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 One function of the Community Land Model (CLM4) is the determination of surface albedo in the Community Earth System Model (CESM1). Because the typical spatial scales of CESM1 simulations are large compared to the scales of variability of surface properties such as snow cover and vegetation, unresolved surface heterogeneity is parameterized. Fractional snow-covered area, or snow-covered fraction (SCF), within a CLM4 grid cell is parameterized as a function of grid cell mean snow depth and snow density. This parameterization is based on an analysis of monthly averaged SCF and snow depth that showed a seasonal shift in the snow depth–SCF relationship. In this paper, we show that this shift is an artifact of the monthly sampling and that the current parameterization does not reflect the relationship observed between snow depth and SCF at the daily time scale. We demonstrate that the snow depth analysis used in the original study exhibits a bias toward early melt when compared to satellite-observed SCF. This bias results in a tendency to overestimate SCF as a function of snow depth. Using a more consistent, higher spatial and temporal resolution snow depth analysis reveals a clear hysteresis between snow accumulation and melt seasons. Here, a new SCF parameterization based on snow water equivalent is developed to capture the observed seasonal snow depth–SCF evolution. The effects of the new SCF parameterization on the surface energy budget are described. In CLM4, surface energy fluxes are calculated assuming a uniform snow cover. To more realistically simulate environments having patchy snow cover, we modify the model by computing the surface fluxes separately for snow-free and snow-covered fractions of a grid cell. In this configuration, the form of the parameterized snow depth–SCF relationship is shown to greatly affect the surface energy budget. The direct exposure of the snow-free surfaces to the atmosphere leads to greater heat loss from the ground during autumn and greater heat gain during spring. The net effect is to reduce annual mean soil temperatures by up to 3°C in snow-affected regions.
 The Community Land Model version 4 (CLM4) [Lawrence et al., 2011], the terrestrial component of the Community Earth System Model (CESM1) [Gent et al., 2011], is a spatially distributed one-dimensional vertical model that provides the lower boundary condition for the atmospheric model. Surface albedo, which strongly influences the net surface energy budget, is calculated by CLM4 as a function of plant functional type, soil texture and moisture, and snow cover [Oleson et al., 2010]. The effect of snow cover on surface albedo leads to a positive feedback on climate called the snow-albedo feedback [Hall and Qu, 2006], whereby increases in surface temperature tend to reduce snow cover, and therefore surface albedo. A reduction in surface albedo tends to increase net surface radiation, thereby amplifying the initial temperature increase; the opposite effect occurs in response to negative temperature perturbations. Thus, the simulation of snow cover plays an important role in predicting the magnitude of future climate change both globally and regionally.
 Climate simulations are typically performed at horizontal spatial scales of hundreds of kilometers and larger, whereas snow cover can vary at spatial scales as small as a few meters. Because of the significant unresolved heterogeneity at subgrid spatial scales, the uniform grid cell snow cover implicit in the vertical one-dimensional model structure tends to overestimate surface albedo. To better simulate surface albedo when snow is present, CLM4 employs a fractional snow-covered area, or snow-covered fraction (SCF), parameterization.
 The fractional snow-covered area parameterization used in CLM4 was developed byNiu and Yang  (hereafter referred to as NY07), who examined the relationship between SCF derived from Advanced Very High Resolution Radiometer (AVHRR) observations and snow depth from the Canadian Meteorological Centre's (CMC) analysis, both at 1° by 1° spatial resolution and monthly temporal resolution. NY07 observed that SCF generally increases as a function of snow depth before saturating at a value of 1 at snow depths of the order of a few tens of centimeters. It was also observed that the average slope of the snow depth–SCF scatterplots was lower during the spring than during the autumn, while the average snow density (based on the CMC analysis) increased. Based on these observations, NY07 proposed the following empirical SCF parameterization:
where d is grid cell average snow depth, z0g = 0.01 is called the ground roughness length, ρsnow is the snow density, ρnew = 100 kg/m3 is the density of new snow, and m is an empirical constant. The value of m given by NY07 is 1.6, while the value of m used in CLM4 is 1.0.
 The NY07fractional snow-covered area parameterization was based on the relationship between monthly averaged snow depth and SCF. Because the melting of the snowpack may occur during periods shorter than a month, we performed a study to assess how well the CLM4 SCF parameterization represented the snow depth–SCF relationship at the daily time scale. Using SCF data derived from the MODIS instrument aboard NASA's TERRA satellite and the CMC snow depth analysis, we examined the snow depth–SCF relationship for both the daily and monthly averages. We were able to reproduce the monthly relationship described byNY07, who noted generally higher values of SCF for a given snow depth during the accumulation season, and generally lower values during the ablation season. However, at the daily time scale, we did not find evidence of this behavior. This apparent contradiction is the result of two things. First, the CMC snow depth analysis exhibits a tendency to melt earlier than indicated by the MODIS observations, leading to a general overestimation of SCF as a function of snow depth. Second, the monthly averaging period is generally shorter than the accumulation period, but longer than the snowmelt period, leading to relatively higher values of monthly averaged SCF in the autumn, and lower values in the spring.
 We then reexamined the snow depth–SCF relationship, by replacing the CMC snow depth analysis with snow depth estimates from NOAA's Snow Data Assimilation System (SNODAS). SNODAS data are limited geographically to the United States, but are based on a higher resolution atmospheric analysis and a more sophisticated snow evolution model than are the CMC analyses. We found higher consistency of melt dates between the SNODAS snow depth data and the MODIS SCF observations, which revealed a much different snow depth–SCF relationship during the ablation season. The SNODAS-MODIS relationship agrees qualitatively with depletion curves derived from field observations. Based on this relationship, we develop an analytical snow depth–SCF parameterization that reproduces the general features of the observed relationship, and is straightforward to implement in a land surface model.
 Changes in surface energy fluxes relative to the control model are shown for two experiments. In the first simulation, surface fluxes are computed with the standard assumption of a uniform snowpack, while in the second simulation surface fluxes are computed separately for snow-covered and snow-free grid cell fractions. In the latter case, snow depth is represented as an average over snow-covered area alone, rather than total grid cell area. When snow depth is defined in this manner, certain numerical properties of the snow depth–SCF parameterization are required to ensure a consistent physical relationship between modeled snow water equivalent, snow depth, and SCF; it is shown that the new parameterization fulfills the required numerical constraint.
 The results presented here are meant to reveal a deficiency in the current snow depth–SCF parameterization, highlight the need for consistent data sets from which to determine the actual relationship between snow depth and SCF, and provide a basis for future model development.
2.1. MODIS Fractional Snow-Covered Area
 The Moderate Resolution Imaging Spectroradiometer (MODIS) measures radiation in the infrared to visible portion of the spectrum (wavelengths from 0.4 μm to 14.4 μm). In this study we use MODIS data from NASA's TERRA satellite. Snow-covered area is based on the normalized snow difference index (NSDI), which exploits differences in the reflectance of snow in the visible (0.55μm) and near-infrared (1.6μm) wavelengths [Hall et al., 2006]. The MOD10C1 product, which has daily temporal resolution and a spatial resolution of 0.05 degrees, incorporates MOD10A1 daily snow cover data at 500 m resolution. Per cent cloud cover is also provided.
2.2. CMC Snow Depth Analysis
 The Canadian Meteorological Centre (CMC) produces an operational snow depth analysis with daily temporal resolution and approximately 24 km spatial resolution for the years 1998 through 2010 [Brown et al., 2003; Brown and Brasnett, 2010]. The CMC analysis, based on the scheme of Brasnett , uses a first-guess field computed by applying a simple snow accumulation and melt model to temperature and precipitation fields provided by the CMC Global Environmental Multiscale (GEM) forecast model. The snow depth field is then updated with observations collected by the World Meteorological Organization (WMO) using the method of optimum interpolation. Snow melt is calculated using a degree day algorithm that removes mass from the snowpack at the rate of 0.15 mm/h K.
2.3. SNODAS Snow Depth Analysis
 The National Operational Hydrologic Remote Sensing Center also produces a snow analysis, called Snow Data Assimilation System (SNODAS) [Barrett, 2003]. The purpose of SNODAS is to merge model estimates of snow cover with snow data from satellite and airborne platforms, and ground stations [Carroll et al., 2001]. The SNODAS snow model is forced by output from the Rapid Update Cycle (RUC2) Numerical Weather Prediction (NWP) model, which is then updated using remotely sensed and ground based observations. The snow model is a physically based, spatially distributed energy and mass balance model, and its output has a spatial resolution of 1 km and a temporal resolution of 1 h. In this study, we use SNODAS data that have been aggregated to daily resolution. The data domain is limited; the analysis covers only the conterminous United States.
2.4. The Community Land Model
 The Community Land Model (CLM4) [Lawrence et al., 2011] is the land component of the Community Earth System Model (CESM1) [Gent et al., 2011]. CLM4 simulates the partitioning of mass and energy from the atmosphere, the redistribution of mass and energy within the land surface, and the export of fresh water and heat to the oceans. To realistically simulate these interactions, CLM4 includes terrestrial hydrological processes such as interception of precipitation by the vegetation canopy, throughfall, infiltration, surface and subsurface runoff, snow and soil moisture evolution, evaporation from soil and vegetation and transpiration [Oleson et al., 2010]. Snowpack in CLM4 is modeled as a one-dimensional vertical column; modeled processes include snow accumulation and melt, compaction, and water transfer between snow layers [Lawrence et al., 2011].
2.5. MERRA Forcing Data
 In this study, we use CLM in offline mode, in which the atmospheric boundary conditions are specified. The observed forcing data, which provide precipitation, air temperature and pressure, specific humidity, shortwave radiation, and wind speed, were obtained from NASA's Modern-Era Retrospective Analysis for Research and Applications (MERRA).
 MERRA was generated with version 5.2.0 of the Goddard Earth Observing System (GEOS) atmospheric model and data assimilation system (DAS) and has a spatial resolution of 1/2° latitude and 2/3° longitude with 72 vertical levels, from the surface to 0.01 hPa [Rienecker et al., 2011]. Land surface state variables were initialized from the land surface state at the end of a multidecade offline CLM4 run, and a simulation was generated for the period 2000–2010 at a spatial resolution 0f 0.9° latitude and 1.25° longitude.
3. Observed SCF–Snow Depth Relationship
 Utilizing recent SCF observations derived from the MODIS instrument, we were able to reproduce the monthly average snow depth–SCF relationship observed by NY07. Monthly averaged snow depth data from the CMC analysis and SCF derived from MODIS for the years 2004 through 2008 were used to create two-dimensional histograms.Figure 1 shows contours of the logarithm of the histograms for each month of the year, excluding June through September. Also plotted is the NY07 SCF parameterization (black line) using the snow densities reported in NY07. Because the snow density varies from about 175 kg/m3 in October to 320 kg/m3 in May, the SCF curves (black lines) are steeper in the fall and less steep in the spring, mimicking the behavior of the observations, which show a tendency toward lower values of SCF for a given snow depth as the snow season progresses. Thus, the use of monthly MODIS data gives a result that is consistent with NY07, who used AVHRR-based SCF data.
 An implicit assumption of NY07 is that this relationship, based on monthly averaged quantities, also holds at shorter time scales, e.g., daily, and can therefore be used to determine changes in SCF that occur at the model time step. When daily averaged data are used, however, we find that this assumption may not be valid. Figure 2 is similar to Figure 1 except that daily SCF and snow depth are used to create the histograms. While the histogram contours are somewhat broader during spring than during fall, the general shape of the histograms varies little; the peak values of the histograms in all months tends toward 1 for snow depths greater than a few cm, and do not follow the evolution of the NY07 SCF curve. Note that contours of the logarithm have been plotted, with higher values indicating much higher histogram values.
Figure 2 indicates that the snow depth–SCF relationship exhibited by the data does not greatly change as a function of season. Why then do the monthly data show snow depth–SCF curves whose slope decreases as time progresses? The evolution of the monthly SCF–snow depth relationship can be explained by seasonal differences in the rate of change of snow depth. During the accumulation season, the snowpack increases over a period of a few months, but in the melt season, the snowpack typically disappears in a few weeks or less [Wang et al., 2008]. This is shown schematically in Figure 3, which depicts a hypothetical snowpack (Figure 3(left), solid line) that steadily increases during the winter months (e.g., November–February) before leveling off and melting over a three week period in the spring. In this scenario, the relationship between SCF, shown as a dashed line during accumulation and a dot-dashed line during melt, and snow depth is the same throughout the year. However, because of the difference in the rates of accumulation and melt, themonthly average snow depth–SCF relationship is more linear during the relatively short melt season. This is shown in Figure 3(right). The actual depth–SCF curve is shown by the solid line, while the monthly average curves are shown as dashed (accumulation) and dot-dashed (melt) lines. This example implies that even if the underlying relationship were time-invariant, one would expect the shift in the monthly snow depth–SCF relationship reported byNY07.
 A superficial interpretation of Figure 2 might be that SCF increases rapidly as a function of snow depth throughout the snow season, with little or no hysteresis. However, a closer analysis of the data reveals that the relationship implied by Figure 2 may actually be due to inconsistencies in the two data sets used to construct the histograms. A comparison of the time series of SCF and CMC snow depth for individual points finds numerous locations where the snowpack estimated by the CMC analysis disappears much earlier than the date implied by the MODIS SCF data. Figure 4 (left) shows contours of the logarithm of a histogram of melt dates, in days from 1 January. The x axis shows the MODIS melt date, while the y axis show the CMC melt date. Melt dates are defined as the first date at which each time series declines to less than 5% of its maximum value.
Figure 4 shows that the CMC analysis systematically melts earlier than MODIS SCF observations indicate. As an example of a location having a large difference in melt date, the time series at a location in the Rocky Mountains, are plotted in Figure 4 (right). MODIS SCF are represented by open circles, while CMC snow depth is depicted with a solid blue line. At this location, the CMC data reach zero about a month before the MODIS data do. Points exhibiting this type of behavior lead to an overestimate of SCF as a function of snow depth, which can be seen in Figure 2 as the large histogram values for SCF values near 1 for all snow depths, and during April and May whose peak histogram values are in the smallest snow depth bin. While Figure 4 implies that there are many locations where the MODIS and CMC data sets are inconsistent, it does not by itself inform us about the relative accuracy of the two data sets. In section 3.2, we examine a second snow depth data set to help understand these discrepancies.
 To complement the CMC snow depth analysis, we also utilize SNODAS snow depth data [Barrett, 2003]. Although SNODAS covers a limited domain (the “lower 48” states of the USA), it possesses a much higher spatial resolution (1 km) than the CMC analysis. A comparison of melt dates from SNODAS and MODIS is shown in Figure 5 (left). The significant early bias of the CMC data relative to the MODIS data observed in Figure 4 is absent from Figure 5, implying a greater consistency between the MODIS and SNODAS data sets.
Figure 5 (right) is the same as Figure 4 (right) except that the SNODAS snow depth time series has been added. The SNODAS peak snow depth is about 60% larger than the CMC peak, while the melt date is later by about a month. The later SNODAS melt date agrees closely with that of the MODIS data.
 Histograms of MODIS SCF and SNODAS snow depth are shown in Figure 6. Like Figure 2, Figure 6 displays contours of the logarithm of the SCF–snow depth histograms for the months October–May. The histograms in Figure 6 for most months are qualitatively similar to those shown in Figure 2, such that after the snowpack has reached a depth of about 20 cm, its SCF is above 0.5 with increasing numbers of points as SCF approaches 1. Differences begin to appear during spring that are especially noticeable during April and May. In these months, the histogram peaks shift away from the SCF = 1 line, and a more linear snow depth–SCF relationship emerges. From Figure 6, we infer that the SCF–snow depth relationship depends not only on the amount of snow, but also on whether it is increasing or decreasing.
 The hyperbolic tangent functional form of the CLM4 SCF parameterization cannot capture the quasi-linear form shown in the April and May histograms ofFigure 6. Furthermore, there is no physical basis for its dependence on snow density. Compaction, for example, is a largely vertical process that results in changes in snow density that are independent of changes in either snow water equivalent or SCF. We therefore propose a new SCF parameterization that better represents the snow depth–SCF relationship depicted in Figure 6 for all months. In addition, the parameterization will include an explicit dependence on the sign of changes in snow mass. Thus, accumulation and melting events of the same magnitude will alter SCF by differing amounts.
4. A New SCF Parameterization
4.1. Accumulation Events
 To parameterize the increase in SCF due to a snowfall event, we assume that precipitation is distributed randomly throughout a region, e.g., a model grid cell or satellite pixel, and that events having greater amounts of snowfall lead to higher SCF. Thus, the fraction of a pixel that is snow covered after a single precipitation event can be expressed as
where s is the probability that a point within the pixel is snow covered after a single snowfall event, w is the snow water equivalent, and k is a scale factor. In principle, k can be estimated by measuring s and wwhen precipitation occurs over an initially snow-free area. After the first snowfall, the probability that a location is snow-free isp1 = 1 − s1. When a second snowfall event occurs, it covers a fraction s2 of the pixel, leaving a snow free fraction p2. Assuming that the areas covered by snowfall events are spatially uncorrelated, the fraction of a pixel that remains snow-free after both events is therefore the product ofp2 and p1, and the snow-covered area isF2 = 1 − p2p1. Generalizing this result to an arbitrary number of snowfall events, the snow-covered fraction due to eventN + 1 is therefore
To update SCF after snowfall event N + 1 using equation (3), one only requires the current SCF, FN, and the amount of snow that falls, wN+1, with which to calculate sN+1.
Figure 7 shows SCF as a function of snow depth using equation (3) for various values of k. In this example, snow accumulates by a series of 100 snowfall events of 3 mm water equivalent each, and snow depth is converted from water equivalent using a constant density of 100 kg/m3. As kincreases from a value of 0.02 (dark blue line) to 0.26 (red line), the probability of a subpixel location being snow-covered increases, and the accumulation curve rises more steeply. Like the hyperbolic tangent function used in the CLM4 parameterization, each curve varies smoothly and saturates at a value of 1. Unlike the CLM4 parameterization, only a single parameter,k, is required, and the functional form of the relationship emerges naturally from its probabilistic formulation.
4.2. Melting Events
4.2.1. Dimensionless Snow Depletion Curves
 The rate at which SCF decreases as snow melt progresses, often called the snow depletion curve, has been studied extensively in field studies. Luce et al.  monitored a 0.26 km2 catchment in Idaho, and found that while peak snow accumulation varied from year to year, the spatial pattern of snow water equivalent (SWE) remained generally constant. This observation led the authors to propose a dimensionless depletion curve, in which the SCF is parameterized as a function of the ratio of current SWE to maximum SWE. In a subsequent study, Luce and Tarboton  showed that the dimensionless depletion curves based on 9 years of observations varied little between years despite differences in maximum accumulation.
Figure 8 (top) shows the MODIS SCF–SNODAS snow depth histograms for February through May using relative depth, i.e., depth divided by that snow season's maximum depth. Figure 8 indicates that the roughly linear relationship displayed in Figure 6 is more pronounced earlier in the melt season when relative depth is used instead of depth.
 Because topography is an important determinant of the spatial variability of SWE and snow depth [Clark et al., 2011], we created histograms based on the variability of topography within each grid cell. Figure 8(middle) shows the histograms computed using only grid cells having a standard deviation of within-grid cell topography of 200 m or less, andFigure 8(bottom) shows histograms based on locations having a standard deviation of topography of greater than 200 m. When this discriminant is used, one can see that the quasi-linear form of the histograms inFigure 8 (top) is mainly due to points having large topographic variation. Relatively flat grid cells (Figure 8, middle) appear to experience nearly uniform snow cover for all values of relative depth greater than about 0.1.
4.2.2. Closed Form SCF Parameterization
 In addition to accurately representing unresolved processes, parameterizations developed for use in Earth System Models like CESM are constrained by the need for computational efficiency and numerical stability. Based on these criteria, we have developed the following empirically derived expression that relates SCF to the dimensionless snow water equivalent during melting events
where Nmeltis a parameter that controls the shape of the snow-covered areaF. As shown below, the inverse cosine function possesses the flexibility to capture the spread in snow depth–SCF trajectories shown in Figure 8, as well as having numerical properties that facilitate an internally consistent description of snow depth, density and SCF. An additional benefit of this SCF parameterization is its computationally efficient closed form.
Figure 9 shows depletion curves defined by equation (4) for values of Nmelt ranging from 0.25 to 8.0. The depletion curves in Figure 9 demonstrate that equation (4) spans the full range of possible SCF trajectories. By varying Nmelt, the new parameterization can model both the steep depletion curves implied in Figure 8 (middle) and the more concave curves depicted in Figure 8 (bottom).
 While the shape parameter Nmelt is not physically based, and therefore cannot be directly measured, the variation of the depletion curves implied by Figure 8 suggests that it should be a function of topographic variability. Figure 10 shows the result of estimating SCF from SNODAS snow depth data using equation (4). The shape parameter, Nmelt, is determined from the standard deviation of topography, σtopo, by
 Thus, depletion curves for regions with high variability have lower values of Nmelt, and those for regions with low variability have higher values of Nmelt. Although the spread of the predicted histograms (Figure 10) is less than the spread of the observations shown in Figure 8 (top), the qualitative agreement is good. The discrepancies are not surprising because topographic variability is not the only determinant of variability in snow melt [Clark et al., 2011; Liston, 2004], and both observational data sets contain errors. Despite these issues, the parameterization captures the different shapes of the depletion curves for regions of low and high topographic variability.
4.2.3. SWE Probability Distributions
 An approach advocated by Clark et al.  is to estimate the depletion curve by assuming an analytical expression for the distribution of snow within a region. Given a probability distribution function (pdf), one can derive functional relationships between amount of melt and SCF, and amount of melt and the mean snow water equivalent of the region [Donald et al., 1995; Luce et al., 1999; Luce and Tarboton, 2004; Liston, 2004]. Given the latter two relationships, one can then determine the SCF for a particular value of mean SWE, i.e., the depletion curve. This approach is appealing because in principal the parameters describing the distribution are measurable. However, two issues arise regarding its use: the appropriateness of the unimodal analytical function used to approximate the distribution, and the numerical properties of the depletion curves for some parameter values. These issues are presented in the remainder of this section.
Figure 11 shows dimensionless depletion curves obtained by assuming a lognormal snow water equivalent distribution (see derivations of Donald et al.  and Liston ). The shape of the depletion curve derived from a lognormal SWE distribution is determined by the coefficient of variation (CV), which is defined as the ratio of the standard deviation to the mean value. Snowpacks characterized by low CV values lead to depletion curves that are very nonlinear; SCF is close to 1 for much of the melt period before rapidly decreasing to zero. In the limit that CV approaches zero, the depletion curve rises abruptly to 1, consistent with a uniform snowpack undergoing uniform melt. As the coefficient of variation increases toward a value of 1, the depletion curves become more linear. For Cv > 1, the depletion curves again become more nonlinear, with a progressively more concave shape.
 Coefficients of variation of premelt SWE from a large number of field studies were collected and reported by Clark et al. . In the reported studies, CV of nonmountainous areas were almost all less than 1, with most values being between 0.1 and 0.5. A comparison of the depletion curves in Figure 11 to the histograms in Figure 8 (middle) indicates that the depletion curves for regions having low topographic variability can be described by CV of about 0.5 or less.
 In areas of higher topographic variability, the histograms shown in Figure 8(bottom) are consistent with depletion curves corresponding to CV of about 1.5 or more. This is problematic for two reasons. First, the functional form of the pdf-derived depletion curves for CV greater than about 0.75, which are concave upward for small values of the abscissa, can cause numerical instabilities when used in a self-consistent model. Consider a grid cell having a unit area, i.e.,A = 1. The volume of snow in the grid cell, V, can be expressed as either
where z is the grid cell mean snow depth, or
where D is the mean snow depth where snow is present and Fis the fractional snow-covered area. The grid cell mean snow depth,z is the grid cell mean snow water equivalent W divided by the bulk snow density ρ. Replacing z and solving for D gives
If we assume that ρ is bounded, e.g., 100 kg/m3 ≤ ρ ≤ 1000 kg/m3, then it becomes apparent that as W decreases, F must approach zero more slowly than W, otherwise D will approach a constant or infinity. Thus, the criterion for numerical stability of equation (8), i.e.,
must be considered when choosing an SCF parameterization and mean snow depth is conditioned on SCF. If one approximates snow-covered area for small values ofW by a power law form:
where c and b are arbitrary constants, then (9) indicates that b must be less than 1, and F must be a sublinear function of W [Luce and Tarboton, 2004]. When the pdf of SWE is approximated with a lognormal distribution, this condition is met for CV having values of about 0.75 or less; larger values will cause F to approach zero more rapidly than W, and D will not approach zero as it should. The inverse cosine based expression in equation (4) satisfies equation (9) for all values of Nmelt, which can be seen by applying L'hôpital's rule.
 The parameterization described by equations (4) and (5) adequately captures the shape of the high topographic variability depletion curves, but a second concern remains. As mentioned above, the curves described by the peak values of the histograms in Figure 8 (middle) resemble pdf derived depletion curves having values of CV of about 1.5 to 2.5. Of the studies reported by Clark et al.  that were classified as taking place in mountainous terrain, only a small fraction estimated CV greater than 1. Thus the depletion curves implied in Figure 8 for regions of high topographic variability, which is highly correlated with high elevation, are not consistent with the field studies reported by Clark et al. .
 An issue that must be considered when using the pdf approach is the spatial scale of the area to be modeled. As pointed out by Essery and Pomeroy , the appropriateness of the assumption of a unimodal SWE distribution will depend on the complexity of the landscape and the spatial scale of the region being examined. Most of the field studies reviewed by Clark et al.  that reported CV were performed at spatial scales of a few square kilometers or less. Global climate and weather models typically simulate regions having areas of hundreds to thousands of square kilometers, and we have therefore analyzed the SCF and snow depth data at these spatial scales. The relatively large spatial scales considered here may explain the differences in the low/high topographic variability histograms; over relatively flat terrain, SWE variability may be well represented by a singly peaked pdf, while regions of higher topographic variability may be better described by multimodal pdfs.
 Consider a grid cell having two regions described by lognormal SWE distributions having the same coefficient of variation, for example CV = 0.25, but different average SWE values. One region has a mean SWE of 0.2 m, while the other has a mean SWE of 1.0 m. Observed separately, both regions will have identical depletion curves, corresponding to CV = 0.25. Observed as an aggregate region, the depletion curve will look quite different. This scenario is illustrated in Figure 12. The pdf of each region is shown in Figures 12a and 12b, while the pdf of their sum is shown in Figure 12c. The depletion curves are plotted in Figure 12d. Because their form depends only on CV, and not on their mean value, the dimensionless depletion curves for the pdfs in Figures 12a and 12b are the same. The aggregate depletion curve, however, has significantly lower SCF values for a given value of W/Wmax. After about 0.4 m of melt has occurred, the first region will be largely snow-free, while the second region will be mostly snow covered. The SCF of the aggregate is therefore about 0.5, while the normalized mean SWE is about 0.67.
 The resemblance of the bimodal depletion curve to the histograms in Figure 8 (bottom) implies that the form of the depletion curves based on the MODIS and SNODAS data is not inconsistent with the field studies reviewed by Clark et al. , but rather is a function of the spatial scale of the observations.
4.2.4. Periods of Mixed Accumulation and Melt
 In nature, a single accumulation season is not always followed by a single melt season; accumulation and melting events occur throughout the cold season, and are often interspersed. In the former scenario, the application of the SCF parameterizations for accumulation and melt is straightforward. As accumulation progresses, SCF rises along the accumulation curve defined by equation (3) until maximum accumulation occurs. Melting then begins, and SCF follows the dimensionless depletion curve described by equation (4) until the modeled snowpack disappears. When an accumulation event follows a melting event, however, the new SCF value, F, is generally no longer consistent with the depletion curve defined by W and Wmax. In this situation, the depletion curve must be adjusted to account for the accumulation event [Luce et al., 1999]. To accomplish this, F, W, and Wmax can be reconciled by redefining Wmax according to
where the values of F and W are evaluated after the accumulation event.
 To assess the effectiveness of the new SCF parameterization at reproducing the observed snow depth–SCF relationship (Figures 6 and 8) and to examine its effect on the surface energy budget, we performed a CLM4 simulation (“NEW_SCF”) forced with observed meteorological forcing from the MERRA reanalysis [Rienecker et al., 2011] for the period 2000–2008 and compared it to a simulation based on the standard version of CLM4 (“Control”). Model output for the period 2004–2008 was used in the following analysis.
5.1. CLM4 Accumulation and Depletion Curves
Figure 13 shows the results of using the new SCF parameterization in CLM4. The NEW_SCF simulation uses an accumulation parameter k (equation (2)) with a value of 0.1 (Figure 7). Figure 13 (top) shows daily averaged SCF from the Control simulation plotted as a function of snow depth during November (Figure 13, left) and April (Figure 13, right). During November in the Control simulation, most grid cells where snow is present experience accumulation and become completely snow covered at depths of 5 to 10 cm. In April, variations in snow density lead to some spread in the snow depth–SCF curves, in accordance with equation (1). Nearly all grid cells with mean snow depth of 20 cm and higher remain completely snow covered.
 The snow depth–SCF relationship that emerges from the NEW_SCF simulation is different in both the accumulation and melt seasons. In November, the snow depth–SCF curves are largely determined by the accumulation parameterization. Due to the choice of accumulation parameter, k= 0.1, most grid cells do not become completely snow-covered until snow depths reach 20 cm. Increasing the value ofk would steepen the accumulation curve, and one could match the Control November curve by using a value of k of about 0.4. Because of inconsistencies in the observations, it is difficult to constrain the value of k; it can be seen in Figure 6, for example, that the histogram shows a number of pixels where the SNODAS snow depth is zero, but MODIS indicates a nonzero SCF. We have chosen the value k = 0.1 as a plausible lower limit.
 In April, when many grid cells experience melt, the melt parameterization used by NEW_SCF (equation (4)) causes a much greater spread in SCF as a function of snow depth. This is accentuated in Figure 14, which shows the April dimensionless depletion curves for the Control (Figure 14, left) and NEW_SCF (Figure 14, right) simulations. Figure 14 shows that most grid cells in the Control simulation follow a very nonlinear trajectory of SCF as a function of relative depth, while many grid cells in the NEW_SCF experience a more nearly linear trajectory; this is more consistent with the SNODAS/MODIS observations plotted in Figure 8. As one would expect from equation (5), grid cells with larger topographic variability exhibit lower SCF values for a given value of relative depth (not shown).
5.2. Effects of New SCF Parameterization on Surface Energy Budget
 The fractional snow-covered area parameterization in CLM4 is used to compute the albedo in grid cells where snow is present, and therefore influences the surface energy budget.Figure 15 shows the differences in SCF, net radiation, turbulent heat fluxes, i.e., the sum of sensible and latent heat, and ground heat flux for the two CLM4 simulations during October. Because the snow depth at most grid cells at this time is less than 20 cm, differences in SCF up to 0.2 exist across much of the high latitudes. The generally lower SCF in the NEW_SCF simulation leads to lower albedos, but because insolation is approaching the boreal winter minimum, net radiation increases by at most 1–2 W/m2. Most of this increase in net radiation is then removed via turbulent heat exchange, and ground heat flux changes are negligible. As a result, temperature in the top soil layer is largely unchanged (Figure 17, left).
 During spring, insolation is higher than in the autumn, and somewhat larger effects can be seen. Figure 16 is the same as Figure 15 except that differences in May average surface energy fluxes are shown. Areas having relatively high topographic variability, e.g., Alaska and eastern Siberia, exhibit lower SCF in the NEW_SCF simulation, leading to increases in net radiation of 15–20 W/m2. This extra energy heats the snowpack, increasing turbulent heat fluxes to the atmosphere and melting the snowpack earlier. Monthly averaged ground heat flux is therefore higher in these areas, and soil temperature increases of 1–2°C result (Figure 17, right).
5.3. Partitioning Snow-Covered and Snow-Free Surface Energy Fluxes
 In CLM4, SCF is used to calculate albedo as the SCF-weighted average of snow-covered and snow-free albedos. SCF therefore affects absorbed solar radiation, but the remaining terms in the surface energy budget (i.e., emitted longwave radiation, sensible and latent heat, and ground heat flux) are calculated based on the assumption of a uniform snowpack. Thus, even for small amounts of snow, the interaction between the atmosphere and the soil column is modulated by the thermal properties of the snowpack. In addition, the modeled snow depth is calculated as though it were spread evenly across the grid cell. For SCF less than 1, this results in a thinner snowpack than would be calculated if the mass of snow were assumed to only occupy the snow-covered portion of the grid cell.
 In reality, areas with patchy snowpack (i.e., SCF < 1) are directly coupled to the atmosphere where snow is absent. Furthermore, the snow depth modifies the flux of heat through the snowpack, the rate of compaction, and the surface albedo through vegetation masking, so that the manner in which snow depth is calculated will influence modeled snowpack evolution. To examine the impact of patchy snow cover on the simulated surface energy budget, we performed a simulation (“SCF_SG”) in which snow depth was calculated as the mean depth of snow averaged over the snow-covered area of the grid cell, rather than the entire grid cell area. One of the motivations for developing a new SCF parameterization for use in CLM4 was to facilitate the calculation of snow depth according toequation (7) rather than (6). As discussed in section 4.2.3, the new SCF parameterization fulfills the requirement that mean snow depth, when conditioned on snow-covered area, goes to zero as snow water equivalent goes to zero.
 The calculation of the surface energy fluxes in SCF_SG is performed separately for snow-free and snow-covered areas, and these fluxes are then aggregated before being passed to the atmospheric model. The aggregation consists of an SCF-weighted average:
where Ei represents each of terms in the surface energy budget, and Fis the fractional snow-covered area.
Figure 18shows the differences between SCF_SG and Control October surface energy fluxes. Like the SCF_NEW simulation, the SCF_SG simulation exhibits widespread SCF decreases. The surface energy fluxes in SCF_SG, however, are quite different. Net radiation, which was slightly higher in SCF_NEW due to increased absorbed solar radiation, is lower in SCF_SG due to an increase in emitted longwave from the snow-free portion of the grid cell. Turbulent heat fluxes to the atmosphere are also higher, again due to higher values calculated over snow-free areas. This is compensated by a greater flux of heat from the ground (ground heat flux is positive into the ground). As a result, soil temperatures in grid cells with patchy snow are lower by 3 degrees or more (Figure 20).
 Once the snowpack accumulates to a depth at which SCF is close to 1, the SCF_SG and Control simulations show similar surface energy budgets. In the SCF_SG simulation, cooler soil temperatures persist until spring, when melt becomes widespread. Figure 19 shows the differences in the surface energy fluxes during May. In regions with high topographic variability, May SCF is generally lower. The effect on the turbulent heat fluxes is similar to that seen in October (Figure 18), when the greater fluxes from the snow-free portion of the grid cell raise the grid cell total. Unlike October, the May net radiation is higher, due to the greater absorbed insolation. Ground heat fluxes are generally higher, leading to higher soil temperatures in areas of incomplete snow cover.
5.4. Relative Effects of Flux Partitioning and New SCF Parameterization
 Based on the comparison of the NEW_SCF and SCF_SG simulations one might conclude that the form of the SCF parameterization has a minor effect relative to the manner in which the surface energy fluxes are calculated, i.e., whether the fluxes are computed separately for snow-covered and snow-free areas. In fact, it is the combination of the two modifications that results in the significant changes seen inFigures 18–20. While the original SCF parameterization cannot be used in the SCF_SG model due to numerical issues, the relatively steep accumulation and depletion curves can be closely reproduced using the new SCF parameterization with suitable parameter values. With values k = 0.4 and nmelt = 10, a simulation with SCF similar to the Control can be obtained. When these values are used in the SCF_SG model configuration, the strong autumn cooling and the spring warming are greatly reduced (not shown). This is because the SCF parameterization rapidly increases as a function of snow depth, and consequently there are relatively fewer periods when incomplete snow cover exists in this simulation. Thus, it is the combination of surface flux partitioning with more widespread patchy snow cover that causes the changes in the first SCF_SG simulation relative to the Control.
6. Discussion and Summary
 A key difference between this study and NY07is the use of daily averaged data in the former and monthly averaged data in the latter. Because the time scale of snowmelt is often shorter than a month, monthly sampling can obscure the relationship between SCF and snow depth. This result implies that model verification of other snow-related variables may benefit from higher temporal resolution analyses.
 One of the simplifications used in the Community Land Model is that snowfall within a model grid cell forms a snowpack of uniform depth that covers the grid cell in its entirety. This assumption is relaxed for the calculation of surface albedo, but remains in place for the calculation of surface energy and moisture fluxes. As part of this study, the assumption of binary snow cover was removed by separately calculating surface energy and moisture fluxes for the snow-covered and snow-free fractions of each grid cell.
 We have shown that partitioning the surface energy fluxes based on snow cover can have a significant impact on interactions between the atmosphere and the land surface, depending on when and how often conditions of incomplete snow cover occur. In the fall, increased direct exposure of the soil column to the atmosphere leads to greater emitted longwave radiation and turbulent heat fluxes, resulting in cooler soil temperatures. In the spring, a similar effect occurs, but is offset by higher absorbed solar radiation, causing the soils to warm more rapidly. These results depend on the relationship between SCF and the amount of snow, i.e., SWE.
 Determining the dependence of SCF on SWE at large spatial scales from observations is a difficult task. Snow water equivalent observations based on passive microwave have been shown to have large biases [Seo et al., 2010]. An alternative to directly measuring snow amount is to use forecast models to estimate precipitation and atmospheric forcing with which to drive snow evolution models. In the case of the CMC snow depth analysis, we find that the time series of snow depth and SWE for many locations exhibit a tendency to melt too soon relative to the MODIS SCF observations. This tendency may be the result of atmospheric forcing data that are too coarse to resolve the effects of topography, or a snow melt model that is too simplistic. SNODAS data, which are derived from a more realistic snow evolution model driven with higher resolution atmospheric output, appear to be more consistent with the SCF observations, but describe a relatively small domain.
 The parameterization described here reproduces the SCF–snow depth relationship exhibited by the MODIS and SNODAS data, but because the SNODAS domain excludes most of the high-latitude area, further research is required to confirm the validity of the parameterization globally. Such research will depend on the future availability of high spatial and temporal resolution snow depth analyses for regions outside the SNODAS domain.
 Observations of SCF provided by MODIS generally compare well to higher resolution sensors, but can be contaminated by the presence of clouds and vegetation [Hall et al., 2002; Klein et al., 1998]. Cloud cover is especially problematic during the accumulation season because snowfall is predicated on the presence of clouds. This makes observations in the visible and near-visible portion of the spectrum, such as those provided by MODIS, poorly suited for examining the rate at which SCF increases with snow accumulation. As a result, the determination of the optimal value of the accumulation parameterk (equation (2)) is subject to uncertainty. The value of k chosen for this study, k = 0.1, provides a reasonable lower bound based on the data. In the future, better constraints on k may be possible using passive microwave, ground based lidar, or very high resolution model simulations.
 During the melt season, a wide range of SCF trajectories as a function of SWE emerge. The differences in these trajectories can be understood in terms of the multiscale distribution of SWE [Clark et al., 2011]. At spatial scales at which unimodal SWE distributions are appropriate, semianalytic depletion curves can be derived from SWE pdfs. However, at spatial scales typical of current and near-future Earth system models, e.g., hundreds to thousands of square kilometers, it appears that the assumption of a unimodel distribution of SWE can break down, especially in regions having large topographic variability. We have therefore eschewed the pdf approach and developed a parameterization that is flexible enough to describe depletion curves that result from either unimodal or multimodal SWE distributions. Furthermore, the functional form of the parameterization is well behaved numerically when used to calculate self-consistent values of SCF and snow depth for small values of SWE. This latter quality is necessary when average snow depth is conditioned on snow-covered area.
 This work is a component of a broad research program whose goal is to improve the ability of CESM to simulate future climate change in high-latitude regions. Thermal and hydrological states in the high latitudes are closely coupled due to the differences in the thermal and hydraulic properties of liquid water and ice [Swenson et al., 2012; Z. M. Subin et al., The influence of soil moisture on the temperature response of freezing soils to climate change, submitted to Journal of Climate, 2012]. We have shown here that the manner in which fractional snow-covered area and surface energy fluxes are calculated affects the model's predictions of subsurface temperatures. Changes in soil temperature lead to changes in the amount and vertical distribution of soil moisture, resulting in feedbacks to surface states by changing the timing of snowmelt, the partitioning of runoff and infiltration, and vegetation dynamics.
 By better capturing these thermal-hydrological interactions, it is expected that the model will simulate a more realistic surface and subsurface climate, and therefore positively impact the simulation of carbon and nutrient cycling. Subsequent research will incorporate these model improvements in CLM along with parameterizations for excess ground ice and thermokarst development, prognostic wetland distributions, methane emissions, and vertically resolved soil biogeochemistry. The resulting model will then be used to study and project the evolution of permafrost under future climate change as well as the potential fate of the extensive perennially frozen soil carbon stores found in high-latitude regions.
 We thank Martyn Clark and Andrew Slater for valuable discussions. The Associate Editor and three reviewers helped to further improve the manuscript with their constructive comments. NCAR and the CESM project is supported by the National Science Foundation. This research was supported by the Office of Science (BER), U.S. Department of Energy, and NSF grant 1048997.