the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 21 Feb 2024
A mathematical model to improve water storage of glacial lakes prediction towards addressing glacial lake outburst floods
Abstract. Moraine-dammed glacial lakes are vital sources of freshwater but also pose a hazard to mountain communities if they drain in sudden glacial lake outburst floods. Accurately measuring the water storage of these lakes is crucial to ensure sustainable use and safeguard mountain communities downstream. However, thousands of glacial lakes still lack a robust estimate of their water storages because bathymetric surveys in remote regions are difficult and expensive. Here we geometrically approximate the shape and depths of moraine-dammed lakes and provide a cost-effective model to improve lake water storage estimation. Our model uses the outline and the terrain surrounding a glacier lake as input data, assuming a parabolic lake bottom and constant hillslope angles. We validate our model using ten new bathymetrically surveyed glacial lakes on the Qinghai-Tibet Plateau, and compiled data from 34 recently measured lakes. Our model overcomes the autocorrelation issue inherent in earlier area/depth-water storage relationships and incorporates an automated calculation process based on the topography and geometrical parameters specific to moraine-dammed lakes. Compared to other models, our model achieved the lowest average relative error of approximately 14 % when analyzing 44 observed data, surpassing the >44 % average relative error from alternative models. Finally, the model is used to calculate the water storage change of moraine-dammed lakes in the past 30 years in High Mountain Asia. The model has been proven to be robust and can be utilized to update the water storage of lake water for conducting further management of glacial lakes with the potential for outburst floods in the world.
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RC1: 'Comment on hess-2024-24', Adam Emmer, 25 Mar 2024
This study approximates glacial lake volume from simplified geometric representations of 4 main sub-types of moraine-dammed lakes, considering glacier-lake relationship (connected vs. unconnected) and lake width to length ratio. The performance of this new approach is reportedly better than the performance of other methods (comparison in Table 5).
However, this is not surprising if the authors used the dataset of 44 Himalayan lakes with measured bathymetries to determine their parameters (section 3.3), and then use the same data to compare the performance of various methods (section 4.2). I hope I understood this correctly since the validation procedure is not described clearly in methods section. If I get it correctly, such performance evaluation is weak. A proper validation would require two independent datasets (training and testing).
And the whole validation procedure is even more confusing since only 4 bathymetries are mentioned as input data for model validation in section 4.1. This is statistically not convincing, considering 4 sub-types of moraine-dammed lakes and number of parameters that are used. Further, a subset of 12 lakes is used in section 5.1 while 4 and 10 lakes are mentioned in Conclusions. This needs to be clarified.
The application section 4.3 is not linked to the methodology. It is not clear what was done and whether (and how?) all 13,166 lakes mapped by Wang et al. (2020) were classified according to the classification scheme used in this study and whether all these are moraine-dammed lakes?
At the end, the importance of this improvement in lake volume estimation for GLOF studies (the main justification throughout the study) is unclear unless other (and much larger) sources of uncertainties in GLOF studies (e,g, coming up with realistic scenarios of GLOF triggers and GLOF mechanism, plausible breach development and dimensions, associated shape of the outburst hydrograph curve, % of lake volume release, etc.) are addressed.
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L39-40: this definition is artificial; moraine-dammed lakes not only trap meltwater (how about water from liquid precipitation?); debris at or near the termini of glacier doesn’t necessarily need to be a moraine
L53: ice- or landslide-dammed lakes may be unstable too
L72-74: the peak discharge is rather linked to the magnitude of triggering event than lake volume
L77: how much was that?
L126-127: this indication is not clear since the ratio is dimensionless (really a width of 1 m?)
Fig. 2: please only display parameters that are further use (remove slope beta, points f and g)
Table 2: please clarify whether alpha is mean or median slope (as mentioned in Table 3); what is the influence of DEM acquisition date on alpha estimation?
Table 4: what is simulated lake depth – a mean? And what do the two values in error column refer to?
Table 5: some of the lakes (e.g. Imja Tsho or Jialong Co) are represented more than once. This may influence performance evaluation; the areas of Jialong Co do not match between Table 4 and 5)
L313-315: R^2 will always be very high (>0.95) for most of the methods
Figure 8: I don’t understand what is the meaning of these box plots unless it is connected to measured data? The XY graph type (inset) is way more meaningful and the authors may consider showing a panel with performance of all methods in XY graphs.
L339: the scaling up of the lake volume estimation procedure to the whole HMA is not properly described in methods.
L367-376: this seems bit out of the context. Clearly, large lakes are frequently considered risky since lake area / volume is commonly used as GLOF susceptibility criteria.
L375: the annual expansion rate +5.6% a^-1 over 32 years does not correspond to a reported growth of 178% over this period
L395: they are not flat (as documented in your Fig. 10)
L433: what is MDLVL?
L453: the term “outburst water storage” is not appropriate. What is estimated here is a lake volume / lake water storage. It doesn’t have much to do with outburst / outburst volume.
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To sum up, this study can help to improve moraine-dammed lake volume estimates in HMA. However, especially the validation process needs to be clarified and treated in statistically convincing way. I recommend major revisions.
Citation: https://doi.org/10.5194/hess-2024-24-RC1 - AC5: 'Reply on RC1', Miaomiao Qi, 22 Sep 2024
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CC1: 'Comment on hess-2024-24', Huayu Zhang, 12 Aug 2024
This paper presents a novel method for estimate the water storage of moraine-dammed glacial lakes (MDLs) by using geometric approximations. This method addresses the significant challenges posed by traditional bathymetric surveys, which are often difficult and costly, particularly in remote and high-altitude regions. By utilizing the outline and surrounding terrain of a glacier lake, and assuming a parabolic lake bottom and constant hillslope angles, the model achieves a lower average relative error (~14%) compared to other models (>44%). The paper validates the model with data from newly surveyed glacial lakes on the Qinghai-Tibet Plateau and other recently measured lakes, proving its robustness and potential for global application in glacial lake management and GLOF risk assessment. Additionally, the study's classification of MDLs into glacier-contacted and glacier-uncontacted lakes, based on their geometric properties, enhances the model's applicability. Overall, this paper offers a significant advancement in glacial lake management, providing crucial insights for hazard mitigation and water resource management in glaciated regions.
Besides, some minor problems in this manuscript should be thought by the authors:
- Line 237 | The author did not mentioned how the threshold of m and n was determined, it should be given more explanations.
- Line 263 | The author mentioned ‘we trained our workflow on ...’. According to this paper, there is no parameter that needs to be trained or optimized, all parameters can be measured through the glacial lakes and their surrounding topology. The author should consider modifying the description of the workflow.
- Line 349 | The total volume of water storage in HMA is not accurate, since this is a method used to estimate the water storage of MDLs, while there are other kinds of glacia lakes such as ice-dammed glacial lakes supraglacial lakes etc., of which volume is accurate using this method.
- Line 433| MDLVL is not mentioned before.
Citation: https://doi.org/10.5194/hess-2024-24-CC1 -
AC2: 'Reply on CC1', Miaomiao Qi, 22 Sep 2024
Response to Reviewer,
This paper presents a novel method for estimate the water storage of moraine-dammed glacial lakes (MDLs) by using geometric approximations. This method addresses the significant challenges posed by traditional bathymetric surveys, which are often difficult and costly, particularly in remote and high-altitude regions. By utilizing the outline and surrounding terrain of a glacier lake, and assuming a parabolic lake bottom and constant hillslope angles, the model achieves a lower average relative error (~14%) compared to other models (>44%). The paper validates the model with data from newly surveyed glacial lakes on the Qinghai-Tibet Plateau and other recently measured lakes, proving its robustness and potential for global application in glacial lake management and GLOF risk assessment. Additionally, the study's classification of MDLs into glacier-contacted and glacier-uncontacted lakes, based on their geometric properties, enhances the model's applicability. Overall, this paper offers a significant advancement in glacial lake management, providing crucial insights for hazard mitigation and water resource management in glaciated regions.
Explanation and revision: Thank you very much for your positive feedback.
Besides, some minor problems in this manuscript should be thought by the authors:
- Line 237 | The author did not mentioned how the threshold of m and n was determined, it should be given more explanations.
Explanation and revision: Thank you very much for your affirmation and questions. We first simplified the model into four equations, with their solutions all dependent on the correct selection of m, n, and r. Based on the geometry of the glacial lake, we established a proportional relationship between m, n, r, and the glacier lake length (l). This proportional relationship is empirically defined and essentially represents a geometric segmentation of the glacial lake. The lake is divided into three sections, and the volume of each section is calculated separately. The total water storage of the lake is then obtained by summing the volumes of these three sections.
Therefore, we first used measured data from four glacial lakes to validate whether this proportional relationship was appropriate. After validation, we found that the empirically derived proportional relationship performed well. Hence, this study adopts this proportional relationship as the standard for the model's input parameters. No calibration or adjustments were made during this process. We have added the following explanation in lines 243 to 248 of the original text: " Based on the geometry of the glacial lake, we established a proportional relationship between m, n, r, and the glacier lake length (l). This proportional relationship is empirically defined and essentially represents a geometric segmentation of the glacial lake. The lake is divided into three sections, and the volume of each section is calculated separately. The total water storage of the lake is then obtained by summing the volumes of these three sections."
- Line 263 | The author mentioned ‘we trained our workflow on ...’. According to this paper, there is no parameter that needs to be trained or optimized, all parameters can be measured through the glacial lakes and their surrounding topology. The author should consider modifying the description of the workflow.
Explanation and revision: Thank you very much for pointing out the issue; we have revised the description of the workflow. The modified sentence is as follows: “We executed our workflow (Figure 6) on 44 MDLs in High Mountain Asia that have known depths and water storages.”
- Line 349 | The total volume of water storage in HMA is not accurate, since this is a method used to estimate the water storage of MDLs, while there are other kinds of glacial lakes such as ice-dammed glacial lakes supraglacial lakes etc., of which volume is accurate using this method.
Explanation and revision: Thank you very much for pointing out the issue; we have revised this sentence: “Over the past three decades, the overall MDL’s water storage increased by 8.94 km3 from 28.24 km3 in 1990, representing a growth of approximately 32%.”
- Line 433| MDLVL is not mentioned before.
Explanation and revision: Thank you very much for pointing out the issue. We have reviewed and revised the entire text, changing "MDLVM" to "our model."
Citation: https://doi.org/10.5194/hess-2024-24-AC2
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CC2: 'Comment on hess-2024-24', Huayu Zhang, 12 Aug 2024
This paper presents a novel method for estimate the water storage of moraine-dammed glacial lakes (MDLs) by using geometric approximations. This method addresses the significant challenges posed by traditional bathymetric surveys, which are often difficult and costly, particularly in remote and high-altitude regions. By utilizing the outline and surrounding terrain of a glacier lake, and assuming a parabolic lake bottom and constant hillslope angles, the model achieves a lower average relative error (~14%) compared to other models (>44%). The paper validates the model with data from newly surveyed glacial lakes on the Qinghai-Tibet Plateau and other recently measured lakes, proving its robustness and potential for global application in glacial lake management and GLOF risk assessment. Additionally, the study's classification of MDLs into glacier-contacted and glacier-uncontacted lakes, based on their geometric properties, enhances the model's applicability. Overall, this paper offers a significant advancement in glacial lake management, providing crucial insights for hazard mitigation and water resource management in glaciated regions.
Besides, some minor problems in this manuscript should be thought by the authors:
- Line 237 | The author did not mentioned how the threshold of m and n was determined, it should be given more explanations.
- Line 263 | The author mentioned ‘we trained our workflow on ...’. According to this paper, there is no parameter that needs to be trained or optimized, all parameters can be measured through the glacial lakes and their surrounding topology. The author should consider modifying the description of the workflow.
- Line 349 | The total volume of water storage in HMA is not accurate, since this is a method used to estimate the water storage of MDLs, while there are other kinds of glacia lakes such as ice-dammed glacial lakes supraglacial lakes etc., of which volume is accurate using this method.
- Line 433| MDLVL is not mentioned before.
Citation: https://doi.org/10.5194/hess-2024-24-CC2 -
AC3: 'Reply on CC2', Miaomiao Qi, 22 Sep 2024
Response to Reviewer,
This paper presents a novel method for estimate the water storage of moraine-dammed glacial lakes (MDLs) by using geometric approximations. This method addresses the significant challenges posed by traditional bathymetric surveys, which are often difficult and costly, particularly in remote and high-altitude regions. By utilizing the outline and surrounding terrain of a glacier lake, and assuming a parabolic lake bottom and constant hillslope angles, the model achieves a lower average relative error (~14%) compared to other models (>44%). The paper validates the model with data from newly surveyed glacial lakes on the Qinghai-Tibet Plateau and other recently measured lakes, proving its robustness and potential for global application in glacial lake management and GLOF risk assessment. Additionally, the study's classification of MDLs into glacier-contacted and glacier-uncontacted lakes, based on their geometric properties, enhances the model's applicability. Overall, this paper offers a significant advancement in glacial lake management, providing crucial insights for hazard mitigation and water resource management in glaciated regions.
Explanation and revision: Thank you very much for your positive feedback.
Besides, some minor problems in this manuscript should be thought by the authors:
- Line 237 | The author did not mentioned how the threshold of m and n was determined, it should be given more explanations.
Explanation and revision: Thank you very much for your affirmation and questions. We first simplified the model into four equations, with their solutions all dependent on the correct selection of m, n, and r. Based on the geometry of the glacial lake, we established a proportional relationship between m, n, r, and the glacier lake length (l). This proportional relationship is empirically defined and essentially represents a geometric segmentation of the glacial lake. The lake is divided into three sections, and the volume of each section is calculated separately. The total water storage of the lake is then obtained by summing the volumes of these three sections.
Therefore, we first used measured data from four glacial lakes to validate whether this proportional relationship was appropriate. After validation, we found that the empirically derived proportional relationship performed well. Hence, this study adopts this proportional relationship as the standard for the model's input parameters. No calibration or adjustments were made during this process. We have added the following explanation in lines 243 to 248 of the original text: " Based on the geometry of the glacial lake, we established a proportional relationship between m, n, r, and the glacier lake length (l). This proportional relationship is empirically defined and essentially represents a geometric segmentation of the glacial lake. The lake is divided into three sections, and the volume of each section is calculated separately. The total water storage of the lake is then obtained by summing the volumes of these three sections."
- Line 263 | The author mentioned ‘we trained our workflow on ...’. According to this paper, there is no parameter that needs to be trained or optimized, all parameters can be measured through the glacial lakes and their surrounding topology. The author should consider modifying the description of the workflow.
Explanation and revision: Thank you very much for pointing out the issue; we have revised the description of the workflow. The modified sentence is as follows: “We executed our workflow (Figure 6) on 44 MDLs in High Mountain Asia that have known depths and water storages.”
- Line 349 | The total volume of water storage in HMA is not accurate, since this is a method used to estimate the water storage of MDLs, while there are other kinds of glacial lakes such as ice-dammed glacial lakes supraglacial lakes etc., of which volume is accurate using this method.
Explanation and revision: Thank you very much for pointing out the issue; we have revised this sentence: “Over the past three decades, the overall MDL’s water storage increased by 8.94 km3 from 28.24 km3 in 1990, representing a growth of approximately 32%.”
- Line 433| MDLVL is not mentioned before.
Explanation and revision: Thank you very much for pointing out the issue. We have reviewed and revised the entire text, changing "MDLVM" to "our model."
Citation: https://doi.org/10.5194/hess-2024-24-AC3
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RC2: 'Comment on hess-2024-24', Anonymous Referee #2, 18 Aug 2024
Comments on “A mathematical model to improve water storage of glacial lakes prediction towards addressing glacial lake outburst floods” by Qi et al.
The authors developed a model to improve lake water estimation by simplifying the shape and depths of moraine-dammed lakes. The model showed better performance compared with other existing models. The author also used this model to calculate the water storage changes of moraine-dammed lakes in the past 30 years in High Mountain Asia. I think this study is a significant advancement in glacial volume estimation and it is worthy of being published after minor revision. Some major comments are shown as below:
1, Four types of moraine-dammed lakes are considered in this study. In fact, as shown in Figure 4 and Figure 10, the shape of MDLs is much more complicated. If the authors can classify the MDLs in more types in the future study, the estimation may be further improved.
2, Line 165-169, the authors gave the core assumption of the model. In the discussion section, the authors add some discussion about this assumption using the available data. It is helpful, but we can also see that the underwater bathymetry is complicated so the model can be further improved in the future study.
Minor comments
1, Line 126-127, the sentence is not clear enough for me, please further rephrase it.
Citation: https://doi.org/10.5194/hess-2024-24-RC2 -
AC1: 'Reply on RC2', Miaomiao Qi, 18 Sep 2024
Response to Reviewer 2,
The authors developed a model to improve lake water estimation by simplifying the shape and depths of moraine-dammed lakes. The model showed better performance compared with other existing models. The author also used this model to calculate the water storage changes of moraine-dammed lakes in the past 30 years in High Mountain Asia. I think this study is a significant advancement in glacial volume estimation and it is worthy of being published after minor revision. Some major comments are shown as below:
1, Four types of moraine-dammed lakes are considered in this study. In fact, as shown in Figure 4 and Figure 10, the shape of MDLs is much more complicated. If the authors can classify the MDLs in more types in the future study, the estimation may be further improved.
Explanation and revision: Thank you very much for your suggestion. We will carefully consider improvements in this regard.
2, Line 165-169, the authors gave the core assumption of the model. In the discussion section, the authors add some discussion about this assumption using the available data. It is helpful, but we can also see that the underwater bathymetry is complicated so the model can be further improved in the future study.
Explanation and revision: Thank you very much for your affirmation and support. As you mentioned, the morphology of moraine dammed lake beds is complex, closely related to glacial erosion processes. In our model, we first proposed a hypothesis and validated it with measured data, striving to approximate the actual basin morphology of moraine-dammed lakes as closely as possible. However, this inevitably comes with certain limitations. We will prioritize improvements to the model in our future work.
Minor comments
1, Line 126-127, the sentence is not clear enough for me, please further rephrase it.
Explanation and revision: I sincerely apologize for the misunderstanding my wording caused during your reading. So, we have revised the sentence as follows:
“If R is less than 0.1, it may indicate the presence of glacial lakes with lengths exceeding 10 meters but widths of approximately 1 meter.”
Citation: https://doi.org/10.5194/hess-2024-24-AC1
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AC1: 'Reply on RC2', Miaomiao Qi, 18 Sep 2024
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RC3: 'Comment on hess-2024-24', Anonymous Referee #3, 18 Aug 2024
The article “A mathematical model to improve water storage of glacial lakes prediction towards addressing glacial lake outburst floods” provides a simple method to estimate Moraine Dam glacier lake volume estimations based on data that can easily be obtained from permanently updated global datasets.
Major comments,
The model calculates the volume for an ideal case when the lake is symmetrical. Although unrealistic, it is a considerable improvement from the empirical equations that look into the area of the lake to estimate the volume. However, this equation relies on the right selection of r, m and n. How confident are you in the estimation of these parameters? I appreciate that the authors provide simple relationships, but I am concerned with the potential overfitting of the empirical thresholds. According to the validation section, they used 40 out of 44 lakes to come up with the thresholds and then validated the method using the 4 lakes that they left behind. Can the authours elavorated on this calibration/validation strategy?
The previous comment applies to Figure 8 and Table 6 as well. If the available equations are compared in the same lakes where this model was calibrated, the exercise is biased, and it should be compared with independent data.
Also, since there are four types of lakes (GCL-1, GCL-2, GUL-1, and GUL-2), the comparison should be shown by type of lake.
Minor comments.
How do you standardize R to make it comparable to other glaciers? Does it provide an indication of potential growth through elongation, for example? For example, in line 129, when you mentioned “newly formed,” does it mean that it has the potential to grow at a higher rate? In that case, how do you define new?
It would be useful to provide a general explanation of why the value of R would go from 0.1-0.6 in a GCL 2 lake to 0.5-1 when it gets detached from the glacier (GUL1) and the glacier continues to retreat. There is an example, but I am thinking as a general definition.
In line 14 “through statistical analysis of glacial lake sizes for different types, we 141 defined the threshold for R”, which method and statistical significant of the values?
Figure 3: A small figure with the axis direction would be appreciated, as would Figure 2, which is in a yz plane according to my interpretation.
Line 199: if r=0 and n=0, m has to be m>0; so this sentence inline 200 “and in most cases, m is not equal to zero” makes no sense. If m =0, after indicating that r=0 and n=0, it means that there is no lake.
How do you account for the potential ice at the bottom of the lake?
Does figure 8-a axis y refer to errors?
Line 395 says, “The underwater landforms of some MDLs are not always completely flat.” Are they ever flat?
Why Jialong Co and Bienong Co are representative of the other 42 lakes for the sensitivity analisys?
Citation: https://doi.org/10.5194/hess-2024-24-RC3 - AC4: 'Reply on RC3', Miaomiao Qi, 22 Sep 2024
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