the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 26 Feb 2026
GlaUnTI: A hybrid physics–machine learning model enables transferable glacier surface mass balance estimation
Abstract. Glacier surface mass balance (SMB) is a key climate indicator and a central driver of glacier change. Direct SMB observations remain sparse and unevenly distributed. Hence, transferable SMB models are essential for large-scale assessments and projections. Here, we propose the GLAcier-UNiversal Temperature Index model (GlaUnTI) for this purpose. This hybrid physics–machine learning model modifies a fully differentiable temperature index (TI) SMB model by introducing a shallow convolutional neural corrector. It learns spatially and temporally varying adjustments to a small set of physically interpretable TI parameters, using glacier geometry and aggregated climate information. We calibrate four models—a basic TI model, a purely data-driven recurrent neural network with no physical inductive bias and two GlaUnTI variants, with and without glacier facies maps as predictors—using a dataset of 65 European glaciers spanning 1995–2024 and covering the Alps, Scandinavia, Iceland, Svalbard and the Pyrenees. Their performance is evaluated on a spatially independent test subset of 13 glaciers across heterogeneous regions. The evaluation uses 793/756/314 (annual/winter/summer) point SMB measurements and 312/235/233 glacier-wide SMB estimates. On the test glaciers, the baseline TI model achieves annual point-level performance with r=0.854 and an RMSE equal to 1.707 m w.e. With GlaUnTI, r increases to 0.940 and the RMSE reduces to 1.068 m w.e. At the glacier-wide scale, the baseline TI model attains an r equal to 0.606 and an RMSE of 0.805 m w.e. With GlaUnTI, r increases to 0.700 and the RMSE reduces to 0.627 m w.e. Including glacier facies maps from the end of the ablation season to the corrector yields moderate benefits in glacier-wide summer (11.0 %) and annual (12.2 %) SMB estimates. We found that the purely data-driven baseline model overall shows the weakest spatial transferability. Also, end-to-end differentiability enables efficient gradient-based calibration, transfer learning, inverse optimisation of effective forcing perturbations, formal model explainability and propagation of forcing-driven aleatoric uncertainty through long SMB trajectories. These results demonstrate that parameter-corrected hybrid models improve SMB transferability across diverse climate regimes while preserving a physically grounded structure, suitable for integration into broader glacier evolution workflows and for informing climate-related policies.
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Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
(4130 KB)
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(4130 KB) - Metadata XML
- BibTeX
- EndNote
- Final revised paper
Journal article(s) based on this preprint
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2026-515', Ritu Anilkumar, 13 Apr 2026
- AC1: 'Reply on RC1', Konstantin Maslov, 19 Jun 2026
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RC2: 'Comment on egusphere-2026-515', Anonymous Referee #2, 22 Apr 2026
General comments
The manuscript describes a hybrid approach that exploits the structure of temperature-index (TI) models to derive a machine learning based model that predicts daily surface mass balance. The approach relies on differentiable programming which allows building a model whose structure is inspired by existing and well established MB models but which allows computing gradients and incorporating learnable components which are trained to account for the spatiotemporal variability in the parameters of these TI models. Two flavors of the model, named GlaUnTI, are presented and they differ by the input features that are used.
The proposed architecture is compared against a differentiable TI model and a fully data-driven architecture which does not integrate a physically driven prior in the model architecture. The four models are trained and compared on various glaciers in Europe to illustrate the transferable properties of the approach.
Finally the studies in the appendices showcase applications of the differentiable approach for inverse modelling, model explainability as well as uncertainty quantification.
The manuscript is well written and very interesting to read. The development of transferable mass balance models that leverage both glaciological and remote sensing based products are timely and quiet new in the literature. Training a model at a daily time step with a term in the loss function that aggregates yearly and at the glacier scale is far from being easy. The comparison of the proposed approach with a fully data-driven model is appreciated, especially since there starts to be multiple works in the literature that build MB models on this class of fully data-driven architectures.
The validation of the trained model is performed in a robust way through independent validation and test sets. Some of the potential caveats in model comparison (models C vs D) are properly treated. However the comparison with other models could be improved.
The illustration of the potential of the approach for inverse modelling thanks to differentiable properties is much appreciated and is part of the contribution of this work, which in the opinion of the reviewer should be more tightly integrated into the main body of the manuscript.
On the overall this is a good contribution that perfectly fits into the scope of The Cryosphere. However there are a few issues that should be addressed and that would strengthen the comparison and the contribution.
Specific comments
1. Major comment: Quantify the impact of ice dynamics on the SWE representation.
The SWE variable represents the accumulated snow within a grid cell but this assumes that there is no ice dynamics and that a column of snow does not move. Incorporating ice dynamics would obviously make the problem much harder but at the same time the initial state of SWE is obtained with a spin-up of 5 years and for some glaciers this can change a lot the topographical and climate conditions. Could the authors quantify the impact of not representing the glacier dynamics?
2. Major comment: Comparison with the autodiff-friendly TI model is not fair.
Usually TI models are calibrated per glacier which is not the case in this study where the parameters are calibrated across different regions. In classical TI models these parameters reflect the various climate conditions which are expected to vary across different glaciers.
While the choice of tuning TI parameters across different glaciers is understandable to be able to assess the performance on an hold-out set, this leads to a poorer performance. A comparison with the autodiff-friendly TI model with parameters tuned per glacier would allow assessing the real performance gain, beyond the intrinsic transferability property machine learning models have over TI models.
3. Major comment: Number of training iterations for the fully data-driven architecture is insufficient.
Given that updates are performed at every epoch there is no stochasticity in the training, which is usually a property that helps to explore the parameter space. The low number of epochs is probably not enough for the fully data-driven architecture to capture the relationship between the inputs and the output. A training with more iterations should be performed.
4. Major comment: Incorporate the appendices more closely into the main body of the paper.
Beyond having a transferable MB model, differentiability is key for inverse modelling. Many scientific questions in glaciology require the representation of both ice dynamics and mass balance over long simulation periods. Having a differentiable MB model is one of the key components to tackle these questions. The authors illustrate this potential in a convincing way through different and complete experiments in the appendices. In the opinion of the reviewer this should be included as one of the main messages of the paper and in the current version the appendices are decorrelated from the rest of the manuscript.
5. Minor comment: More details should be provided on the glacier facies maps.
More information should be given in section 3.5 about the glacier facies maps. The classes, which are ice, snow, debris, firn and refrozen-like according to Maslov et al. (2026)), are not defined and giving them would make it clearer for the reader of what information these maps carry.
6. Minor comment: Clarify the two “regimes” that produce SMB predictions.
According to section 4 the models have two “regimes” to produce SMB predictions but how these two regimes are obtained in practice is not detailed. Are the authors referring to the aggregation in the loss function over a time window like in Eq (5)? If so the statement “considerably reducing the memory footprint” (L193) is only partially true since even though the predictions can be aggregated recursively the inputs still needs to be stored somehow.
7. Minor comment: Clarify if normalization is applied to the inputs of the deep learning corrector.
This is not detailed in the manuscript, but there is probably a normalization given the heterogeneity of the input data ranges. The bounds should also be given for reproducibility.
8. Minor comment: Explain how the folds are constructed.
Given the small number of groups, the performance of the model depends a lot on the construction of the folds. Are they randomly split or is there a smarter construction strategy that leads to better representation in the train, validation and test sets?
Technical corrections:
- L308: “stands for the mean value” → “stands for the spatial mean value” to be more precise
- L312 “so that the training dynamics is optimal at the beginning”: in which sense? Loss function value? On the training or the validation set?
- Section 3.1: The glacier-wide MB targets are model outputs. Since the approach mixes glaciological point measurement with glacier-wide MB values, this should be clearly stated.
- Section A1: Emphasize that in a perfect modelling framework we would have to change also the distribution of precipitations since in a changing climate we expect the distribution of climate variables to change.
Citation: https://doi.org/10.5194/egusphere-2026-515-RC2 - AC2: 'Reply on RC2', Konstantin Maslov, 19 Jun 2026
Peer review completion
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2026-515', Ritu Anilkumar, 13 Apr 2026
- AC1: 'Reply on RC1', Konstantin Maslov, 19 Jun 2026
-
RC2: 'Comment on egusphere-2026-515', Anonymous Referee #2, 22 Apr 2026
General comments
The manuscript describes a hybrid approach that exploits the structure of temperature-index (TI) models to derive a machine learning based model that predicts daily surface mass balance. The approach relies on differentiable programming which allows building a model whose structure is inspired by existing and well established MB models but which allows computing gradients and incorporating learnable components which are trained to account for the spatiotemporal variability in the parameters of these TI models. Two flavors of the model, named GlaUnTI, are presented and they differ by the input features that are used.
The proposed architecture is compared against a differentiable TI model and a fully data-driven architecture which does not integrate a physically driven prior in the model architecture. The four models are trained and compared on various glaciers in Europe to illustrate the transferable properties of the approach.
Finally the studies in the appendices showcase applications of the differentiable approach for inverse modelling, model explainability as well as uncertainty quantification.
The manuscript is well written and very interesting to read. The development of transferable mass balance models that leverage both glaciological and remote sensing based products are timely and quiet new in the literature. Training a model at a daily time step with a term in the loss function that aggregates yearly and at the glacier scale is far from being easy. The comparison of the proposed approach with a fully data-driven model is appreciated, especially since there starts to be multiple works in the literature that build MB models on this class of fully data-driven architectures.
The validation of the trained model is performed in a robust way through independent validation and test sets. Some of the potential caveats in model comparison (models C vs D) are properly treated. However the comparison with other models could be improved.
The illustration of the potential of the approach for inverse modelling thanks to differentiable properties is much appreciated and is part of the contribution of this work, which in the opinion of the reviewer should be more tightly integrated into the main body of the manuscript.
On the overall this is a good contribution that perfectly fits into the scope of The Cryosphere. However there are a few issues that should be addressed and that would strengthen the comparison and the contribution.
Specific comments
1. Major comment: Quantify the impact of ice dynamics on the SWE representation.
The SWE variable represents the accumulated snow within a grid cell but this assumes that there is no ice dynamics and that a column of snow does not move. Incorporating ice dynamics would obviously make the problem much harder but at the same time the initial state of SWE is obtained with a spin-up of 5 years and for some glaciers this can change a lot the topographical and climate conditions. Could the authors quantify the impact of not representing the glacier dynamics?
2. Major comment: Comparison with the autodiff-friendly TI model is not fair.
Usually TI models are calibrated per glacier which is not the case in this study where the parameters are calibrated across different regions. In classical TI models these parameters reflect the various climate conditions which are expected to vary across different glaciers.
While the choice of tuning TI parameters across different glaciers is understandable to be able to assess the performance on an hold-out set, this leads to a poorer performance. A comparison with the autodiff-friendly TI model with parameters tuned per glacier would allow assessing the real performance gain, beyond the intrinsic transferability property machine learning models have over TI models.
3. Major comment: Number of training iterations for the fully data-driven architecture is insufficient.
Given that updates are performed at every epoch there is no stochasticity in the training, which is usually a property that helps to explore the parameter space. The low number of epochs is probably not enough for the fully data-driven architecture to capture the relationship between the inputs and the output. A training with more iterations should be performed.
4. Major comment: Incorporate the appendices more closely into the main body of the paper.
Beyond having a transferable MB model, differentiability is key for inverse modelling. Many scientific questions in glaciology require the representation of both ice dynamics and mass balance over long simulation periods. Having a differentiable MB model is one of the key components to tackle these questions. The authors illustrate this potential in a convincing way through different and complete experiments in the appendices. In the opinion of the reviewer this should be included as one of the main messages of the paper and in the current version the appendices are decorrelated from the rest of the manuscript.
5. Minor comment: More details should be provided on the glacier facies maps.
More information should be given in section 3.5 about the glacier facies maps. The classes, which are ice, snow, debris, firn and refrozen-like according to Maslov et al. (2026)), are not defined and giving them would make it clearer for the reader of what information these maps carry.
6. Minor comment: Clarify the two “regimes” that produce SMB predictions.
According to section 4 the models have two “regimes” to produce SMB predictions but how these two regimes are obtained in practice is not detailed. Are the authors referring to the aggregation in the loss function over a time window like in Eq (5)? If so the statement “considerably reducing the memory footprint” (L193) is only partially true since even though the predictions can be aggregated recursively the inputs still needs to be stored somehow.
7. Minor comment: Clarify if normalization is applied to the inputs of the deep learning corrector.
This is not detailed in the manuscript, but there is probably a normalization given the heterogeneity of the input data ranges. The bounds should also be given for reproducibility.
8. Minor comment: Explain how the folds are constructed.
Given the small number of groups, the performance of the model depends a lot on the construction of the folds. Are they randomly split or is there a smarter construction strategy that leads to better representation in the train, validation and test sets?
Technical corrections:
- L308: “stands for the mean value” → “stands for the spatial mean value” to be more precise
- L312 “so that the training dynamics is optimal at the beginning”: in which sense? Loss function value? On the training or the validation set?
- Section 3.1: The glacier-wide MB targets are model outputs. Since the approach mixes glaciological point measurement with glacier-wide MB values, this should be clearly stated.
- Section A1: Emphasize that in a perfect modelling framework we would have to change also the distribution of precipitations since in a changing climate we expect the distribution of climate variables to change.
Citation: https://doi.org/10.5194/egusphere-2026-515-RC2 - AC2: 'Reply on RC2', Konstantin Maslov, 19 Jun 2026
Peer review completion
Journal article(s) based on this preprint
Data sets
GlaUnTI: Glacier surface mass balance dataset Konstantin A. Maslov et al. https://doi.org/10.4121/5ea53bc3-2c85-42bb-89d1-606c8ed1d80a
Model code and software
GlaUnTI: GLAcier-UNiversal Temperature Index model Konstantin A. Maslov et al. https://github.com/konstantin-a-maslov/glaunti
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Konstantin A. Maslov
Thomas Schellenberger
Claudio Persello
Alfred Stein
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(4130 KB) - Metadata XML