the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Estimating degree-day factors based on energy flux components
- 1TUM School of Engineering and Design, Technical University of Munich, Germany
- 2Department of Civil Engineering, Koblenz University of Applied Sciences, Germany
- 3Faculty of Agriculture, Yamagata University, Tsuruoka, Japan
- 1TUM School of Engineering and Design, Technical University of Munich, Germany
- 2Department of Civil Engineering, Koblenz University of Applied Sciences, Germany
- 3Faculty of Agriculture, Yamagata University, Tsuruoka, Japan
Abstract. Melt water from snow and ice dominated mountainous catchments is a valuable source of fresh water in many regions. Seasonal snow cover and glaciers act like a natural reservoir by storing precipitation during winter and releasing it in spring and summer. Snowmelt runoff is usually modelled either by energy balance or by temperature-index approaches. The energy balance approach is process-based and more sophisticated but requires extensive input data, while the temperature-index approach uses the degree-day factor (DDF) as key parameter to estimate melt merely from air temperature. Despite its simplicity, the temperature-index approach has proved to be a powerful tool for simulating the melt process especially in large and data scarce catchments. The present study attempts to quantify the effects of spatial, temporal, and climatic conditions on the DDF, in order to gain a better understanding which influencing factors are decisive under which conditions. The analysis is physically based on the individual energy flux components, however approximate formulas for estimating the DDF are presented to account for situations where observed data is limited. A detailed comparison between observed and estimated DDF values yielded a fair agreement with BIAS= 0.2 mm °C-1 d-1 and RMSE=1.1 mm °C-1 d-1. The analysis of the energy balance processes controlling snowmelt indicates that cloud cover and under clear sky snow albedo are the most decisive factors for estimating the DDF. The results of this study further underline that the DDF changes as the melt season progresses and thus also with altitude, since melting conditions arrive later at higher elevations. A brief analysis of the DDF under the influence of climate change shows that the DDFs are expected to decrease when comparing periods of similar degree-days, as melt will occur earlier in the year and albedo is then likely to be higher. Therefore, the DDF cannot be treated as a constant parameter especially when using temperature-index models for forecasting present or predicting future water availability.
Muhammad Fraz Ismail et al.
Status: final response (author comments only)
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RC1: 'Comment on tc-2022-64', Roger Braithwaite, 12 May 2022
SUBSTANTIVE COMMENTS.
Izmail and others (submitted) is an interesting article and is very timely as many of us are concerned about the increased melting of snow and ice and its effects on streamflow. The basic premises and ambitions of the study are well presented in the ABSTRACT and in the INTRODUCTION.
The basic idea is to explain the empirical degreeâday approach in terms of energy fluxes. This was done by Braithwaite (1995a) and the present paper does something similar but with a broader array of methods and models.
Modern workers have access to detailed measurements from sophisticated monitoring systems and should use them where they can. Workers modelling historic data may have to use obsolete variables such as maximum and minimum temperatures and sunshine duration if these were measured with simple instruments. Izmail and others (submitted) address both communities.
Izmail and others (submitted) discuss the basic formulation of degreeâday sums on lines 150-159. They are correct that several methods have been used in the past, but the common method of equating daily degree-day sum to the daily mean temperature if positive (or greater than the reference temperature if not 0 ° C) is open to the criticism that there may be melt in part of the day even with daily mean temperature below zero (Arnold and McKay, 1964). Workers should calculate their degree-day sums from a sum of positive temperatures throughout the whole day if they have a modern datalogger. Braithwaite and Hughes (2022) suggest a new way of calculating degree-days if you only have maximum and minimum temperatures. This takes account of the daily temperature range, which can be quite large at lower latitudes, e.g., the Himalaya, and may cause degree-day factors to vary with latitude, as mentioned in line 146.
Izmail and others (submitted) is exceptional well-referenced, but I would like them to cite a ‘senior’ degree-day publication by Zinng (1951) that has stood the test of time.
USE OF ENGLISH AND RERENCING
Izmail and others (submitted) is well written, but I wish they would use active verbs more often, and they do overuse ‘however’. The text may be about 25% too long and they should remove padding and re-arrange text, so any issue is only addressed once. The reference list is accurate except for leaving out names of journals in some places, which may be an artefact of citing on-line journals.
DETAILED POINTS
Line 24: define BIAS and RMSE the first time they occur.
Lines 25-26: Better to say ‘cloud cover and snow albedo under clear sky’
Lines 30-32: Good point!
Line 36: ‘main’ is better than ‘unique’
Line 41: ‘more’ is better than ‘most’.
Line 52: add citation to Braithwaite (1995a) here
Line 88: According to Braithwaite (1995a) degree-day factors depend on mean temperatures
Lines 105-115: Good!
Table 1. Some variables should be defined in caption or in a foot note
Figure 2: Is ‘Wolfgang Bogacki, 2016’ reference to a publication?
Line 147: ‘following’ is better than ‘along’
Lines 151-159: I already mentioned this
Lines 168-169: They should not have done this! From my own thinking about the data used by Braithwaite (1995a) I am quite sure that degree-day factors are only valid for periods of many days, e.g., 10-20 days when you might expect a combination of different weather conditions and when day-to-day measurement errors may compensate.
Line 180: much better to say ‘largest’ and not ‘most important’ as this has caused lots of problems in the literature since about 1952.
Line 192: ‘although’ is better than ‘however’. This occurs in a few places.
Line 196: ‘rigorous’ is better than ‘rigid’ and ‘but’ is better than ‘however’
Line 200: ‘Day of the year’ is a modern muddle as 1 January is day 1 in the usual counting. This means that 12:00 on 1 January is day=1.5, which is obviously wrong! Sorry!
Line 209: Should ‘attenuation’ be ‘reflection’?
Line 215-218. The Prescott equation is useful for historic data but not needed for modern instruments
Line 226: ‘when’ is better than ‘that’
Lines 233-239: Very comprehensive!
Line 319: Better is ‘the sensible heat component depends mainly on high wind speed and temperature’ because it uses an active verb
Line 321: better ‘is smaller on average than…’. The point is that sensible heat flux is generally smaller than the radiation components in most snowmelt situations, but sensible heat fluxes changes by a greater amount if you change temperature by 1 °C.
Lines 352-353. Latent heat flux is generally a heat source to the ice/snow surface in South Greenland and a heat sink in North Greenland. This is explained by variations in vapour pressure and temperature.
Line 374: do you mean ‘… such events are rare…’?
Lines 392-394: Is this a small limitation?
Line 415: I was confused by the start of a new chapter here. You probably mean ‘Results from Brunnenkopfhütte’. This brings me to a small concern. I accept this paper is much more than a data report from a single location, and I applaud this, but it is difficult to keep track of what material relates to which. Location. Please consider restructuring, e.g., you could discuss ALL results from Brunnenkopfhütte either before or after discussion of the more general modelling.
Line 419: You should base your degree-day sum on hourly data (if positive) from your nice AWS in Fig. 2. See Braithwaite and Hughes (2022).
Line 430: That confusing ‘most important’ again.
Chapters 4 and 5: I am confused by all the examples given. Could you not define a few ‘typical’ cases and give energy flux values for each case? In general, I think both chapters would benefit from some smoothing. This is something you can do more easily 1-2 months after you have written the original text.
Lines 557-361: I think this is correct, but you could phrase it better!
Lie 563: I know what RMSE means but what is BIAS? You should define all acronyms first time you use them.
Figure 8: I like it. Braithwaite (1995a) should have done this for all the months in his study rather than just comparing grand-means of measured and simulate degree-day factors. I am thinking about a new paper on my old data and I will certainly make a figure like this.
Chapter 5. I like this. Braithwaite )1995b) looked in detail at the effect of stability on sensible heat flux model used by Braithwaite (1995a). The sensible (and latent heat) fluxes depend the density of air at the altitude in question so the degree-day factor should depend on altitude, and on latitude as lower latitude glaciers occur at greater altitude. There should be a greater latitude effect on degree-day factors than we have discovered so far. If not, why not?
Lines 612-617: Interesting!
Line 630-633. Walter Ambach is the master of albedo under overcast conditions. In Braithwaite (1995a) this is one factor that reduces the time-variability of the net radiation flux.
Line 654: this should be ‘breaking in’.
Line 665-19. I think you well explain here the importance of rain on snow.
Section 5.6. Ingenious!
Section 5.7: Although Braithwaite (1995a) clearly showed the change of degree-day factor with changing energy balances, he assumed constant degree-day factors for climate change projections in his later papers. (I am not going to give references here as you already have too many!)
Acknowledgements
Was there no funding? No good advice from somebody?
REFERENCES CITED IN THIS REVIEW
Arnold, KC and DK MacKay. 1964. Different methods of calculating mean daily temperatures, their effects on degree-day totals in the high Arctic and their significance to glaciology. Geographical Bulletin 21, 123-129.
Braithwaite RJ 1995a. Positive degree-day factors for ablation on the Greenland ice sheet studied by energy-balance modelling. Journal of Glaciology 41, 137, 153-160.
Braithwaite RJ 1995b. Aerodynamic stability and turbulent sensible-heat flux over a melting ice surface, the Greenland ice sheet. Journal of Glaciology 41, 139, 562-570.
Braithwaite RJ and PD Hughes 2022. Positive degree-day sums in the Alps: a direct link between glacier melt and international climate policy. Journal of Glaciology 1-11. http://doi.org/10.1017/jog.2021.140
Izmail MF, W Bogacki, M Disse, M Schäfer and L Kirschbauer. 2022. Estimating degree-day factors based on energy flux components. The Cryosphere Discussion.
Zinng T 1951. Beziehung zwischen Temperatur und Schmelzwasser und Bedeutung für Niederschlags- und Abflüssfragen. International Association of Scientific Hydrology Publications 32, 1, 266-269.
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RC2: 'Comment on tc-2022-64', Lander Van Tricht, 14 Jun 2022
Review “Estimating degree-day factors based on energy flux components”
The manuscript describes the possibility to estimate degree-day factors based on energy flux components. It studies in detail the contribution of each component as well as the variation (spatial, temporal, climate change). Consequently, the study is a valuable contribution in the context of calibrating DDF in temperature-index models to better represent melting. This is relevant given the importance of correctly calibrated models to assess (future) snowmelt.
The paper is well written, and the methods/formulations are clearly described. Further, the main ideas are very well presented in the introduction which ensures that the reader is immediately introduced in the topic and knows what the study focuses on. The study also contains an enormous number of references and (explanations of) parametrisations that sometimes make it read like a literature review, especially in the method section. The study is not particularly "innovative", but it does contribute to a better understanding of DDF and the implementation/calibration of these factors in models that can be used to determine snowmelt.
In conclusion, I think the study is worth publishing with some smaller (technical) revisions. Further, the authors may consider making the structure/division of method - results - discussion a bit clearer. Now it is not entirely clear what certain datasets are used for in this study (BrunnenkopfhuÌtte, Upper Indus Basin, etc.). Furthermore, it could be an option to do an analysis with the hourly temperature data instead of just looking at the average, as this data is available from the meteorological station.
Specific comments
- Line 23: yields <-> yielded
- Line 24: mm w.e.? If water equivalent is used, use this abbreviation
- Line 24: What is BIAS? RSMe is clear for most readers. Use the full notation, especially the first time.
- Line 45: Odd use of however in this sentence
- Line 61-66: Some repetition with previous paragraphs. Consider integrating this a little more in other paragraphs. That way, the text can also become a bit shorter.
- Line 88: Why does albedo decrease with increasing altitude?
- Line 96: .. and topographic settings?
- Line 117: a part of “the” Isar River system “lying” in the …
- Line 122: made up sounds a bit strange. Is mainly composed/characterised?
- Line 123: A reference here is not essential.
- Line 128: Have <-> has
- Line 130: Sometimes British – American English is used (parametrise – parametrize etc.)
- Line 130: Summarizes
- Table 1: Some variables need explanation. What is Kt? SRin?
- Figure 1: Snow station or meteorological station?
- Line 151: Units are in water equivalent?
- Line 155: What is the difference between part 1 and part 2 of this sentence? “T is set to 0°C” vs “The freezing point is chosen.”.
- Line 193-194: Which value is used in this study?
- Line 252: Odd use of however. Use a different word or rephrase the (part of the) sentence
- Line 294: Parametrise vs parametrize
- Line 304: Parametrise vs parametrize
- Line 324: It would be interesting to also mention a typical value for these conditions (W m-2)
- Line 391: Analysed vs analyzed
- Line 419-420: This is based on data of the Hutt? How is the mean calculated?
- Line 470: I think it is clearer o put the panel letter before the sentence.
- Line 474: Snow station or meteorological station?
- Line 488-489: An average temperature of 20°C, it is not very common in early spring, wright?
- Line 492 and Figure 5: for selected cloudiness and average air temperatures?
- Line 510-512: Would it be an option to derive an average using the average hourly windspeeds?
- Line 539: In prefer “refreezes” <-> is refrozen
- Line 554: meteorological station <-> snow station
- Line 563: What is BIAS? Use full notation the first time
- Line 563-565: The snowmelt periods which are neglected, are these particular days? Or 10-day periods?
- Line 582: “is” or “to be”
- Line 704-706: Where does this data come from? The area of Indus Basin is not elsewhere introduced or mentioned.)
- Line 734: Parametrizes <-> parametrises (probably I have missed other ones)
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RC3: 'Comment on tc-2022-64', Rijan Kayastha, 15 Jun 2022
This paper tries to do something new on the positive degree-day factor by analyzing different previous research which is very good. It is good that the authors still agree that the conventional degree-day approach is still good to use where data are insufficient. I have found the paper deals with the shortwave radiation calculation in detail which is very good for data insufficient regions. But the others such as the need of using different degree-day factor for space and time has already been applied in many previous researches and need to mention in this study. I also like to comment on the symbol used for a degree-day factor; in the past papers degree-day factor is denoted by the letter “k or K” but nowadays DDF is being used. The authors should also think about this issue. About the use of the degree-day factor in a climate change study, if we consider all parameters which affect the degree-day factor and assign the degree-day factor accordingly, it will still give a good result. Authors should also think about it.
A few other line-wise comments are as follows:
Line 118: Need to mention the name of the country (Germany) after Ammergauer Alps.
Line 260: It should be “The net longwave radiation flux …….
Line 161: Equation (20) should be at line 164 instead of line 161 at present. The sentence does not look good at present.
Line 233: Need to use a different letter for a coefficient other than k. Because k is used as Von Karmann constant on line 310.
Line 404: should be degree-day models instead of “degree-day factor models.”
Line 461-463: The result stated in those lines “All of these models show the same tendency of linear increase by altitude, with the altitude factor being comparatively smaller under clear sky compared to overcast conditions” is to some extent is different from the results which we have received on a Glacier AX010 in Nepal (Kayastha et al., 2000). Figure 10 shows that the degree-day factor at higher altitudes is higher in a comparative clear sky (in June) compared to July and August (peak monsoon season with a highly overcast period in Nepal). We assumed that due to the overcast situation, air temp. does not change much and hence degree-day factors too do not change much. Why in the present study is the altitude factor comparatively smaller under the clear sky?
Line 639 -640: This statement “Under overcast conditions, however, the DDF is virtually stable ranging from 4.4 to 4.5 mm °C-1 d-1 in the same period” is in agreement with what was shown in Figure 10 in Kayastha et al. (2000).
Line 760-761: The message of this statement “Therefore, and as pointed out by many researchers, the DDF cannot be considered a constant model parameter. Rather, its spatial and temporal variability must be taken into account ….” Has already been implemented in Kayastha et al. (2020; Table 3) in which we have used two sets of degree-day factors; lower degree-day factor at lower attitudes (lower than 5000 m) and higher degree-day factor for higher altitudes (above 5000 m). Also, monthly degree-day factors are used to incorporate the seasonality of degree-day factors.
References:
Kayastha, R. B., Ageta, Y. & Nakawo, M. (2000). Positive degree-day factors for ablation on glaciers in the Nepalese Himalayas: case study on Glacier AX010 in Shorong Himal, Nepal. Bulletin of Glaciological Research, 17, 1-10.
Kayastha, R. B. & Kayastha, R. (2020). Glacio-Hydrological Degree-Day Model (GDM) Useful for the Himalayan River Basins. In: Dimri A., Bookhagen B., Stoffel M., Yasunari T. (eds) Himalayan Weather and Climate and their Impact on the Environment. Springer, Cham, Doi: 10.1007/978-3-030-29684-1_19.
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RC4: 'Comment on tc-2022-64', Álvaro Ayala, 23 Jun 2022
PAPER SUMMARY AND RECOMMENDATION
Ismail and co-authors investigate how degree-day factors (DDFs) depend on the components of the snowpack energy balance. Assuming a snowpack close to melting conditions and a negligible cold content, the authors connect DDFs to the variations of each energy balance component by means of a set of widely used equations. In this way, DDFs are related with different characteristics and conditions, such as elevation, latitude and meteorological variables. The authors provide several summary tables and figures that can be used by other researchers to estimate DDFs in poorly monitored regions using minimum data requirements. Additionally, the authors estimate the impact of climate change on DDFs. They conclude that cloud cover and snow albedo are the main processes controlling DDFs and that DDFs cannot be treated as constant parameters.
The study is appropriate for The Cryosphere. The article is well written, but some parts describing the equations can be shortened. I think that the authors do a valuable contribution. Having tools to estimate DDFs is a good idea, and it can be useful for researchers working on the snow hydrology of poorly monitored regions. However, I think that the article needs to be improved before being suitable for publication. Please see my main comments.
MAJOR COMMENTS
1. Presentation and role of the datasets
- Field dataset: The purpose of including the datasets from Brunnenkopfhütte and Naran stations is not clearly presented. The authors should mention in the Introduction what is the role of these datasets in their study. Are they used as validation, or test sites? Do the authors make tests at the catchment or point scales? Importantly, the use of the Naran dataset comes a surprise in the middle of the discussion section.
- Climate change dataset: Please provide more details about this dataset and add this analysis to the objectives of the study.
2. Discussion section
In this section, the authors continue their analysis and calculations, but they provide almost no comparisons with the results of other studies. The authors should discuss their results using the literature presented in the Introduction. Additionally, I recommend the inclusion of some other references regarding the spatial and temporal transferability of degree-day factors (or temperature factors) and melt parameters that, in my opinion, are missing (Ohmura, 2001; Carenzo et al., 2009; MacDougall and Flowers, 2011; MacDougall et al., 2011; Gabbi et al., 2014) . The limitations of the approach proposed by the authors and the assumptions made through the article should be more discussed. For example, the authors validate their approach using only one monitoring station, can the authors include more data? There are certainly more datasets available for which DDFs have been derived. Otherwise, this is an important limitation of the study that should be discussed.
3. Conclusions and recommendations
As the aim of the study is to “quantify the effects of spatial, temporal, and climatic conditions on the DDFs” and the conclusion is that “DDF cannot be treated as a constant parameter”, what are the recommendations of the authors to a researcher modelling the snow hydrology of poorly monitored catchments? Should that researcher use a range of parameters from your equations? How large should be the variability of DDFs in space and time? Different DDFs for each sub-catchment, slope or elevation band? How often should the DDFs change in time? Every week, month or season? I think that the article would benefit from such discussions and recommendations.
MINOR COMMENTS FOR THE AUTHORS
12-13: I would add “At mid-latitudes, seasonal snow …” because this seasonal pattern is not necessarily found on every snow and ice dominated mountain catchment (e.g. tropical glaciers).
13: I think that the concept of snowmelt runoff is wider than what the authors are describing. The authors are describing only the process of melt whereas snowmelt runoff include other processes controlling the movement of excess meltwater through a catchment.
21: is physically based -> is based
22: I don’t think that the formulas are “approximate”, they just have limitations and assumptions.
23: observed -> field-derived. DDFs cannot be measured in the field because they are not a physical quantity.
30: “albedo is likely to be higher”, there are also other reasons, such as lower radiation and temperatures, aren’t they?
35: It would be interesting to mention somewhere in the Introduction that researchers usually select DDFs values from other studies and that the spatial transferability is not always good [e.g. Carenzo et al., 2009; Wheler, 2009].
35: The authors should briefly mention at the end of the Introduction what is the role of the Study area in the article as Section 2 “Study area” comes as a surprise. See my main comment.
79: “longer time periods” Can the authors be more precise? Weeks, months, years?
81-82: Also, the spatial variability of air temperature does not fully describe the spatial variability of the energy balance.
118: Please mention the country
123: The Kopp reference is not necessary here as the authors also have a DEM of the catchment.
171-172: “The balance of the energy fluxes over the surface of the snowpack”. Please note that Q_G (ground heat) is not a surface flux. By including DeltaQ and Q_G, the authors are describing the energy balance of the entire snowpack and not only the surface, which has not heat capacity [den Broeke et al., 2011]. Otherwise please clearly define what is the control volume considered by the authors.
179: The length of this section can be reduced.
182: No reference is needed for equation 5
241: I’m a bit confused, when the authors correct by elevation, what is the term that goes in eq. 6, K_z or K_T?
277: Please clarify at what height above the surface are Pv and Ta measured.
300: What do the authors mean by “a probabilistic reasoning”?
344: I think a step or equation is missing here and it should be that relating RH and p0. Or how do the authors calculate pv? Also, are the authors assuming saturated conditions at the snow surface?
354/375: Sections 3.2.5 and 3.2.7 don’t read as “Methods”. They seem a review on the subject. As both terms (Q_G and DeltaQ) are neglected by the authors, I suggest the shortening of these sections and to move them to the beginning of Section 3.2 where a suitable justification to neglect them can be provided.
422: Delete “approximate”.
431: higher altitudes, as well as dry climates.
504: As wind speed is highly variable in space and time, I don’t think that the authors can refer to “typical values”. It would be better to write something such as: “… can be roughly estimated based on the topographic and climate characteristics of the study site”.
551: I think that this is the first time that the authors mention the goal of these dataset. Please see my main comments.
579: I believe that this is not clearly a discussion section because there are almost no comparisons against other studies (and almost no references). Instead, the authors present more results and analysis. Please my main comments.
592: This is the first time that the authors mention these data. Please properly introduce this site and the dataset in section 2. Also explain what is the purpose of including this dataset.
598: Please change the word “altitude” by “elevation” throughout the article. Altitude is the vertical distance between an object and the earth’s surface.
606: Why does the solar angle change with altitude?
693-695: Not clear, please reword.
702-705: This belongs to methods. The climate change analysis should be introduced earlier in the manuscript. Provide more details about these data, are those values an average of different GCMs?
697: Musselman et al. [2017] is an excellent article regarding slower melt rates in climate change scenarios.
SUGGESTED TECHNICAL CORRECTIONS FOR THE AUTHORS
11: Meltwater
11: Consider: “Meltwater from mountainous catchments dominated by snow and ice is a …”
36: Meltwater
42: Delete “for the prediction”
44: Delete “runoff”. The authors discuss only the process of melt.
59: Add “using runoff” after DDF
61: Delete “runoff”
68: by the inclusion
72: the position
95: Since melt depends …
117: system and lies
119: delete about or ~
127: a standard
128: Brunnenkopfhütte site
146: Delete “concrete”, or maybe use “actual”.
221: Delete “,” after disadvantage
283: the above relation
284: … snowpack amounts to
294: Add “,” after parameterize
294-295: and their effects on radiation depend…
329: the snow and the snow surface
371: even during extreme weather conditions
588-590: Please rewrite these lines for clarity.
591: “The example”, what example?
638: see Table
650: in Table
FIGURES
Figure 1: I think that m (instead of cm) are enough for “High” and “Low” in the legend.
Figure 4: Why is the clearness index (K_T) at a given elevation larger for overcast conditions than for clear sky? Shouldn’t be the opposite? Please clarify
Figure 10: Why exactly do DDFs on each panel (present and scenarios) decrease as the season progresses but in Figure 9 DDFs increase as the season progresses?
TABLES
Table 1: Explain the name of the variables.
Table 1: Please provide SRm in W/m2
REFERENCES
den Broeke, M., X. Fettweis, and T. Mölg (2011), Surface Energy Balance, in Encyclopedia of Snow, Ice and Glaciers, edited by V. P. Singh, P. Singh, and U. K. Haritashya, pp. 1112–1123, Springer Netherlands, Dordrecht.Carenzo, M., F. Pellicciotti, S. Rimkus, and P. Burlando (2009), Assessing the transferability and robustness of an enhanced temperature-index glacier-melt model, J. Glaciol., 55(190), 258–274, doi:10.3189/002214309788608804.
Gabbi, J., M. Carenzo, F. Pellicciotti, A. Bauder, and M. Funk (2014), A comparison of empirical and physically based glacier surface melt models for long-term simulations of glacier response, J. Glaciol., 60(224), 1199–1207, doi:10.3189/2014JoG14J011.
MacDougall, A. H., and G. E. Flowers (2011), Spatial and temporal transferability of a distributed energy-balance glacier melt model, J. Clim., 24(5), 1480–1498, doi:10.1175/2010JCLI3821.1.
MacDougall, A. H., B. A. Wheler, and G. E. Flowers (2011), A preliminary assessment of glacier melt-model parameter sensitivity and transferability in a dry subarctic environment, Cryosph., 5(4), 1011–1028, doi:10.5194/tc-5-1011-2011.
Musselman, K. N., M. P. Clark, C. Liu, K. Ikeda, and R. Rasmussen (2017), Slower snowmelt in a warmer world, Nat. Clim. Chang., 7(3), 214–219, doi:10.1038/nclimate3225.
Ohmura, A. (2001), Physical Basis for the Temperature-Based Melt-Index Method, J. Appl. Meteorol., 40(4), 753–761, doi:10.1175/1520-0450(2001)040<0753:PBFTTB>2.0.CO;2.
Wheler, B., 2009: Glacier melt modelling in the Donjek Range, St. Elias Mountains, Yukon Territory. M.S. thesis, Dept. of Earth Sciences, Simon Fraser University, 283 pp
Muhammad Fraz Ismail et al.
Muhammad Fraz Ismail et al.
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