WATER RESOURCES RESEARCH, VOL. 36, NO. 1, PAGES 13-26, JANUARY 2000

Combining binary decision tree and geostatistical methods to estimate snow distribution in a mountain watershed

Benjamin Balk I and Kelly Elder Department of Earth Resources, Colorado State University, Fort Collins

Abstract. We model the spatial distribution of snow across a mountain basin using an approach that combines binary decision tree and geostatistical techniques. In April 1997 and 1998, intensive snow surveys were conducted in the 6.9-km 2 Loch Vale watershed (LVWS), Rocky Mountain National Park, Colorado. Binary decision trees were used to model the large-scale variations in snow depth, while the small-scale variations were modeled through kriging interpolation methods. Binary decision trees related depth to the physically based independent variables of net solar radiation, elevation, slope, and vegetation cover type. These decision tree models explained 54-65% of the observed variance in the depth measurements. The tree-based modeled depths were then subtracted from the measured depths, and the resulting residuals were spatially distributed across LVWS through kriging techniques. The kriged estimates of the residuals were added to the tree-based modeled depths to produce a combined depth model. The combined depth estimates explained 60-85% of the variance in the measured depths. Snow densities were mapped across LVWS using regression analysis. Snow-covered area was determined from high-resolution aerial photographs. Combining the modeled depths and densities with a snow cover map produced estimates of the spatial distribution of snow water equivalence (SWE). This modeling approach offers improvement over previous methods of estimating SWE distribution in mountain basins.

1. Introduction and Background

Ensuring adequate water supplies is the crux of many polit- ical, economic, social, and environmental debates, particularly in the arid American West. A major source of water in the American West is the seasonal accumulation of snow in moun-

tain basins. Snowmelt runoff from mountainous regions con- tributes about 75% of the streamflow in the western United

States [Doesken and Judson, 1996]. Downstream agricultural, municipal, industrial, and recreational interests greatly depend on this snowmelt runoff to meet their demands.

Traditional methods of monitoring the seasonal snowpack through snow courses and snowpack telemetry (SNOTEL) sites provide little knowledge of the spatial distribution of water stored as snow. Most snow courses and SNOTEL sites

are established to obtain an index of snow water equivalence (SWE) over an area for use in predicting snowmelt runoff volumes through regression analysis [Goodison et al., 1981]. Such point measurements prove satisfactory for normal pre- cipitation years; however, the errors in streamflow prediction dramatically increase for abnormally wet and dry years. These years are precisely the times of most concern to flood and water supply forecasters [Elder and Dozier, 1990].

The development of physically based, spatially distributed snowmelt models requires better estimates of the spatial dis- tribution of SWE. BlOschl et al. [1991b] and Luce et al. [1998] demonstrated that a solid understanding of SWE distribution

1Now at National Weather Service, Alaska River Forecast Center, Anchorage.

Copyright 2000 by the American Geophysical Union.

Paper number 1999WR900251. 0043-1397/00/1999WR900251 $09.00

13

is crucial for realistic predictions of the spatial characteristics of the snowmelt process. The distribution of SWE is one of the controlling factors in the timing of runoff as certain areas of mountain basins may generate snowmelt more rapidly than other areas. The spatial distribution of snow is also important to landscape ecology and surface water chemistry. Bowman [1992] found potential nitrogen inputs to an alpine ecosystem varied primarily according to differences in snow depth, while plant species distribution has been correlated to varying snow depths at the landscape level [Evans et al., 1989; Walker et al., 1993]. The temporal variability in the chemical composition of snowmelt is partially blurred by spatial variability in snowmelt runoff generation and snow distribution [Campbell et al., 1995]. Energy exchanges, topographical characteristics, surface roughness, and wind redistribution contribute to the spatial heterogeneity of mountain snowpacks, particularly snow depth and SWE [Elder et al., 1991], and complicate efforts to model snow distribution.

Recent emphasis in snow distribution modeling has focused on statistical relationships between snow cover properties and terrain features or land cover parameters. Terrain character- istics are often known in a continuous, spatially distributed fashion and thus prove favorable when spatially modeling snow. Elevation, slope, curvature, terrain classes, and terrain- based modeled radiation have been used to distribute snow

measurements across small catchments [Woo and Marsh, 1978; BlOschl et al., 1991 a; Elder et al., 1991].

The spatial extent of snow cover is an important component of snow storage estimations. When snow cover is discontinu- ous, total SWE is more sensitive to snow-covered area (SCA) than to average snow depth [Elder et al., 1998]. Snow cover mapping from satellite imagery or high-resolution aerial pho- tographs can be accomplished by binary mapping or spectral

14 BALK AND ELDER: COMBINING STATISTICAL METHODS TO ESTIMATE SNOW DISTRIBUTION

mixture analysis. Binary mapping assigns either 100% or 0% snow cover to a pixel based on a brightness threshold. Spectral mixture analysis estimates fractional SCA at subpixel resolu- tion by quantifying the degree of mixing between snow and other surface materials in each pixel. The spectral mixture algorithm of Rosenthal and Dozier [1996] has been used to map SCA in Sierra Nevada basins [Cline et al., 1998; Elder et al., 1998].

Although SCA can be mapped from operational remote sensing platforms, the use of remote sensing alone to directly measure SWE is not feasible. Measurements of the spatial distribution of basin-wide SWE must be performed by inten- sive field sampling to capture the large spatial variability of mountain snowpacks [Cline et al., 1998]. The interpolation of ground-based point measurements therefore becomes neces- sary to explain and understand the spatial distribution of SWE over an entire drainage basin.

In any spatial modeling attempt the spatial variation of the data can be decomposed into two components, large-scale and small-scale variation. There is no standard as to what lengths constitute large scale or small scale. The difference between the two scales depends upon the study objective, data accuracy, and spatial locations [Reich and Davis, 1998]. In snow model- ing, large-scale and small-scale variability are relative to the geographical scale. Modeling snow cover over Colorado's Front Range has different scale implications than modeling snow distribution in small headwater basins, and large-scale and small-scale variability will have different meanings in each case. For this study, factors such as elevation, terrain features, wind redistribution, and broad physiographic or vegetation cover types influence the distribution of snow at the large scale (100 m to 1 km lengths) [Gray and Prowse, 1993]. At the small-scale (10 to 100 m lengths), surface roughness largely affects the variability in snow cover [Steppuhn and Dyck, 1974].

Spatial analyses are often conducted through geostatistical methods that have an underlying assumption across the study region of constant mean and variance in the variable to be estimated. When the spatial heterogeneity in the variable of interest does not warrant such assumption of stationarity, then it is common to detrend the data before applying geostatistical methods [Chua and Bras, 1982]. Trend surfaces and general linear regression models are often used to remove large-scale trends, while the residuals, or small-scale trends, are modeled through kriging interpolation. In the field of hydrology, linear regression models and kriging techniques have been combined to map aquifer transmissivity [Ahmed and de Marsily, 1987], mean areal precipitation [Chua and Bras, 1982; Phillips et al., 1992], and SWE distribution [Hosang and Dettwiler, 1991; Car- roll and Cressie, 1996].

However, some variables used in snow models, such as ele- vation, exhibit nonlinear relationships with snow distribution. Observations of this nonlinear behavior motivated the hypoth- esis of this study: Improved estimates of snow distribution can be obtained by detrending nonlinear large-scale variations in snow depth and then interpolating the first-order model resid- uals. We test this hypothesis by combining binary decision tree and geostatistical methods to estimate snow distribution in a Colorado Front Range watershed.

Research efforts using binary decision tree methods to non- linearly distribute measured values of SWE over mountain watersheds have proved promising [e.g., Elder, 1995a; Elder et al., 1995, 1998]. Geostatistical techniques have also been ap- plied to estimate basin-wide SWE [e.g., Carroll and Cressie,

1997; Balk et al., 1998]. However, no previous attempt has been made to combine binary decision trees and geostatistical meth- ods to model snow distribution. For this study in a small mid- latitude mountain basin we reduced the large-scale trends in snow depth with binary decision tree methods. The tree- modeled residuals, or small-scale variations, were then inter- polated using geostatistical methods. Snow densities were dis- tributed across the basin through regression analysis. Combining the modeled depths and densities with SCA pro- duced spatially distributed SWE estimates near peak accumu- lation in 1997 and 1998. By improving spatial estimates of SWE distribution, we believe this combined deterministic-stochastic modeling approach would be a significant contribution to the field of snow hydrology.

2. Field Methods

2.1. Study Site

Loch Vale watershed (LVWS) lies in Colorado's Front Range immediately east of the Continental Divide at 40ø17'N, 105ø40'W (Figure 1). This glacially scoured basin in Rocky Mountain National Park has an area of approximately 6.9 km 2 and elevations between 3091 and 4003 m. The basin has a

general east-northeast aspect and is flanked by steep cliffs on most margins. Slopes in LVWS range from a minimum of 0 ø to a maximum of 85 ø with a mean slope of 33 ø. Both Andrews Creek and Icy Brook drain the watershed and eventually flow into the Big Thompson River, a major tributary of the South Platte River. Baron and Mast [1992] provide a more detailed description of the watershed.

2.2. Field Measurements

Intensive snow surveys were completed in LVWS during April 15-18, 1997, and April 7-10, 1998. Snow depths were measured at 197 points in 1997 and 173 points in 1998. Snow densities were measured in seven snow pits in 1997 and six snow pits in 1998. Sample depth and density locations were chosen to be representative of the range of elevations, slopes, and aspects of the watershed, considering safety limitations. The sample locations for 1997 (Figure 2) were transcribed onto the U.S. Geological Survey (USGS) 7.5' McHenrys Peak Quadrangle, while the sample locations for 1998 were pin- pointed on 5-m interval contour maps. The respective univer- sal transverse Mercator coordinates were then obtained for

each sampled site and registered to a 10-m resolution digital elevation model (DEM) of LVWS. The 5-m interval contour maps were derived directly from the 10-m DEM, which was constructed in August 1997.

The watershed was divided into three subbasins, Andrews Creek, Sky Pond, and Lower Loch. Maps of each subbasin were enlarged from the USGS 7.5' McHenrys Peak Quadran- gle for 1997 and the 10-m DEM for 1998. Fifty-meter grids were aligned with the valley walls of Andrews Creek and Sky Pond subbasins, while a 100-m grid was aligned with the valley walls of Lower Loch. The finer-resolution grids were estab- lished to obtain a higher concentration of sample points for the upper drainages of LVWS where the heterogeneity in snow depths was apparent.

At each sample location a central depth measurement was taken with aluminum probe poles. Two additional depths were measured at 5-m spacing from the central point in directions aligned with the grid. To minimize local variation in depth, the three measurements were averaged and recorded to the near-

BALK AND ELDER: COMBINING STATISTICAL METHODS TO ESTIMATE SNOW DISTRIBUTION 15

105 ø 40' W

I

40 ø 17' 30"N --

*LVWS

Colorado

N

0 0.5 I Kilometers

Figure 1. Location and topographic map of the Loch Vale watershed (LVWS). Contour interval is 50 m.

est 0.05 m. However, to ensure adequate spatial coverage in the upper drainages, only one measurement was made at most sample points, and the sampling was shifted to 100-m spacing because of probing difficulties and time constraints. The spac- ing between sample points was measured with probe poles.

Snow pits were dug in the watershed to obtain density and temperature profiles. Densities were measured with a 1-L stainless steel wedge-shaped cutter and an electronic digital scale or with a 0.25-L cutter and a spring scale. Continuous density profiles were sampled in 0.10-m increments on the shaded snow pit wall. The density samples were then averaged to obtain one representative density for each snow pit.

2.3. Snowpack Conditions

The 1997 water year was abnormally wet for the state of Colorado. The Natural Resource Conservation Service re-

ported the South Platte basin's snowpack to be at a record level for May 1 at 156% of the 30-year average. Within the South Platte basin, snowpack for the Big Thompson basin was about 130% of average for April 1 and was about 148% of the May 1 average [Natural Resource Conservation Service, 1997]. Com- pared to a typical continental snowpack, the season snowpack was uncharacteristically stable.

The 1998 snowpack was near average for the South Platte basin. The April 1 and May 1 averages for the basin were about 91% and 111%, respectively. The Big Thompson basin's snow- pack was about 98% of the May 1 average [Natural Resource Conservation Service, 1998]. During the 1998 snow survey, av- alanche danger was considerable. Continuous wind loading on steeper slopes prevented depth sampling in the higher drain-

ages of LVWS. As a result, the higher elevations had less spatial coverage of depth measurements.

Snow pit temperature profiles indicated the cold content of the snowpack had not been entirely removed as snowpack temperatures ranged from near 0øC to -9øC for 1997 and from near 0øC to -6.5øC for 1998. There was no significant snow- melt in LVWS before either snow survey.

3. Modeling Methods 3.1. Binary Decision Trees

Binary decision trees relate independent variables with a dependent variable in a nonlinear or hierarchical manner through a series of binary decisions or splits. A binary decision tree effectively delineates similar values of the dependent vari- able through a progressive subdivision of a heterogeneous sample. On the basis of a learning sample X containing a set of predictor variables and a response variable y, binary decision trees estimate values for y. A binary tree is grown by repeated splits of subsets of X into two descendent subsets or nodes. The splits begin with X itself, the root node, and end in a series of terminal nodes. At each node the splits are conditionally cho- sen from values in a vector of independent variables X(Xm, where m = 1, 2,... ) and based on the value of a single predictor variable in x. Considering all possible parent nodes, the algorithm selects the split at a particular parent node that maximizes the reduction in total model deviance in y. Thus the learning sample X is recursively partitioned to minimize model deviance in y.

For each ordinal or ratio scale variable, Xm, the decision

16 BALK AND ELDER: COMBINING STATISTICAL METHODS TO ESTIMATE SNOW DISTRIBUTION

alized variables. These regionalized variables are functions of their spatial coordinates. A representation of the spatial vari- ability, the variogram, can be used to estimate the value Z(Xo) at a point x o at which no data are available [Journel and Hui- jbregts, 1978]. All possible data pairs are grouped by distance classes. One half the variance of the difference in values, the semivariance, is then graphed versus the distance class. Thus the experimental semivariogram, usually referred to as a "var- iogram," can be estimated by

7(h) = •nn • [z(x,) - z(xi + h)] 2, (la) i=1

where Z(Xi) and z(xi + h) are samples taken at locations xi and x• + h, respectively, and n is the number of data pairs separated by the vector h [Herzfeld et al., 1990]. The cross variogram for a dependent variable z • and one auxiliary vari- able z2 can be estimated by

N

0 0.5 1 Kilometers

Figure 2. The 1997 depth sampling sites (n = 197) indi- cated with dots. Contour interval is 50 m.

follows from the question, Is x m ( ½ .9 Here the values of c are within the domain of x m. The decisions for categorical vari- ables follow from the question, Is X m G S ? Here S ranges over all subsets of the categories of X m. Decisions are continued until the algorithm encounters stopping criteria. Typically, de- cisions cease when node membership size or the reduction in model deviance is less than a critical number or value. Cross

validation can then be used to help determine optimal tree size. The mean values of the observations ofy in each terminal node can then be used to map the dependent variable across the area of interest. For a more detailed discussion of binary decision trees, refer to Breiman et al. [1984]. Details of the decision tree modeling software used in this study are ex- plained by Clark and Pregibon [1992].

Binary decision trees offer many advantages over other methods used to distribute snow across mountainous terrain.

The dependent variable, snow depth, is included in the model development through the stepwise reduction of model devi- ance. Decision trees can explain nonlinear relationships through multiple splits on the same independent variable. Un- like many multiple linear regression models, decision trees are easy to interpret. Individual splits can also be checked for physical justification. Does the split make intuitive sense, or is it a statistical artifact? The major disadvantage associated with decision trees is the requirement of a large data set. Although the need for a large data set may hinder the widespread applicability of binary decision trees for modeling snow distribution, they have been found to be superior to most attempts to distribute snow over complex terrain [Elder, 1995a; Elder et al., 1995].

3.2. Kriging

Natural phenomena can often be characterized by the dis- tribution of one or more spatially variable quantities or region-

3q2(h) = •nn • {[z•(x,) - z•(x, + h)][z2(xi) -- Z2(Xi + h)]}, t=l

(lb)

where n is the number of data pairs separated by the vector h. Unlike semivariances, the cross semivariances can be negative if the two variables are negatively correlated [Trangmar et al., 1986]. The vector h separates the two locations x and x + h in both distance and direction. The ordered set of values obtained

by increasing h constitutes the experimental variogram or cross variogram.

To capture the basic structure of spatial dependence, it is necessary to fit a theoretical model to the sample values to represent the true continuous variogram. A theoretical model is fit to the sample data by ordinary least squares regression or by optimizing through cross-validation procedures. Commonly used variogram models, such as the Gaussian, spherical, and exponential, are explained in detail by Journel and Huijbregts [1978] and Oliver et al. [1989].

Once a theoretical model has been established, kriging in- terpolation can be performed. At its simplest, kriging is a method of weighted averaging of the observed values of a measurement space Z, within a neighborhood V, from the measured values of z(x•) of Z at n sites, x i, i = 1, 2, ..., n [Oliver et al., 1989]. To calculate an interpolation value at a point (ordinary kriging), the weights of the neighboring mea- surement sites are determined by solving a system of equations that consider the theoretical variogram model. For ordinary kriging, interpolations are made using only the variogram for the variable to be estimated. For ordinary cokriging a cokriging matrix containing additional variograms for correlated vari- ables and cross variograms for their interactions are used [Phil- lips et al., 1992]. Refer to Journel and Huijbregts [1978] and Cressie [1991] for more detailed discussions on kriging and cokriging.

3.3. Independent Variables

Previous research has determined a link between some pa- rameters and the processes controlling the distribution of snow. Radiative fluxes and physiographic features, such as elevation, slope, aspect, surface roughness, and the optical and thermal properties of the substrate, rationally and demonstra- bly relate to snow cover variations [Meiman, 1968; Young, 1975;

BALK AND ELDER: COMBINING STATISTICAL METHODS TO ESTIMATE SNOW DISTRIBUTION 17

Table 1. Summary of Net Solar Radiation Indices Where Standard Deviation, Coefficient of Variation, and Sample Size Are Included

Table 2. Summary of Elevations Where Standard Deviation, Coefficient of Variation, and Sample Size Are Included

Min, Max, Mean, Radiation W m -2 W m -2 W m -2 W m -2 CV n

Min, Max, Mean, tr, Elevation m m m m CV

LVWS 37 592 365 146 0.40 69188

1997 depth sites 221 571 278 72 0.26 197 1998 depth sites 223 572 272 69 0.25 173

Definitions are as follows: tr, standard deviation; CV, coefficient of variation; n, sample size; Min, minimum; Max, maximum; and LVWS, Loch Vale watershed. Sample size for LVWS is the total number of 10 m cells found in the basin. Values are summed from calculations

made for the fifteenth of each month from December through April.

McKay and Gray, 1981]. Net solar radiation, elevation, slope, and vegetation cover were used as physically based indepen- dent variables in our modeling attempts to spatially distribute SWE across LVWS.

Net solar radiation has been shown to be the largest energy source for melting alpine snowpacks [Megahan et al., 1967; Zuzel and Cox, 1975] and plays a controlling role in the accu- mulation and redistribution of snow. Elder [1995a] found that using a spatially distributed net solar radiation index over a portion of the snow accumulation season may be sufficient for statistical models of SWE distribution in alpine areas. When using net solar radiation, elevation, slope angle, and vegetation cover type to model SWE with binary decision trees in a similar small alpine basin, net solar radiation was the most important variable in almost all model results [Elder, 1995a].

Net solar radiation was spatially modeled in Image Process- ing Workbench (IPW) [Frew and Dozier, 1986], using the al- gorithm of Dozier [1980] and following the method of Elder [1995a] except for the parameterization of snow surface al- bedo. Temporal variations of albedo throughout the accumu- lation season are too great to model absolutely, so we param- eterized albedo based on reasonable values found in literature

[e.g., Wiscombe and Warren, 1980; Marks and Dozier, 1992]. The method uses a simple linear decay of albedo as a function of time [Marks and Dozier, 1992; Elder, 1995a]. These albedo estimates were applied to snow-covered areas, whereas a con- stant albedo of 0.25 was applied to snow-free areas. This snow- free albedo value proved sufficient in modeling efforts con- ducted by Olyphant [1986] in nearby alpine sites. Snow-covered and snow-free areas were determined using a mask of a SCA image of LVWS. Using the 10-m DEM for LVWS and assuming clear-sky conditions, net solar radiation was calculated for all grid cells in the basin to obtain one radiation image representing an index of net solar radiation through a portion of the snow accu- mulation season. The modeled radiation values at the 1997 and

1998 sampled depth sites are summarized in Table 1. The ranges of radiation values are not fully representative of the distributed radiation in LVWS because safety and logistics predominantly constrained sampling sites to the valley floors.

Elevation has been shown to be an important factor in snow distribution primarily through orographic effects [e.g., Rhea and Grant, 1974; Caine, 1975]. A linear relationship is com- monly assumed between elevation and SWE accumulation [U.S. Army Corps of Engineers, 1956]; however, quite often the relationship is nonlinear [McKay and Gray, 1981] as wind re- distribution at the highest elevations can decrease SWE accu- mulation. Elevations for this study (Table 2) were obtained

LVWS 3091 4003 3491 213 0.06 69188

1997 depth sites 3097 3634 3288 124 0.04 197 1998 depth sites 3093 3477 3242 92 0.03 173

Sample size for LVWS is the total number of 10-m cells found in the basin.

from the 10-m DEM of LVWS, and vertical errors for the elevations were less than 1 m.

Slope values were calculated for each grid cell in the DEM using an algorithm in ARC/INFO Version 7.2.1 geographic information system. Slopes for the basin and the depth sam- pling locations are summarized in Table 3. Snow is often re- distributed from steeper slopes to more gentle slopes through avalanching or sloughing. Avalanching does not change the total mass of SWE in a basin, but it does affect SWE distribu- tion as large volumes of snow can be transported to and con- centrated in lower-elevation runout zones [Elder et al., 1991; de Scally, 1996]. In alpine areas with extreme topography some slopes are steep enough such that they will continually ava- lanche or slough and greatly enhance accumulation in lower- angle slopes below [Bl6schl et al., 1991a; Elder et al., 1991; Elder, 1995b].

Vegetation type and size influence snow distribution through the interception of snowfall, subsequent losses due to evaporation, redistribution of snow, and surface energy ex- change. Vegetation influences the surface roughness and wind velocity thereby affecting the erosional, transport, and deposi- tional characteristics of the surface [McKay and Gray, 1981]. Numerous examples of vegetation effects on snow accumula- tion have been documented [e.g., Steppuhn and Dyck, 1974; Adams, 1976]. The vegetation coverage used in this study was simplified from a detailed vegetation classification for Rocky Mountain National Park. The vegetation classes include rock/ alpine grassland, glacier, open water, spruce/fir, and krum- mholz/willow (Table 4). Glacier and open water are physi- ographic classes that have different snow accumulation patterns.

3.4. Dependent Variables

3.4.1. Snow depth. Snow depths were distributed across LVWS with a combination of binary decision tree and geostatistical methods. The large-scale variability of snow depth was modeled with binary decision trees, while the small- scale variability was modeled with kriging techniques. Rugged terrain, variable winds, and complex radiation inputs contrib-

Table 3. Summary of Slopes Where Standard Deviation, Coefficient of Variation, and Sample Size Are Included

Min, Max, Mean, Slope deg deg deg deg CV

LVWS 0 85 33 17 0.52 69188

1997 depth sites 2 72 20 11 0.54 197 1998 depth sites 0 55 17 10 0.62 173

Sample size for LVWS is the total number of 10-m cells found in the basin.

18 BALK AND ELDER: COMBINING STATISTICAL METHODS TO ESTIMATE SNOW DISTRIBUTION

Table 4. Summary of Vegetation Cover Type Including the Number of Depth Measurements Made in Each Vegetation Type

Number Number

Area, LVWS of 1997 of 1998 Vegetation Type km 2 Area, % Depth Sites Depth Sites

Glacier 0.10 1.4 12 1

Open water 0.12 1.7 1 16 Spruce/fir 0.71 10.2 63 61 Krummholz/willow 0.39 5.6 21 21 Rock/alpine grassland 5.60 81.1 100 74 Total 6.92 NA 197 173

NA indicates not applicable.

ute to the large heterogeneity in snow depth found in mountain environs. Measured snow depths in LVWS often varied by a few meters between sampling locations with the 1997 extreme case of a 10-m difference (10.25 m and 1.25 m) over a 50-m spacing just to the lee of the Continental Divide. This large heterogeneity in snow depth complicated initial modeling ef- forts of distributing snow depths by the kriging interpolation process alone [Balk et al., 1998] since the large-scale variations invalidated the stationarity assumption of constant mean and constant variance across the study region. Binary decision trees can accurately handle abrupt changes in the dependent vari- able, snow depth, and thus were used to detrend the large-scale variations in snow depth. (In this study, abrupt describes how adjacent grid cells can differ in tree-modeled depth by a few meters as well as field observations of drifts several meters

deep adjacent to bare ground.) The estimated snow depths from the binary decision trees were then subtracted from the measured snow depths, and the resulting residuals were mod- eled through kriging interpolation. The modeled snow depths from the binary decision trees and modeled residuals from geostatistical techniques were then added together to produce estimates of the spatial distribution of snow depth in LVWS. Trend surfaces were also used to model the large-scale vari- ability of snow depth, but these models were less satisfactory compared to the binary decision trees.

Multiple binary decision trees were grown on the 1997 and 1998 data sets using tree-based model implementation in the S-Plus mathematical language [Chambers and Hastie, 1992]. For both years, binary decision trees were grown with various combinations of net solar radiation, elevation, slope, and veg- etation cover as independent variables. As a measure of the overall goodness of fit of each decision tree, the coefficient of determination (R 2) was calculated using standard statistical procedures [Bailey and Gatrell, 1995]. The default stopping criteria in the S-Plus mathematical language grew trees be- tween 25 and 30 terminal nodes prior to pruning. The best fit decision tree model for snow depth in 1997 was grown with net solar radiation, elevation, and slope. Using net solar radiation, elevation, slope, and vegetation cover type yielded the best fit tree model for 1998.

Selection of the optimal binary decision trees for 1997 and 1998 was made through cross validation and pruning. The decision tree algorithm pruned the 1997 and 1998 best fit tree models to the respective upper range of terminal nodes deter- mined through cross validation. Optimal tree sizes were found where the cross-validated deviance was minimized when plot- ted versus tree size. We further pruned these decision trees based on physical justification of the individual splits. For ex-

ample, an individual split was pruned if the split dictated greater snow depth accumulations with greater radiation indi- ces. The pruning of the implausible splits was interactively done through the tree-based software in S-Plus. The optimal decision trees for both years coincidentally had 18 terminal nodes. The 1998 optimal tree is shown in Figure 3.

Examination of the tree-modeled residuals and their spatial distribution indicated the means, variances, and large-scale trends in snow depth were reduced. The residual depths were then spatially distributed across LVWS through kriging inter- polation techniques conducted in S-Plus [Chambers and Hastie, 1992] with software developed by Reich and Davis [1998]. The residuals were estimated by ordinary kriging using only the primary variable, the residual depth. The residuals were also cokriged using net solar radiation, elevation, and slope as sep- arate auxiliary variables. To ensure that the auxiliary variables and residuals were in similar units, it was necessary to rescale the variables by dividing each variable by the maximum value in their respective data set. The cokriged surface was then scaled back to the original units by multiplying each cokriged estimate by the maximum residual value. Interpolations were made at 10-m intervals to correspond with the 10-m DEM.

Experimental variograms and cross variograms were con- structed with maximum distance parameters and a certain number of distance classes, or bins, to best represent the spa- tial variability of the residuals, auxiliary variables, and their interactions. A stationarity assumption was made where the means and variances of each regionalized variable were as- sumed to be independent of location and constant throughout the watershed. Isotropic variance structure was also assumed for each regionalized variable since no significant directional differences were found in the experimental variograms. The experimental variograms were then fitted with Gaussian, spherical, and exponential variogram models. Properties asso- ciated with these variogram models satisfy the positive definite criterion of the cokriging matrix (R. M. Reich, Colorado State University, personal communication, 1999). The best fit theo- retical variograms used in the kriging interpolations were se- lected through ordinary least squares regression. The spherical theoretical model was frequently used in this study and is described by

0

3'(h) = 0 + (c - Co) ( -- - -- h=0

3h h 3 ) 2a •-• 0 < h -< a (2) otherwise,

where a is the range, c is the sill variance, and c o is the nugget effect. The experimental and theoretical variograms for 1997 radiation and 1998 residual depths as well as the 1997 exper- imental and theoretical cross variogram for radiation and re- sidual depths are shown in Figure 4.

Cross validation was used to evaluate the statistical proper- ties of the kriged and cokriged surfaces and to determine the ideal number of nearest neighbors, or residual values, to be used in each model. Each observed residual depth z i was sup- pressed, in turn, and its value was predicted from the remain- ing data using the selected theoretical variogram. The ob- served residual depths minus the estimated residual depths yielded a new set of residuals, hereafter referred to as the residual errors. These residual errors were analyzed to evalu- ate the underlying model assumptions. All models had approx- imately normally distributed residual errors with a mean of zero and constant variance. To assess the overall goodness of

BALK AND ELDER: COMBINING STATISTICAL METHODS TO ESTIMATE SNOW DISTRIBUTION 19

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11 I Figure 3. Binary decision tree for 1998 measured snow depths. Values in each node are mean depths (meters) for all members in that node. Vegetation type classifications are as follows: a, glacier; b, rock/alpine grassland; c, spruce/fir; d, open water; and e, krummholz/willow.

fit of each model, the coefficient of determination (R 2) was calculated by

rt

t=l

R 2= 1- , (3)

(zi- i=1

where s i is the residual error associated with the response variable, z/[Reich and Davis, 1998]. This R 2 value has an upper limit of one and no lower limit. A negative R 2 value indicates the variance in errors of the model is larger than the variance in the observed data. To test spatial autocorrelation of the residuals, the Moran's I was calculated for spatial proximity matrices of both years [Bailey and Gattell, 1995].

3.4.2. Snow density. Snow density measurements involve excavating snow pits and sampling the shaded snow pit wall. Snow depth measurements simply require probing and there- fore are much less labor-intensive and time-consuming. In alpine areas the major source of variation in SWE is variation in depth [Logan, 1973; Elder et al., 1991]. The conservative variation in density enables field sampling as a few density profiles can supplement many more easily obtained depth mea- surements. Although density is conservative in relation to other snow properties such as depth, it does exhibit some spatial variation [Elder, 1995a], especially before the onset of melt. Thus the measured densities need to be spatially distrib- uted across the watershed.

Net solar radiation, elevation, and slope were considered as independent variables in snow density modeling. Elevation and slope did not show a significant relationship with snow density;

20 BALK AND ELDER: COMBINING STATISTICAL METHODS TO ESTIMATE SNOW DISTRIBUTION

8

1997 Radiation Variograms

sill

n Spherical fit

s6o ran,'ge

,,

lO0O

Distance (m)

b)

1998 Residual Variograms

sill

nugget effect Spherical fit

range

0 500 1000 1500

Distance (m)

c)

1997 Cross Variogram

nugget effect

. ß Gaussian fit •

..si!! .............................................................................................

6 s6o obo sbo Distance (m)

Figure 4. Experimental variograms and theoretical models for (a) 1997 radiation, (b) 1998 residuals (measured depth minus tree-modeled depth), and (c) 1997 cross variogram of radiation and residual depths. The nugget effect, sill, and range are identified.

therefore only an index of net solar radiation was used to model density. For 1997 a general linear regression model explaining 47% of the observed variance in field measurements of snow density was used to distribute density (R 2 = 0.466, n - 7, and p - 0.091). A general linear regression model

Table 5. Summary of Field-Measured and Modeled Snow Densities Where Standard Deviation, Coefficient of Variation, and Sample Size Are Included

Min, Max, Mean, Density kg m -3 kg m -3 kg m -3 kg m -3 CV n

1997 measured 297 406 375 37 0.10 7 1997 modeled 290 397 354 25 0.07 39022 1998 measured 262 363 320 36 0.11 6 1998 modeled 260 392 308 36 0.12 39022

Sample size for modeled densities is the total number of 10-m cells with snow cover in LVWS.

explaining 80% of the observed variance in 1998 sampled den- sities was used to distribute density for that year (R 2 = 0.796, n = 6, and p = 0.017). These general linear regression models proved sufficient and were more ideal than merely applying a mean density that would ignore the observed spatial variations in density. Snow pit and basin-modeled densities are summarized in Table 5.

3.4.3. Snow-covered area. SCA was determined using high-resolution aerial photographs of LVWS taken on April 9, 1996. No aerial flight was made in 1997, while one flight was made on May 18, 1998. However, the 1998 aerial photographs were taken almost 6 weeks after the snow survey and did not adequately capture the snow cover at the time of the survey. Owing to the extreme topography of LVWS it is assumed that snow cover near peak accumulation will be fairly consistent from year to year since most of the steeper slopes in the watershed cannot maintain snow cover, even during heavy snowfall years. The April 9, 1996, aerial photographs were orthorectified to the 10-m DEM for the watershed to produce one orthoimage. A binary threshold was then chosen to most accurately delineate snow cover for the orthoimage. The binary threshold was determined such that pixels with a digital num- ber (DN) value less than the threshold were assigned a DN of 0 (black), and pixels with a DN value greater than the thresh- old were assigned a DN of 255 (white). This procedure did not distinguish forested and shaded areas from cliff bands or wind- scoured areas that were snow-free. Therefore a mask of the

forested and shaded areas that were snow-covered was created

and combined with the binary threshold image to produce the final snow-covered image. In the final snow-covered image, areas with no snow were assigned a zero value, while areas with snow cover were assigned a value of one. The SCA for LVWS was calculated to be 56% of the watershed area.

3.4.4. Snow water equivalence. Estimates of SWE distri- bution for each year were calculated by multiplying the com- bined depth surfaces by the modeled density and SCA map. There were areas (between 0.1 and 0.7% of the total basin area) with negative SWE values because of kriged residuals with large negative values. For the cells with a negative SWE output we arbitrarily assigned SWE values by multiplying the decision tree-modeled depths by the modeled density and SCA. These cells were determined to have snow cover on the

SCA map and therefore were assigned positive SWE values modeled without the kriged or cokriged estimates of the re- siduals.

4. Results

4.1. Snow Depth

Considering the heterogeneous snow distribution found in complex alpine terrain, the model results proved favorable.

BALK AND ELDER: COMBINING STATISTICAL METHODS TO ESTIMATE SNOW DISTRIBUTION 21

The results for the 1997 tree (R 2 = 0.539) and the 1998 tree (R 2 = 0.648) are similar to the results found for other alpine basins [Elder, 1995a; Elder et al., 1995, 1998]. The higher R 2 value for the 1998 tree is probably the partial result of more accurate registration of the sample points to the DEM. Most splits in the 1997 tree were made on the independent variable of elevation. An initial split on elevation separates the high- elevation, deep, wind-deposited snows from the remainder of the basin. For the 1998 tree, there was greater representation of each independent variable in the binary decisions. The 1998 tree generates a mosaic of discrete depth classes that effec- tively capture the abrupt changes in snow depth.

For the 1997 and 1998 kriged residual models all R 2 values were less than zero. The negative R 2 values indicate that no additional information is gained by kriging the residuals. The Moran's I for the 1997 residuals (p = 0.073) and 1998 re- siduals (p = 0.077) were not significant at the 0.05 level. This lack of significant spatial autocorrelation (p value > 0.05) can explain the poor results of the ordinary kriging models. With the poor results of the kriged surfaces, subsequent analysis only considered the ordinary cokriging models.

The stability of the cokriged models for each year was ex- amined. Stability in this study refers to the characteristics of plotting R 2 versus the number of nearest neighbors used in the interpolation process. The R 2 values for a stable model would asymptotically approach a positive value with increasing near- est neighbors, whereas an unstable model would have rapidly fluctuating R 2 values, often dropping to large negative values. The elevation cokriging models for both years were unstable and were not used in further analysis. Both the radiation and slope cokriging models had similar stability and R 2 values for various nearest neighbors. However, the radiation cokriging models were chosen since radiation tends to exhibit a stronger correlation with snow distribution.

Combining the decision tree-modeled depths with the radi- ation cokriged estimates of the residuals offers improvement on the tree-modeled depth surface. The R 2 values for the combined model depths were calculated by subtracting from one the variance that is still unexplained by the combined models. The R 2 values are essentially calculated using (3) except here the response variable z i refers to the measured snow depths. The combined model depths explained 60-85% of the observed variance in measured depths (Figure 5).

The dependence of R 2 values on the number of nearest neighbors can vary when cokriging residuals of first-order mod- els as compared to cokriging the variable of interest. The averaging of positive and negative residual depths over larger areas reduces the impact of increasing the number of nearest neighbors, such that less spatial averaging increases the amount of variability explained by the cokriging models (R. M. Reich, Colorado State University, personal communication, 1999). This averaging effect causes the R 2 values to decrease asymptotically (Figure 5). In this study the cokriged residual models do not gain additional information from more than seven nearest-neighbor samples.

Summaries of field-measured depths and modeled depths using two nearest neighbors in the interpolation process are given in Table 6. Statistics were computed for only the snow- covered areas in the "SCA" model, whereas the "basin" model statistics were generated for the entire watershed. Since inter- polations were made at 10-m intervals and did not necessarily correspond precisely with the sampled locations, the estimated residual depths often did not correspond to the observed re-

0.9

0.8

.-.- 1997 combined

0.7 - -' -'-•-.• -' .• ,--- ._._1998combined ß ß ß ß ß ß ß ß ß , 1997tree

•:: 0.6 :•.-,--,--,... ß 1998 tree ß ß ß ß ß ß ß

0.5

0.4

0.3 .... 2 4 6 8 10 12 14

Number of Nearest Neighbors

Figure 5. Snow depth models. Coefficient of determination (R 2) versus number of nearest neighbors used in cokriging the residuals is shown. Curves with symbols indicate combined models of snow depth (i.e., cokriged estimates of residuals added to the tree-modeled depths). Horizontal rows of sym- bols indicate R 2 values for only the binary decision tree mod- els.

sidual depths. With nugget effects present, the variograms are discontinuous at zero distance, and the kriged or cokriged estimate at a small distance from a sampled location can be quite different than the sampled measurement [Kitanidis, 1993].

4.2. Snow Water Equivalence

The 1997 and 1998 SWE distributions were generated using two to seven nearest neighbors in the cokriging interpolation of the depth residuals. This range of nearest neighbors was cho- sen because it represented the highest R 2 values of the 1997 and 1998 combined depth models (Figure 5). Hereafter these models will be referred to as SWEYY_N, where YY denotes the year and N denotes the number of nearest neighbors. For example, SWE97_6 refers to the 1997 model with six nearest neighbors.

Quantitative comparisons between final SWE distributions were referenced to the SWE97_2 and SWE98_2 maps. These SWE maps had the highest model fit of snow depth (Figure 5). Expressed as depths, SWE values for SWE97_2 and SWE98_2 are summarized in Table 7, where "SCA" and "basin" have the

Table 6. Summary of Field-Measured and Model- Estimated Snow Depths Where Standard Deviation, Coefficient of Variation, and Sample Size Are Included

Min, Max, Mean, o-, Depth m m m m CV

1997 measured 0.05 10.25 3.03 2.13 0.70 197

1997 modeled, 0 11.90 4.46 2.53 0.57 39022 SCA

1997 modeled, 0 11.90 2.52 NA NA 69188 basin

1998 measured 0.05 6.05 2.09 1.41 0.67 173

1998 modeled, 0 6.05 2.39 1.17 0.49 39022 SCA

1998 modeled, 0 6.05 1.35 NA NA 69188 basin

Sample size for modeled depths is the total number of 10-m cells, where "SCA" denotes snow-covered areas in LVWS and "basin" de-

notes the entire watershed. Modeled depths are from the two nearest- neighbor combined models.

22 BALK AND ELDER: COMBINING STATISTICAL METHODS TO ESTIMATE SNOW DISTRIBUTION

Table 7. Summary of Modeled Snow Water Equivalence (SWE) Expressed as Depth From the Two Nearest-Neighbor Combined Models Where Standard Deviation, Coefficient of Variation, and Sample Size Are Included

SWE Min, Max, Mean, Model m m m m CV

1997, SCA 0 4.20 1.55 0.84 0.54 39022 1997, basin 0 4.20 0.88 NA NA 69188 1998, SCA 0 1.95 0.72 0.34 0.47 39022 1998, basin 0 1.95 0.41 NA NA 69188

Sample size is the total number of 10-m cells, where "SCA" denotes snow-covered areas in LVWS and "basin" denotes the entire water- shed.

same meanings as above. The mean 1997 modeled SWE depth within only the snow-covered areas is 1.55 m compared to a mean 1997 modeled SWE depth of 0.88 m over the entire basin. For 1998 the mean modeled depth within snow-covered areas is 0.72 m, while the mean modeled depth is 0.41 m for LVWS (Table 7). Both mean depths over the entire basin are 56% of the mean depths in the snow-covered areas. This dif- ference reflects the 56% of LVWS that is snow-covered and

demonstrates the sensitivity of mean SWE to SCA. The SWE97_2 and SWE98_2 maps are shown in Figures 6

and 7, respectively. Figures 6 and 7 show relatively high SWE accumulations in the sheltered Andrews Creek subbasin and

relatively low SWE accumulations in the broader Sky Pond subbasin. The effects of wind redistribution of snow can be

seen along the western margins of LVWS. These areas, just to the lee of the Continental Divide, accumulate large amounts of SWE. Noticeable elevation banding of SWE is present in the SWE97_2 map. This banding reflects elevation as the domi- nant split in the 1997 decision tree of snow depth.

Of note in the SWE98_2 map is the lack of higher SWE values around Andrews glacier at the headwaters of Andrews Creek and the high SWE accumulations near the LVWS out- let. The lower SWE values around Andrews glacier can be explained by the inability to sample that region during the survey. Andrews glacier is expected to have large seasonal deposits of SWE [Outcalt, 1965]. The few depth samples taken at high elevations in the Sky Pond subbasin were not repre- sentative of the conditions, particularly depth and radiation, around Andrews glacier. As a result, the 1998 decision tree had difficulty modeling the snow distribution in this small region. The high anomaly near the LVWS outlet represents a region of wind-deposited snow. The SWE values in this area are pro- gressively smoothed as the number of nearest neighbors used in interpolation increases. Using only a small number of near- est neighbors in the interpolation process inhibits the ability to interpolate the depth residuals at distances far away from the sampled sites. This characteristic explains a more prevalent polygonal structure underlying the SWE maps with fewer near- est neighbors. Consideration of this characteristic should be

SWE, rn :•0

o.o- o,5

._..;:• 0,5 - 1,0 ..... 1.0 1.5

1.5 2.0 L._.•.• 2,0 2,5 2..5 4.2

N

0 0.5 I Kilometers

Figure 6. Combined model snow water equivalence expressed as depth for 1997 (SWE97_2) Dark regions indicate shallow SWE accumulations; light regions indicate deeper SWE accumulations. Depths are shown in meters.

BALK AND ELDER: COMBINING STATISTICAL METHODS TO ESTIMATE SNOW DISTRIBUTION 23

.&

.•ig: •i:: ' ii..-'•:'*: .-'?•' .,-,•.....--.,....'•i•,. ::.•:;•:•: ;•%: ..... - .... ".-. . -' .•- .• •-*.-•-.-..:.•..•:•.•.r •:;.• ..... .:-•. • -- • ....--•- -•> . .--:a:•:.:as•:a::•a•;ab _x... - -J½:.• -2:::- ..:½;::•;.;.•:.,•::::(,•:-:";'.....::: i,::-:::z;•x:as2, .,y;•'•,g..½*a•.•Sd.Z. "-*::.---::. ,,":":. ;..:,½:--.' -,::,:.:*•*,;a;:•:::•;' - '½ '•{,• ........... ....... .....

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-.• %' . -: •....•:::.; '.... •;;:r:::.•:'-:•--"-',; .-: ":•/ffi..•'"•*•'•?:E;•½•a•-,•-• •?•2a•B•4? ':": '•'":':'•?• ...... : •...•....:.?;, "::;•'a•.•...';.•!•:'q•-•( a ............... ;:-;??/";.•::;:;:'szs?¾5%%;B• ": $(½?'•:---•?:-----:½-c-•'"-a• .... ..:• :?--:•:' :•'" .... . .. ':a•g* :g½:X-$::.:::•a -:•' ..-•-•..m:z•½;q•::7•:•.[;•;•;•,. -- •:;:•;•....;½•:':•::•S•88/:4•;:;½: ..:'a ß "•'•½*•: :••• ::E'•Xa'*?:•½•$•;$?." * ß ß .'

.:.,•, ......... m-•..• ........ •:, -,:*:'•%•:•--•2 ...... ½•:- ' ß ' :::'.•-:• ;q:?½..:-'-:

-. ..... i---:-:.. ". -:L.: ß ,• --..::•:•::-:•.::.7• -.--•:•½;•:::•:-:-, •:g:• -:L. < ' • • ":.:::•;?:•-•;•:-a•::½::•.: , ..•};• * •, • ß '.•;:..- •.-.:.: .• ..... - •::...:..•::.•:•- .......... • ..... -•: ½-.....:-.::y.:•, • * • •.- ......... ..,•::•,::2:4-:-a•½..•: .:x.---..4.--- ß -.--. ß •:..- ..................... •.•,. ß ...... ,.•.•.•.:•,...,a ................. •: .•:,½ :: •, • -•, •:., ,, ....•½.•.......•:::•.•::,:,::..•.::z•:½;:,.: .• •- .....

'"•'•'*':'•8:':•'•-::::•.'::... - . .-..½-:•-•%•;::•.:•::.•.:., . • q'-•.:•.a'.•-..-::-'--':'"'•½;?-•'•;'•: • , . .• g•'•,

•.•:'• '•:::•:•:':' ': ".- "":":::':'½ ½::.• . • --• .... -2' -;:;": . ::-%-½* -:-' "-.('::' - ½,•½,4' .?" {* '?" .: ..... i• ............... ,."•::•-' ..%:: ½ ..... • , '

• ....... , / ß

' ' ,..A"?,"'" • . ........ ½-•,; ..

/:-:?.½ ß •;..-

N

SWE, m lllO

0.0- 0.3 -'• 0.3 - 0.6

:'-•:•-:'-B 0.6 - 0.9 0.9 - 1.2 1,2-1,6

-'• 1.6- 2.0.

Figure 7. Combined model snow water equivalence expressed as depth for 1998 (SWE98_2). Dark regions indicate shallow SWE accumulations; light regions indicate deeper SWE accumulations. Depths are shown in meters.

weighed along with the quantitative model fit when selecting a "best" model.

For the two through seven nearest-neighbor SWE maps the mean SWE depth, SWE, and total SWE volume, SWEtot, are shown in Table 8. As an indicator of SWE distribution differ-

ences, the total SWE volume displaced, SWEdisp , between two model surfaces was calculated by

M N

SWmdisp: E E ISWEi,- SWE,,,I, i=0 j=O

(4)

where SWE x and SWEy are model values at cell locations i, j for an M by N grid [Elder, 1995a]. Displaced SWE volumes calculated between the two nearest-neighbor SWE maps and

Table 8. Summary of Modeled SWE for 1997 and 1998

SWEto t ASWEto t From SWEdisp ASWEdisp From Model SWE, m Volume, m 3 SWEYY_2, % Volume, m 3 SWEYY_2, %

SWE97 2 0.88 6,050,000 NA NA NA SWE97_3 0.86 5,950,000 1.7 657,000 10.9 S WE 97_4 0.85 5,900,000 2.5 789,000 13.0 SWE97_5 0.85 5,870,000 3.0 899,000 14.9 SWE 97_6 0.85 5,860,000 3.1 962,000 15.9 SWE97_7 0.85 5,870,000 3.0 1,010,000 16.7

SWE98 2 0.41 2,810,000 NA NA NA SWE98_3 0.40 2,790,000 0.7 353,000 12.6 SWE 98_4 0.40 2, 750,000 2.1 443,000 15.8 SWE98_5 0.40 2,730,000 2.8 484,000 17.2 SWE98_6 0.39 2,730,000 2.8 512,000 18.2 SWE98_7 0.40 2,750,000 2.1 519,000 18.5

SWE represents mean SWE expressed as depth for the basin. SWEto t represents total SWE in the basin. SWEdisp refers to the displaced SWE in the basin calculated between the two nearest-neighbor models and the three through seven nearest-neighbor models.

24 BALK AND ELDER: COMBINING STATISTICAL METHODS TO ESTIMATE SNOW DISTRIBUTION

the three through seven nearest-neighbor SWE maps are in- dicated in Table 8.

The changes in SWEto t between the two nearest-neighbor SWE maps and the other maps are less than 3.1% (Table 8). These small differences indicate that all combined models es-

timate similar volumes of snow. However, the large changes in SWEdisp , 10.9-18.5% (Table 8), suggest a larger change in distribution of snow between the SWE maps. Such snow dis- tribution differences have little effect on SWEto t estimates but do have important implications in snowmelt modeling. For example, the model relocation of 15 % of the total snow volume to areas receiving greater amounts of incoming radiation may substantially alter the resultant snowmelt hydrograph both in terms of timing and peak runoff.

5. Discussion

The extreme topography of LVWS plays a dominant role in the snow distribution. Owing to its location just to the east of the Continental Divide, LVWS is subject to strong winds that are funneled by the glacial terrain. The rugged terrain ampli- fies the complexity of the wind patterns resulting in a mosaic of wind-scoured and depositional areas. The rugged terrain also leads to very heterogeneous radiation inputs across the water- shed. Steep topography, variable winds, and uneven energy balance are primary contributors to the large heterogeneity in SWE found in LVWS.

Binary decision trees were used to model the large-scale variability of snow depth, that is, to reduce the large-scale trends. The residuals from the decision tree models were then

interpolated across the watershed through geostatistical meth- ods. Separately, binary decision trees and geostatistical meth- ods have been previously used to model snow distribution; however, they have never been combined into one approach. The combined deterministic-stochastic modeling approach dis- cussed in this paper is thus unique to the field of snow hydrol- ogy.

Using the independent variables of net solar radiation, ele- vation, slope, and vegetation cover type, the binary decision trees explained 54-65% of the observed variance in snow depths in LVWS. The results of the decision tree models are similar to results found using similar independent variables for other alpine basins in the American West [Elder, 1995a; Elder et al., 1995, 1998]. Because this statistical technique can de- scribe nonlinear relationships between independent variables and dependent variables, the binary decision tree method ex- plained a greater portion of the variance in field data from the Emerald Lake watershed than any other method explored [El- der et al., 1995]. In this study we took the binary decision tree method one step further by adding spatially interpolated re- sidual depths to the tree-modeled output.

Geostatistical methods have also been used to spatially model SWE. Before geostatistical techniques are applied to map SWE distribution, snow data have been detrended to remove large-scale variations and to validate the stationarity assumption of constant mean and constant variance across the study region. Hosang and Dettwiler [1991] modeled SWE mea- surements over a small catchment in Switzerland by multiple linear regression using elevation, presence or absence of forest, and potential direct solar radiation as independent variables. The spatially distributed residuals from the regression equa- tion were then mapped across the catchment by kriging inter- polation. Using SNOTEL and snow course data in a northern

Idaho basin, Carroll and Cressie [1996] compared the accuracy of SWE estimates produced by two geostatistical methods (simple kriging and elevation-detrended kriging). The eleva- tion-detrended kriging approach is an attempt to detrend the data by subtracting an orographic effect, which is modeled as a linear function of site elevation, from the SWE measurements. Carroll and Cressie [1996] found simple kriging to be superior in terms of prediction accuracy. Balk et al. [1998] attempted to krige LVWS snow depth data without using a first-order model to detrend the large-scale variations. Problems occurred in this approach since the stationarity assumption was invalid.

There are a few distinct differences between our combined

modeling approach and the detrended kriging approaches mentioned above. Both Hosang and Dettwiler [1991] and Car- roll and Cressie [1996] used linear regression to detrend the SWE measurements before kriging the residuals. Our com- bined deterministic-stochastic approach offers improvement upon these methods since binary decision trees can account for many observed nonlinear relationships between the dependent and independent variables. Snow cover properties are known to vary in response to changes in terrain, energy balance, veg- etation, and surface roughness patterns, and often the varia- tions do not behave linearly. The decision trees were not con- strained to linear first-order models and thus could explain more of the observed variances in snow depth. The 33 mea- surement points used by Hosang and Dettwiler [1991] were basically distributed along one main axis in the drainage basin, while Carroll and Cressie [1996] only had 12 data sites. In comparison, our larger data sets were more representative of the study region enabling the application of binary decision trees. The binary decision trees sufficiently modeled abrupt large-scale variations in depth, highlighting a mosaic of wind- scoured and depositional areas. By adding the cokriged resid- ual surfaces to the tree models, the transitions between areas of high and low accumulations are smoothed. We feel the combined model depth surfaces adequately represent the snow depth surfaces found in the field. The application of geostatis- tical methods to the tree-modeled residuals then is unique and improves the spatial modeling of snow distribution in moun- tain basins.

Knowledge of the spatial distribution of SWE is crucial for the accurate prediction of the timing and magnitude of snow- melt runoff. For example, the same volume of SWE modeled across a basin by two separate methods would yield different snowmelt simulations assuming all other variables are con- stant, such as meteorological and soil moisture conditions. Simply applying the mean SWE value evenly across the basin would yield a snowmelt hydrograph that would peak earlier and slightly higher, while the late-season runoff would be lower than a simulated hydrograph using our spatially distributed SWE maps. The earlier and slightly higher peak runoff would largely result from greater amounts of snow on solar-loaded, south facing slopes, and the late-season depletion of flows would stem from the underestimation of SWE in sheltered, north facing aspects.

Future work could be done with the differencing of various SWE distribution estimates. Since most snow distribution

modeling efforts are quantitatively assessed only by the coef- ficient of determination (R2), choosing a "best" model be- comes difficult and may be arbitrary. For a watershed, SWE distribution models could be compared by subtracting the SWE depth surfaces. Such differencing could be used to de- termine relative "hot" and "cold" areas in terms of snowmelt

BALK AND ELDER: COMBINING STATISTICAL METHODS TO ESTIMATE SNOW DISTRIBUTION 25

prediction. The SWE distribution estimates could then be ap- plied to snowmelt models to gain a certain level of confidence between upper and lower bounds in snowmelt volume. Dis- charge simulations could be computed from physically based, spatially distributed snowmelt models and compared to mea- sured discharge to assess relative accuracy of SWE distribution models. If the simulated discharge was similar to the observed discharge, then the SWE distribution estimate would be con- sidered relatively accurate.

Although this study focused on a small watershed, we be- lieve the methods presented could be applied in some form to snow modeling over larger, regional scales. Ultimately, re- gional scale models are needed for accurate water supply and flood forecasting for regions such as the Colorado Rockies, the Sierra Nevadas, and the Cascades. Regional watershed assess- ments drive the water supply forecasts for many of the major metropolitan areas in the American West. In the future we need to find ways to extrapolate small watershed modeling efforts over a regional scale. Such extrapolation may be done through energy balance calculations and remote sensing of SCA over large regions and coupling intense snow surveys in representative small catchmerits with the existing network of SNOTEL and snow course sites.

6. Conclusions

Difficulties were encountered in modeling efforts to inter- polate snow depths across LVWS through kriging interpola- tion alone [Balk et al., 1998]. These complications motivated interest to model the large-scale variations in snow depth with binary decision trees and to model the small-scale variations through geostatistical techniques. The combined models of snow depth are improvements over using decision tree or geostatistical methods alone to estimate snow distribution in complex mountain environs. Between 60 and 85% of the ob- served variance in depth measurements was explained by the combined models of snow depth. Binary decision trees delin- eate similar snow accumulations by relating snow depth with independent variables based on topography, energy balance, and vegetation. By modeling depth in a nonlinear, hierarchical fashion, decision trees can detect the abrupt changes in snow depth found in heterogeneous mountain snowpacks. Using net SOlar radiation as an auxiliary variable, cokriging interpolation of the residuals effectively smoothes the transitions between these abrupt changes in depth. Estimating the residual depths with ordinary kriging did not provide additional information to the decision tree models.

Through regression analysis, snow densities were spatially modeled through LVWS. The combination of the modeled depth surfaces, modeled densities, and SCA delivered spatially distributed estimates of SWE. Such spatial estimates of SWE are necessary for accurate prediction of the magnitude and timing of snowmelt runoff that is critical to water supply and flood forecasting. This combined deterministic-stochastic ap- proach to modeling snow distribution would greatly enhance the accuracy and precision of physically based, spatially dis- tributed snowmelt models.

Acknowledgments. Jill Baron provided financial and editorial as- sistance. Don Campbell, George Ingersoll, and numerous other USGS personnel from Denver along with Joe Stock, Eric Allstott, and Doug Bopray provided help with the snow surveys. Robin Reich provided a great deal of help with geostatistical modeling efforts. Don Cline,

National Operational Hydrologic Remote Sensing Center, NWS, NOAA, coordinated the preprocessing of the aerial photos that were used to construct the 10 m DEM of LVWS and the SCA map. Ralph Root and Larry Fairbank, USGS/BRD, Center for Biological Infor- matics, provided assistance in constructing the DEM. The manuscript was substantially improved by comments from G/inter B16schl, Charles Luce, and Mark Williams. This research was funded by the Colorado Rockies Global Climate Change Program (NPS-1268-2-9004 and COLR-R92-0201,140), U.S. Geological Survey, Loch Vale Watershed project. The first author was partially supported by the Natural Re- source Ecology Laboratory Graduate Fellowship.

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(Received March 10, 1999; revised August 12, 1999; accepted August 12, 1999.)