Thermal resistances in the Everest Area ( Nepal Himalaya ) derived from satellite imagery using a nonlinear energy balance model

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Introduction
Debris-covered glaciers are common in the Everest area of the Himalayas.The debris cover has a large impact on the sub-debris ablation rate and hence the evolution of the glacier.A thin debris layer may enhance ablation by reducing the albedo causing the surface to absorb more radiation compared to clean ice, while a thicker debris layer will Figures insulate the glacier causing the ablation rate to decrease.The critical thickness at which the debris cover reduces ablation is around 2 cm (Ostrem, 1959;Mattson et al., 1993;Kayastha et al., 2000).Field studies have supported these results showing that beyond this critical thickness, the melt rate greatly decreases (Nakawo and Young, 1981;Conway and Rasmussen, 2000;Nicholson and Benn, 2006;Reid and Brock, 2010;Reid et al., 2012).The role that debris cover has on the evolution of glaciers in the Everest area is summarized well by Benn et al. (2012).In short, the debris cover increases towards the tongue of the glacier, where the slopes are gentler.The spatial variation of debris cover causes the ablation to be predominately focused in areas of thinner debris behind the tongue of the glacier.The differential melting causes the tongue of the glacier to become stagnant and promotes the development of supraglacial lakes.
The sub-debris ablation rate is controlled by the debris thickness, the thermal properties of the debris, and meteorological conditions.The debris thickness may be measured by surveying exposed ice faces (Nicholson and Benn, 2012) or via manual excavation (Reid et al., 2012).Surveying exposed ice faces greatly reduces the amount of labor involved in measuring the debris thickness, but may not be representative of the entire glacier and is limited to regions with significant differential melting.Due to the labor-intensive nature of this work, few other surveys of debris thickness have been performed in the Everest area (Nakawo et al., 1986).The thermal property associated with describing the debris cover is the effective thermal conductivity.Studies have found the thermal conductivity of debris cover in the Khumbu to range from 0.85 to 1.29 W m −1 K −1 (Conway and Rasmussen, 2000; Nicholson and Benn, 2012).The water content and lithology of the debris cover may partly explain the variation in thermal conductivity as the water content will change the effective thermal conductivity of the debris (Nicholson and Benn, 2006) and the lithology will influence the bulk volumetric heat capacity, which is used to derive the thermal conductivity (Nicholson and Benn, 2012).
In addition to the properties of the debris cover, the meteorological conditions will affect the sub-debris ablation rate.The net solar radiation has been found to be the Figures main source of energy responsible for ablation on debris-covered glaciers (Inoue and Yoshida, 1980;Kayastha et al., 2000;Takeuchi et al., 2000); however, the turbulent heat fluxes are still significant (Brock et al., 2010).Many studies have modeled the energy balance on debris-covered glaciers with varying levels of success (Nakawo and Young, 1982;Nakawo et al., 1999;Han et al., 2006;Nicholson and Benn, 2006;Mihalcea et al., 2008b;Reid and Brock, 2010;Reid et al., 2012).These models integrate meteorological data from automatic weather stations with knowledge of the debris cover to solve for the surface temperature of the debris, which may then be used to calculate the subdebris ablation rates.These models are limited by their knowledge of how the debris cover varies over the glacier or they require a great deal of site-specific information.
This has led other studies to use satellite imagery to derive the properties of the debris cover.These studies use surface temperature data from Aster or Landsat satellite imagery in conjunction with an energy balance model to solve for the thermal resistance, which is the debris thickness divided by the thermal conductivity (Nakawo and Rana, 1999;Nakawo et al., 1999;Suzuki et al., 2007;Zhang et al., 2011).If the thermal conductivity of the debris is known, the model can solve directly for debris thickness (Foster et al., 2012).Mihalcea et al. (2008a) used a different approach by deriving debris thickness from linear relationships between surface temperature and debris thickness for different elevation bands.
One problem associated with the studies that solved for the thermal resistance is that while the spatial distribution of thermal resistances typically agreed well, the actual values of thermal resistances were significantly lower than those derived from field studies.Suzuki et al. (2007) attributed their low thermal resistances to the mixed pixel effect, which refers to the pixels in the satellite imagery comprising supraglacial ponds, ice cliffs, and bare ice areas.Nakawo and Rana (1999)  altitude, aspect, and shading in different areas, as well as the unknown nature of water content in the debris.The mixed pixel effect and the spatial variation in meteorological conditions may reduce the thermal resistances, but it is unlikely to cause the satellitederived thermal resistances to be one or two orders of a magnitude lower than those found in the field.Foster et al. (2012) is the first study, to the authors' knowledge, that accurately derives debris thickness from satellite imagery.The model uses a DEM generated from an airborne lidar survey and compares the results of a sloped model, which accounts for variations in topography, and a flat model.The sloped model resulted in thicker debris areas when compared to the flat model, but also identified some pixels as having unrealistically high or negative debris thicknesses.These errors occurred in pixels with steep slopes and high surface temperatures and were replaced with the values from the flat model.Unfortunately, the model is difficult to transfer to other glaciers because a great deal of site-specific data was used.Their modifications to their energy balance include the addition of a heat storage term that is a fraction of the ground heat flux and an empirical relationship between the surface temperature and air temperature.
We report a method for deriving the thermal resistances of debris-covered glaciers using an energy balance model with Landsat7 ETM+ satellite imagery and apply the method in the Everest region of Nepal.The performance of various models is assessed via comparison with field data.First, the use of a correction factor that accounts for the nonlinear temperature gradient in the debris cover is investigated.This simple nonlinear energy balance model is then used to compare a flat model with a sloped model, which accounts for the variations in topography.The affect of the quality of the DEM is then explored by comparing DEMs of different resolutions.Lastly, the applicability of this model to other areas is discussed.Introduction

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Remotely sensed data
Landsat7 ETM+ (hereon referred to as Landsat7) satellite imagery over the same period as the meteorological data was used to derive the thermal resistance of the debris.All clear-sky images from the same period of time that meteorological data are available in the melt season were used.The melt season was defined as 15 May to 15 October, which is the time period where the temperature in the debris was above freezing (Nicholson, 2005).Twelve Landsat7 images met this criterion (Table 1).All scenes were downloaded from the NASA Land Processes Distributed Active Archive Center (NASA LP DAAC, 2011).The processing level of the Landsat7 images were all L1T indicating the images were all geometrically rectified using ground control points (GCPs) from the 2005 Global Land Survey in conjunction with the 90 m global DEM generated by the Shuttle Radar Topographic Mission (SRTM).Landsat7 satellite imagery comprises 8 different bandwidths with various resolutions.The two bands of interest here are the thermal band (Band 6) and the panchromatic band (Band 8).The thermal band has a resolution of 60 m, but is automatically resampled to 30 m and was used to derive surface temperature according to NASA (2011).It was atmospherically corrected using the methods described by Coll et al. (2010) co-registration for Landsat7 is 7.3 m and the uncertainty of the derived surface temperature data is estimated to be ±1.0K (Barsi et al., 2003;Coll et al., 2010Coll et al., , 2012)).The panchromatic band has a horizontal resolution of 15 m and was used to co-register the images.
The high resolution DEM used in this study was generated by Lamsal et al. (2011) from Advanced Land Observing Satellite (ALOS) PRISM images.The generated DEM has a horizontal resolution of 5 m and relative error of ±4 m.In order to co-register the DEM with the panchromatic band from the Landsat7 imagery, a shaded version of the DEM was generated using the Hillshade tool in ArcGIS 10.3.The swipe visualization tool in PCI Geomatica 2013 showed that the images were properly coregistered without any further processing.The coarser resolution global DEM used in this study was the ASTER GDEM, which is composed of automatically generated DEMs from the Advanced Spaceborne Emission and Reflection radiometer (ASTER) stereo scenes acquired from 2000-present (METI/NASA/USGS, 2009).Nuth and Kaab (2011) found the accuracy of the ASTER GDEM to be similar to the validation summary (METI/NASA/USGS, 2009) when applied to debris-covered glaciers in New Zealand.They found the ASTER GDEM to have biases up to 10 m and RMSE of 5-50 m.The horizontal resolution of the ASTER GDEM has been found to be better than 50 m (Fujisada et al., 2005).The swipe visualization tool in PCI Geomatica 2013 was used with a shaded version of the ASTER GDEM to confirm that the images were properly co-registered.While residual anomalies and artifacts may exist in this experimental/research grade product, it has been used in this study to develop an understanding of how the quality of DEM will affect the thermal resistances.

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Full the debris/ice interface.Holes were excavated to the debris/ice interface and as the thermistors were installed, the debris was replaced in its original position as best as possible.The thermistors recorded temperature at hourly intervals from 13 September to 24 September.The first 48 h of data for each thermistor was discarded to allow the thermistors to equilibrate with the debris.One of the surface thermistors malfunctioned on 23 September, so the data from this thermistor beyond this date was discarded.Debris thickness measurements were performed at 25 locations and were concentrated in one melt basin that appeared to be formed by differential melting and backwasting (Fig. 1).The melt basin was selected as the focus area of this study because it appeared to be representative of the hummocky terrain on Lhotse Shar/Imja glacier and was relatively easy to access.To the best ability of the authors, the measurements were performed randomly throughout the melt basin.Of the 25 sites, 23 were measured via manual excavation using a tape measure.This process involved digging holes to the ice surface and measuring the perpendicular distance from the ice surface to the surface of the debris.The other two sites were the debris on top of an ice face, which was measured using a laser range finder (TruPulse 360B) because the ice face could not be accessed safely on foot.Twelve debris thickness measurements were also performed outside of the melt basin to understand if the melt basin was representative of the debris-covered glacier.More debris thickness measurements were unable to be made due to time and labor restraints.Furthermore, the maximum depth of excavation was 1 m because further excavation was too physically demanding.Introduction

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Energy balance model
The energy balance model developed for the debris cover is a steady state energy balance similar to Nakawo and Young (1982) where R n is the net radiation flux, H is the sensible heat flux, LE is the latent heat flux, and Q c is the ground heat flux (all in W m −2 ).The control volume for this energy balance is the upper 10 cm of the debris (Fig. 2) and is assumed to be in steady state.
The net radiation flux includes the shortwave radiation flux and the longwave radiation flux where S↓ is the incoming shortwave radiation (W m −2 ), α is the albedo (0.30), ε is the emissivity (assumed to be 0.95), L↓ is the incoming longwave radiation (W m −2 ), σ is the ), and T S is the surface temperature (K).For the sloped model, incoming shortwave radiation was corrected for the effects of topography, altitude, and shading similar to the methods of Hock and Noetzli (1997).
The flat model assumes that each pixel has a slope of 0 • and does not correct for the topography.The incoming longwave radiation and surface albedo were assumed to be constant over the entire debris cover.The albedo used in this study (0.30) was the average albedo of the debris cover on Ngozumpa glacier (Nicholson and Benn, 2012).Introduction

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Full The turbulent fluxes were calculated according to Nicholson and Benn (2006) with a modification to the computation of the surface vapor pressure where where ρ air is the density of air (1.29 kg m −3 ), P is the atmospheric pressure computed using the barometric pressure formula, P 0 is the atmospheric pressure at sea level (101 325 Pa), c is the specific heat capacity of air (1010 J kg −1 K −1 ), A is the dimensionless transfer coefficient, u is the wind speed at Pyramid Station, T air is the air temperature two meters above the surface calculated using a lapse rate of 0.0065 K m −1 , L v is the latent heat of vaporization of water (2.49× 10 6 J kg −1 K −1 ), e air is the vapor pressure two meters above the surface, e s is the surface vapor pressure, k vk is Von Karman's constant (0.41), z is the height of meteorological measurements (2 m), z 0 is the surface roughness length, RH is the relative humidity at Pyramid Station, R is the gas constant (461 J kg −1 K −1 ), and T 10 cm is the temperature 10 cm below the surface (K).The temperature 10 cm below the surface was used because it is the depth at which the debris transitions from being dry to wet based on results from the thermal conductivity, which will be discussed later.T 10 cm was approximated by the average Introduction

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Full slope in the temperature profiles in the upper 10 cm of the debris.The surface roughness length was assumed to be 0.016 m, which was the surface roughness length measured on a different debris-covered glacier, Miage Glacier, by Brock et al. (2010).
The ground heat flux is different for the linear and the nonlinear models Linear Model: where TR is the thermal resistance (m 2 K −1 W −1 ), and G ratio is the nonlinear correction factor.The linear model assumes the temperature gradient in the debris is linear from the surface temperature to the debris/ice interface, which is assumed to be at 273.15 K.
At the time that Landsat7 images are acquired (10:15 LT), this linear assumption is not accurate (Fig. 3).G ratio is used to approximate the nonlinear temperature gradient in the debris by assuming the temperature gradient in the top 10 cm of the debris is linear.This is a more reasonable assumption (Fig. 3).G ratio is therefore defined as the ratio of the nonlinear temperature gradient to the linear temperature gradient

Thermal conductivity
The effective thermal conductivity, k, of the debris cover was computed following the methods in Conway and Rasmussen (2000) assuming a density (ρ = 2700 kg m −3 ) and a specific heat capacity (c = 750 J kg −1 K −1 ) of rock.The average effective thermal conductivity was calculated to be 0.96 (±0.33)W m −1 K −1 .This effective thermal conductivity agrees well with other thermal conductivities computed in this area, which range from 0.85 to 1.29 W m −1 K −1 (Conway and Rasmussen, 2000; Nicholson and Benn, 2012).The thermal conductivity was greatly influenced by depth as the average values above and below 10 cm were 0.60 and 1.20 W m −1 K −1 , respectively.
The drastic difference in thermal conductivity above and below 10 cm is likely due to the amount of water content in the debris.Nicholson and Benn (2006) found the thermal conductivity of wet debris (assuming the pores were saturated with water) to be two to three times larger than dry debris.These results indicate that the top 10 cm of the debris is dry, while 15 cm and lower is wet.This is consistent with observations in the field.These results lend confidence to the use of T 10 cm for computing the vapor pressure at the surface of the debris as the thermal conductivities indicate the interface of dry and wet debris is between 10 and 15 cm.

Debris thickness and thermal resistance
The debris thickness in this melt basin ranged from bare ice (0 cm) to depths greater than 1 m.The average debris thickness, assuming a maximum thickness of 1 m, was 42 (±29) cm.These debris thicknesses are consistent with the debris thickness of other debris-covered glaciers in the Everest region (Nakawo et al., 1986;Nicholson and Benn, 2012).Debris thicknesses greater than 1 m were found in the bottom of these melt basins where the debris had likely accumulated over time due to differential melting and backwasting of the debris cover.Areas of thin debris cover were located on Introduction

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Full the slopes of the melt basin.These trends were also found at the 12 other sites outside of the melt basin and are identical to those found by Nicholson and Benn (2012).There did not appear to be any trends in debris thickness with respect to aspect.The thermal resistance, TR, was calculated by dividing the debris thickness, d , by the effective thermal conductivity, k.The thermal resistances in the melt basin range from 0 to 1.04 m 2 K W −1 with an average of 0.44 (±0.30) m 2 K W −1 .Ideally, debris thickness would be sampled over the entire debris-covered glacier along with measurements of the thermal conductivity to derive a thermal resistance map that could be used to validate the modeled thermal resistances.As this was not feasible due to restraints on time and labor, the modeled results within this melt basin and the adjacent cells will constitute the focus area of the satellite imagery that will be compared to the measured thermal resistances to assess the validity of the modeled results.

Nonlinear correction factor, G ratio
G ratio was computed from all the temperature profiles based on the interpolated temperatures at 10:15. Figure 4 shows the temperature profiles at site LT3 and a schematic of the temperature gradients used to compute G ratio .The average value of G ratio for the melt basin was 2.66 (±0.45).Nicholson (2005) is the only other study that has measured temperature profiles in the Everest area with a small enough spacing between thermistors to compute G ratio .The value of G ratio derived from their temperature profile was 2.55.Based on these results, it appears that the derived value of G ratio on Lhotse Shar/Imja glacier may be transferable to other debris-covered glaciers in the Everest region.However, this should be verified in future studies.Introduction

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Nonlinear vs. linear model
The nonlinear model accounts for the nonlinear temperature gradient in the debris cover using the G ratio correction factor.Figure 5a and b show the modeled thermal resistance maps for the nonlinear and linear models.For both models, the thermal resistance is greater on the terminal moraine and directly behind Imja Lake and becomes smaller upglacier.These trends indicate thicker debris on the moraine and thinner debris upglacier, which agrees well with debris-covered thickness surveys performed on the Khumbu glacier (Nakawo et al., 1986) and Ngozumpa glacier (Nicholson and Benn, 2012).While the trends are apparent in both models, the linear model derives significantly smaller thermal resistances.In the focus area, the average thermal resistance is 0.41 (±0.23) and 0.15 (±0.09) m 2 K W −1 for the nonlinear and linear models, respectively.The nonlinear model agrees very well with the measured thermal resistances, while the linear model severely underestimates the thermal resistances.One limitation associated with these models is that steep north and west facing pixels are undefined, which means the derived thermal resistances are negative.In the focus area, four pixels are undefined.Table 2 shows that steep slopes on average have a lower value of net radiation due to the topographic correction.Pixels with north and west aspects also have a lower value of net radiation.The topographic correction reduces the amount of incoming shortwave radiation on steeper slopes as well as north and west facing slopes.This reduces the net radiation, which lowers the net energy flux (net radiation and turbulent heat fluxes) used to derive the thermal resistance.In some cases, the net energy flux is negative, which causes a pixel to be undefined.Otherwise, the net energy flux is positive, but small, which results in large thermal resistances.A minimum threshold for the net energy flux of 10 W m −2 was set, such that unrealistically high thermal resistances would be classified as undefined.
Table 2 clearly shows that the topographic correction causes a trend in the thermal resistance with respect to slope, with steeper slopes having higher thermal resistances.900 Figures

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Full This is the opposite trend with respect to slope that was observed in the field.There is also a clear trend with respect to aspect, with north and west facing slopes having higher thermal resistances.This trend was not observed in the field.Therefore, despite the magnitude of thermal resistances agreeing well with the measured values, the model is incapable of capturing the fine local variations.
Derived thermal resistances do not capture local variations with respect to slope and aspect due to the poor resolution of the thermal band.The thermal band has a resolution of 60 m, which causes the surface temperatures over the 60 m pixel to be combined.This is referred to as the mixed-pixel effect.Conventionally, the mixed-pixel effect has been used to explain how bare ice faces reduce the surface temperature of the pixel causing the derived thermal resistances to be low.While this may be true, the mixed-pixel effect also explains how local variations in surface temperature are not properly accounted for.Table 2 reveals that the average surface temperature in the focus area is almost constant and does not vary with respect to slope or aspect.A higher resolution thermal band would show higher surface temperatures on south and east facing slopes, since their orientation allows them to receive more incoming shortwave radiation throughout the morning.North and west facing slopes that do not receive radiation would have lower surface temperatures, which would reduce the thermal resistance.The mixed-pixel effect explains why the modeled results agree well with the average measured values, but do not capture the local variations.Therefore, after the thermal resistance maps have been derived, they must be resampled to 60 m since this is the level of their accuracy.

Sloped vs. flat model
One method to fill in the undefined pixels from the sloped model is by using the flat model (Foster et al., 2012).Figure 5a and c show the thermal resistance maps derived from the nonlinear sloped and flat models.The flat model captures the trends of higher thermal resistances in the terminal moraine and directly behind the glacier with smaller thermal resistances upglacier.However, these trends are not as prominent as in the 901 Introduction

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Full  2 reveals that the thermal resistances are underestimated due to the net radiation being overestimated since the flat model does not correct for topography.This is important because if the flat model values are used to fill in the undefined pixels in the sloped model, one must understand that the thermal resistances will be lower.Furthermore, the main discrepancy between the sloped model and the flat model arises for steep north and west facing slopes, which are the pixels that are classified as undefined.
A preferable alternative may be to use the average thermal resistance from the sloped model in Table 2 based on its slope and aspect.When this alternative is performed on the focus area (Fig. 5d), the average thermal resistance for the nonlinear sloped model changes slightly to 0.41 (±0.24) m 2 K W −1 .

High resolution vs. poor resolution DEM
The requirement of a high resolution DEM limits the ability to transfer these models to other regions where this may not be available.The ASTER GDEM was used to assess the importance of DEM resolution.The average thermal resistance in the focus area derived using the ASTER GDEM was 0.36 (±0.16) m 2 K W −1 .These thermal resistances underestimate the measured values and those derived using the DEM generated from ALOS PRISM (Lamsal et al., 2011).However, these thermal resistances are slightly better than those derived from the flat model despite its poorer resolution.
Therefore, while a high-resolution DEM will yield better results, we recommend the ASTER GDEM should be used instead of using a flat model to get an estimate of the thermal resistances in an unknown area.

Thermal resistances for all glaciers in the Everest region
The ASTER GDEM was used to derive the thermal resistances for the debris-covered glaciers in the Everest region (Fig. 6).The Ngozumpa and Khumbu glaciers have Introduction

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Full been outlined using the Global Land Ice Measurements from Space (GLIMS) database to compare the derived thermal resistances with previous debris thickness measurements (Nakawo et al., 1986;Nicholson and Benn, 2012).The results on the Khumbu glacier show good agreement with the debris thickness map generated by Nakawo et al. (1986).The thermal resistance is higher towards the terminal moraine and decreases upglacier.The same trend applies on Ngozumpa glacier, which agrees with the debris thickness measurements by Nicholson and Benn (2012).Thermal resistances on both glaciers range from above 0.50 m 2 K W −1 near the terminal moraines to less than 0.20 m 2 K W −1 upglacier.

Sensitivity analysis
The model developed in this study relies heavily upon meteorological inputs and assumed values for parameters associated with the debris cover.The meteorological inputs are subject to instrument error and may not be directly transferable from the site of the automatic weather station to the debris-covered glaciers.The particular meteorological parameters of interest are wind speed (u) and air temperature (T air ).The parameters associated with the debris cover that may affect results are the surface roughness length (Z 0 ) and the albedo (α).In addition, there is uncertainty associated with the nonlinear correction factor (G ratio ).Lastly, the scenarios of assuming zero latent heat flux (LE) or calculating the latent heat flux (LE) as a function of surface temperature are analyzed.A sensitivity analysis with respect to these parameters and scenarios was performed on the focus area to identify those that affect the derived thermal resistances (TR) the most.With respect to meteorological and debris cover parameters, the model is most sensitive to the wind speed and albedo and is moderately sensitive to the surface temperature, the nonlinear correction factor, air temperature, and lower values of the surface roughness length (Table 3).The model's sensitivity to wind speed is concerning because the automatic weather station is located 10 km away from the glacier.The model assumes the wind is the same on the glacier as it is at the automatic weather station, 903 Introduction

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Full but no data on this exists.Future research should determine if the wind speed at the automatic weather station is representative of the wind speed on the glacier.However, the derived thermal resistances are the average of 12 Landsat7 images, which should account for variations in the meteorological conditions at the automatic weather station and on the glacier.
The assumption of a constant albedo over the debris-covered glacier is another limitation of this model, especially since the model is sensitive to albedo.Methods exist to use other Landsat7 bands to estimate albedo (Liang, 2001); however, the authors had no way of validating these results.Furthermore, since the albedo affects the amount of incoming shortwave radiation absorbed by the debris, higher surface temperatures would likely indicate any large differences in the albedo.The lack of variation in surface temperatures over the debris-covered glacier suggests that assuming a constant albedo is reasonable.Therefore, the average value of albedo of 0.30 determined by a previous study on Ngozumpa glacier (Nicholson and Benn, 2012) was used in this study.Future work should explore deriving the albedo from satellite imagery in conjunction with field measurements to validate these results.
The model was most sensitive to how the latent heat flux term was defined.Previous results have assumed the latent heat flux to be zero based on the assumption that the debris cover is dry (Nakawo and Young, 1982;Foster et al., 2012).The sensitivity analysis reveals that assuming the latent heat flux is zero greatly underestimates the thermal resistance.The problem with this assumption is that the latent heat flux associated with the bare ice faces and melt ponds, which exist throughout the debris cover, is not accounted for.Ideally, the ice faces and melt ponds could be identified from satellite imagery and the latent heat flux could be applied to these pixels, but the resolution of the satellite imagery is too coarse.Other studies have calculated the latent heat flux as a function of the surface temperature (Nakawo and Young, 1982;Nakawo et al., 1999;Nicholson and Benn, 2006;Zhang et al., 2011).The sensitivity analysis shows this assumption greatly overestimates the thermal resistance.This is likely due to the surface vapor pressure in the latent heat flux term being overestimated by the assumption Introduction

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Full that the surface vapor pressure is saturated, which is not the case at the time of day when the satellite image is acquired.This study calculates the surface vapor pressure as a function of T 10 cm , which results in accurate thermal resistances and reasonable values for the latent heat flux term (results not shown).The use of T 10 cm is supported by the thermal conductivity results, which indicate the debris cover changes from dry to wet between 10 and 15 cm.Conceptually, the use of T 10 cm calculates the latent heat flux associated with the water content in the debris at the wet/dry interface evaporating.
The assumption that the latent heat flux may accurately be estimated using T 10 cm is another limitation of this work.However, the thermal resistance, latent heat flux, and thermal conductivity results appear to justify its use in this study.Future work should seek to measure the vapor pressure and water content throughout the debris cover to accurately estimate the latent heat flux.

Conclusions
The model described in this paper allows thermal resistances on debris-covered glaciers to be derived from Landsat7 satellite imagery in conjunction with meteorological data from a nearby automatic weather station.The model was applied to glaciers in the Everest region and the resulting thermal resistances were validated with field measurements.The model accounts for the nonlinear temperature gradient in the debris through the use of a nonlinear correction factor.Furthermore, the use of a high resolution DEM greatly improves the results of the derived thermal resistances.In the event that a high resolution DEM is not available, the authors recommend using a lower resolution global DEM to estimate thermal resistances as opposed to using a flat model.A sensitivity analysis reveals that the model is very sensitive to the latent heat flux.This model uses T 10 cm to estimate the latent heat flux, which yields accurate results.Field measurements of the latent heat flux over debris-covered glaciers in this region would allow these estimates to be properly validated.With regards to the meteorological and debris cover parameters, the model is most sensitive to the wind speed and Introduction

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Full  Full  Full  Full Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | also commented on areas with exposed ice cliffs reducing the surface temperature of the pixel, thereby lowering the calculated thermal resistances.Zhang et al. (2011) did not address the low values of thermal resistances, but did attribute the small disagreement between modeled and observed melt rates to the unknown variations in meteorological conditions caused by Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | . The required meteorological data for the MODTRAN 4 model used by Coll et al. (2010) was taken from Pyramid Station.The image-to-image Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | the debris thickness (cm).As the Landsat7 images are acquired at 10:15 and the thermistors recorded hourly temperatures, the temperatures in the debris at 10:15 were computed by linearly interpolating between 10:00 and 11:00.These interpolated temperatures were used to compute G ratio .
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | sloped model.The focus area reveals the flat model slightly underestimates the thermal resistances on the glacier as it has an average value of 0.34 (±0.10) m 2 K W −1 .Table Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | the surface albedo.Future work should explore how the wind speed varies spatially and derive the albedo from satellite imagery.The main limitation of this work is the poor resolution of the Landsat7 thermal band.The derived thermal resistances must be resampled to the resolution of the thermal band before they are used in melt models or other applications.Discussion Paper | Discussion Paper | Discussion Paper | Coll, C., Caselles, V., Valor, E., and Niclos, R.: Comparison between different sources of atmospheric profiles for land surface temperature retrieval from single channel thermal infrared data, Remote Sens. Environ., 117, 199-210, 2012.Conway, H. and Rasmussen, L. A.: Summer temperature profiles within supraglacial debris on Khumbu Glacier, Nepal, debris-covered glaciers, in: Proceedings of a Workshop held at Discussion Paper | Discussion Paper | Discussion Paper | measurements to derive supraglacial debris cover and thickness patterns on Miage Glacier (Mont Blanc Massif, Italy), Cold Reg.Sci.Technol., 52, 341-354, 2008a.Mihalcea, C., Mayer, C., Diolaiuti, G., D'Agata, C., Smiraglia, C., Lambrecht, A., Vuillermoz, E., and Tartari, G.: Spatial distribution of debris thickness and melting from remote-sensing and meteorological data, at debris-covered Baltoro glacier, Karakoram, Pakistan, Ann.Glaciol.Discussion Paper | Discussion Paper | Discussion Paper | Reid, T. D. and Brock, B. W.: An energy-balance model for debris-covered glaciers including heat conduction through the debris layer, J. Glaciol., 56, 903-916, 2010.Reid, T. D., Carenzo, M., Pellicciotti, F., and Brock, B. W.: Including debris cover effects in a distributed model of glacier ablation, J. Geophys.Res., 117, D18105, doi:10.1029/2012JD017795,2012.
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Table 1 .
Overview of satellite imagery used in this study.

Table 2 .
Trends in thermal resistance, surface temperature, and net radiation with respect to slope and aspect in the focus area.