1. Introduction
The snow surface temperature (SST) is an important variable in energy balance calculations of snowpack energetics and as a lower boundary condition for the atmosphere over snow-covered surfaces (King et al. 2008). The SST is defined here as the temperature responsible for longwave exitance and is not the temperature of the uppermost few centimeters of the snowpack. It forms the basis for calculations of longwave emission from the snow cover and a lower reference condition for calculations of sensible and latent heat flux (Kondo and Yamazaki 1990; Marks and Dozier 1992; Fierz et al. 2003). These calculations govern the coupled energy and mass budget equations that determine snow dynamics, particularly the energy state of snow, surface sublimation, and snowmelt. Various methods exist to estimate SST, including the assumption that it is at 0°C when melting occurs and is otherwise related to air temperature when net radiation is positive (Jordan 1991; Marsh and Pomeroy 1996), modified force–restore techniques (e.g., Luce and Tarboton 2010), heat conduction equations (e.g., Tarboton and Luce 1996; Raj Singh and Yew Gan 2005), dewpoint methods (Andreas 1986; Raleigh et al. 2013), and methods that employ the coupled mass and energy balance equations including radiation to snow (Kondo and Yamazaki 1990; Jordan 1991; Lehning et al. 2002; Ellis et al. 2010). Many land surface schemes (LSSs) for atmospheric models include explicit SST calculations; these are usually coupled energy and mass balance calculations for an infinitesimal “skin” layer of snow [e.g., Canadian Land Surface Scheme (CLASS; Verseghy 1991; Verseghy et al. 1993), Community Land Model (CLM; Oleson et al. 2008), and Joint UK Land Environment Simulator (JULES; Best et al. 2011)], though a version of the Interactions between Soil, Biosphere, and Atmosphere (ISBA) model uses the force–restore method (Douville et al. 1995). Evaluation of LSS performance over snow has suggested that most LSSs become too cold over the winter, and this could be partly due to an overestimation of longwave energy loss from snowpacks in some of these models (Slater et al. 2001). Energy balance snow models used for hydrology and snow dynamics vary from single-layer models such as the energy-budget snowmelt model (EBSM; Gray and Landine 1988) to more physically detailed layered models such as SNOBAL (Marks et al. 1999, 2008), SNTHERM (Jordan 1991), Crocus (Brun et al. 1989, 1992; Vionnet et al. 2012), and SNOWPACK (Bartelt and Lehning 2002). Marks et al. (2008) have shown that the performance of physically based layered snowmelt models is very sensitive to how the upper model snow layers are parameterized. A recent snow model intercomparison study found that many of the models had significant discrepancies in their longwave exitance when compared to observations (Rutter et al. 2009).
What is not always appreciated in process or modeling studies of SST is the strong difference between the temperature at the snow surface and the temperature just below or near the snow surface. A recent study (Helgason and Pomeroy 2012b) including detailed fine-wire thermocouple measurements of temperatures just below the snow surface (0–10 cm) found that they were strongly related to the 1.5-m air temperature because of convection through porous media; in contrast, radiometrically measured surface temperatures were up to 4°C colder than the snow just below. This is consistent with microwave observations of wet snow under freezing snow surfaces (Koh and Jordan 1995) and the rapid change in SST upon exposure in a snowpit wall (Schirmer and Jamieson 2014). It is therefore important to define the snow surface temperature as that occurring on the upper boundary of the snowpack, the boundary that is responsible for longwave exitance. Because longwave radiation is not transmitted through snow or water and has a very low reflectance (Dozier and Warren 1982), this boundary is likely to be exceedingly thin and will lay above the physical layers that can be measured with fine-wire thermocouple thermometry.
The wide variety of methods and apparent deficiencies in land surface scheme and snow model estimates of longwave exitance suggest a need to more fully understand the major energy and mass fluxes that control the SST and how these might be reliably calculated outside of full mass and energy balance models. Some methods focus on the radiometric cooling of the snowpack (Marsh and Pomeroy 1996), some on conduction from the snowpack (Luce and Tarboton 2010), while others focus on the aerodynamic considerations (Andreas 1986). It would be advantageous for calculating SST if methods could avoid relying on uncertain prognostic state variables such as the internal energy of the snowpack or the albedo of the snowpack. This avoids accumulation of biases in estimating snow surface and internal energy state that are a large source of error in snow models (Essery et al. 2013).
The purpose of this paper is to document observations of SST in a wide variety of environments and to attempt to relate these observations in a tractable to the main driving aerodynamic and radiative energy fluxes via a simple predictive model with minimal driving variable and parameter requirements. Parameter uncertainty and optimality are examined to derive a robust predictive model of SST. By doing so, the relative importance of aerodynamic and radiative transfer in controlling the SST under various environmental conditions can be diagnosed and the applicability of the model for estimating SST can evaluated for global applications.
2. Theory
A snowpack surface cross-sectional photograph taken in April 2003 in Wolf Creek Research Basin, Yukon, Canada. The cold snowpack has poorly bonded surface crystals and displays light penetration indicative of its porous medium nature.
Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0226.1
The RPM requires knowledge of air temperature, humidity, wind speed, incoming longwave and shortwave radiation, aerodynamic roughness, and atmospheric pressure (which can be measured or found from site elevation). Its parameters are snow aerodynamic roughness length and shortwave absorption factor. Aerodynamic roughness can be measured or estimated from published values. Estimation of the shortwave absorption factor at the surface requires information on the spectral distribution of shortwave radiation, the spectral albedo of the snow surface, angular reflectance, and the extinction of NIR in snow. All of these factors vary in complex ways: the spectral distribution of radiation with atmospheric conditions and multiple reflections by vegetation and terrain; the spectral albedo of snow with surface grain size, contaminants, and liquid water content; and radiation extinction with snow structure and contamination (Pomeroy and Brun 2001). It is possible to estimate snow radiative absorption using the calculations described by Warren and Wiscombe (1980) with recent adjustments for contaminants (Dang et al. 2015), but such estimates will depend on uncertain assumptions of surface layer thickness, dust, black carbon or organic matter contamination, grain size, wetness, and site-specific spectral irradiance. This factor is expected to be small because NIR is less than half of shortwave radiation and not all NIR is extinguished at the snow surface. As such, it should be less than (1 − albedo) and so should be less than 0.1 for fresh, clean snow and less than 0.3 for dirty, wet snow.
3. Sensitivity analysis
The RPM was investigated initially with a sensitivity analysis of its driving variables using fixed parameters in order to demonstrate how wind speed influences the ventilation factor and how temperature, humidity, wind speed, and radiation influence the snow surface temperature. Figure 2 shows that the ventilation factor increases initially rapidly from 0 as wind speed increases and approaches 1 asymptotically as wind speeds become high. Wind speed, temperature, and humidity for this example are from a reference height of 2 m above the snow surface, and relative humidity is with respect to ice. Example conditions are relative humidity = 80%, incoming longwave radiation = 250 W m−2, and wind speed = 2 m s−1. The rapid rate of change in 
Sensitivity of the ventilation factor 
Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0226.1
Sensitivity of simulated SST (solid line), aerodynamic equilibrium temperature (dashed line), and radiative equilibrium temperature (dotted line) to variations in (a) air temperature, (b) relative humidity, (c) incoming longwave radiation, and (d) wind speed for an aerodynamic roughness length of 3 × 10−3 m and no incoming shortwave radiation. Reference heights for atmospheric variables are 2 m above the snow surface. As each variable is changed, the others are kept fixed at an air temperature of −10°C, relative humidity of 80%, incoming longwave radiation of 250 W m−2, and wind speed of 2 m s−1.
Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0226.1
4. Observations
Observations of driving meteorology and snow surface temperatures to parameterize and test the RPM were taken at mountain pasture, lake, glacier, prairie pasture, and agricultural field sites in North and South America and obtained from data carefully collected by Météo-France in a large forest clearing mountain site in Europe. Data collection at the Americas sites was during periods of frequent site visits, which included frequent radiometer checking and cleaning. Kipp and Zonen (KZ) net radiometers (CNR1) were heated to reduce frost and snow accumulation. Data collection at the Météo-France site involved hourly cleaning of radiometers to ensure high-quality measurements over a long time period. All sites except for the French site had uniform, level fetches of at least 100 m with short or nonexistent vegetation. Site locations and photographs are shown in Fig. 4 and site descriptions follow. Table 1 lists instrumentation used to measure snow surface temperature and the driving meteorological variables.
Location of field sites and site photographs.
Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0226.1
Instrumentation used at various sites.
a. Pomeroy Acreage
The Pomeroy Acreage site is located in Saskatchewan, Canada (52°02′N, 106°38′W; 508 m MSL). Measurements were taken every 15 min over an undulating, snow-covered prairie grassland with greater than 100 m of open fetch in central Saskatchewan, Canada, 6 km south of the city of Saskatoon, from 15 February to 19 March 2004. The region sustains a subhumid continental climate with cold, dry winters. The site was snow covered with at least 25 cm snow depth throughout the experiment, but a small amount of grass was exposed above the snow surface.
b. Kernen Farm
The Kernen Farm site is located in Saskatchewan, Canada (52.09°N, 106.31°W; 512 m MSL). Measurements were taken every 15 min as part of a study published by Helgason and Pomeroy (2012b) over a level cultivated fallow field with greater than 100 m of fetch, 2.5 km east of the city of Saskatoon, from 23 January to 2 March 2007. Climate is similar to the Pomeroy Acreage. The site was snow covered throughout the experiment with a depth of approximately 42 cm.
c. Mud Lake
The Mud Lake site is located in Alberta, Canada (50°47′N, 115°18′W; 1896 m MSL). Measurements were taken every 30 min on a frozen lake surface with greater than 100 m of fetch in the Spray Valley, Canadian Rockies, from 24 to 30 January 2006. This is a cold continental site with deep, even snow covering the lake with at least 80 cm depth. The site experiences significant shading from surrounding mountains in January.
d. Zongo Glacier
The Zongo Glacier site is located in Bolivia (16°15′S, 68°10′W; 5150 m MSL). Measurements were taken every 30 min from 8 to 16 August 2004 as part of a joint France–Canada study at a site with more than 100 m of fetch described by Sicart et al. (2005) on a flat, snow-covered lower lobe of the Zongo Glacier, Huayna Potosi Massif, Cordillera Real, Bolivia. Climate is typical of tropical glaciers, and the austral winter was cool with occasional snowfall. The surface was primarily covered with a shallow snow cover, but glacier ice patches were exposed during the measurement period.
e. Col de Porte
The Col de Porte site is located in France (45.30°N, 5.77°E; 1325 m MSL). Measurements as part of a study published by Morin et al. (2012) were taken every 60 min by Météo-France over a mown grass surface in a forest clearing in a mountain pass, Chartreuse mountain range, French Alps from 1993 to 2011. The forest edge on three sides was initially 25–50 m from the instruments, and a large building was 50 m away on the fourth side. Forest clearing after 1999 left forest on two sides and the large building on the other. Shading by trees and mountains occurs at this site in winter. The climate is temperate humid continental with substantial snowfall and mild winter temperatures. Snow depth exceeds 50 cm for much of the winter, and shallow snow periods were excluded from our analysis.
f. Hay Meadow
The Hay Meadow site is located at Marmot Creek, Alberta, Canada (50°56′N, 115°08′W; 1436 m MSL). Measurements as part of a study by Helgason and Pomeroy (2005, 2012a) were taken every 30 min from a large, gently sloping, grass-covered clearing with at least 60 m fetch in a mixed-wood forest in Marmot Creek Research Basin, Kananaskis Valley, Canadian Rocky Mountains, from 13 February to 5 March 2005. The site was snow covered throughout the experiment with a depth greater than 15 cm, but a small amount of sparse grass was exposed above the snowpack.
5. Analysis
The RPM was run with the time step available from the dataset (15–60 min) over the six sites, five of them with observations available for one snow season and one site for 18 seasons, depending on data availability. To investigate sensitivity to model parameters, the model was run 1681 times for each of the sites with 41 values of the surface shortwave radiation absorption factor in linear increments from 0 to 1 and 41 values of the aerodynamic roughness length in logarithmic increments from 10−4 to 1 m. A 35-day calibration and demonstration season (January 2006) was chosen from the large Col de Porte dataset. Figure 5 shows contour plots of root-mean-square (RMS) differences between simulated and measured surface temperatures from these runs. For each site, a unique parameter combination that minimizes the root-mean-square error (RMSE) without equifinality was found; these parameter values, along with minimum RMSEs and corresponding average errors (bias) in surface temperature, are given in Table 2. The optimized shortwave absorption factor was small (<15%) for all sites, and from very small (<5%) to zero at two sites. The smaller absorption factors occurred at the higher-latitude midwinter sites in Canada where there were no local sources of dust or organic material (Hay Meadow sometimes had some sparse exposed grass above and on the snow and was near a gravel road, which was a source of dust), suggesting that NIR absorption effects on SST are primarily important for conditions where dust, organic material, and black carbon deposition may occur. Dust deposition is more common on snow in temperate and tropical mountain environments where there are nearby geological sources. The optimized roughness length was quite variable between sites, varying from 0.001 m for Mud Lake to 0.063 m for Col de Porte. The optimal roughness length for the four flat, long fetch sites in the Canadian Prairies and mountains was small, averaging 0.004 m, while higher roughness lengths on the Zongo Glacier (0.032 m) and Col de Porte (0.063 m) may reflect local boundary layer characteristics on a rough glacier and near a forest edge, respectively.
Sensitivity of model RMSE (contour interval 0.2°C) to variations in shortwave radiation absorption and aerodynamic roughness parameters.
Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0226.1
Parameters, biases, and RMSEs for optimized snow surface temperature simulations.
Figure 6 shows RPM simulations and observations of SST at single seasons for the six sites with the optimal parameters (Table 2) for each site. The figure illustrates the generally good fit (Table 2) of the optimized RPM to observations for a wide range of environments (from prairies to mountains to glaciers) and SST (from 0° to −40°C). The same model runs can be used to examine the behavior of the ventilation factor at the various sites (Fig. 7). The prairie sites were usually well ventilated with high 
SST measured (black dots) and modeled (red lines) with RPM at the six sites using optimized parameters from Table 2.
Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0226.1
Ventilation factor 
Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0226.1
To evaluate potential model performance with global parameters and the necessity of using shortwave radiation to drive the RPM, the model was run with 10% and 0% shortwave radiation absorption for smooth (0.003 m) and rough (0.03 m) aerodynamic roughness lengths for the complete dataset at all sites. The results are plotted as observed versus modeled data in Fig. 8, and the statistics for these simulations are listed in Table 3. The best global parameter simulations based on RMSE were for Mud Lake, Kernen Farm, Pomeroy Acreage, and Zongo Glacier—all sites with long open fetch, good wind exposure, and RMSEs <1.3 K. The best simulations based on bias were Pomeroy Acreage, Col de Porte, and Zongo Glacier. The poorest simulations based on RMSE and/or bias were for Hay Meadow and Col de Porte, which had forests nearby and RMSEs ranging from 2.3 to 3.5 K for the best set of global parameters. The only site with notably larger errors than the others is Hay Meadow. This site has an extremely gusty turbulent regime (Helgason and Pomeroy 2005) and sometimes had exposed grass above the snow. The gustiness of the site might have degraded the aerodynamic calculations, and the exposed grass may have affected surface temperature measurements. The parameter combination of smooth with 10% shortwave absorption provided the best simulations (RMSE) for the relatively level prairie and Hay Meadow mountain valley bottom sites while the rough and 10% shortwave absorption combination was optimal for the complex terrain sites of Col de Porte and Zongo Glacier. For the Mud Lake simulations (frozen lake snowpack, very clean snow, and low insolation period in midwinter), the optimal parameters were for zero shortwave absorption and a smooth aerodynamic roughness reflecting its extremely smooth and high albedo condition. There was no benefit to using shortwave radiation data to run the model for Mud Lake and little benefit at the prairie and mountain valley sites in Canada, as small differences in bias and RMSE show; however, RMSEs increased appreciably by from 0.85 to 1.66 K when radiation absorption was not included at the tropical and temperate mountain sites in Bolivia and France, where both high insolation and contamination of snow are more probable. It is clear that there is no one global parameter set, but that site information can be used to choose parameters from the set shown in boldface in Table 3 and demonstrated in Fig. 8. High-latitude sites where snowpacks are normally clean with relatively little dust deposition do not require consideration of shortwave absorption, while lower-latitude sites do. Sites on frozen lakes, in open valley bottoms, and on prairies are best served with a small aerodynamic roughness length, while those on glaciers and near forests and complex terrain should use a larger length. It is likely that a dynamical model of shortwave absorption would provide improved values for the absorption parameter and its seasonal evolution, but at the expense of a substantial increase in RPM complexity.
Scatterplot of measured and RPM-simulated SST for the six sites with four sets of global parameters (0% or 10% SW absorption, 0.03 or 0.003 m roughness length).
Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0226.1
RMSEs (K) and bias (K) for simulations with global parameters (0% or 10% SW absorption, 0.03 m or 0.003 m roughness length). Smallest RMSEs and bias are in boldface. The global parameter sets selected are italicized.
Any new model needs a test of its transferability to datasets not involved in its optimization or selection of global parameters. To test the RPM, the full 18-yr dataset from Col de Porte was used with the global parameter set for a rough aerodynamic surface with 10% shortwave absorption (Table 3), and results are shown in Fig. 9. The RMSE of 2.56 K and bias of −0.81 K are similar to the January 2006 data shown in Table 3 for the same global parameter set, suggesting model predictive stability despite climate variability and changes in site conditions and instrumentation over 18 years.
Scatterplot using tonal density to show the number of points in small hexagons representing measured and modeled SST for Col de Porte over 18 years using the shortwave absorption factor of 0.1 and aerodynamic roughness height of 0.03 m.
Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0226.1
Methods to estimate the SST that use the air temperature, dewpoint temperature, or ice bulb temperature (e.g., Raleigh et al. 2013) are attractive in that they only require information on atmospheric temperature and humidity and so have a requirement for fewer driving variables and parameters than the RPM. Unfortunately, these methods lack a physical basis to predict SST and so may not be able to accurately estimate it. To evaluate how well these methods could predict the SST over this dataset, their outputs were compared to observations and the results shown in Table 4. The RPM more accurately estimated SST than any of these approaches, with RMSE improvements ranging from 1.15 to 6.33 K. The more accurate of the simple methods were the ice bulb and dewpoint approaches with RMS difference with RPM of only 2.53 and 2.67 K, respectively. Errors from assuming the SST was equal to the air temperature were large, and the RPM improved these simulations by an RMSE change of 4.19 K.
RMSEs (K) for approximating snow surface temperature by air temperature Ta, dewpoint temperature Td, or wet bulb temperature Tw and the increase in RMSE (in parentheses) compared with those for the selected global parameters for IPM (selected set is italicized in Table 3).
6. Conclusions
The SST is the critically important upper boundary condition for the snowpack and lower boundary condition for the atmosphere and so of great interest to snow scientists, hydrologists, and atmospheric scientists. Various methods have been used in snow, land surface, and hydrological models to estimate SST, and they principally include air temperature, force–restore, heat conduction, dewpoint, ice bulb, and coupled energy and mass balance calculations. The physically based coupled energy and mass balance methods require a greater number of driving variables and parameters and so have larger uncertainty due to these inputs than do the other methods, despite their physical correctness.
In an effort to reconcile model complexity, uncertainty, physical correctness, and simplicity to create a robust model for estimating SST, the primary driving processes that influence snow surface energetics were identified as aerodynamic (sensible and latent heat transfer) and radiative (thermal and near-infrared radiation). A new SST model, the radiative psychrometric model (RPM) was devised based on this understanding and written so that the radiative and aerodynamic factors controlling SST could be clearly identified. The RPM was tested against careful SST measurements at six sites in North America, South America, and Europe that span prairie, mountain, frozen lake, and glacier surfaces with various wind exposures and fetch characteristics and was found to perform very well in estimating the SST with optimized parameters for shortwave radiation absorption and aerodynamic roughness length. Global parameters for shortwave absorption and roughness length were identified and applied based on a site classification. High-latitude sites with clean snow remote from sources of dust and pollution do not need to consider shortwave absorption in RPM, while lower- and midlatitude sites that are proximal to particulate sources do. Sites on frozen lakes, in open valley bottoms, and on prairies are best served with a small aerodynamic roughness length, while those on glaciers and near forests and complex terrain should use a larger length. A test of the RPM with site-selected global parameters for a longer time span at Col de Porte showed good temporal transferability. A comparison of the RPM with recently proposed SST estimation methods shows that the RPM provides superior predictions of SST when compared to air temperature, dewpoint, or ice bulb calculation approaches.
Acknowledgments
The authors would like to acknowledge funding from NSERC, NERC, CFI, CFCAS, Canada Research Chairs, Global Institute for Water Security, Alberta Agriculture and Forestry, and IRD France. The assistance of many students and staff of the Centre for Hydrology over the years was essential to high quality data collection. Data provided by IRD France from Dr. J.E. Sicart and Météo-France from Dr. S. Morin is gratefully acknowledged.
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