Modelling snow accumulation with a geographic information system

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1 int. j. geographical information science, 2000, vol. 14, no. 7, 693± 707 Research Article Modelling snow accumulation with a geographic information system KANG-TSUNG CHANG* Department of Geography, University of Idaho, Moscow, ID 83843, USA; chang@uidaho.edu and ZHAOXING LI Texas Rehabilitation Commission, Austin, Texas 78758, USA (Received 15 August 1997; accepted 15 March 1999) Abstract. Snow courses that measure snow water equivalent (SWE) are clustered and limited in areal coverage in Idaho. This study used a cell-based geographic information system and multiple regression models to construct SWE surfaces from the snow course data by month (January to May) and by watershed. SWE was the dependent variable and location and topographic variables derived from a digital elevation model were used as the independent variables. Multiple regression performed better than the traditional interpolation methods for SWE estimation. The estimated SWE surface can be displayed at diœerent spatial scales through neighbourhood operations, or used directly as a map layer for hydrologic modelling. 1. Introduction Snowpack provides approximately 75% of the volume of seasonal stream ow in the western United States (Palmer 1988), and accounts for more than 60% of the annual variability in stream ow (Doesken et al. 1989). Point snowpack measurements of snow water equivalent (SWE) are critical for snowpack accumulation variability studies and for water supply forecast models. The purpose of this paper is to create a SWE surface from these irregularly distributed data points. Common methods for interpolating a statistical surface from given point data include Thiessen polygons, linear interpolation, inverse distance weighted, and, more recently, kriging (Creutin and Obled 1982, Lam 1983, Tabios and Salas 1985, Burrough and McDonnell 1998). These methods mainly consider distances from known points or spatial dependence models among data points in spatial interpolation. The reliability of an interpolated surface therefore depends to a large extent on the distribution of data points and the areal coverage of data points. The interpolated surface is much more reliable if the points are uniformly distributed within the study area than if the points are clustered and limited in their areal coverage. Snow courses, at which snowpack accumulation data are measured, form clustered patterns in mountainous areas of Idaho. The traditional methods for interpolating SWE surfaces from clustered snow courses typically result in irregular isoline Internationa l Journal of Geographica l Information Science ISSN print/issn online 2000 Taylor & Francis Ltd

2 694 K-T. Chang and Z. L i patterns in areas with no point data and circular isolines around snow courses ( gure 1). A diœerent approach is needed to construct SWE surfaces. In this paper, we compare multiple regression with the traditional interpolation methods for SWE estimation, and model snowpack accumulation with a geographical information system (GIS) and multiple regression analysis. We also discuss the display of SWE surfaces at diœerent spatial scales. 2. Snowpack accumulation Snowpack accumulation (and conversely, ablation) is in uenced by meteorological and topographic factors such as air temperature, wind, incoming solar radiation, elevation, and aspect (McKay and Gray 1981, Gerrard 1990). These two groups of factors act together and are related to each other (Barry 1981). For example, slope and aspect in uence incoming solar radiation, and elevation controls air temperature and ground temperature. Past studies have shown that elevation, aspect, slope, and location, either individually or in combination, signi cantly in uence precipitation in western mountainous regions (Caine 1975, Aguado 1986, Doesken et al. 1989, Changnon et al. 1991, Troendle et al. 1993, Johnson and Hanson 1995). Location and topographic variables have been used as predictor variables in modelling mean annual precipitation and in modelling precipitation over large mountain areas (Winters et al. 1989, Daly et al ). This study assumes that location and topographic variables in uence snowpack accumulation in Idaho and, by understanding relationships between SWE and these variables at snow courses, it is possible for us to predict SWE at unknown points and to construct SWE surfaces. Figure 1. An isoline map of maximum snow water equivalent (SWE) for region Derived from linear interpolation in Arc/Info GRID, the map uses a 3-inch interval. See text about region 1704 and snow courses within the region.

3 Modelling snow accumulation with a GIS Data 3.1. Snow course data The Natural Resources Conservation Service (NRCS, formerly the Soil Conservation Service) has conducted snow surveys in western mountainous regions since Manual surveys take 5 to 10 measurements at regular intervals along a snow course and records the average measurement in SWE. In 1977, the NRCS began developing a network of SNOTELs (SNOwpack TELemetry data stations). Each SNOTEL station is equipped with automatic measuring devices and remote communication devices. Over 250 snow courses are maintained in Idaho. For this study, we removed those snow courses that did not have continuous 30-year ( ) records, and those that had apparent positional errors. We ended with 194 courses. For each course, we computed the average monthly SWEs from January through May and the average maximum SWE. The maximum SWE referred to the highest SWE during a snow season. The month in which the maximum SWE occurred varied between snow courses Digital Elevation Models We used 3 arc-second Digital Elevation Models ( DEMs) from the US Geological Survey ( USGS) to derive location and topographic variables for our study. After obtaining all the DEMs for Idaho, we projected them into the Lambert conformal conic projection, merged them, and removed sinks from them. Sinks in DEMs are commonly due to errors in the data and should be removed (Jenson and Domingue 1988, Marks 1988). The nal single DEM grid measured 5966 columns and 8229 rows, with a cell size of m. Of the total cells, cells covered the state of Idaho. 4. Location and topographi c variables A set of location and topographic variables were created for this study. Elevation, elevation derivatives such as slope and aspect, and secondary or compound variables that measured land surface curvatures or the combined eœect of slope and aspect (Moore et al. 1993) were generated using functions in the Arc/Info GRID module (ESRI 1997). 1. EASTING, SOUTHING: The easting and southing of a cell corresponded to the column number and the row number in the DEM grid. They were like map projection coordinates in X and Y. 2. Elevation (ELEV): Elevation was the Z-value,in metres, of a cell in the DEM grid. 3. SLOPE, SLOPE2: Slope measured the rate of maximum change in elevation from a cell, in degrees among 0 and 90. SLOPE was derived by tting a plane surface within a 3Ö 3 window around the cell, whereas SLOPE2 was derived by tting a 4th-order polynomial equation. 4. ASPECT, ASPECT2: Aspect identi ed the direction of maximum rate of change in elevation from a cell and was measured in degrees clockwise from the north. The variable ASPECT was derived by tting a plane surface within a 3Ö 3 window around the cell, whereas ASPECT2 was derived by tting a 4th-order polynomial equation. The circular measure of aspect could not be used directly in statistical analysis. Aspect values were therefore converted from degrees to radians. 5. Transformed aspect variables: We used data transformation to create aspect variables that directly addressed the eight principal directions. To capture the N S

4 696 K-T. Chang and Z. L i principal direction, we set 0 degree at north, 180 degrees at south, and 90 degrees at both west and east ( gure 2). We named the transformed aspect variable ASP0. We did similar transformations to ASPECT to capture the E W (ASP90), NE SW ASP45), and NW SE (ASP135) principal directions. The same transformations were performed on ASPECT2 (ASPI0, ASPI90, ASPI45, ASPI135). 6. Slope-aspect index (SAINDEX): A compound surface measure for relief shading, the slope-aspect index ranged in value from 0 to SHADOW: A compound surface measure, SHADOW considered the illumination angle. SHADOW had values ranging from 0 for areas in the shadow to 255 for areas with the brightest light. In our study, we assumed the source of illumination coming from the south at an elevation angle of about 70 degrees, the maximum sun angle at any point in Idaho. 8. Curvature measures: Curvature measures included the overall curvature of an elevation surface (CUR1000), the curvature in the slope direction (PROF1000), and the curvature in the direction perpendicular to the slope direction (PLAN1000). A positive curvature at a cell indicated an upwardly convex surface, a negative curvature an upwardly concave surface, and 0 a at surface. The unit of these curvature measures from Arc/Info GRID was 1/100 metres; we multiplied the curvature values by 1000 to avoid truncation problems in data processing. 5. Regionalization Regionalization, or dividing the study area into diœerent regions for separate analyses, was a tool we used to capture diœerent climatic and topographic characteristics within the state of Idaho. We experimented with regionalization using cluster analysis of the location and topographic variables but the result did not provide clear-cut regional boundaries. We also considered climatic regions for Idaho used by the National Climatic Data Center but settled for watersheds for two reasons. Watersheds have clear, physical boundaries and are organized hierarchically; this allowed us to easily combine watersheds if necessary. Furthermore, the watershed boundary delineates the border of orographic in uence, meaning that all sample points in a watershed would fall on the same side of a given orographic feature (Phillips et al ). Idaho contains 88 basic, 8-digit hydrologic units according to the USGS Water Resource Division. We combined these basic units at two spatial levels for our analysis. The rst level simply followed the 4-digit hydrologic units: 1701, 1706, 1705, Figure 2. Transformed aspect variables. ASP0 and ASPI0 were designed to capture the N S principal direction; ASP45 and ASPI45, the NE SW direction; ASP90 and ASPI90, the E W direction; and ASP135 and ASPI135, the NW SE direction. ASP and ASPI were both aspect measures, but based on diœerent computing algorithms in Arc/ Info GRID.

5 Modelling snow accumulation with a GIS , 1601, and 1602 ( gure 3). Of the six regions, only 1704, 1705, and 1706 had 30 or more snow courses. We started with 6-digit hydrologic units at the second level of regionalization, and reorganized them into 15 watersheds. Eight of these 15 watersheds had a minimum of 13 snow courses ( gure 4). The other seven watersheds were smaller and had either less than 13 or no snow courses; these watersheds were combined with the eight larger watersheds for data analysis. 6. Analysis We used Arc/Info for derivation of the location and topographic variables, data management, watershed delineation, SWE estimation, and data display. To stay with Figure 3. Distribution of regional watersheds in Idaho and snow courses within regions 1704, 1705, and Region 1704 covers the middle and upper Snake River Basin. Region 1705 covers the lower Snake River Basin and the basins of the Payette, Boise, and Owyhee. Region 1706 includes the basins of the Clearwater and Salmon.

6 698 K-T. Chang and Z. L i Figure 4. Distribution of watersheds in Idaho and snow courses. 1. Kootenai; 2. Pend Oreille; 3. Coeur d Alene and St. Joe; 4. Palouse; 5. Clearwater; 6. Salmon; 7. Snake; 8. Payette, Boise, and Owyhee; 9. Big Wood and Big Lost; 10. Snake; 11. Snake; 12. Deep Creek; 13. Bear; 14. Thomas Fork; 15. Henry s Fork and Camas Creek. the DEM data structure and to facilitate computation, we conducted our analysis in the raster data format Comparison of multiple regression with traditional interpolation methods The rst analysis was a comparative analysis of multiple regression modelling and the traditional interpolation methods for SWE estimation. Our approach was to split snow courses into two equal, independent samples: one sample for developing regression models or for spatial interpolation, and the other sample for testing the accuracy of the models or interpolation. Using an independent data set for model evaluation is more rigorous than the re-sampling technique such as the jackknife method. Sample splitting, however, requires a large number of snow courses in the

7 Modelling snow accumulation with a GIS 699 original sample. For that reason, our analysis was performed at the regional level for 1704, 1705, and Region 1704 had 71 snow courses, 1705 had 31, and 1706 had 37. The maximum SWE was the dependent variable in multiple regression modelling, and the independent variables included EASTING, SOUTHING, ELEV, SLOPE, and ASPECT (in radians). We used the stepwise regression method. The spatial interpolation methods included Thiessen polygons, linear interpolation, and the inverse distance squared method. From the computed multiple regression model and the interpolated surface, we derived the estimated maximum SWE for those snow courses in the validation group. We then computed the mean squared error (MSE) between the observed and estimated maximum SWE for comparison: MSE 5 (sum of the squared errors)/number of snow courses (1) MSE, or the sum of squared errors, is commonly used for testing the accuracy of precipitation estimation procedures (Chua and Bras 1982, Tabios and Salas 1985) Multiple regression analysis at the watershed level We conducted the second analysis at a ner spatial scale, with more independent variables, and for more SWE data sets than the rst. The snow courses in each of the eight watersheds were overlaid with the DEM data grids to extract their location and topographic values. We then ran separate multiple regression analyses using each monthly SWE and the maximum SWE as the dependent variable and all the location and topographic variables, except aspect in radians, as the independent variables. Several steps were taken to ensure the validity of our SWE models. We identi ed those independent variables with high inter-correlations from a simple correlation matrix. As expected, SLOPE and SLOPE2, ASPECT and ASPECT2, SLOPE and SAINDEX, SLOPE2 and SAINDEX, transformed aspect variables, and surface curvature variables all had high correlations. We made sure that variables with high inter-correlations would not be included in the same model. We limited the number of independent variables in a model to ve or less because of our small sample sizes (13 to 31) at the watershed level. The stepwise method was used to initially select independent variables into a model; the level of signi cance for adding a variable was set at Results from the stepwise method were modi ed, if necessary, by considering the collinearity between variables, the number of variables in a model, and the diœerence in the R-square value between the stepwise method and the maximum R-square method. We also applied the jackknife re-sampling method to avoid the overspeci cation of our SWE models. Although the number of snow courses in each watershed was 13 or more, some snow courses did not have SWE readings for January or May. This resulted in smaller sample sizes for some models. We ran the jackknife resampling method for those models with sample sizes of less than 15, each with diœerent subsets of three to ve independent variables. The re-sampling method produced the PRESS statistic (Ott 1988): the sum of squares of predicted residual errors, for each subset of independent variables. The subset that had the smallest PRESS statistic was considered the best- tting model.

8 700 K-T. Chang and Z. L i 6.3. SW E surfaces Using the regression models, we were able to predict the SWE value for each cell in each of the eight watersheds. For those small watersheds that were not analysed separately, the estimation of SWE was based on models for combined watersheds or models from nearby watersheds. We then pieced together state-level surfaces for each monthly SWE and the maximum SWE from the watershed data. 7. Results 7.1. Comparison of multiple regression with traditional interpolation methods Table 1 summarizes the R-square, root mean squared error (RMSE), mean of the dependent variable, sample size, and selection of independent variables of the multiple regression models for regions 1704, 1705, and Table 2 compares the MSE errors between the multiple regression models and the traditional interpolation methods. The regression model has the lowest MSE error for regions 1705 and It has a MSE error that is better than Thiessen polygons but slightly higher than linear interpolation and the inverse distance squared method for region Multiple regression models Table 3 summarizes by watershed the R-square, root mean square error (RMSE), sample size, and selection of independent variables for each monthly SWE model and the maximum SWE model. As shown in table 3, diœerent monthly models for the same watershed all have the same or similar independent variables. Watershed 6 has the same independent variables (EASTING, ELEV, and SAINDEX) for all six models; watershed 9 has ELEV and EASTING in most models; watershed 8 includes ELEV, EASTING, and SOUTHING in every model; and watershed 5 include EASTING, SOUTHING, and ELEV in most models. Table 1. Region, R-square, root mean square error, mean of dependent variable, sample size, and independent variables of maximum SWE models, by region. Region R-sq. RMSE Dep-Mean N Independent variables & sign of coe cient EASTING ELEV -SLOPE EASTING SOUTHING ELEV EASTING SOUTHING ELEV Table 2. Comparison of mean squared errors from multiple regression models, Thiessen polygons, linear interpolation, and the inverse distance squared method. Thiessen Linear Inverse. Region Reg. model polygons interpolation distance squared

9 Modelling snow accumulation with a GIS 701 Table 3. Dependent variable, R-square, root mean square error, mean of dependent variable, sample size, and independent variables of SWE models, by month and by watershed. Dep- Dep-V R2 RMSE Mean N Independent variables & sign of coe cient WATERSHED 3 SWE EASTING SOUTHING ELEV ASP90 SWE EASTING ELEV -ASP135 PLAN1000 -SHADOW SWE ELEV SWE ELEV SWE EASTING SOUTHING SWEM ELEV WATERSHED 5 SWE EASTING SOUTHING ELEV ASPI45 -SHADOW SWE EASTING SOUTHING ELEV ASPI45 -SHADOW SWE SOUTHING ELEV ASPI45 SHADOW SWE EASTING SOUTHING ELEV ASPI45 SHADOW SWE SOUTHING ELEV ASPI45 SHADOW SWEM SOUTHING ELEV ASPI45 SHADOW WATERSHED 6 SWE EASTING ELEV SAINDEX SWE EASTING ELEV SAINDEX SWE EASTING ELEV SAINDEX SWE EASTING ELEV SAINDEX SWE EASTING ELEV SAINDEX SWEM EASTING ELEV SAINDEX WATERSHED 8 SWE EASTING -SOUTHING ELEV PLAN1000 SWE EASTING -SOUTHING ELEV -ASPI90 PLAN1000 SWE EASTING -SOUTHING ELEV -ASPI90 PLAN1000 SWE EASTING -SOUTHING ELEV -ASPI90 PLAN1000 SWE EASTING SOUTHING ELEV -ASPI90 CUR1000 SWEM EASTING -SOUTHING ELEV ASPI90 PLAN1000 WATERSHED 9 SWE EASTING ELEV SWE EASTING ELEV SWE EASTING ELEV SWE EASTING ELEV SWE EASTING ELEV ASPI135 PLAN1000 SWEM EASTING ELEV WATERSHED 10 SWE ELEV ASPI45 PLAN1000 SWE ELEV SAINDEX SWE ELEV SAINDEX SWE ELEV SAINDEX SWE ELEV ASPI45 SWEM ELEV SAINDEX WATERSHED 13 SWE EASTING SOUTHING ELEV SHADOW SWE SOUTHING ELEV -ASPI0 SWE SOUTHING ELEV -ASP0 PROF1000 SWE SOUTHING ELEV -ASP0 PROF1000 SWE EASTING -SOUTHING ELEV ASPI45 SAINDEX SWEM SOUTHING ELEV ASP0 PROF1000

10 702 K-T. Chang and Z. L i Table 3. (Continued ). Dep-V R2 RMSE Mean N Independent variables & sign of coe cient WATERSHED 15 SWE EASTING SOUTHING ELEV ASPI135 SWE EASTING SOUTHING ELEV ASPI135 SWE EASTING SOUTHING ELEV ASPI135 SWE EASTING SOUTHING ELEV ASPI135 SWE EASTING ELEV CUR1000 SWEM EASTING SOUTHING ELEV PLAN SW E surfaces Figure 5 shows the SWE surface for April: the bright areas indicate areas of snowpack accumulation. Figure 5 represents a grid created from the original 95 m grid using a neighbourhood, or ltering, operation (Tomlin 1990). The new grid had Figure 5. The SWE surface map of Idaho for April, re-sampled at the 1 km resolution from the 95 m grid.

11 Modelling snow accumulation with a GIS 703 a cell size of 1km, and each cell in the grid had the average value of about 100 cells from the original grid. The neighbourhood operation tended to blur the eœect of derivative topographic variables such as aspect. Figure 5, however, corresponds well with an elevation map of Idaho: the high concentrations are in the Salmon River Mountains in Central Idaho, the Clearwater Mountains near the Montana border, and along the Canadian border. Neighbourhood operations on the original grid were also desirable for making isoline maps at the state or watershed level. Figure 6 shows an SWE isoline map for April in Idaho, derived from a grid with a cell size of 10 km. Figure 7 shows an SWE isoline map for April for a watershed (the Clearwater River Basin), derived from a grid with a cell size of 5 km. While neighbourhood operations, which are essentially for data generalization, are desirable for data display, the original 95 m grid can be used directly in data analysis. One example is to use it for computing the total amount of water storage Figure 6. The SWE isoline map of Idaho for April, re-sampled at the 10 km resolution from the 95 m grid and compiled with the 5-inch interval.

12 704 K-T. Chang and Z. L i Figure 7. The SWE isoline map of watershed 5 for April, re-sampled at the 5 km resolution from the 95 m grid and compiled with the 5-inch interval. at the watershed level for water supply forecast. Another example is to use the 95 m grid as a map layer in hydrologic modelling involving snowpack. 8. Discussion Regression modelling with cell data is not new in geographic research. Robinson et al. ( 1969), for example, used it three decades ago in studying rural farm population densities in the Great Plains. We turned to regression modelling, generally considered an aspatial technique (Fotheringham et al. 1996), after we found the traditional spatial interpolation methods inadequate for SWE estimation. The multiple regression models performed better than the interpolation methods in two out of three regions in the comparative analysis. Because many snow courses from the validation group were spatially close to those from the model group ( gure 3), the interpolation methods were expected to perform well in this kind of test. That advantage, however, was not evident in our test. Our result has certainly con rmed the validity of using regression models for SWE estimation. The literature suggests that SWE is related to location and topographic variables and can be explained by a set of predictor variables. Our results support the literature: the R-square value exceeded 60% in every model at the watershed level and reached 90% in half of all models (table 3). Moreover, our study has found that

13 Modelling snow accumulation with a GIS 705 the physical factors in uencing a watershed s snow accumulation appear to remain eœective throughout the snow season. The traditional interpolation methods do not make use of relationships between SWE and topographic variables. Recent studies of precipitation interpolation using elevation-detrended kriging or co-kriging have included elevation as an additional variable to the X, Y location, and have shown better results than kriging (Phillips et al. 1992, Garen et al. 1994, Carroll and Cressie 1996, Martinez-Cob 1996). The same bene t of including elevation has also been reported in interpolation using thin-plate splines ( Hutchinson 1995). There are, however, potential limitations in using more topographic variables in precipitation interpolation. One limitation is the computational complexity involved, and the other is the uncertainty in assuming global relationships between precipitation and topographic variables over large mountain areas (Phillips et al. 1992, Daly et al ). Similar to our study, PRISM (Precipitation-elevation Regression on Independent Slopes Model) combines a cell-based GIS, local regression, and DEM-derived variables to distribute point measurements of precipitation to grid cells over mountainous terrain ( Daly et al. 1994). There are, however, major diœerences in methodology. PRISM requires the initial division of the study area into a mosaic of topographic facets based on consistency of aspect (north, south, east, west, and at). A local regression model is then developed for each facet by relating precipitation to elevation. To obtain a minimum of three stations for local regression modelling, PRISM prepares diœerent facet grids by subjecting the DEM to diœerent levels of ltering. In our study, we divided Idaho into watersheds and built regression models directly at the watershed level. Instead of preparing diœerent facet grids, we used transformed aspect variables directly in multiple regression. Through the transformed aspect variables, our models were sensitive to eight principal directions rather than four as in PRISM. Our method was more straightforward than PRISM, primarily because we did not have to rst develop diœerent topographic facet grids. The National Operational Hydrologic Remote Sensing Center of the National Weather Service has developed the Snow Estimation and Updating System (SEUS) to oœer real-time estimates of gridded SWEs (McManamon et al. 1993). The system is GIS-based, and uses data of elevation, slope, aspect, forest cover, mean areal temperature, and mean areal precipitation, with a series of linear and non-linear models. More recently, SEUS has incorporated airborne snow survey data and remotely sensed data (McManamon et al. 1995). Our study is diœerent from SEUS: we based our study on historical SWE data, whereas SEUS is designed for dynamic estimation of SWE. The use of remotely sensed data is interesting, however, because it can help delineate snow-covered areas for SWE estimation. 9. Conclusion Using a cell-based GIS, we have successfully modeled snowpack accumulation in watersheds of Idaho with the location and topographic variables derived from the USGS 3 arc-second DEMs as the predictor variables in multiple regressions. Watershed SWE models can be pieced together to provide a synoptic view at the state level. They can also be displayed at diœerent spatial scales or used directly as map layers for hydrologic modelling. GIS is well suited to multiple regression modelling because of its capabilities in integrating and analyzing diverse spatial data sets.

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