Radar Equivalent Snowpack: reducing the number of snow layers while retaining its microwave properties and bulk snow mass

Meloche, Julien; Leroux, Nicolas R.; Montpetit, Benoit; Vionnet, Vincent; Derksen, Chris

doi:https://doi.org/10.5194/egusphere-2024-3169

Preprints

https://doi.org/10.5194/egusphere-2024-3169

Preprints

25 Oct 2024

| 25 Oct 2024

Radar Equivalent Snowpack: reducing the number of snow layers while retaining its microwave properties and bulk snow mass

Julien Meloche, Nicolas R. Leroux, Benoit Montpetit, Vincent Vionnet, and Chris Derksen

Abstract. Snow water equivalent (SWE) retrieval from Ku-band radar measurements is possible with complex retrieval algorithms involving prior information on the snowpack microstructure and a microwave radiative transfer model to link backscatter measurements to snow properties. A key variable in a retrieval is the number of snow layers, with more complex layering yielding richer information but at increased computational cost. Here, we show the capabilities of a new method to simplify a complex multilayered snowpack to less than or equal to 3 layers, while preserving the microwave scattering behavior of the snowpack and conserving the bulk snow water equivalent. The method is based on a K-means clustering algorithm to group the snow layers based on the extinction coefficient and the height of the layer. Then, a weighted average using the extinction coefficient and the thickness was applied to the snow properties. We evaluated our method using snow properties from simulations of the SVS-2/Crocus physical snow model at 11 sites spanning a large variety of climates across the world and the Snow Microwave Radiative Transfer model to calculate backscatter at 17.25 GHz. Grouping and averaging snow stratigraphy into 3 layers effectively reproduced the total snowpack backscatter of multi-layered snowpacks with overall root mean squared error = 0.5 dB and R² = 0.98. Using this methodology, SWE retrievals can be applied to simplified snowpacks, while maintaining similar scattering behavior, without compromising the modeled snowpack properties. Reduction in the mathematical complexity of SWE retrieval cost functions and reduction in computation of up to 80 % can be gained by using fewer layers in the SWE retrieval.

Received: 11 Oct 2024 – Discussion started: 25 Oct 2024

Competing interests: At least one of the (co-)authors is a member of the editorial board of The Cryosphere.

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Journal article(s) based on this preprint

07 Aug 2025

Radar-equivalent snowpack: reducing the number of snow layers while retaining their microwave properties and bulk snow mass

Julien Meloche, Nicolas R. Leroux, Benoit Montpetit, Vincent Vionnet, and Chris Derksen

The Cryosphere, 19, 2949–2962, https://doi.org/10.5194/tc-19-2949-2025,https://doi.org/10.5194/tc-19-2949-2025, 2025

Short summary

Julien Meloche, Nicolas R. Leroux, Benoit Montpetit, Vincent Vionnet, and Chris Derksen

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-3169', Anonymous Referee #1, 03 Jan 2025
Dear authors,
The manuscript presents a novel method for simplifying a multi-layer snowpack to a simplified 2- or 3-layer snowpack, while preserving snow mass and backscatter, in order to
improve the computational cost of the forward model, and

reduce the mathematical complexity of the cost function used in the associated inverse problem of retrieving snow properties (primarily SWE) from remote sensing observations.

The method is applied to several sites with different environmental conditions reflected in the meteorological input data used. Otherwise, the study is based on simulations. The proposed method is compared with the complex multi-layered snowpack and other simpler methods used in previous studies. The manuscript is well written and concise and addresses a well-defined problem in snow remote sensing. However, the manuscript does not discuss how the proposed method would be used in retrieval problems. As this is the motivating problem for the study, such a discussion is necessary. In addition, I would like to see a more detailed description and/or discussion of some aspects of the paper (see specific comments below).

Specific Comments
Lines 35-38, end of paragraph “…, but can prove challenging for remote sensing applications.” Could you elaborate on some of these challenges?

Your proposed method is based on K-means clustering. Being a central part of the proposed method, could you add more detailed description of the method? (sections 2.4. and 3.1. of the manuscript).

The K-means clustering algorithm is known to converge to a local minimum, which is not necessarily a global one. Therefore, the results can depend on the chosen initial points of the cluster means. The initial points should be mentioned in the manuscript.

It would be interesting to see an example of the plane with the data points, initial mean points, and the converged clusters. If you don’t consider this figure informative, it does not have to appear in the manuscript, but perhaps you could produce it in the comments?

You are using a two-dimensional parameter space in the K-means clustering. Could you comment on the choice of the parameter space? Is there a reason to consider the extinction coefficient instead of both scattering and absorption coefficients in a three-dimensional parameter space?

In section 2.4. of the manuscript, you describe the K-means clustering and the equal thickness grouping in rather equal terms, although the proposed method uses the K-means clustering. If you find it appropriate, please consider reorganising the section so that the K-means is described first as the primary method, and the equal thickness grouping is described then as a secondary method that is used for comparison. In my opinion, this would make the section easier to read.

In section 2.4. of the manuscript, you describe the weighting of the more complex snowpack into the simplified layers. You use the thickness based weighting (h-weighting) for the equal thickness grouping and optical thickness based weighting (τ-weighting) for the 3- and 2-means clusters. Does this not make the comparison between the two clustering methods unfair?

The last paragraph of section 2.4. (lines 162-170) describes the problem of removing interfaces when merging layers. This is a very important and an interesting point. However, I found the paragraph somewhat difficult to read. If you agree, please consider reorganising the paragraph so that the problem is introduced first, and then the solution.

Paragraph on lines 221-226 discusses the case when K-means finds clusters of layers that are not connected. In the averaging of such layers, scattering and absorption properties of the layers are not the only things affecting, but also the (optical) depth of the layers in the snowpack. For example, if two layers with equal thickness and extinction coefficient are placed at the top and bottom of the snowpack respectively, their effect to the total backscatter is not the same. In such a case moving one of the layers either from top to bottom or from bottom to up might not make sense. Could you comment on this?

The proposed method is based on an idea of running a snow process model with a high number of layers. However, this can also be computationally challenging. Can you comment on how computationally demanding it is to run the snow process model compared to the radiative transfer model? How does the computational cost increase with increasing number of layers (e.g. is running a 4-layer simulation much less expensive than running a 40-layer simulation?)

Your proposed method merges the snow layers based on their microwave properties, whereas the snow process model merges them (when the maximum number of layers is reached) based on their physical properties. The latter preserves the physical properties of the snowpack, while in the former does not. In terms of the computational cost of the radiative transfer simulation and the complexity of the cost function, the same benefits are obtained by running a snow process model with the maximum number of layers set to three. How does the simulated backscatter from such a snow process model run compare with that simulated using your proposed method? Does your method have a clear advantage in terms of the backscatter when both options are compared with the 50-layer run?

How could your proposed method be applied to different snow retrieval problems?

Is it only suitable for retrieving SWE or can it also be used to retrieve other snow variables? When simplifying the snowpack, the averaging of the physical snow variables is done using extinction coefficient. Therefore, the effective values don’t have the same physical meaning. Does this limit the potential application of the proposed method?

In the Bayesian retrieval, the cost function is defined as the sum of the mismatch term (between the observed and the modelled microwave signature) and the prior term. Your proposed method affects both terms; the mismatch term through the forward model configuration (how many layers are assumed) and the prior term. Can you comment on the use of the proposed method in this case?

On the other hand, in the data assimilation approach (e.g. particle filter), your method seems to have a more straightforward application to simplify the snowpack before the radiative transfer simulation to save computation time in the radiative transfer simulation. However, this approach would require an ensemble of snow process model runs, which can be computationally expensive. Can you comment on the use of the proposed method in this case?

Technical Comments
Line 151: ”We investigated three different ways of averageing: …”. Only two are mentioned in the text.

Line 26-27: “More typically, observations are unavailable, so snowpack information must come from …”. “must” is perhaps too strong word?

Line 104: “The choice of the electromagnetic model does not influence the final result of this method, …”. This could be wrongly understood to mean that the used electromagnetic model does affect the simulated backscatter (the result) and through that, the conclusions of the manuscript. Could you use another expression?

Line 119: “where p_11= … and p_22 = … are defined from the phase function.” repeats line 116.

Line 121: Reference should be (Sihvola,1999)?

Lines 145,149,150,etc.: I don’t think symbols after text require parenthesis. E.g. “extinction coefficient (κ_e)” --> “extinction coefficient κ_e” and “layer thickness (h_i)” --> “layer thickness h_i”.

Should section 3 of the manuscript be “Results and Discussions”?

Lines 176,177: The word “scattering” is used but is supposed to be “extinction”?

Suggestion: Consider using “h-weighting” and “τ-weighting” (instead of keD) for the two layer-averaging methods.

Line 190-191: “3-Kmeans-hi” can be slightly confusing. “K” stands for the number of clusters; would it make more sense to write “3-means”? Also, in my opinion separating the clustering and averaging methods could make the expression clearer. Suggestion: “3-means clustering with h-weighting.” Apply as you see fit.

Similarly, “3-equal-h_i” could be “3-equal-thickness” or “3-equal-h” omitting the subindex i as it is used to refer to the 50-layer snowpack. Using “3-equal-thickness” for clustering and “h-weighting” for the weighting could help to avoid confusion between the two. Apply as you see fit.

Lines 202,205: Same suggestion; consider changing “3-Kmeans-keD” to something along the lines “3-means clustering with τ-weighting” or “3-means clustering with τ-averaging”. Apply as you see fit.

Figure 3: Y label of the bottom figures should be “difference” as “bias” represents systematic or average difference. Same in figure caption.

Line 230: “Frequency analysis” has other meanings. Consider different expression.
Citation: https://doi.org/10.5194/egusphere-2024-3169-RC1
- AC1: 'Reply on RC1', Julien Meloche, 31 Mar 2025
  
  Thank you for your constructive comments. See the attached document for a detailed list of the response to your comments.
  
  Citation: https://doi.org/10.5194/egusphere-2024-3169-AC1
RC2:
'Comment on egusphere-2024-3169', Anonymous Referee #2, 17 Feb 2025

In this study, the author analyzed the number of layers of snowpacks needed to retain the same radar signals as the measurements.

One question for this topic is whether the layer simplification was utilized directly in the snow process simulation (e.g., using Crocus) or it was solely utilized in the microwave simulation. A clarification may be needed in the abs.
Another point is that the authors did experiments for 11 sites globally, whereas only results from three sites were explicitly presented in Fig. 3-5. However, I may be more interested in knowing how snow differs in different sites and what is the key characteristic (wind slab, melting?) that results in the requirement for 2-3 snow layers.
The authors didn't clearly state whether they focus mainly on dry snow.

More detailed comments are as follows:

1. Lines 35-38: Actually, multiple layers provide a continuous, high-resolution temperature profile inside the snowpack, which benefits most to the accuracy of snow microstructure simulation. To retain snow mass, people don't need a fine-resolution snow process model. In addition, this kind of model has great potential to accurately simulate the snowmelt process, because the fine-resolution snow layers can have different snow temperatures, for the model to evaluate whether the melting point has met.
2. Line 40: In Pan et al. (2017), only two snow layers were retrieved. Therefore, it does not have the "large numbers of layers" problem mentioned in the lines 42–43 followed. This part needs a revision.
3. Lines 44-49: It is suggested to separate the retrieval studies that considered only one layer and two layers. The use of two layers had been a tradeoff between simulation accuracy and retrieval feasibility. It is ok that the authors think they should at least consider one more layer, i.e., to consider 3 layers in total. However, the following sentence sounds really strange in the current manuscript:

"This is why current retrievals typically employ a one or two-layer model (Saberi et al., 2021; Durand et al., 2024; Pan et al., 2017). However, neglecting stratigraphy by using a small number of layers model reduces the performance of the retrieval (Durand et al., 2011) because layering strongly influences the backscattering properties of snow (Rutter et al., 2016)."

Actually, the studies in Saberi et al., 2021 and Pan et al., 2017 have considered the snow stratigraphy in some sense.

Line 56: It is suggested to consider this reference:

Yu, Y., Pan, J., Shi, J., 2021. Evaluation of the effective microstructure parameter of the microwave emission model of layered snowpack for multiple-layer snow. Remote Sens. 13. https://doi.org/10.3390/rs13102012.

By retaining air-snow and snow-soil boundary reflectivities using the topmost and bottom-most snow density instead of the profile-average density, the idea of Yu et al. (2021) becomes an indirect supporter of your 3-layer configuration suggestion.
Line 56: The snow type in Pan et al. (2017) was taiga snow. It does not have a wind-slab layer.
Line 5: According to line 65, it is better to say, "2-3 layers" directly in the abs.
Line 88: Could you add more details for the wind-slab and basal vegetation consideration and state why they are important in your study?
Line 127: The use of "presented" is confusing here. Better say, could be different for different snow grain types, according to Picard et al., 2022.
Line 130: However, the key point and importance for evaluating using different frequencies is that one can use a single profile to simulate multiple frequencies correctly, despite different penetration depths.
Section 2.4: A K-means classification will not work well directly, because you usually will not be allowed to combine layers that are not adjacent. More details are needed here.
Line 153: To preserve mass and snow depth, one can also actually only tune the snow microstructure parameter, because any change you make will not go into the subsequent CROCUS simulation. The snow layer simplification for RT calculation was operated stand-alone.
Figure 2 shows two examples where the existence of surface wind slabs and intermediate melt-freeze crusts inside the snowpacks can be two reasons to suggest using 3 layers, respectively. Are the first and second rows for the same site?
Line 189: Which does the "keD method" refer to in the context of Section 2.4?
From Figure 2, did you compare the differences in efficiency between applying K-means grouping on extinction coefficients and on original snow physical parameters (SSA, density, etc.)?

Lines 195-196: The error increases with increasing snow depth and increasing SSA, too, as the snowpack evolves with time. From my experience, your results shown in Fig. 3 have not included the snowmelt period yet. Therefore, it should not be caused by warmer snow temperatures, at least solely. By the way, it could be better to add subplots showing snow depth and air temperature time series together in Fig. 3.

Lines 214: What does "transparent layer simulation" mean again? Did you mean mandatorily setting the interface transmissivities to 1 and interface reflectivities to 0, although there are 3 layers with different refraction indexes?

Citation: https://doi.org/10.5194/egusphere-2024-3169-RC2
- AC1: 'Reply on RC1', Julien Meloche, 31 Mar 2025
  
  Thank you for your constructive comments. See the attached document for a detailed list of the response to your comments.
  
  Citation: https://doi.org/10.5194/egusphere-2024-3169-AC1

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-3169', Anonymous Referee #1, 03 Jan 2025
Dear authors,
The manuscript presents a novel method for simplifying a multi-layer snowpack to a simplified 2- or 3-layer snowpack, while preserving snow mass and backscatter, in order to
improve the computational cost of the forward model, and

reduce the mathematical complexity of the cost function used in the associated inverse problem of retrieving snow properties (primarily SWE) from remote sensing observations.

The method is applied to several sites with different environmental conditions reflected in the meteorological input data used. Otherwise, the study is based on simulations. The proposed method is compared with the complex multi-layered snowpack and other simpler methods used in previous studies. The manuscript is well written and concise and addresses a well-defined problem in snow remote sensing. However, the manuscript does not discuss how the proposed method would be used in retrieval problems. As this is the motivating problem for the study, such a discussion is necessary. In addition, I would like to see a more detailed description and/or discussion of some aspects of the paper (see specific comments below).

Specific Comments
Lines 35-38, end of paragraph “…, but can prove challenging for remote sensing applications.” Could you elaborate on some of these challenges?

Your proposed method is based on K-means clustering. Being a central part of the proposed method, could you add more detailed description of the method? (sections 2.4. and 3.1. of the manuscript).

The K-means clustering algorithm is known to converge to a local minimum, which is not necessarily a global one. Therefore, the results can depend on the chosen initial points of the cluster means. The initial points should be mentioned in the manuscript.

It would be interesting to see an example of the plane with the data points, initial mean points, and the converged clusters. If you don’t consider this figure informative, it does not have to appear in the manuscript, but perhaps you could produce it in the comments?

You are using a two-dimensional parameter space in the K-means clustering. Could you comment on the choice of the parameter space? Is there a reason to consider the extinction coefficient instead of both scattering and absorption coefficients in a three-dimensional parameter space?

In section 2.4. of the manuscript, you describe the K-means clustering and the equal thickness grouping in rather equal terms, although the proposed method uses the K-means clustering. If you find it appropriate, please consider reorganising the section so that the K-means is described first as the primary method, and the equal thickness grouping is described then as a secondary method that is used for comparison. In my opinion, this would make the section easier to read.

In section 2.4. of the manuscript, you describe the weighting of the more complex snowpack into the simplified layers. You use the thickness based weighting (h-weighting) for the equal thickness grouping and optical thickness based weighting (τ-weighting) for the 3- and 2-means clusters. Does this not make the comparison between the two clustering methods unfair?

The last paragraph of section 2.4. (lines 162-170) describes the problem of removing interfaces when merging layers. This is a very important and an interesting point. However, I found the paragraph somewhat difficult to read. If you agree, please consider reorganising the paragraph so that the problem is introduced first, and then the solution.

Paragraph on lines 221-226 discusses the case when K-means finds clusters of layers that are not connected. In the averaging of such layers, scattering and absorption properties of the layers are not the only things affecting, but also the (optical) depth of the layers in the snowpack. For example, if two layers with equal thickness and extinction coefficient are placed at the top and bottom of the snowpack respectively, their effect to the total backscatter is not the same. In such a case moving one of the layers either from top to bottom or from bottom to up might not make sense. Could you comment on this?

The proposed method is based on an idea of running a snow process model with a high number of layers. However, this can also be computationally challenging. Can you comment on how computationally demanding it is to run the snow process model compared to the radiative transfer model? How does the computational cost increase with increasing number of layers (e.g. is running a 4-layer simulation much less expensive than running a 40-layer simulation?)

Your proposed method merges the snow layers based on their microwave properties, whereas the snow process model merges them (when the maximum number of layers is reached) based on their physical properties. The latter preserves the physical properties of the snowpack, while in the former does not. In terms of the computational cost of the radiative transfer simulation and the complexity of the cost function, the same benefits are obtained by running a snow process model with the maximum number of layers set to three. How does the simulated backscatter from such a snow process model run compare with that simulated using your proposed method? Does your method have a clear advantage in terms of the backscatter when both options are compared with the 50-layer run?

How could your proposed method be applied to different snow retrieval problems?

Is it only suitable for retrieving SWE or can it also be used to retrieve other snow variables? When simplifying the snowpack, the averaging of the physical snow variables is done using extinction coefficient. Therefore, the effective values don’t have the same physical meaning. Does this limit the potential application of the proposed method?

In the Bayesian retrieval, the cost function is defined as the sum of the mismatch term (between the observed and the modelled microwave signature) and the prior term. Your proposed method affects both terms; the mismatch term through the forward model configuration (how many layers are assumed) and the prior term. Can you comment on the use of the proposed method in this case?

On the other hand, in the data assimilation approach (e.g. particle filter), your method seems to have a more straightforward application to simplify the snowpack before the radiative transfer simulation to save computation time in the radiative transfer simulation. However, this approach would require an ensemble of snow process model runs, which can be computationally expensive. Can you comment on the use of the proposed method in this case?

Technical Comments
Line 151: ”We investigated three different ways of averageing: …”. Only two are mentioned in the text.

Line 26-27: “More typically, observations are unavailable, so snowpack information must come from …”. “must” is perhaps too strong word?

Line 104: “The choice of the electromagnetic model does not influence the final result of this method, …”. This could be wrongly understood to mean that the used electromagnetic model does affect the simulated backscatter (the result) and through that, the conclusions of the manuscript. Could you use another expression?

Line 119: “where p_11= … and p_22 = … are defined from the phase function.” repeats line 116.

Line 121: Reference should be (Sihvola,1999)?

Lines 145,149,150,etc.: I don’t think symbols after text require parenthesis. E.g. “extinction coefficient (κ_e)” --> “extinction coefficient κ_e” and “layer thickness (h_i)” --> “layer thickness h_i”.

Should section 3 of the manuscript be “Results and Discussions”?

Lines 176,177: The word “scattering” is used but is supposed to be “extinction”?

Suggestion: Consider using “h-weighting” and “τ-weighting” (instead of keD) for the two layer-averaging methods.

Line 190-191: “3-Kmeans-hi” can be slightly confusing. “K” stands for the number of clusters; would it make more sense to write “3-means”? Also, in my opinion separating the clustering and averaging methods could make the expression clearer. Suggestion: “3-means clustering with h-weighting.” Apply as you see fit.

Similarly, “3-equal-h_i” could be “3-equal-thickness” or “3-equal-h” omitting the subindex i as it is used to refer to the 50-layer snowpack. Using “3-equal-thickness” for clustering and “h-weighting” for the weighting could help to avoid confusion between the two. Apply as you see fit.

Lines 202,205: Same suggestion; consider changing “3-Kmeans-keD” to something along the lines “3-means clustering with τ-weighting” or “3-means clustering with τ-averaging”. Apply as you see fit.

Figure 3: Y label of the bottom figures should be “difference” as “bias” represents systematic or average difference. Same in figure caption.

Line 230: “Frequency analysis” has other meanings. Consider different expression.
Citation: https://doi.org/10.5194/egusphere-2024-3169-RC1
- AC1: 'Reply on RC1', Julien Meloche, 31 Mar 2025
  
  Thank you for your constructive comments. See the attached document for a detailed list of the response to your comments.
  
  Citation: https://doi.org/10.5194/egusphere-2024-3169-AC1
RC2:
'Comment on egusphere-2024-3169', Anonymous Referee #2, 17 Feb 2025

In this study, the author analyzed the number of layers of snowpacks needed to retain the same radar signals as the measurements.

One question for this topic is whether the layer simplification was utilized directly in the snow process simulation (e.g., using Crocus) or it was solely utilized in the microwave simulation. A clarification may be needed in the abs.
Another point is that the authors did experiments for 11 sites globally, whereas only results from three sites were explicitly presented in Fig. 3-5. However, I may be more interested in knowing how snow differs in different sites and what is the key characteristic (wind slab, melting?) that results in the requirement for 2-3 snow layers.
The authors didn't clearly state whether they focus mainly on dry snow.

More detailed comments are as follows:

1. Lines 35-38: Actually, multiple layers provide a continuous, high-resolution temperature profile inside the snowpack, which benefits most to the accuracy of snow microstructure simulation. To retain snow mass, people don't need a fine-resolution snow process model. In addition, this kind of model has great potential to accurately simulate the snowmelt process, because the fine-resolution snow layers can have different snow temperatures, for the model to evaluate whether the melting point has met.
2. Line 40: In Pan et al. (2017), only two snow layers were retrieved. Therefore, it does not have the "large numbers of layers" problem mentioned in the lines 42–43 followed. This part needs a revision.
3. Lines 44-49: It is suggested to separate the retrieval studies that considered only one layer and two layers. The use of two layers had been a tradeoff between simulation accuracy and retrieval feasibility. It is ok that the authors think they should at least consider one more layer, i.e., to consider 3 layers in total. However, the following sentence sounds really strange in the current manuscript:

"This is why current retrievals typically employ a one or two-layer model (Saberi et al., 2021; Durand et al., 2024; Pan et al., 2017). However, neglecting stratigraphy by using a small number of layers model reduces the performance of the retrieval (Durand et al., 2011) because layering strongly influences the backscattering properties of snow (Rutter et al., 2016)."

Actually, the studies in Saberi et al., 2021 and Pan et al., 2017 have considered the snow stratigraphy in some sense.

Line 56: It is suggested to consider this reference:

Yu, Y., Pan, J., Shi, J., 2021. Evaluation of the effective microstructure parameter of the microwave emission model of layered snowpack for multiple-layer snow. Remote Sens. 13. https://doi.org/10.3390/rs13102012.

By retaining air-snow and snow-soil boundary reflectivities using the topmost and bottom-most snow density instead of the profile-average density, the idea of Yu et al. (2021) becomes an indirect supporter of your 3-layer configuration suggestion.
Line 56: The snow type in Pan et al. (2017) was taiga snow. It does not have a wind-slab layer.
Line 5: According to line 65, it is better to say, "2-3 layers" directly in the abs.
Line 88: Could you add more details for the wind-slab and basal vegetation consideration and state why they are important in your study?
Line 127: The use of "presented" is confusing here. Better say, could be different for different snow grain types, according to Picard et al., 2022.
Line 130: However, the key point and importance for evaluating using different frequencies is that one can use a single profile to simulate multiple frequencies correctly, despite different penetration depths.
Section 2.4: A K-means classification will not work well directly, because you usually will not be allowed to combine layers that are not adjacent. More details are needed here.
Line 153: To preserve mass and snow depth, one can also actually only tune the snow microstructure parameter, because any change you make will not go into the subsequent CROCUS simulation. The snow layer simplification for RT calculation was operated stand-alone.
Figure 2 shows two examples where the existence of surface wind slabs and intermediate melt-freeze crusts inside the snowpacks can be two reasons to suggest using 3 layers, respectively. Are the first and second rows for the same site?
Line 189: Which does the "keD method" refer to in the context of Section 2.4?
From Figure 2, did you compare the differences in efficiency between applying K-means grouping on extinction coefficients and on original snow physical parameters (SSA, density, etc.)?

Lines 195-196: The error increases with increasing snow depth and increasing SSA, too, as the snowpack evolves with time. From my experience, your results shown in Fig. 3 have not included the snowmelt period yet. Therefore, it should not be caused by warmer snow temperatures, at least solely. By the way, it could be better to add subplots showing snow depth and air temperature time series together in Fig. 3.

Lines 214: What does "transparent layer simulation" mean again? Did you mean mandatorily setting the interface transmissivities to 1 and interface reflectivities to 0, although there are 3 layers with different refraction indexes?

Citation: https://doi.org/10.5194/egusphere-2024-3169-RC2
- AC1: 'Reply on RC1', Julien Meloche, 31 Mar 2025
  
  Thank you for your constructive comments. See the attached document for a detailed list of the response to your comments.
  
  Citation: https://doi.org/10.5194/egusphere-2024-3169-AC1

Peer review completion

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

ED: Publish subject to revisions (further review by editor and referees) (01 Apr 2025) by Jürg Schweizer

AR by Julien Meloche on behalf of the Authors (01 Apr 2025) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (06 Apr 2025) by Jürg Schweizer

RR by Anonymous Referee #2 (08 Apr 2025)

RR by Anonymous Referee #1 (25 Apr 2025)

Suggestions for revision or reasons for rejection

The changes that the authors have made to the manuscript have improved it. However, I believe that the paper would benefit from some further considerations. In the following comments the line numbers refer to the “track-changes” document.

General comments
• The core idea in the paper, the application of the K-clustering method to reduce the complexity of the snowpack while retaining the backscattering properties and snow mass, is nice and straightforward. However, compared to this, the paper feels a bit cluttered. I think the manuscript would benefit from more clarity.
• For example, are all the 6 reference methods needed? I recognize that together they show how much things can go wrong if one considers the simplest possible case (single layer snowpack), but on the other hand, is this essential to the main point of the manuscript? Is this additional information more important than the distraction that it adds to the results?
• In some instances, the notation is a bit confusing (see specific comments for examples).
• I hope that the authors would check the manuscript again and look for places to improve the clarity beyond the specific comments listed below.

Specific comments
1. Lines 3-4: “…, with more complex layering yielding richer information but at increased computational cost.” It should mention also the increased number of unknowns, since the context is retrieval process.
2. Line 5: “…, while preserving the microwave scattering …” Perhaps “preserving” is too strong term here, since the scattering does change in the process.
3. Line 7: “Then, a weighted average…” This sentence seems disconnected from the previous sentence.
4. Lines 13-14: “SWE retrievals can be applied to simplified snowpacks, while maintaining similar scattering behavior without compromising the modeled snowpack properties”: The use of the method for retrievals was not demonstrated in the paper.
5. Lines 39-40: “… or the identification of weak layers in the context of …” is this part relevant?
6. Line 44: “SWE (depth and density)” -> “SWE (function of depth and density)”
7. Lines 69-71: Sentence starting with “If a method that …” sounds bit weird, please check.
8. Lines 119-142 contain quite specific description of the RT modeling. Is this necessary for this paper?
9. Line 153-154: Ikotun et al. is referred twice.
10. Line 163: Unnecessary parenthesis, (\kappa_e) -> \kappa_e, please fix elsewhere also.
11. Line 166: “(referred to equal thickness)” -> to as
12. Lines 165-170: h_norm is not defined here? Also, the paragraph discussion h_norm seems to cut between two parts of the text that both discuss K-means clustering. Please consider clarifying and restructuring.
13. Also, this paragraph could be otherwise improved. E.g. the sentence “The 1-layer was created by grouping all layers into 1 group” is a bit weird.
14. Lines 171-173: Use of h_group is a bit confusing. Is “group” used here as an index, i.e. group = 1,2,3? On the other hand, is the symbol used anywhere in the text after (is it necessary)?
15. Line 174: “(referred to as h_i)” -> “(… h_i-average/averaging)”
16. Line 179: “… whole snowpack h_n …”: Confusing to use n as a subindex, how does it differ from h_i?. E.g. h_snow could be better.
17. Schematics in Figure 1 could be improved to clarify the process. At the moment it is difficult to interpret. I think what should be highlighted is the fact that the process in this paper has two levels, the grouping method and the averaging method, both of which have multiple options, and the different combinations of these two produce the “methods” that are being compared. In addition, some technical points: 0.33 and 0.66 or 1/3 and 2/3? The word “group” appears both as italic and non-italic text. Also, please make the equations less vague, e.g. h_group = \sum h_i, what is summer over?
18. Line 184: re-defines RMSE, why? It’s not used in that section again.
19. Line 195: The last sentence in the paragraph (starting with “The difference in backscatter was estimated…”) is unclear.
20. Line 206: “… layers classified as 2 at …”. Please make it more clear, e.g.: “classified/assigned as/in/under group/category 2.
21. Line 210: “… with averaging using the h_i method” This is referred later as h_i-average method? Please use consistent terms for clarity.
22. Line 211: “(1 to 3)” -> “(from 1 to 3)”
23. Line 213: Please clarify sentence “Using a cluster is not always better …”. E.g. “… Using a K-means clustering method is not …”
24. Line 214: “\tau-method”. This is later referred to as \tau-average? Please use consistent terms for clarity.
25. Line 216-217: “Figure 4 shows biases … as a function of snow properties”. Seems a bit awkward sentence, add “methods” after mentioning the two methods?
26. Figure 3. is a bit cluttered with the number of time series. Is it necessary to show the differences in its own plots? How would the results look in a scatterplot?
27. Line 231-233 “The 3-equal with \tau-average achieved similar performance to the …” This seems like an important result. Why is it not mentioned in the conclusions? Is it not so that due to its simplicity, the 3-equal grouping is more reliable compared to K means?
28. Could Table 2 be replaced with a graph representing the information, and the Table moved to appendix? This could help to convey the message more clearly.
29. Section 3.2. is interesting but on the other hand the manuscript is quite long. Is this section necessary for the main point of the manuscript? If it is, then some technical comments: “> 10 GHz, up to 30 GHz” please use words instead of symbols, e.g. “from 10 GHz, up to 30 GHz”. Same comment for “> 40 GHz”. Sentence “…3-equal with h-average because…” add “method” after h-average.
30. Table 3: Change column name “layering” -> “number of layers” and remove repetition of “layer” from the data rows. Similarly, column name “\sigma_0 computation time” could include the unit (s) and avoid repeating it on the data rows. In my opinion the grouping times and associated \sigma_0 computation times (the last 4 numbers in the table) could be given in the caption of the table to reduce its size and make it clearer.
31. Line 274: Similar to previous comment, sentences “… while preserving their microwave scattering behavior …” expression “preserving” is perhaps too strong. Please consider another expression, e.g. “aiming to preserve”.
32. The last paragraph in conclusions is a good addition. However, I found the paragraph too vague. The application of the method to SWE retrievals is one of the main motivations of the study and it is mentioned on many occasions in the manuscript. Therefore, I think it requires more attention and concrete detail. Also, I think this part should be moved to discussions and summarized in conclusions in shorter format.

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ED: Publish subject to minor revisions (review by editor) (30 Apr 2025) by Jürg Schweizer

AR by Julien Meloche on behalf of the Authors (14 May 2025) Author's response Author's tracked changes Manuscript

ED: Publish as is (22 May 2025) by Jürg Schweizer

AR by Julien Meloche on behalf of the Authors (23 May 2025) Manuscript

Journal article(s) based on this preprint

07 Aug 2025

Radar-equivalent snowpack: reducing the number of snow layers while retaining their microwave properties and bulk snow mass

Julien Meloche, Nicolas R. Leroux, Benoit Montpetit, Vincent Vionnet, and Chris Derksen

The Cryosphere, 19, 2949–2962, https://doi.org/10.5194/tc-19-2949-2025,https://doi.org/10.5194/tc-19-2949-2025, 2025

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Julien Meloche, Nicolas R. Leroux, Benoit Montpetit, Vincent Vionnet, and Chris Derksen

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Measuring the snow mass from radar measurements is possible with information on the snow and a radar model to link the measurements to snow. A key variable in a retrieval is the number of snow layers, with more layer yielding richer information but at increased computational cost. Here, we show the capabilities of a new method to simplify a complex snowpack, while preserving the scattering behavior of the snowpack and conserving the mass.


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