the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
A framework for timedependent Ice Sheet Uncertainty Quantification, applied to three West Antarctic ice streams
Daniel Goldberg
James R. Maddison
Joe Todd
Abstract. Ice sheet models are the main tool to generate forecasts of ice sheet mass loss; a significant contributor to sealevel rise, thus knowing the likelihood of such projections is of critical societal importance. However, to capture the complete range of possible projections of mass loss, ice sheet models need efficient methods to quantify the forecast uncertainty. Uncertainties originate from the model structure, from the climate and ocean forcing used to run the model and from model calibration. Here we quantify the latter, applying an error propagation framework to a realistic setting in West Antarctica. As in many other icesheet modelling studies we use a control method to calibrate gridscale flow parameters (parameters describing the basal drag and ice stiffness) with remotelysensed observations. Yet our framework augments the control method with a Hessianbased Bayesian approach that estimates the posterior covariance of the inverted parameters. This enables us to quantify the impact of the calibration uncertainty on forecasts of sealevel rise contribution or volume above flotation (VAF), due to the choice of different regularisation strengths (prior strengths), sliding laws and velocity inputs. We find that by choosing different satellite ice velocity products our model leads to different estimates of VAF after 40 years. We use this difference in model output to quantify the variance that projections of VAF are expected to have after 40 years and identify prior strengths that can reproduce that variability. We demonstrate that if we use prior strengths suggested by Lcurve analysis, as is typically done in icesheet calibration studies, our uncertainty quantification is not able to reproduce that same variability. The regularisation suggested by the Lcurves is too strong and thus propagating the observational error through to VAF uncertainties under this choice of prior leads to errors that are smaller than those suggested by our 2member “sample” of observed velocity fields. Additionally, our experiments suggest that large amounts of data points may be redundant, with implications for the error propagation of VAF.
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Beatriz Recinos et al.
Status: closed

RC1: 'Comment on tc202327', Anonymous Referee #1, 13 Apr 2023
This paper is the application to a real case study of a framework to quantify the uncertainty in ice sheetmodel projections arising from model calibration.
This paper address a very relevant question and the results are well presented and convincing and I have mainly minor comments.
My main comment is that the conclusion indicates that the regularisation weights suggested by the LCurve analysis seem to lead to priors that are too confident, suppressing the propagation of the uncertainty from the velocity datasets used for the calibration. However, I found that the method for the LCurve is not very well described as there is 4 parameters to calibrate, and it is not not clear if they are chosen independently?, and there is a high level of userjudgement in the selection of these parameters; Comparing the values given in section 4.1 to those used in Table 1, it appears that the main differences are on the $\delta$ parameters for which the results are not shown. I am also wondering part of the issue cannot come from wrong priors as they are particluarly poorly constrained and here, the prior for the friction parameter $\alpha$ is 0, so that pure sliding everywhere? So maybe the conclusion could be revisited a little to not put too much attention on the LCurve?
Additional minor comments:
 Line 124: “constant surface mass balance”; Is it constant and uniform; or is there spatial variability?
 6 : QT here is defined as the VAF while is it use as the difference of VAF from t=0 in the manuscript. What is the meaning of the “+” symbol?
 Line 142: Hf=max(0,R(\rho_w/\rho_i))
 Line 196: if the prior is strong, $\gamma$ is “large” not “small”? (in agreement to line 342check for consistency everywhere)
 Sec 4.1 would be interesting to discuss the smoothing parameters in terms of variance and correlation length scales (Eq. 1314) as it appears that the parameters used here lead to a very small variance compared to the values used in Table 1.
 Line 389: $J^c$ should be $J^c_{miss}$? Check for consistency everywhere. I don’t understand why it does not change with the number of observations as according to Eq.8 it should depend on the number of observations?
 Line 477: “as the basal stress does not scale with effective stress in the interior”. I don’t understand the argument here.
 Lie 486: “is due to insensitivity of basal stress to $\alpha$ when the ice is near flotation”. The WeertmanBudd relation Eq. 1 is also insensitive to $/alpha$ near flotation as it depends on $N$; main difference is that Eq.3 tends to a Coulomb regime, independent of $\alpha$, for high velocity and low effective pressure. However using eq. 12 for N tends to restrict this domain to the close vicinity of the grounding line (Joughin et al., 2019)
References:
 Joughin, I., Smith, B.E., Schoof, C.G., 2019. Regularized Coulomb Friction Laws for Ice Sheet Sliding: Application to Pine Island Glacier, Antarctica. Geophys. Res. Lett.
Citation: https://doi.org/10.5194/tc202327RC1  AC1: 'Reply on RC1', Beatriz Recinos, 14 Jun 2023

RC2: 'Comment on tc202327', Anonymous Referee #2, 19 Apr 2023
This paper quantifies the impact of the calibration uncertainty on forecasts of sealevel rise contribution or volume above flotation, due to the choice of different regularization strengths, sliding laws and velocity inputs. To that aim, the authors first apply a classical optimizationbased inversion of an ice sheet/shelf model applied to the ice streams of the West Antarctica,
and second, apply Bayesian inference to estimate the impact of uncertainties on control parameters (i.e. to estimate the propagation of errors). Using these techniques, the authors comparatively measure the impact of uncertainties on regularization strengths, sliding laws and RS velocity products on volume above flotation, which is of most interest for SLR forecasting.
The paper treats a very important topic  Uncertainty Quantification in Ice Sheet Modelling  and combine stateoftheart ice flow modelling and bayesian inversion to allow such a quantification. I think this is the first time this is done in a timedependent model setting. The method and the results are extremely relevant to quantify uncertainties on SLR projection wrt to the most uncertain model parameters. Therefore, I believe this is a promising study that may inspire future articles in the community. My main concern here is on the method description.
The authors assume the readers to be familiar with Bayesian methods, and use the wording associated to it. As a result, and despite some time spent, I found it hard to understand (especially 2.4.1 and 2.5) within a reasonable time frame with the current version of the manuscript. I believe this work has a great potential, but I think the authors should make more efforts to make their paper more accessible to the community as most of readers have no or little prior knowledge on Bayesian inversion. There is room to open further and provide more insights on the methods in a narrative way rather than technical. It would be very beneficial to illustrate and explain the ideas behind some important concepts (prior / posterior), which are not known from all. Therefore, most of my comments below are requests to be more educational for people who don't have the background. I hope this will help the authors to improve their manuscript, especially to reach a larger pool of glaciologists.Main Comments
+The 3 sentences (l5256) are not enough to introduce basic concepts of Bayesian inference to the community, and especially to connect to the ice sheet model present study. To elaborate, please define clearly here what you mean by prior/posterior/covariance, link it directly to glaciological quantities, and give some intuition on the method. Also, it could be better motivated. If I understand, l 5455, propagation of errors between uncertain control parameters, and VAF could be obtained by proceeding to a massive amount of model realization, which is prohibitively expensive due to the costs of Stokes solving, right? This is what motivates you to take another approach? If yes, I suggest to restructure your paragraph starting from this motivation statement, and then elaborating (substantially) on Bayesian approach, and what this means in the context of your problem.
+ Despite several passes, Section 2.4.1 and 2.5 remains unclear to me, probably because I have no prior experience in Bayesian approaches, and I have not looked at the references. Here, I would expect to at least get a rough idea from these sections without having to go to references. E.g. where do the finite element matrices use to define $\Gamma_{prior}$ come from? What is the role of the operator (11) in the story? Justifications and explanations would be very welcome to explain all equations given in 2.4.1. As this is central in the paper, this part must be selfexplained (i.e. referencing if not enough). Similarly, Eq. (16) and (17) are highly important, but underexplained, please elaborate, give some intuition, and connect to what this means in the context of your glaciological problem. Several sentences could be founded an other articles on using Bayesian approach for a completely different problem. Therefore, there is room to better connect the approach and the application.
+ Following my last point, several times in the paper, one refers to "priors" or "posteriors" in a generic way, without specifying the meaning (regularization strength). E.g. a number of sentences are general statements with Bayesian vocabulary and unspecific to the ice flow problem considered here, and this contributes to making the paper abstract for nonspecialists. Efforts are required to make the paper further "educational", and the choice of words really matters in that respect.
+ The large amounts of data points in TSinferred velocity max be redundant, with implications for error propagation of VOF: This is not surprising. RS products may be very dense as the efficiency of featuretracking algorithms has improved. Therefore, trying to fit densely covered (poss. noisy) observation fields is probably more difficult than if we were selecting only a sparse version of the data, with the result of relaxing / giving room to the optimization. This is an interesting outcome, however, I find it a bit distracting to find this technical point coming back several times in other sensitivity experiments. Why not simply taking 1.6% of the data in all the paper explaining this choice somewhere. I don't feel this is sufficiently important finding to be part of the abstract.
+ I'm not sure to understand when you say that the regularization is too strong (l 506512): Would you have expected the VAF error (propagated) due to regularization higher than the one induced by different observed velocity products? What are the implications, and your recommendations for regularizing in future studies?
+ Have you tried to include ice thickness as part of the control parameters? or is this not justified in the special case of ice streams?
Specific comments:
+ l73: "overly informative prior" is an example of Bayesian wording for which I have no intuition. Please try to find other words, or better connect with glaciological words.
+ Can you explain you mean by "lowdimension" / "highdimensional" or *lowrank" in several places in the text (l47, 226, ...)
+ The norm $\Vert \cdot \Vert_{\Gamma_{obs}}$ is not defined here, I understood later than$\Gamma_{obs}$ are STD weighting the field in the norm computation, but I don't think this is clearly said, or defined at this point.
+ The method section is not well structured (e.g. the section "notation", 1 subsection 2.4.1).
+ l 502: "different estimates" : please quantify it in percentage.
+ In general, one refrains from starting sentences with mathematical symbols.
+ The paper introduces an impressive number of symbols without any reason, also it would greatly help the reader not to refer to symbols (as it requires the reader to memorize it), but instead to its meaning.
Citation: https://doi.org/10.5194/tc202327RC2 
AC2: 'Reply on RC2', Beatriz Recinos, 14 Jun 2023
Dear Anonymous reviewer #2,
Thank you for taking the time to review and improve our manuscript. Please find all our comments in the supplement PDF, below the comments to Reviewer #1 (page 4) together with a diff report between our latest version of the manuscript and original submission.
Regards

AC2: 'Reply on RC2', Beatriz Recinos, 14 Jun 2023
Status: closed

RC1: 'Comment on tc202327', Anonymous Referee #1, 13 Apr 2023
This paper is the application to a real case study of a framework to quantify the uncertainty in ice sheetmodel projections arising from model calibration.
This paper address a very relevant question and the results are well presented and convincing and I have mainly minor comments.
My main comment is that the conclusion indicates that the regularisation weights suggested by the LCurve analysis seem to lead to priors that are too confident, suppressing the propagation of the uncertainty from the velocity datasets used for the calibration. However, I found that the method for the LCurve is not very well described as there is 4 parameters to calibrate, and it is not not clear if they are chosen independently?, and there is a high level of userjudgement in the selection of these parameters; Comparing the values given in section 4.1 to those used in Table 1, it appears that the main differences are on the $\delta$ parameters for which the results are not shown. I am also wondering part of the issue cannot come from wrong priors as they are particluarly poorly constrained and here, the prior for the friction parameter $\alpha$ is 0, so that pure sliding everywhere? So maybe the conclusion could be revisited a little to not put too much attention on the LCurve?
Additional minor comments:
 Line 124: “constant surface mass balance”; Is it constant and uniform; or is there spatial variability?
 6 : QT here is defined as the VAF while is it use as the difference of VAF from t=0 in the manuscript. What is the meaning of the “+” symbol?
 Line 142: Hf=max(0,R(\rho_w/\rho_i))
 Line 196: if the prior is strong, $\gamma$ is “large” not “small”? (in agreement to line 342check for consistency everywhere)
 Sec 4.1 would be interesting to discuss the smoothing parameters in terms of variance and correlation length scales (Eq. 1314) as it appears that the parameters used here lead to a very small variance compared to the values used in Table 1.
 Line 389: $J^c$ should be $J^c_{miss}$? Check for consistency everywhere. I don’t understand why it does not change with the number of observations as according to Eq.8 it should depend on the number of observations?
 Line 477: “as the basal stress does not scale with effective stress in the interior”. I don’t understand the argument here.
 Lie 486: “is due to insensitivity of basal stress to $\alpha$ when the ice is near flotation”. The WeertmanBudd relation Eq. 1 is also insensitive to $/alpha$ near flotation as it depends on $N$; main difference is that Eq.3 tends to a Coulomb regime, independent of $\alpha$, for high velocity and low effective pressure. However using eq. 12 for N tends to restrict this domain to the close vicinity of the grounding line (Joughin et al., 2019)
References:
 Joughin, I., Smith, B.E., Schoof, C.G., 2019. Regularized Coulomb Friction Laws for Ice Sheet Sliding: Application to Pine Island Glacier, Antarctica. Geophys. Res. Lett.
Citation: https://doi.org/10.5194/tc202327RC1  AC1: 'Reply on RC1', Beatriz Recinos, 14 Jun 2023

RC2: 'Comment on tc202327', Anonymous Referee #2, 19 Apr 2023
This paper quantifies the impact of the calibration uncertainty on forecasts of sealevel rise contribution or volume above flotation, due to the choice of different regularization strengths, sliding laws and velocity inputs. To that aim, the authors first apply a classical optimizationbased inversion of an ice sheet/shelf model applied to the ice streams of the West Antarctica,
and second, apply Bayesian inference to estimate the impact of uncertainties on control parameters (i.e. to estimate the propagation of errors). Using these techniques, the authors comparatively measure the impact of uncertainties on regularization strengths, sliding laws and RS velocity products on volume above flotation, which is of most interest for SLR forecasting.
The paper treats a very important topic  Uncertainty Quantification in Ice Sheet Modelling  and combine stateoftheart ice flow modelling and bayesian inversion to allow such a quantification. I think this is the first time this is done in a timedependent model setting. The method and the results are extremely relevant to quantify uncertainties on SLR projection wrt to the most uncertain model parameters. Therefore, I believe this is a promising study that may inspire future articles in the community. My main concern here is on the method description.
The authors assume the readers to be familiar with Bayesian methods, and use the wording associated to it. As a result, and despite some time spent, I found it hard to understand (especially 2.4.1 and 2.5) within a reasonable time frame with the current version of the manuscript. I believe this work has a great potential, but I think the authors should make more efforts to make their paper more accessible to the community as most of readers have no or little prior knowledge on Bayesian inversion. There is room to open further and provide more insights on the methods in a narrative way rather than technical. It would be very beneficial to illustrate and explain the ideas behind some important concepts (prior / posterior), which are not known from all. Therefore, most of my comments below are requests to be more educational for people who don't have the background. I hope this will help the authors to improve their manuscript, especially to reach a larger pool of glaciologists.Main Comments
+The 3 sentences (l5256) are not enough to introduce basic concepts of Bayesian inference to the community, and especially to connect to the ice sheet model present study. To elaborate, please define clearly here what you mean by prior/posterior/covariance, link it directly to glaciological quantities, and give some intuition on the method. Also, it could be better motivated. If I understand, l 5455, propagation of errors between uncertain control parameters, and VAF could be obtained by proceeding to a massive amount of model realization, which is prohibitively expensive due to the costs of Stokes solving, right? This is what motivates you to take another approach? If yes, I suggest to restructure your paragraph starting from this motivation statement, and then elaborating (substantially) on Bayesian approach, and what this means in the context of your problem.
+ Despite several passes, Section 2.4.1 and 2.5 remains unclear to me, probably because I have no prior experience in Bayesian approaches, and I have not looked at the references. Here, I would expect to at least get a rough idea from these sections without having to go to references. E.g. where do the finite element matrices use to define $\Gamma_{prior}$ come from? What is the role of the operator (11) in the story? Justifications and explanations would be very welcome to explain all equations given in 2.4.1. As this is central in the paper, this part must be selfexplained (i.e. referencing if not enough). Similarly, Eq. (16) and (17) are highly important, but underexplained, please elaborate, give some intuition, and connect to what this means in the context of your glaciological problem. Several sentences could be founded an other articles on using Bayesian approach for a completely different problem. Therefore, there is room to better connect the approach and the application.
+ Following my last point, several times in the paper, one refers to "priors" or "posteriors" in a generic way, without specifying the meaning (regularization strength). E.g. a number of sentences are general statements with Bayesian vocabulary and unspecific to the ice flow problem considered here, and this contributes to making the paper abstract for nonspecialists. Efforts are required to make the paper further "educational", and the choice of words really matters in that respect.
+ The large amounts of data points in TSinferred velocity max be redundant, with implications for error propagation of VOF: This is not surprising. RS products may be very dense as the efficiency of featuretracking algorithms has improved. Therefore, trying to fit densely covered (poss. noisy) observation fields is probably more difficult than if we were selecting only a sparse version of the data, with the result of relaxing / giving room to the optimization. This is an interesting outcome, however, I find it a bit distracting to find this technical point coming back several times in other sensitivity experiments. Why not simply taking 1.6% of the data in all the paper explaining this choice somewhere. I don't feel this is sufficiently important finding to be part of the abstract.
+ I'm not sure to understand when you say that the regularization is too strong (l 506512): Would you have expected the VAF error (propagated) due to regularization higher than the one induced by different observed velocity products? What are the implications, and your recommendations for regularizing in future studies?
+ Have you tried to include ice thickness as part of the control parameters? or is this not justified in the special case of ice streams?
Specific comments:
+ l73: "overly informative prior" is an example of Bayesian wording for which I have no intuition. Please try to find other words, or better connect with glaciological words.
+ Can you explain you mean by "lowdimension" / "highdimensional" or *lowrank" in several places in the text (l47, 226, ...)
+ The norm $\Vert \cdot \Vert_{\Gamma_{obs}}$ is not defined here, I understood later than$\Gamma_{obs}$ are STD weighting the field in the norm computation, but I don't think this is clearly said, or defined at this point.
+ The method section is not well structured (e.g. the section "notation", 1 subsection 2.4.1).
+ l 502: "different estimates" : please quantify it in percentage.
+ In general, one refrains from starting sentences with mathematical symbols.
+ The paper introduces an impressive number of symbols without any reason, also it would greatly help the reader not to refer to symbols (as it requires the reader to memorize it), but instead to its meaning.
Citation: https://doi.org/10.5194/tc202327RC2 
AC2: 'Reply on RC2', Beatriz Recinos, 14 Jun 2023
Dear Anonymous reviewer #2,
Thank you for taking the time to review and improve our manuscript. Please find all our comments in the supplement PDF, below the comments to Reviewer #1 (page 4) together with a diff report between our latest version of the manuscript and original submission.
Regards

AC2: 'Reply on RC2', Beatriz Recinos, 14 Jun 2023
Beatriz Recinos et al.
Data sets
Output of several experiments with Fenics_ice over Smith, Pope, and Kohler Glaciers Beatriz Recinos, Daniel Goldberg, and James R. Maddison https://doi.org/10.5281/zenodo.7612243
Model code and software
Experiments with Fenics_ice applied to three West Antarctic ice streams Beatriz Recinos, Daniel Goldberg, James R. Maddison, and Joe Todd https://doi.org/10.5281/zenodo.7615259
Beatriz Recinos et al.
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