Can streamflow observations constrain snow mass reconstructions? Lessons from two synthetic numerical experiments

Wiersma, Pau; Magnusson, Jan; Peleg, Nadav; Schaefli, Bettina; Mariéthoz, Grégoire

doi:10.5194/egusphere-2025-3610

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https://doi.org/10.5194/egusphere-2025-3610

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15 Oct 2025

| 15 Oct 2025

Can streamflow observations constrain snow mass reconstructions? Lessons from two synthetic numerical experiments

Pau Wiersma, Jan Magnusson, Nadav Peleg, Bettina Schaefli, and Grégoire Mariéthoz

Abstract. Historical snow mass estimates are key to understanding snowmelt-driven streamflow and climate change impacts on snow water resources. However, snow mass observations are scarce, and SWE reconstructions rely largely on snow models forced with meteorological inputs. Ground-based and satellite observations are often used to constrain the typically high uncertainty of modeled snow mass reconstructions, but their constraining potential is limited in data-scarce regions and prior to the onset of satellite monitoring. Here, we suggest using streamflow information as an additional information source to better reconstruct snow mass. We introduce an inverse hydrological modeling framework that selects realistic snow mass realizations based on the accuracy of their streamflow response. Before real-world application, we test the framework in two synthetic experiments. Our results demonstrate that streamflow has the potential to constrain snow mass reconstructions, but that non-uniqueness in the snow-streamflow relationship and uncertainties in the inverse modelling chain can easily stand in the way. We also show that streamflow is most helpful in estimating catchment-aggregated properties of snow mass reconstructions, in particular catchment-aggregated melt rates. Future work should assess the potential of streamflow-constrained snow mass reconstruction under real-world conditions and investigate the added value of streamflow when combined with other snow data sources.

Received: 25 Jul 2025 – Discussion started: 15 Oct 2025

Competing interests: At least one of the (co-)authors is a member of the editorial board of Hydrology and Earth System Sciences.

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Pau Wiersma, Jan Magnusson, Nadav Peleg, Bettina Schaefli, and Grégoire Mariéthoz

Status: final response (author comments only)

RC1:
'Comment on egusphere-2025-3610', Joschka Geissler, 08 Nov 2025
Summary:
The submitted manuscript investigates how streamflow observations can help reconstruct historical snow mass. Therefore, the authors formulate the problem as a Bayesian inversion (i.e. estimating the posterior distribution of SWE given streamflow observations). Two complementary numerical experiments are conducted: a fully synthetic (FS) case, in which the reference SWE is generated by the same model used in the ensemble, and a semi-synthetic (SS) case, where the reference SWE is taken from OSHD data. While the selection of the posterior ensemble was carried out using NSE between observed and modelled streamflow, the evaluation of how well the selected posterior SWE maps match with the SWEprior is performed using grid-based and catchment-aggregated snow metrics. The results demonstrate that streamflow does contain meaningful information about seasonal snow dynamics, can narrow the range of plausible SWE scenarios, and is especially useful for reconstructing catchment-aggregated melt. However, the study’s results also suggest that SWE scenarios with systematic biases can still produce excellent streamflow performance. Moreover, the authors state that there is a large degree of non-uniqueness in the SWE–streamflow relationship. The semi-synthetic experiment further highlights that realistic model and forcing uncertainties amplify this equifinality. For future work, the authors recommend incorporating complementary snow observations (e.g. snow depth, SCA) to reduce equifinality and improve SWE reconstruction.
Overall Feedback:
The manuscript is of high quality in both methodology and writing and will likely be a valuable contribution to the snow-hydrology community. The authors provide a clear motivation, a well-structured experimental design, and a thoughtful discussion of relevant uncertainties (e.g., performance metric choice, model structural limitations, and forcing errors). Overall, the work is comprehensive, clearly presented, and addresses a timely question within hydrologic modeling and cryospheric science. I suggest to return the manuscript to the authors for a minor revision. I would appreciate if the authors would address my comments below during this revision.
General Comments:
In my opinion, the manuscript could benefit from a deeper methodological integration and/or theoretical discussion of recurring spatial patterns of snow accumulation, which the authors also acknowledge in the introduction (L46–56). Numerous studies show that snow distribution is controlled by (relatively static) topographic controls (elevation gradients, wind exposure, canopy effects) and tends to vary primarily in magnitude rather than in relative spatial differences. This raises an important question for the inversion framework:
Can the recurring nature of snow distribution patterns be used to reduce the effective dimensionality of the prior space or to inform the posterior selection procedure?
I would encourage the authors to reflect on how recurring snow patterns could be incorporated into different parts of their framework, for example:
1. Method (Sect. 2.1): Dimensionality reduction of the prior (Could be discussed for future applications..)
Currently, the prior is defined through independent parameter ranges, which leads to a large prior state space. Considering the reoccurring behaviour of snow distribution, do the authors think that this creates an unnecessarily large prior space and hence potentially amplifies non-uniqueness? One possibility would be to represent SWE as a scaled version of a known distribution pattern (Pflug & Lundquist, 2020; Vögeli et al., 2016; Ylönen et al., 2025), e.g. H_SWE(x,y,t) = α(t)⋅H_pattern(x,y)+ϵ(x,y,t)
2. Posterior ensemble selection (Sect. 2.4): Plausibility
Posterior filtering could penalize SWE fields that violate well-known topographic dependencies (e.g., monotonic elevation gradients), thereby favoring more physically meaningful inversions. Figure 4 and especially Figure 5 show that the NSE-selected posterior contains some of the best SWE simulations of the prior – even for the SS experiment. I wonder whether a second-stage selection step, based on physically realistic spatial patterns, could further refine the posterior and help exclude SWE fields that match streamflow but are unlikely given known snow distribution processes. Such plausibility-based filtering (e.g., constraining elevation–SWE relationships or enforcing typical accumulation gradients) may help reduce non-uniqueness and improve the interpretability of the resulting posterior ensemble.
3. Discussion (Section 4.2.):
It may be worthwhile to discuss whether, in real-world applications, the existence of spatial patterns may actually help reduce non-uniqueness compared to the synthetic inversion setups presented here.
Specific Comments:
L21: I would argue that snow patterns can also be rather consistent – I suggest writing: In addition, snowfall and snowmelt vary spatially…

Section 2.2.3 or 2.2.4: I am missing a sentence here to what years your method is applied…

L213: Note that this...

L293: Space between =2500

L319: missing space after the

L384: missing spaces..

L452: In your introduction you introduced data assimilation products but no real data-driven approaches. This is why I find this sentence feels a bit isolated, but still interesting. Maybe it would be worth introducing data driven approaches (e.g. Daudt et al. (2023), Zheng et al. (2018), Ylönen et al. (2025)…).

L510ff: What about catchments with lower elevation gradient, less topographic roughness, e.g. boreal catchments? Would it make sense to perform future experiments here?

References
Daudt, R. C., Wulf, H., Hafner, E. D., Bühler, Y., Schindler, K., & Wegner, J. D. (2023). Snow depth estimation at country-scale with high spatial and temporal resolution. ISPRS Journal of Photogrammetry and Remote Sensing, 197, 105–121. https://doi.org/10.1016/j.isprsjprs.2023.01.017
Pflug, J. M., & Lundquist, J. D. (2020). Inferring Distributed Snow Depth by Leveraging Snow Pattern Repeatability: Investigation Using 47 Lidar Observations in the Tuolumne Watershed, Sierra Nevada, California. Water Resources Research, 56(9). https://doi.org/10.1029/2020WR027243
Vögeli, C., Lehning, M., Wever, N., & Bavay, M. (2016). Scaling Precipitation Input to Spatially Distributed Hydrological Models by Measured Snow Distribution. Frontiers in Earth Science, 4.https://doi.org/10.3389/feart.2016.00108
Ylönen, M., Marttila, H., Kuzmin, A., Korpelainen, P., Kumpula, T., & Ala-aho, P. (2025). UAV LiDAR surveys and machine learning improves snow depth and water equivalent estimates in the boreal landscapes.https://doi.org/10.5194/egusphere-2025-1297
Zheng, Z., Molotch, N. P., Oroza, C. A., Conklin, M. H., & Bales, R. C. (2018). Spatial snow water equivalent estimation for mountainous areas using wireless-sensor networks and remote-sensing products. Remote Sensing of Environment, 215, 44–56. https://doi.org/10.1016/j.rse.2018.05.029
Citation: https://doi.org/10.5194/egusphere-2025-3610-RC1
- AC1:
  'Reply on RC1', Pau Wiersma, 13 Jan 2026
  We thank Joschka Geissler for the constructive feedback and appreciation of our work, and for taking the time to review our paper. We would also like to apologize for the late response. Please find our comments in italics below.
  Summary:
  The submitted manuscript investigates how streamflow observations can help reconstruct historical snow mass. Therefore, the authors formulate the problem as a Bayesian inversion (i.e. estimating the posterior distribution of SWE given streamflow observations). Two complementary numerical experiments are conducted: a fully synthetic (FS) case, in which the reference SWE is generated by the same model used in the ensemble, and a semi-synthetic (SS) case, where the reference SWE is taken from OSHD data. While the selection of the posterior ensemble was carried out using NSE between observed and modelled streamflow, the evaluation of how well the selected posterior SWE maps match with the SWEprior is performed using grid-based and catchment-aggregated snow metrics. The results demonstrate that streamflow does contain meaningful information about seasonal snow dynamics, can narrow the range of plausible SWE scenarios, and is especially useful for reconstructing catchment-aggregated melt. However, the study’s results also suggest that SWE scenarios with systematic biases can still produce excellent streamflow performance. Moreover, the authors state that there is a large degree of non-uniqueness in the SWE–streamflow relationship. The semi-synthetic experiment further highlights that realistic model and forcing uncertainties amplify this equifinality. For future work, the authors recommend incorporating complementary snow observations (e.g. snow depth, SCA) to reduce equifinality and improve SWE reconstruction.
  Overall Feedback:
  The manuscript is of high quality in both methodology and writing and will likely be a valuable contribution to the snow-hydrology community. The authors provide a clear motivation, a well-structured experimental design, and a thoughtful discussion of relevant uncertainties (e.g., performance metric choice, model structural limitations, and forcing errors). Overall, the work is comprehensive, clearly presented, and addresses a timely question within hydrologic modeling and cryospheric science. I suggest to return the manuscript to the authors for a minor revision. I would appreciate if the authors would address my comments below during this revision.
  General Comments:
  In my opinion, the manuscript could benefit from a deeper methodological integration and/or theoretical discussion of recurring spatial patterns of snow accumulation, which the authors also acknowledge in the introduction (L46–56). Numerous studies show that snow distribution is controlled by (relatively static) topographic controls (elevation gradients, wind exposure, canopy effects) and tends to vary primarily in magnitude rather than in relative spatial differences. This raises an important question for the inversion framework:
  Can the recurring nature of snow distribution patterns be used to reduce the effective dimensionality of the prior space or to inform the posterior selection procedure?
  This is a very good point that we have indeed not given sufficient attention. Recurring snow patterns should in fact be particularly compatible with streamflow-based SWE reconstruction, since they are valid for any year in the past once they have been mapped. In the revised manuscript, we will strengthen the link to the recurring-pattern literature in the Introduction and clarify its implications for our inversion framework. In the Discussion, we will reflect more explicitly on how streamflow is likely most effective when used alongside other snow information and we will highlight recurring spatial patterns as a promising form of catchment-specific prior knowledge. In particular, we will discuss how prior SWE ensembles could be formulated as scaled versions of empirically derived spatial SWE patterns, thereby reducing the number of parameters to be inferred and mitigating part of the physical non-uniqueness identified in this study. We will also note that such patterns have so far been documented in a limited number of catchments, but recent advances in automated UAV/LiDAR observations are likely to expand this basis substantially, enabling longer, pattern-consistent SWE reconstructions when combined with long streamflow records.
  I would encourage the authors to reflect on how recurring snow patterns could be incorporated into different parts of their framework, for example:
  Method (Sect. 2.1): Dimensionality reduction of the prior (Could be discussed for future applications..)
  
  Currently, the prior is defined through independent parameter ranges, which leads to a large prior state space. Considering the reoccurring behaviour of snow distribution, do the authors think that this creates an unnecessarily large prior space and hence potentially amplifies non-uniqueness? One possibility would be to represent SWE as a scaled version of a known distribution pattern (Pflug & Lundquist, 2020; Vögeli et al., 2016; Ylönen et al., 2025), e.g. H_SWE(x,y,t) = α(t)⋅H_pattern(x,y)+ϵ(x,y,t)
  Posterior ensemble selection (Sect. 2.4): Plausibility
  
  Posterior filtering could penalize SWE fields that violate well-known topographic dependencies (e.g., monotonic elevation gradients), thereby favoring more physically meaningful inversions. Figure 4 and especially Figure 5 show that the NSE-selected posterior contains some of the best SWE simulations of the prior – even for the SS experiment. I wonder whether a second-stage selection step, based on physically realistic spatial patterns, could further refine the posterior and help exclude SWE fields that match streamflow but are unlikely given known snow distribution processes. Such plausibility-based filtering (e.g., constraining elevation–SWE relationships or enforcing typical accumulation gradients) may help reduce non-uniqueness and improve the interpretability of the resulting posterior ensemble.
  Discussion (Section 4.2.):
  
  It may be worthwhile to discuss whether, in real-world applications, the existence of spatial patterns may actually help reduce non-uniqueness compared to the synthetic inversion setups presented here.
  These are all great points and we will incorporate them in the discussion chapter.
  Specific Comments:
  L21: I would argue that snow patterns can also be rather consistent – I suggest writing: In addition, snowfall and snowmelt vary spatially…
  
  Section 2.2.3 or 2.2.4: I am missing a sentence here to what years your method is applied…
  
  L213: Note that this...
  
  L293: Space between =2500
  
  L319: missing space after the
  
  L384: missing spaces..
  
  L452: In your introduction you introduced data assimilation products but no real data-driven approaches. This is why I find this sentence feels a bit isolated, but still interesting. Maybe it would be worth introducing data driven approaches (e.g. Daudt et al. (2023), Zheng et al. (2018), Ylönen et al. (2025)…).
  
  L510ff: What about catchments with lower elevation gradient, less topographic roughness, e.g. boreal catchments? Would it make sense to perform future experiments here?
  
  Thank you for these minor suggestions. We will incorporate them in the manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2025-3610-AC1
RC2:
'Comment on egusphere-2025-3610', Simon Gascoin, 04 Jan 2026

Review
First, I would like to apologize for the long delay in providing my review. I struggled to find time to read this manuscript due to some unforeseen academic duties.
I was attracted by the title. After reading the entire paper, I can see that it does not violate Betteridge’s law of headlines.
The study is well conducted, the methodology is sound, and the results are well illustrated. However, in some parts I found the manuscript difficult to follow. In particular, the Results section is full of acronyms and symbols and would benefit from more concise and synthetic writing. The five-page Discussion is also very long; frankly, I ended up skimming parts of it. I understand that the authors wish to be very honest about the limitations of their results, but a single paper will never be sufficient to discuss all the issues related to inverse modelling in hydrology. I would suggest focusing on key issues that are specific to this study and referring to previous articles for the rest. For example, in my opinion, Section 4.5 of the Discussion is out of scope, as it deviates from the scientific objective of the study as announced in the Introduction.
In the Introduction, the authors could better justify the motivation for their work. What is a typical use case that justifies the need to generate posterior gridded SWE from streamflow data in a gauged catchment? Doing hydrology “backward” is useful if the inferred variable has an application outside the scope of the hydrological model itself. I think that a deeper reflection on the potential applications of this approach could help to better focus the Discussion.
If the idea is to reconstruct recent SWE, then readily available remote-sensing information (or their synthetic surrogates), such as annual snow cover duration maps, could be incorporated into the model evaluation to better constrain the likelihood. This echoes Referee 1’s suggestion to consider prior knowledge on the spatial distribution of snow cover to narrow the posterior spread.
Minor comments
L27: “SCA and wet snow measurements provide only binary information.” This is true at the product resolution (20 m, 100 m); however, both variables can be expressed as fractional values after spatial aggregation at the model resolution.
L75: I feel that the Introduction should consider the work of Le Moine et al. (2015).
L155: What is the rationale for having a different correction factor for liquid versus solid precipitation?
Precipitation is obtained from a 2 km grid. The authors refer to gauge undercatch to explain why catchment-scale precipitation is lower than streamflow (i.e., a runoff coefficient greater than 1). However, this discrepancy may also be due to the smoothed topography of the 2 km dataset used in the gridded precipitation interpolation scheme, which may be biased relative to the actual elevation distribution of the catchment. I did not clearly understand how precipitation was interpolated from this 2 km source to the 900 m × 700 m resolution. This processing involves two DEMs. The authors should be more specific about the source and/or resampling methods used for both DEMs. I also recommend explicitly writing the equations used for temperature and precipitation interpolation. Indeed, I do not understand why $SFCF$ is referred to as a “multiplicative correction factor,” whereas $SFCF_{ELE}$ is described as “a linear elevation lapse rate.” Precipitation cannot be interpolated using a temperature-like lapse rate, as this could lead to negative precipitation values. Moreover, linear scaling with elevation is not always chosen to interpolate precipitation, as it does not represent the observed capping of precipitation at high elevations due to atmospheric moisture depletion (e.g., Liston and Elder, 2006).
L245: Why is “air temperature capped at a minimum of 0 °C”?
L247: The authors mention “some physical inconsistencies, such as the omission of refreezing.” What are the other inconsistencies?
L450: The lack of “temporal continuity” of the ASO data is not a convincing argument for rejecting these data. ASO provides SWE maps every two weeks in several gauged catchments over multiple years. ASO catchments would actually form an ideal dataset to test the inversion of SWE using the proposed framework.
Formatting: I think that the acronyms (“SCFCF”) or text (“snow”) in the equations should not be italicized (e.g. using \mathrm).
References
Le Moine, N., Hendrickx, F., Gailhard, J., Garçon, R., Gottardi, F., 2015. Hydrologically Aided Interpolation of Daily Precipitation and Temperature Fields in a Mesoscale Alpine Catchment. Journal of Hydrometeorology 16, 2595–2618. https://doi.org/10.1175/JHM-D-14-0162.1
Liston, G.E., Elder, K., 2006. A Meteorological Distribution System for High-Resolution Terrestrial Modeling (MicroMet). Journal of Hydrometeorology 7, 217–234. https://doi.org/10.1175/JHM486.1

Citation: https://doi.org/10.5194/egusphere-2025-3610-RC2
- AC2:
  'Reply on RC2', Pau Wiersma, 13 Jan 2026
  First, I would like to apologize for the long delay in providing my review. I struggled to find time to read this manuscript due to some unforeseen academic duties.
  We thank Simon Gascoin for his helpful review, we understand that things can get in the way unexpectedly so we appreciate that he still took the time. Please find our comments in italics below.
  I was attracted by the title. After reading the entire paper, I can see that it does not violate Betteridge’s law of headlines.
  That is a funny reference. I would say the answer to the title question is not a definitive no, but it is certainly not a definitive yes either.
  The study is well conducted, the methodology is sound, and the results are well illustrated. However, in some parts I found the manuscript difficult to follow. In particular, the Results section is full of acronyms and symbols and would benefit from more concise and synthetic writing. The five-page Discussion is also very long; frankly, I ended up skimming parts of it. I understand that the authors wish to be very honest about the limitations of their results, but a single paper will never be sufficient to discuss all the issues related to inverse modelling in hydrology. I would suggest focusing on key issues that are specific to this study and referring to previous articles for the rest. For example, in my opinion, Section 4.5 of the Discussion is out of scope, as it deviates from the scientific objective of the study as announced in the Introduction.
  Thank you for the feedback and the useful comments. We propose to address them as follows:
  Rewrite the results section more concisely and synthetically, which would lead to a reduction of approximately one page and a good number of symbols and acronyms. We also plan to update figure 3, which contained an error, and rewrite section 3.1 accordingly. It would not change our interpretation of the results.
  
  Rewrite much of the discussion and leave section 4.5 out, leading to a reduction of approximately two pages.
  
  In the Introduction, the authors could better justify the motivation for their work. What is a typical use case that justifies the need to generate posterior gridded SWE from streamflow data in a gauged catchment? Doing hydrology “backward” is useful if the inferred variable has an application outside the scope of the hydrological model itself. I think that a deeper reflection on the potential applications of this approach could help to better focus the Discussion.
  If the idea is to reconstruct recent SWE, then readily available remote-sensing information (or their synthetic surrogates), such as annual snow cover duration maps, could be incorporated into the model evaluation to better constrain the likelihood. This echoes Referee 1’s suggestion to consider prior knowledge on the spatial distribution of snow cover to narrow the posterior spread.
  In response to this comment, we plan to put more emphasis on the potential of streamflow for pre-satellite era SWE reconstructions in the 4^th paragraph of the introduction, and to add a clarifying sentence on the overall goal of our study at the end of the introduction. We also plan to feed back to the potential for pre-satellite era reconstructions in the Discussion, by suggesting that information on recurring snow patterns also lend themselves well for pre-satellite era SWE reconstructions, and by proposing the joint application of streamflow and recurring snow patterns in pre-satellite era SWE reconstructions.
  Minor comments
  L27: “SCA and wet snow measurements provide only binary information.” This is true at the product resolution (20 m, 100 m); however, both variables can be expressed as fractional values after spatial aggregation at the model resolution.
  L75: I feel that the Introduction should consider the work of Le Moine et al. (2015).
  L155: What is the rationale for having a different correction factor for liquid versus solid precipitation?
  Precipitation is obtained from a 2 km grid. The authors refer to gauge undercatch to explain why catchment-scale precipitation is lower than streamflow (i.e., a runoff coefficient greater than 1). However, this discrepancy may also be due to the smoothed topography of the 2 km dataset used in the gridded precipitation interpolation scheme, which may be biased relative to the actual elevation distribution of the catchment. I did not clearly understand how precipitation was interpolated from this 2 km source to the 900 m × 700 m resolution. This processing involves two DEMs. The authors should be more specific about the source and/or resampling methods used for both DEMs. I also recommend explicitly writing the equations used for temperature and precipitation interpolation. Indeed, I do not understand why $SFCF$ is referred to as a “multiplicative correction factor,” whereas $SFCF_{ELE}$ is described as “a linear elevation lapse rate.” Precipitation cannot be interpolated using a temperature-like lapse rate, as this could lead to negative precipitation values. Moreover, linear scaling with elevation is not always chosen to interpolate precipitation, as it does not represent the observed capping of precipitation at high elevations due to atmospheric moisture depletion (e.g., Liston and Elder, 2006).
  L245: Why is “air temperature capped at a minimum of 0 °C”?
  L247: The authors mention “some physical inconsistencies, such as the omission of refreezing.” What are the other inconsistencies?
  L450: The lack of “temporal continuity” of the ASO data is not a convincing argument for rejecting these data. ASO provides SWE maps every two weeks in several gauged catchments over multiple years. ASO catchments would actually form an ideal dataset to test the inversion of SWE using the proposed framework.
  Formatting: I think that the acronyms (“SCFCF”) or text (“snow”) in the equations should not be italicized (e.g. using \mathrm).
  Thank you for these minor suggestions. We will incorporate them in the manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2025-3610-AC2

Pau Wiersma, Jan Magnusson, Nadav Peleg, Bettina Schaefli, and Grégoire Mariéthoz

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Short summary

Using a newly introduced inverse hydrological modeling framework, we demonstrate that streamflow observations have the potential to improve snow mass reconstructions, but that non-uniqueness in the snow-streamflow relationship and uncertainties in the inverse modeling chain can easily stand in the way. We also show that streamflow is most helpful in estimating catchment-aggregated properties of snow mass reconstructions, in particular catchment-aggregated melt rates.


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