The spatial distribution of frozen soil and snow cover at the start of the spring melt season plays an important role in the generation of spring runoff and in the exchange of energy between the land surface and the atmosphere. Field observations were made at the University of Minnesota's Rosemount Agricultural Experiment Station to identify statistical distributions that can be used to describe the spatial variability of frozen soil and snow in macroscale hydrology models. These probability distributions are used to develop algorithms that simulate the subgrid spatial variability of snow and soil ice content for application within the framework of the variable infiltration capacity macroscale hydrologic model. Point simulations show that the new snow algorithm increases the melt rate for thin snowpacks, and the new soil frost algorithm allows more drainage through the soil during the winter. Simulations of the Minnesota River show that the new snow algorithm makes little difference to regional streamflow but does play an important role in the regional energy balance, especially during the spring snowmelt season. The new soil frost algorithm has a larger impact on spring streamflow and plays a minor role in the surface energy balance during the spring soil thaw season.
 The spatial distribution of frozen soil and snow cover at the start of the spring melt season plays an important role in the generation of spring runoff and in the exchange of energy between the land surface and the atmosphere. Ice in the soil reduces its infiltration capacity, resulting in a higher percentage of snowmelt and spring precipitation being partitioned into surface runoff. Cherkauer and Lettenmaier  describe a frozen soil algorithm that was developed for use with the Variable Infiltration Capacity (VIC) macroscale hydrologic model to represent the point effects of frozen soils on the surface energy balance and runoff generation. The algorithm solves for soil ice contents separately within each vegetation type and can therefore represent different frozen soil characteristics in forested and open areas, as have been documented by field observations [Kienholz, 1940; Pierce et al., 1958; Dingman, 1975; Shanley and Chalmers, 1999]. The algorithm does not, however, simulate the spatial variability of ice content within each vegetation type. Stahli et al.  showed that infiltration into frozen soils is underestimated by models that simulate uniformly frozen soil. Surface water tends to find areas of higher infiltration capacity as it flows across a frozen surface; thus the response to frozen soils is likely to be very different than would be predicted for a uniformly frozen (or thawed) landscape. This observation is consistent with the general tendency of the VIC model to overestimate catchment runoff.
 Because soil freezing is closely linked with the properties of any overlying snowpack, we also describe in this paper a generalization of the snow accumulation and ablation used by the VIC model [Cherkauer and Lettenmaier, 1999]. As with the frozen soil algorithm, the VIC snow algorithm as implemented to date does not account for the spatial variability of snow properties within vegetation categories. While new snow is often fairly uniform in its spatial coverage, drifting and differences in melt caused by topography and local shading can result in highly variable snow coverage by the onset of the melt season [Gray and Male, 1981; Marsh and Pomeroy, 1996; Shook and Gray, 1997; Luce et al., 1998]. The melt rate of partially snow covered areas has been observed to be significantly higher than that of fully covered regions because of the advection of sensible heat from bare to snow covered areas [Marsh and Pomeroy, 1996; Marsh et al., 1997]. Snow-free soil also warms faster than snow covered regions producing higher sensible heat fluxes. In models that do not represent spatial variability of snow cover extent such as the VIC snow accumulation and ablation algorithm as used to date, there is an abrupt change in sensible and latent heat when the (spatially uniform) snow cover is finally completely ablated. Such abrupt transitions over large areas are rarely observed in practice, as snow cover transitions through a mix of snow covered and snow-free areas during the melt season. The changes we describe to the VIC snow cover algorithm are intended to represent this process more realistically.
 The algorithms developed to represent subgrid spatial variability in snow extent and frozen soil are both based on derived distribution approaches, wherein the probability distribution of one or more forcing variables is prescribed, and the probability distribution of related variables is derived. The underlying spatial probability distributions for snow cover and soil temperatures are estimated from field observations conducted at the University of Minnesota's Rosemount Agricultural Experiment Station (K. A. Cherkauer et al., Field observations of the spatial distribution of snow and frozen soil, submitted to Water Resources Research, 2003, hereinafter referred to as Cherkauer et al., submitted manuscript, 2003) during the winters of 1997–1998 and 1998–1999.
2. VIC Model Structure
Liang et al.  describe in detail the formulation of the VIC model with two soil layers; therefore only a summary is provided here. A third soil layer was added by Liang et al. [1996b], who determined that a thin top layer (5–15 cm) significantly improved evapotranspiration predictions in arid climates. The VIC model uses a mosaic-type representation of surface cover, which allows specification of N vegetation types within each grid cell as shown in Figure 1. Vegetation can intercept precipitation, or precipitation can reach the ground surface as throughfall. Water can also reach the soil surface as snowmelt.
 When the model is implemented over a grid mesh (i.e., watershed), evaporation, energy fluxes, runoff and base flow are predicted independently for each grid cell. Infiltration and runoff are controlled by the variable infiltration capacity curve, which approximates the spatial variability of soil moisture based on the area-averaged soil moisture content of the upper two soil layers. Base flow is generated on the basis of the soil moisture in the bottom layer using the empirical Arno base flow curve [Francini and Pacciani, 1991]. Streamflow is then simulated at a specified location by routing runoff and base flow from each grid cell using the method of Lohmann et al. [1998a].
 The snow algorithm is a minor variation of the two-layer model used in the Distributed Hydrology-Soil-Vegetation Model (DHSVM) [Storck and Lettenmaier, 1999]. The DHSVM algorithm employs two snow layers of variable thickness. A thin surface layer is used to solve energy exchange with the atmosphere, while the lower or pack layer is used as storage to simulate deeper snowpacks. A snow interception algorithm is also incorporated to allow snow to be retained in an overstory, if present. The snow algorithm was modified to exchange heat energy with the soil surface, thus allowing it to be coupled with the frozen soil algorithm [Cherkauer and Lettenmaier, 1999]. Snow surface temperature is now propagated through the pack assuming a linear temperature profile and is iteratively balanced with the ground heat flux. Snow depth, required for determining the heat flux through the pack, is estimated by compressing the pack under the weight of new snow and through densification as the snowpack ages.
2.2. Frozen Soil Algorithm
 The motivation for development of the Cherkauer and Lettenmaier  frozen soil algorithm was to represent the effects of seasonally frozen ground on surface water and energy fluxes, at a level of complexity consistent with the previously developed VIC algorithms [Liang et al., 1994, 1999, 2001]. In this spirit, the VIC soil moisture transport scheme was retained and the thermal and moisture fluxes are solved separately. The decoupling of the moisture and thermal flux solutions means that soil thermal fluxes can be solved at any number of nodes through the soil column, whereas soil moisture values are computed as averages over the three soil moisture layers. Soil thermal nodes can be specified at any depth. The bottom boundary condition is prescribed as either a constant temperature or a constant flux.
 At each time step, thermal fluxes through the soil column are solved first. The soil node temperatures are then used to predict the soil layer ice content. Subsequently, moisture fluxes are computed using the updated ice contents. Total water content (ice plus liquid water) is used in the calculation of infiltration. Soil moisture drainage, on the other hand, is based only on the liquid water content of each layer; hence drainage is greatly reduced under frozen soil conditions. Finally, soil thermal node properties for the next time step are estimated from the updated soil layer moisture and ice contents.
3. Spatial Variability of Snow and Frozen Soil
 As a part of this project, observations of the spatial variability of snow depth and soil temperature were made during the winters of 1997–1998 and 1998–1999 at the University of Minnesota's Rosemount Agricultural Experiment Station (Cherkauer et al., submitted manuscript, 2003). Located about 20 km south of St. Paul, Minnesota, the Experiment Station provided the opportunity to make observations in an agricultural setting typical of large parts of the upper Mississippi River basin. Three statistical distributions were fit to observed snow depths and soil temperatures: the three-parameter lognormal distribution, the three-parameter gamma distribution, and the uniform distribution. The lognormal distribution was shown by Donald et al.  to provide a reasonable approximation to the spatial variability of snow depths at a site in southern Ontario, Canada. The gamma distribution was used by Koren et al.  to represent the spatial variability of soil ice content in the Noah land-surface scheme. In this work we favor the uniform distribution as a simple method of capturing most of the effects of observed spatial variability; however, for comparison purposes we estimate the properties of the other two distributions as well.
 Gamma and lognormal distributions were fit to all observations for each day of field measurements, but the distribution tails (Cherkauer et al., submitted manuscript, 2003) were censored in the fitting of the uniform distribution. By eliminating the extreme maximum and minimum observed values (P < 0.1 and P > 0.9), the fitted uniform distribution was able to represent most of the variability in each day's observations and generally performed at least as well as the other distributions. This is largely due to the limited spatial variability in the observations. The average standard deviation of observations of snow depth on days with full snow cover was less than 4 cm, so the effects of the distribution tails are minor. While the algorithms developed below make use of the uniform distribution derived from the Rosemount Experiment Station data, they are not restricted to either the uniform distribution or the particular parameters estimated from the Rosemount data. These algorithms represent spatial variability in snow and soil freezing and we refer to them as the distributed snow and distributed frost algorithms. We refer to the algorithms previously developed for point applications [Cherkauer and Lettenmaier, 1999] as the uniform snow and uniform frost algorithms to indicate that snow cover and soil frost do not vary for a given vegetation type. It is important to note that “uniform” in this sense is equivalent to “constant”, as opposed to spatially variable as represented by a uniform (or any other) probability distribution.
4. Subgrid Variation in Spatial Snow Properties
 Field observations show that the maximum rate of snowmelt occurs when a basin is partially snow covered [Shook et al., 1993]. As the snowpack melts and becomes discontinuous, sensible heat becomes a major source of turbulent melt energy at the snowpack edges [Shook et al., 1993], thereby increasing the melt rate relative to that predicted from consideration of radiative energy alone. The two-layer snow algorithm implemented in the VIC model assumes that the snow cover is spatially uniform within each vegetation type and elevation band. This implies that the model should underestimate the melt rate of thin, discontinuous snowpacks. Observations of snowmelt conditions at Rosemount, Minnesota (Cherkauer et al., submitted manuscript, 2003), clearly showed that shallow snowpacks in open fields become discontinuous during melt and after redistribution by wind. Runoff and meltwater ponding were also observed to increase at Rosemount once the snowpack became discontinuous.
 To improve the VIC model's simulation of melt events the algorithm was modified to represent the effects of partial coverage of the snowpack. The new algorithm affects the surface energy balance in two ways: (1) it allows for an increasing fraction of snow-free ground to participate in the surface energy balance, and (2) it represents the advection of sensible heat from the snow-free areas in the energy balance of the snowpack. This section describes the development, testing and evaluation of the distributed snow cover algorithm.
4.1. Algorithm Development
4.1.1. Partial Snow Cover
Figure 2a shows a series of areal depletion curves which represent the spatial probability distribution of snow depth around the mean snowpack depth for three cases. Case I is a deep snowpack where the area is 100% snow covered. Because the algorithm is based on the uniform probability distribution, the snow is distributed linearly around the mean depth in the figure. Case II shows a snowpack where the mean snow depth is equal to the critical depth, . This is the shallowest mean depth for which the snowpack will completely cover the area. Case III shows the snowpack during melt, when the mean snow depth is less than , and snow-free areas exist. Figure 2b shows the same three cases on a snow ablation curve, so that coverage can be discussed in terms of the snow cover fraction, Ps. The dashed vertical line indicates that snow is assumed to accumulate with Ps equal to 1. Ps will only be less than one when the mean snow depth of a melting snowpack is less than .
 The distributed snow algorithm also handles two special cases: (1) melting of thin accumulating snowpacks and (2) fresh snow on a partially snow free grid cell. If mean snowpack depth does not exceed prior to a melt event, the depletion curve is adjusted by setting to the current mean snow depth (see Figure 2c). This eliminates a potentially rapid change in Ps, as the pack switches from accumulating to ablating. The second special case is the accumulation of new snow over a partially covering snowpack. In this case, the depth distribution has already been established by a previous melt event, but new snow falls evenly across the cell area covering snow covered and snow-free areas alike (Figure 2d). If another melt event occurs before the mean snow depth exceeds , it is assumed that the new snow has not been redistributed so the previous spatial distribution of snow cover is preserved. This means that the algorithm first melts off the new snow cover with a constant Ps of 1; then it returns to the previous distribution. If, however, accumulation continues until the mean pack depth equals or exceeds , then the difference between new and old snow is forgotten, and the algorithm returns to case II.
 The snowpack energy balance is given by:
where ME is the energy available for melt, Rns is the net radiation, Hs is the sensible heat flux, Ls is the latent heat flux, Gs is the ground heat flux, HA is the advected heat flux, and ΔCC is the change in cold content. The surface energy balance is computed on the basis of the mean snowpack. The spatial distribution is applied around the mean snow depth (as seen in Figure 2a) and only impacts the energy balance by controlling the mix of snow and snow-free areas. While snow coverage is 100%, i.e., mean snow depth is greater or equal to , equation (1) represents the entire surface energy balance. If Ps is less than 1, the snow-free region makes use of the standard energy equation:
where Rnb is the net radiation, Hb is the sensible heat flux, Lb is the latent heat flux, and Gb is the ground heat flux.
 Average atmospheric energy fluxes are found by combining equations (1) and (2), such that Ps controls the ratio of snow-free to snow covered surface energy fluxes. To avoid the difficulties of preserving the energy balance when variable fractions of the soil column have different ground heat fluxes, and therefore different temperature profiles, the combined equation is solved to find an average ground heat flux, G:
This equation is solved iteratively to find a surface temperature where the average ground heat flux balances the other surface energy terms.
4.1.2. Advection of Sensible Heat
 The advection of sensible heat to snow covered areas from neighboring snow-free areas plays a major role in melting the edges of patchy snowpacks. The snow-free region generally is warmer than the snowpack during the day, as it absorbs more incoming radiation. The warm air above the snow-free surface is transported to the neighboring snow covered areas by local winds. This energy transfer is referred to as the turbulent or advected sensible heat flux.
 From observations of the advected sensible heat flux in the Arctic during snowmelt, Marsh and Pomeroy  proposed that the portion of the sensible heat flux that is advected to the snow patches (HA) can be represented as:
where Hb is the sensible heat from the bare patches and Fs is the fraction of the snow-free area that is advected.
Fs is a function of the connectivity between snow covered and snow-free areas. When snow-free areas begin to emerge they are surrounded by snow, so nearly all of the sensible heat generated from these areas will affect the melt process. As melt continues, the snow-free areas become larger, and what were previously patches of bare ground become connected. This increases the distance between some of the snow-free areas and the remaining snow. As the distance increases, the likelihood diminishes that sensible heat from a given snow-free area will contribute to snow melt. Once the snowpack has been reduced to a few deep drifts, the snow-free area contributing to the melt process constitutes a small part of the total snow-free area.
 Observations by Marsh et al.  confirm this process. They observed that the snowmelt rate peaks when snow coverage is about 60%, at which time nearly 100% of the sensible heat flux from the bare patches is advected to the snow covered patches. As the snow-free area increased, they saw an exponential decrease in the contributing area, which changed Fs from 1.0 with snow coverage at 60%, to 0.48 with 50% snow cover to 0.01 with a snow cover of 20%. Values for Fs were not available for coverage fractions less than 20%. These observations were used to develop an equation for Fs:
4.2. Point Tests
 Testing of the modified snow algorithm was conducted using observations from the University of Minnesota's Rosemount Agricultural Experiment Station [Cherkauer and Lettenmaier, 1999]. Two simulation cases were run: the first used the original (uniform) snow algorithm and the second used the distributed snow algorithm described above. Parameter calibration was conducted using the uniform snow algorithm.
4.2.1. Parameter Estimation
 The VIC soil parameters, including Ksat, bulk density, percent sand and clay, wilting point and critical point, were estimated from soil surveys conducted at the Rosemount site. During the winter it was assumed that the field was bare; that is, the effects of tillage and residue were neglected. However, to predict the antecedent soil moisture conditions correctly for each successive winter, knowledge of summer vegetation cover (corn in this case) was required. Vegetation type controls evapotranspiration through the summer months, and antecedent soil moisture is important to the development of early season soil frost. The vegetation height was assumed to increase each month starting in April, until a maximum height was reached in August. Harvest (the removal of vegetation) was assumed to take place in October. The LAI was also assumed to increase each month from April to August, and then decrease slightly in the fall as the crops dry out before harvest.
 Snow depth was measured in the field using a circle of snow sticks. The number of snow sticks used in the circle varied each winter, as did their location within the field. For comparisons with model predictions, the mean depth and standard deviation for snow at all snow sticks were computed for each day measurements were collected.
 The VIC model simulation was started on 12 December 1994 using observations of the snowpack, and soil column water content and temperature for the first 1.2 meters. Soil moisture solutions were computed using the standard VIC model configuration of three soil layers. Layer thicknesses were set to 0.1, 0.9 and 0.75 meters from top to bottom. Soil thermal fluxes were computed using 31 nodes with a spacing of 0.1 meters to a damping depth of 3 meters. A constant temperature bottom boundary condition was used, with the bottom node set to 15°C on the basis of annual average air temperature.
 Calibration of the VIC model was conducted using data for the winter of 1994–1995, with precipitation records adjusted for gauge undercatch using results of the WMO solid precipitation measurement intercomparison [Yang et al., 1998]. Snow roughness, minimum rainfall temperature and maximum snowfall temperature were adjusted to calibrate the accumulation of the snowpack. Soil column moisture and thermal fluxes were calibrated using the infiltration parameter, the pore size distribution, the bubbling pressure and the quartz content. Figures 3 and 4compare simulated snow and frost depths for the calibration period and the validation period, respectively. Temperatures for 5 cm soil depths agree with observations for most of both winters (Figures 3a and 4a). Simulated temperatures do not have the same extremes in variability as the observations, but the missed extremes are typically associated with overestimations of the snowpack depth (Figures 3c and 4c). Deeper snowpacks provide more insulation to the ground surface, thus reducing the variability of the surface temperature. Differences in the 5 cm soil temperature are also reflected in the development of the freezing and thawing fronts (Figures 3d and 4d). The freezing front remains shallower than observed in Figure 3 because of the thicker snowpack. The thick snowpack in February 1996 means that the freezing front does not show the rapid growth seen in the observations (Figure 4). Surface soil moisture is well represented in both winter simulations, but the 10–100 cm layer enters winter a little too dry in Figure 4.
 Point model simulations were run through the winter of 1997–1998. Comparisons with observations indicate that the model performed reasonably well over longer simulation times. All of the following discussions regarding point simulations focus on the winter of 1994–1995. This winter had the thinnest average snowpack and therefore showed the greatest differences between the distributed and uniform algorithms.
Figure 5 compares melt period simulations for the winter of 1994–1995. Snow cover during the winter of 1994–1995 was very thin, and coverage became fractional several times during the winter. This can be seen in Figure 5a where the uniform algorithm covers 100% of the vegetation until the snowpack has completely melted. Snowmelts completely in the distributed algorithm in late December and again on 19 February 1995. In early February snow coverage reaches a low of 30% before new snowfall (see Figure 3c) temporarily returns coverage to 100%.
 The distributed snow algorithm yields a thinner snowpack than the uniform algorithm (Figure 5b). This is due primarily to the increased rates of melt, seen in Figure 5c, corresponding to periods of time where the snow cover fraction is less than 100%. During these periods advection of sensible heat from snow-free areas can contribute to the snowmelt process. The magnitude of the advected sensible heat is shown in Figure 5d. Negative advection values correspond to days with rapid melt seen in Figure 5c.
5. Spatially Variable Frozen Soil Algorithm
 Simulations using the Cherkauer and Lettenmaier  frozen soil algorithm had a tendency to overestimate peak flow response to frozen soil. One of the reasons for this is the spatial variability in soil freezing, which has the effect of increasing average infiltration over an area. Other possible reasons include differences in the vegetation cover [Shanley and Chalmers, 1999] or the redistribution of snow. The influence of frozen soil can also be affected by the distribution of ice content, either by antecedent precipitation events or by melt or rain inputs once the soil has frozen. In any case, the distribution of ice in the soil has a large effect on infiltration. Water from rain or melting snow will pond or flow off impermeable soil, but over a patch of more permeable soil nearby it will infiltrate. Thus by having a uniform frost layer within each vegetation type, the VIC model point algorithm is likely to overestimate the effects of frozen soil on peak flows.
5.1. Algorithm Development
 As with the modification of the snow algorithm to represent spatial variability (see section 4), the frozen soil spatial variability algorithm makes use of the observed spatial distribution of soil temperatures from the Rosemount field site (Cherkauer et al., submitted manuscript, 2003). The VIC model frozen soil algorithm is used to compute the average soil column temperature profile, and the distribution is applied around it to produce a range of temperatures (Figure 6a).
where Wi is the liquid water content of layer i (mm), Wic is the maximum water content of layer i (mm), g is the acceleration due to gravity (m s−2), Lf if the latent heat of fusion (Jkg−1), T is the soil temperature (°C), ψe is the air entry potential (m), and Bp is the pore size distribution index.
 To account for the spatial variability in soil ice, each layer is divided into a series of separate bins for which ice content is computed (Figure 6b). Liquid water is distributed evenly through all of the bins and allowed to drain proportional to the ice content in that bin. Bins with higher ice contents will allow less water to drain, so the total soil moisture drainage for the layer will be reduced. With representation of the spatial distribution of ice content, some bins can restrict almost all drainage, while others can drain virtually unrestricted. Variations in ice content area are also accounted for in infiltration computations. Bins with higher ice contents have higher total moisture contents and therefore less infiltration. Tests of the algorithm showed that 10 bins provided adequate approximations to the distributions, while minimizing computational requirements.
5.2. Point Tests
 As with the snow spatial variability algorithm, the frost spatial variability algorithm was tested in point mode at the Rosemount field site. Two simulations were run: one with the uniform frost algorithm and one with the distributed frost algorithm. The spatially distributed snow algorithm was not used for either case. Results for the winter of 1994–1995 are plotted in Figure 7.
Figure 7a shows the precipitation and snowpack during the melt season. There is no visible difference in the snowpack between the uniform and spatial frost algorithms, indicating that any changes in the surface energy balance between the algorithms is small. Figure 7b shows that changes in the simulated freezing and thawing fronts are also minimal. Where the distributed frost algorithm does play an important role is in determination of soil moisture fluxes. Figures 7c and 7d show that the distributed frost algorithm increases base flow through March, while reducing runoff response in the second half of the month. This is also reflected in soil moisture differences between the two models (Figures 7e and 7f). The distributed algorithm allows for more soil moisture drainage during the winter, yielding a slightly drier soil in the spring. This in turn allows for increased infiltration, which can be seen from the response to precipitation around 27 March 1995. By mid-April soil moisture differences are gone, indicating that the effect of the distributed frost algorithm is focused on melt season.
Figure 8 compares the VIC model simulation using both the distributed snow and distributed frost algorithm to the observations from the Rosemount Experiment Station. With the addition of the distributed algorithms, the soil temperature shows increased variability, matching and in some places exceeding the observations. This is coupled with the reduced thickness of the snowpack (Figure 8c), which now simulates depths closer to observations in January and early February but underestimates the pack thickness in the spring. The freezing front penetrates more deeply (Figure 8d) with decreased surface insulation, but the date of thaw changes only slightly.
6. Catchment-Scale Implementation
 To further test the spatial snow algorithm, it was applied to the Minnesota River above Jordan, Minnesota (see Figure 9). This 41,958 km2 watershed drains most of southwestern Minnesota along with small parts of South Dakota and Iowa. Though the Rosemount Agricultural Experiment Station lies just outside this catchment soil type, topography and land use practices are very similar.
 On average, snow begins to accumulate in the watershed in October and November. Accumulations reach a maximum in January and February with a mean annual monthly simulated snow water equivalent of 32 mm from 1980 to 1997. Melt occurs rapidly through March and April, but it is not uncommon to have significant, but short duration, accumulations in late April and early May.
 Soils in the watershed begin to freeze in October and November. As with snow cover, maximum freezing depth occurs in February, with a maximum simulated depth of about 50 cm. However, because of the insulating effects of the snowpack, frozen soils do not really begin to melt until late March, and they can persist into early May.
 Simulations were conducted on the period between 1980 and 1997, using both the distributed snow and frozen soil algorithms. Simulations were forced using maximum and minimum daily air temperature, reanalysis winds and daily precipitation corrected for gauge undercatch [Cherkauer, 2001]. Daily precipitation was disaggregated into three hour time steps using a statistical method described by Maurer et al. . Soils data were obtained from gridded STATSGO data [Miller and White, 1998], and maps of land use type were obtained from the University of Maryland [Hansen et al., 2000].
 Six model parameters were used to calibrate the model: the infiltration parameter, the thicknesses of the middle and bottom layers, and the three parameters that control the Arno base flow curve. Calibration simulations were run on the period from 1990 to 1997 using the model with distributed snow and distributed frost algorithms. Three soil moisture layers were used, with the top layer set to a thickness of 10 cm. Soil thermal fluxes were computed using seven nodes with a constant temperature bottom boundary at a depth of 3 meters. Monthly and weekly simulated discharge was compared to observations to identify the best representation of both. The calibrated parameters were also used for four other simulations: uniform snow with no frozen soil representation; uniform snow and uniform frozen soil; distributed snow and uniform frozen soil; and uniform snow and distributed frozen soil. Using a single set of parameters allows the effects of the various algorithms to be compared directly but does not guarantee that any of the other model simulations are making use of their best parameter sets.
 Annual average monthly flow for all five simulations are shown in Figure 10 for both the calibration and validation periods. Statistics describing each simulation's ability to represent the observations for average annual monthly flows during the calibration period are shown in Table 1. Table 2 shows the same statistics as computed for the validation period. The Nash-Sutcliffe R2 is a measurement of how well the simulated and observed discharge records correspond to one another. The peak difference is the absolute value of the difference in magnitude between the highest observed flow and the highest simulated flow.
Table 1. Test Statistics for Monthly Minnesota River Simulated Flows During the Calibration Period of Water Years 1990 to 1996
Table 2. Test Statistics for Monthly Minnesota River Simulated Flows During the Validation Period of Water Years 1980 to 1989
Distributed snow and frozen soil
No frozen soil
Uniform snow and frozen soil
Distributed frozen soil
 As can been seen from both Figure 10 and the statistics, the five simulations are quite similar. Summer and early fall discharge is nearly identical among the simulations. The largest differences among the various model representations are in the spring when both the state of the soil and the timing of snowmelt play a key role in developing spring flows. The limited scale of these differences yields very small variability in Nash-Sutcliffe R2 values between the simulations. However, the peak difference statistic shows very large differences, especially during the calibration period. Annual average monthly flows are much higher during the calibration period than during the validation period, and the simulation without frozen soils falls about 20% short in predicting average May discharge.
 It can be seen from the figure that the spatially distributed snow algorithm increases the spring flow response; however, the presence of frozen soil, either distributed or uniform, has a bigger impact on monthly discharge. The distributed frozen soil simulations predict average monthly flows in May that are nearly identical to those predicted by the uniform frozen soil simulations. In June, the discharge simulations made with the spatially distributed soil frost algorithm fall almost in the middle of the discharge for simulations without frozen soil and with uniform frozen soil. This is most noticeable in the calibration period (Figure 10b), because of the greater difference in flows with and without frozen soil representation.
 Calibration period mean monthly flows are higher than the validation period flows largely because of two major events: summer flooding along the Mississippi River in 1993 and a large spring melt event in 1997. Weekly simulated and observed flows from March through August during these two periods are shown in Figure 11.
 The flood of 1993 was primarily caused by summer precipitation. By the week of 21 June 1993, when basin discharge peaks at the mouth of the Minnesota River, there is no longer any snow cover or soil frost. Therefore the simulated peak flow response of all five simulations are virtually identical. Response to spring melt is late in all of the simulations, indicating that the simulated snowpack did not melt early enough.
 Peak flow in 1997 is caused by the spring snowmelt. Unlike the 1993 flood peak, this flood is underpredicted by all of the simulations. A major issue in the simulation of cold regions is the accurate prediction of snowmelt. Figures 3, 4, and 8 demonstrate that the VIC model snow algorithm can simulate the snowpack at a point location where precipitation, wind speed and solar radiation are well measured. However, only daily temperature and precipitation observations are available for the Minnesota River catchment simulations; solar radiation and wind must be estimated, and this may be the source of some of the error. The representation of annual average snow accumulation was improved by using the precipitation correction method described by Cherkauer . Ablation of the pack, however, is controlled by solar radiation and wind speed. The peak spring response, seen in Figure 11, is clearly early by about a week. Calibration of runoff can yield a better prediction of the peak flow rate but only accentuates the early snowmelt.
 Statistics calculated for weekly discharge are shown in Table 3. Once again, there is very little difference between Nash-Sutcliffe R2 statistics. These are lower than for the monthly flows, which indicates that the simulations are less able to represent the observed short-term variability. Peak flow differences are, however, smaller for the cases that use the distributed snow and soil frost algorithms.
Table 3. Simulation Statistics for Weekly Minnesota River Simulations
Distributed snow and frozen soil
No frozen soil
Uniform snow and frozen soil
Distributed frozen soil
6.2. Sensible Heat Flux
 As shown in the previous section, the spatially distributed frozen soil algorithm has a larger effect on streamflow than does the spatially distributed snow algorithm. Where the distributed snow algorithm makes a larger impact is in the simulation of the surface energy balance. As noted in the development section, discontinuous snow cover results in higher sensible heat exchange with the atmosphere as the snow-free regions with their lower albedos warm to a much higher temperature than the neighboring snow covered regions.
Figure 12b compares basin averaged values of the average annual monthly sensible heat flux for the VIC model with uniform snow and frozen soil with that of the VIC model with distributed snow and uniform frozen soil. On a monthly basis the increase in sensible heat from the distributed snow algorithm can reach about 6 W/m2. Changes to the sensible heat flux are limited to the months between October and May when snow is likely to be present on the ground. The maximum difference occurs in March, when snowmelt typically is at its peak. Basin average daily sensible heat flux differences for the water year 1996 are shown in Figure 12a. Positive difference values indicate an increase in predicted sensible heat using the spatially variable snow algorithm. As with the monthly values, the differences are mostly positive and limited to the winter. The maximum difference is about 18 W/m2 and occurs in early April during snowmelt. Positive differences in October and November are caused by early accumulations of thin snow that melted at least partially before the onset of winter. Changes related to the distributed frost algorithm are about half the magnitude of those for the distributed snow algorithm. They reach a monthly maximum for the Minnesota River basin of 2 W/m2 in April, the month after the peak response to the spatial snow algorithm.
Figure 13 shows the spatial distribution of annual average winter sensible heat flux in the Minnesota River basin. In the winter, seasonal average sensible heat fluxes range from −79 to −7 W/m2, with no clear spatial pattern. Variations are caused by the accumulation and ablation of the snowpack as well as differences in the mix of vegetation types in each grid cell. Differences between sensible heat fluxes predicted by the VIC model with distributed snow and frost and the VIC model with uniform snow and frost are shown in Figure 13c. Except for a few cells in the northern part of the basin, the differences are positive indicating higher sensible heat fluxes from the distributed snow and frost algorithms. Negative differences are caused entirely by the distributed frost algorithm and are due to the slightly higher ice contents developed by the distribution of soil temperature. Higher ice contents reduce the magnitude of the predicted sensible heat flux, meaning that in the late fall and early winter the uniform frozen soil algorithm predicts higher sensible heat fluxes.
 The addition of the distributed snow algorithm changes the timing of the melt season, especially for thin snowpacks. Once snow-free areas begin to emerge, the addition of advected sensible heat to the energy balance of the snowpack increases the rate of melt. Because of the limitations of the uniform distribution used to define the distribution of snow, thicker snowpacks that melt quickly show little or no impact from the algorithm, as bare soil appears only late in the melt process.
 When the spatially distributed snow algorithm is applied to a large watershed, its effects on simulated discharge are muted and second order in magnitude when compared to the effects of the distributed and uniform frost algorithms. The distributed snow algorithm does, however, increase sensible heat fluxes during the winter and spring throughout the Minnesota River basin. This indicates that it enhances the exchange of heat fluxes between the ground surface and atmosphere.
 Representation of spatial variation in soil frost increases infiltration and base flow through the winter and spring when compared with the uniform frost algorithm. It also maintains higher spring snowmelt peak flows as compared with predictions that do not use the frozen soil algorithm. The spatially distributed soil frost algorithm also impacts the surface energy balance, though its impact is smaller than that from distributed snow and limited to periods when the ground is snow-free.
 The inclusion of both spatial distribution algorithms enhances the VIC model's ability to represent observed subgrid cell physical processes. Enhancements are limited to periods and locations where cold season processes dominate. However, with the increasing interest in high-latitude land areas and their role in climate change, improving the understanding of the spatial variability of cold season processes takes on added importance.
 This publication was supported by National Science Foundation Grant OPP-0230372 to the University of Washington.