyze stability and accuracy properties of a particular method, and I support efforts like CalvingMIP mentioned by the authors. However, these studies need to be representative of the targeted applications and I don’t think it is the case of the work in this manuscript. In fact, stabilizations that work well with the simplified prescribed velocities used in this work, might not work well when the ice front is driven by realistic ice velocity and calving. It is fine to focus on simplified geometries, but the velocity of the front should respect the physical problem.

I’ve pointed out that one significant aspect that is absent in their choices of velocity profiles, is that the ice front velocity typically has a term (the calving term) that is normal to the ice front, whereas the velocity profile proposed does not depend on the front evolution. I appreciate that the paper is not about calving, and in fact I was not asking to study the impact of different calving laws to the ice front evolution. What I asked was to consider an additional term to the velocity that is normal to the ice front and that would mimic the action of calving. I understand that in this case it would be extremely hard to find an analytic solution, but that’s the case of most benchmarks in ice sheet modeling and an analytic solution is not necessarily needed to verify the accuracy and stability of the methods. To my knowledge, the first paper modeling ice front migration using level-set methods (Bondzio et al, Cryosphere, 2016) used the ISSM model, so the authors already have a code-base to perform a more impactful study on ice front migration.

I asked the author to describe the reinitialization used, but the authors only wrote two sentences that are not detailed enough. My understanding is that the level set function is reinitialized by computing the signed distance from the zero level-set. It is not clear to me how they compute it and if they use any strategy/approximation to make the computation more tractable.There are in fact several approaches in the literature both for accelerating the reinitialization and for using level set evolution schemes that reduce or eliminate the need of reinitialization. These approaches are not even mentioned.
In this regard, I’ve brought to the attention of the authors that the manuscript does not point out that one of the goals of the penalization in the forward-backward diffusion stabilization is to avoid the need of performing the reinitialization, which I think is relevant for the discussion of the paper results. The authors pointed me to the documentation of the Úa code and didn’t add the discussion to the paper.

Some of the paper results have been updated by switching to an unstructured mesh. In particular, the results in Figure 2 do not present the non-symmetric artifacts observed in the original paper. While it is reassuring that with an unstructured mesh you do not observe asymmetries, it would be good to show that the asymmetry vanishes as the (structured) mesh is refined. In fact, it was a bit concerning to see relatively large asymmetries in a relatively fine mesh. It might be good also to show the unstructured mesh used for the revised results in the paper.

Regarding the level-set literature, I’m providing some references below. As the authors point out, a lot of the level-set papers in the literature focus on particular applications, but I would argue that most of the methods presented to target specific applications would behave well on simplified problems with prescribed front velocity like the ones targeted in this paper. The literature on level-sets methods is vast. Below I list some references that might be useful and worth citing. The first two include stabilization methods presented in this work, the third is about reinitiliazation methods, the fourth is a book that also deals with problems where the front velocity is prescribed (as in this manuscript), and the last is a recent review that might contain useful references. I’m not an expert in level-set methods and I would expect that it is possible to find references that are more relevant to this work.

1. Mamadou Kabirou Touré, Azzeddine Soulaïmani, Stabilized finite element methods for solving the level set equation without reinitialization, Computers & Mathematics with Applications, Volume 71, Issue 8, 2016
2. C. Li, C. Xu, C. Gui and M. D. Fox, "Distance Regularized Level Set Evolution and Its Application to Image Segmentation," in IEEE Transactions on Image Processing, vol. 19, no. 12, 2010
3. Félix Henri, Mathieu Coquerelle, Pierre Lubin, Geometrical level set reinitialization using closest points method and kink detection for thin filaments, topology changes and two-phase flows, Journal of Computational Physics, Volume 448, 2022
4. Stanley Osher, Ronald Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, 2003
5. Frederic Gibou, Ronald Fedkiw, Stanley Osher, A review of level-set methods and some recent applications, Journal of Computational Physics, Volume 353, 2018

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ED: Reconsider after major revisions (13 Feb 2024) by Ludovic Räss

Dear authors,

Thank you for submitting a revised version of your work. Although the revised version addressed some of the minor comments from the first round of revision, I still believe that some major concerns are still not addressed as pointed out by the second assessment of reviewer 2. The suggestions from reviewer 1 are mostly addressed, but it feels that overall not much effort was placed into providing an in-depth improvement of the study.

The revised paper improves upon the original version by adding numerical experiments showcasing the impact of time step and mesh resolution on the solution. However, important un-addressed concerns are left:
- The main issue regarding the oversimplification of the ice front migration problem remains unaddressed.
- The study's scenarios and stabilisations may not accurately represent realistic ice velocity and calving dynamics.
- The manuscript's velocity profiles lack a term normal to the ice front (calving term) which may represent a crucial component.

Further, some valuable recommendations from reviewer 2 remained partly ignored:
- Incorporating a term mimicking the action of calving.
- The response to the query about reinitialisation of the level set function is still insufficient and lacks detail.
- The current paper draft still overlooks discussing how the penalisation in the forward-backward diffusion stabilization could negate the need for reinitialisation.

The revised manuscript addresses an important point raised earlier in the revision, namely the asymmetry in the results obtained when using a regular grid. It is interesting to see that using an unstructured mesh improves the situation making the asymmetry to vanish. However, this does not address the underlying question on why there is an asymmetry for the simplest considered mesh, the regular one. It would be important to show that either there is no asymmetry in any kind of grid considered, or that some minor asymmetry vanishes upon grid refinement, and discuss that further in order to make sure one does not overlook a bug in the implementation.

Finally, it seems that relevant references are not handled in an optimal way, and it should be the role of the authors and not the reviewers to perform the literature search.

Based on the previous important points and the concerns raised by the reviewers, the manuscript draft would need some major changes and updates in order to be publishable.

As I see the potential of this study, I could reconsider the manuscript after major revisions. Hereafter, some recommendations on what should be addressed:
1. Address Oversimplification: Revisit the model to ensure it better represents realistic scenarios in ice front migration, testing maybe for another geometry and addressing the oversimplifies velocity field.
2. Add a calving term to the velocity profiles, as it is crucial for the realism of the model. Or at least make some systematic investigation on the front dynamic for similar settings.
3. Provide a more detailed explanation of the level set function reinitialisation process, possibly exploring different approaches mentioned in literature.
4. Include a discussion on how the penalization in the stabilization technique might reduce or eliminate the need for reinitialisation.
5. Perform additional experiments to show how asymmetry in results reduces with the refinement of the structured mesh, or figure out why there is an asymmetry.
6. Review and integrate the provided level-set literature into the study to enhance the methodological approach and discussion.

Best regards,
Ludovic Räss

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AR by Gong Cheng on behalf of the Authors (23 Mar 2024) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (07 Apr 2024) by Ludovic Räss

RR by Anonymous Referee #2 (22 Apr 2024)

Suggestions for revision or reasons for rejection

In this current iteration, the authors only marginally improved the paper but avoided the major concerns I raised, therefore my recommendation is the same.

I’m glad that the authors improved the description of the re-initialization method used and added relevant references. The authors also made some minimal improvement to the stabilization section. In my understanding, the FAB diffusion method is a variation on the energy penalization method (C. Li, et al. Level set evolution without re-initialization: A new variational formulation
2005 IEEE), so in addition to the Gudmundsson paper I think it would be fair to cite also the Li paper.

The authors also added two numerical studies in the appendix. In Appendix D, the authors show the differences in the level-set evolution when using different grids, and in Appendix E they consider a sinusoidal forcing term to mimic seasonal variations.

While it can be useful for the reader to show how the front evolution changes as a function of the mesh used, this experiment doesn’t address my concern. I still would like to see whether the asymmetry vanishes on the triangular (diagonally aligned) mesh, as you refine the mesh.

I don’t find the experiment in Appendix E useful and it does not address the request of having more realistic (or less simplified) problems. The prescribed front velocity is directed along the x axis, which is an oversimplification. Adding yet another modulation in time doesn’t make it more realistic.

For the second time, the authors refused to even add a calving-like term. Let alone considering velocities that come from an ice sheet model. Their defense is that the front velocity “implicitly incorporates the effects of calving or calving rate”. However, the main point, that I already made in the previous reviews, is that the calving velocity is typically orthogonal to the front, which is at odds with the fact that their prescribed front velocity is aligned with x (unless we consider some nonphysical ice velocity that perfectly counteracts the calving effect).

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ED: Publish subject to minor revisions (review by editor) (19 Jun 2024) by Ludovic Räss

AR by Gong Cheng on behalf of the Authors (28 Jun 2024) Author's response Author's tracked changes Manuscript

ED: Publish subject to minor revisions (review by editor) (03 Jul 2024) by Ludovic Räss

AR by Gong Cheng on behalf of the Authors (04 Jul 2024) Author's response Author's tracked changes Manuscript

ED: Publish as is (09 Jul 2024) by Ludovic Räss

AR by Gong Cheng on behalf of the Authors (10 Jul 2024) Manuscript

Short summary

We conducted a comprehensive analysis of the stabilization and reinitialization techniques currently employed in ISSM and Úa for solving level-set equations, specifically those related to the dynamic representation of moving ice fronts within numerical ice sheet models. Our results demonstrate that the streamline upwind Petrov–Galerkin (SUPG) method outperforms the other approaches. We found that excessively frequent reinitialization can lead to exceptionally high errors in simulations.