Interpolating local snow depth data: an evaluation of methods
J. I. Lopez-Moreno ´ 1* and D. Nogues-Bravo ´ 2
1 Instituto Pirenaico de Ecolog´ıa, CSIC, Campus de Aula Dei, Apartado 202, 50080 Zaragoza, Spain
2 Macroecology and Conservation Unit, University of Evora, Estrada dos Leoes, 7000-730 ´ Evora, Portugal
Hydrol. Process. 20, 2217–2232 (2006)
Abstract:
Snow depth measurements have been taken since 1986 at 106 snow poles distributed in the Spanish Pyrenees. Here, we compared the capacity of several local, geostatistical and global interpolator methods for mapping the spatial distribution of averaged snowpack (1986–2000) and the snowpack distribution in two single years with different climatic conditions. The error estimators indicate that the terrain complexity of the area makes it difficult to apply local and geostatistical methods satisfactorily. Regression-tree models provide an accurate description of the data set used (the calibration phase), but they show a relatively low predictive capability for the study case (the validation phase). Using linear regression and generalized additive models (GAMs), we achieved more robust estimations than by means of a regression-tree model. The GAMs give the most accurate prediction because they consider the non-linear relationships between snowpack and the external characteristics (physical features) of the sampling points.
INTRODUCTION
Snow accumulation in the Spanish Pyrenees determines fluvial regime and explains the interannual variability of spring discharge (Lopez-Moreno and Garc ´ ´ıa-Ruiz, 2004). Given that melting flow contributes to filling reservoirs, thereby ensuring water supply to irrigated areas in lowlands (Lopez-Moreno, 2005; L ´ opez-Moreno ´ et al., 2004), accurate estimates of the snow accumulated in the basins improve water resource management. Furthermore, snow depth maps are useful tools for other fields, such as avalanche- and flood-risk estimation, the planning of tourist activities, climate variability assessment, etc. (Carroll and Cressie, 1996; Haefner et al., 1997).
One of the most reliable methods for mapping the spatial distribution of snow is the collection of data on local snow depth and density and subsequent interpolation. Therefore, in the last years several studies have focused on testing methods to map snowpack and snow water equivalent (Elder et al., 1998; Yang and Woo, 1999; Balk and Elder, 2000; Chang and Li, 2000; Erxleben et al., 2002; Anderton et al., 2004). Recently, Lopez-Moreno and Nogu ´ es-Bravo (2005) applied a generali ´ zed additive model (GAM) (Hastie and Tibshirani, 1987) to map snow depth in the Pyrenees. They proposed that GAMs (used mainly in other fields, such as ecology and biogeography) could be a promising tool to interpolate snow data and other climatic or environmental variables.
Here, we compare the most commonly used interpolation methods in snow studies in order to assess their capacity to predict the snow distribution at the end of April for the average of the period 1986–2000 in the central Spanish Pyrenees. In addition to the analysis of a multiyear average data set, single-year data sets were used for interpolating. Thus, a better understanding of the predictive capability of the models is obtained, and it allows for a more balanced comparison with prior studies that usually consider single-year data sets. Two years, the 1994 and 1995 data sets, were chosen since they represent the most contrasting winters relating to the amount of snow accumulated (Lopez-Moreno, 2005). Our approach is featured by a limited number of obser- ´ vations to model snowpack at regional scales. The scarcity of available observations at the large scale in snow studies is a frequent drawback owing to the difficulty in carrying out data collection surveys during wintertime.
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Correspondence to: J. I. Lopez-Moreno, Instituto Pirenaico de Ecolog ´ ´ıa, Campus de Aula Dei, Apartado 202, E-50080 Zaragoza, Spain. E-mail: nlopez@ipe.csic.es
THE STUDY AREA
The Pyrenees is an alpine range located in the northeast of the Iberian Peninsula (Figure 1). Altitude increases eastward, exceeding 3000 m a.s.l. in the headwaters of the Gallego, Cinca, ´ Esera and Noguera Ribagorzana ´ rivers. The relief is organized in parallel bands following a west–east axis, which causes a strong barrier effect against Atlantic fronts, thereby enhancing the transition from northwest (Atlantic) to southeast (Mediterranean) influences. Precipitation ranges between 1000 and 2500 mm year 1 (Garc´ıa-Ruiz et al., 2001). Temperature is mainly governed by the altitudinal gradient (Del Barrio et al., 1990). During winter, the 0 °C isotherm is located around 1600 m a.s.l. (Garc´ıa Ruiz et al., 1986). This isotherm allows the persistence of an extensive snowpack until May and isolated snow patches until the end of summer. However, the snowpack shows noticeable interannual variability, which constitutes a source of uncertainty when forecasting the availability of spring water resources (Lopez-Moreno, 2005).
DATA AND METHODS
Data
In 1985, interest in assessing snow accumulation in the Pyrenees led the Office of Hydraulic Works (Spanish Ministry of Public Works, Transports and Environment) to fund the ERHIN programme (Estudio de los Recursos H´ıdricos producidos por la Innivacion [Study of Water Resources from Snow Accumulation]). This ´ programme involved the installation of 106 snow poles in the Pyrenees to collect three measurements per year on snow depth and, occasionally, on density. The programme was later extended to other mountain ranges in Spain (Cantabrian range and Sierra Nevada). The snow poles are located in flat areas, without shrub or tree cover, and they are relatively sheltered from wind-drift processes. Of all the sampling points, those with a full data record were selected (74) for this study. Samplings were carried out in January, at the end of March and at the end of April. This study used data from April because of their hydrological implications during the melting period (Lopez-Moreno and Garc ´ ´ıa-Ruiz, 2004).
Figure 1. Study area
Interpolation methods
The interpolation methods compared in this study can be classified as detailed below (Burrough and McDonnell, 1998).
DISCUSSION AND CONCLUSIONS
Here, we assessed the capacity of several local, geostatistical and global interpolator methods to predict and map snow depth in unsampled areas of a large mountainous sector of the central Spanish Pyrenees.
Our results indicate a low predictive capacity of the local and geostatistical methods. Several studies report a low percentage of explained variance of snow distribution using these methods in mountainous areas (Erxleben et al., 2002). This low ability of local methods to predict snowpack values in our study is mainly related to the limited number of snowpack measurements in the central Pyrenees and their topographical complexity. Local interpolators based on the continuous change of response variable (temperature, rainfall, etc.) across space do not fit adequately in mountain areas with important changes of controlling variables such as altitude, aspect or slope in a reduced zone. However, other climatic variables, such as temperature and precipitation (both related to snow accumulation and melting), have been interpolated successfully using the closest observations or probability functions in areas where a dense network of observations is available and where the variable to be interpolated changes according to a marked spatial pattern (Burrough and McDonnell, 1998; Vicente-Serrano et al., 2003; Diodato and Ceccarelli, 2005).
Figure 10. Predicted snow depth by the three global methods in two north–south profiles, and their relation with altitude and solar radiation: (a) western profile; (a) eastern profile
In this study, global methods provided better results than local and geostatistical methods, since the predictions are based on the response of the snowpack to variables that summarize the terrain complexity and the climate conditions of the study area (Balk and Elder, 2000; Anderton et al., 2004; Lopez-Moreno and ´ Nogues-Bravo, 2005). The linear regression model considers t ´ he massivity of a sector (in contrast to isolated reliefs) to explain snow distribution. However, the regression tree and GAM do not include this variable as significant. The three global methods consider altitude, distance to the main divide, exposure to Atlantic or Mediterranean climatic influences and incident solar radiation as predictor variables of snowpack.
In recent years, regression-tree models have been commonly used for interpolating snow depth data. Here, we have shown that this method is useful for identifying the variables that explain snow accumulation and their non-linear responses with the dependent variable. Also, the graphical representations of trees facilitate the understanding of the relationships between controlling factors and response variable. However, we have identified several drawbacks of this method in unsampled areas:
1. Snow cover predictions were highly overfitted to the observations even though the complexity of the tree model was reduced to six nodes. Consequently, we observed a noticeable decrease in the explained variance when the predictions were obtained by means of cross-validation. The tendency of the tree models to overfit their predictions has been reported elsewhere (Chambers and Hastie, 1993; Guisan and Zimmerman, 2000; Willfried et al., 2003). In the upper Marble Fork basin, California (Leydecker and Sickman, 1999), the variance of snow distribution explained decreased from 60% to only 15% when non-selected points were modelled.
2. The snowpack map provided was poor, since it had only six discrete classes, which correspond to the terminal nodes of the tree regression. A greater number of nodes to improve cartography would lead to an increase in overfitting to the observations and meaningless relationships between predictors and response variable.
3. It cannot extrapolate predicted values above or below the observed values. This regression makes it difficult to distinguish between the snow-covered and non-snow-covered areas. Furthermore, the snow depth of the sectors with the greatest tendency to accumulate snow was underestimated. This is especially relevant, since the highest observation is below 2800 m, and the method used should also have the capacity to extrapolate a deeper snowpack in the highest and more favourable sectors.
The drawbacks commented here affect strongly our results since the specific conditions of this study: regional scale with a low number of observations. These constraints could be minimized with a sampling strategy that covers a wider range of locations and topographic characteristics. Nevertheless, measurement surveys are determined, in many cases, by the inaccessibility of some mountain areas.
In spite of the assumption of a linear response of the independent variables to snow depth, the stepwise linear regression model showed a high capacity to predict snow distribution and a low level of overfit to the observations. The use of linear regression for interpolating climatic data (Ninyerola et al., 2000; VicenteSerrano et al., 2003) or snow depth data (Anderton, 2000; Chang and Li, 2000) has also provided good predictive and cartographic results.
The GAM explained 73% of the variance of snow depth (1986–2000). The increase in the accuracy of the prediction with regard to the linear model is due to the capacity of the GAM to include the non-linear relationships between the predictive variables and the snow distribution in the study area. Furthermore, the slight decrease in explained variance confirms the robustness of the model produced when the prediction is obtained by means of cross-validation. Also, GAMs provided the most accurate predictions of snow depth distribution for climatically different single years (1994 and 1995).
According to the field observations and available meteorological data in our study area, the amount of snow predicted by the GAM in the areas most prone to snow accumulation is more realistic than that predicted by the linear model, which clearly underestimates the results. The latter does not provide estimations greater than 350 cm, whereas field records confirm the occurrence of deeper accumulations. Thus, in Balneario de Panticosa, which is located at 1600 m a.s.l. and influenced by an oceanic climate, snowfall in the summit areas usually begins in October and precipitation from December to April exceeds 700 mm. Rijckborst (1967) reported 400 cm of snow accumulated around 3000 m a.s.l. in the headwater of the Esera River during the three months with highest accumulation of snow (i.e. December, February and April). Davy (1978) estimated that the average (1955–1965) snowfall over the Aneto Glacier during winter and early spring was 802 cm.
The opposite occurs with the snow depth predicted by the GAM in the southernmost sectors of the study area, where it introduces a remarkable overestimation. This overestimation may be an artefact, because the slight increase in snow depths observed furthest from the main divide leads to the GAM overestimating depths in some mountainous sectors in the south of the study area, where no observations are available. Therefore, this type of study requires a set of measurements that are well distributed and that extensively cover the topographic and locational situations of the study area.
On the basis of our results, we conclude that:
1. Given the density of available snow depth observations and the topographic complexity of the study area, local and geostatistical methods do not provide proper predictions of snowpack in unsampled areas.
2. Global methods show a noticeable increase in the capacity to predict snow distribution and allow the study of the effect of predictor variables on snow accumulation.
3. Regression trees tend to overfit their predictions to observations. On the contrary, linear regression and the GAM provide more robust predictions.
4. Tree-regression models cannot extrapolate their predictions in areas where the independent variables are out of the range of the observations, unlike linear regressions and the GAM. However, important degrees of uncertainty must be considered in extrapolated values out of the observed range of snow depth measurements. It is highly recommended to have a data set of observations covering the extent of the area of interest and the wider range of topographic conditions.
5. GAMs are useful to predict and to study the spatial distribution of snow and other climate variables. GAMs explain variance better and provide the most robust predictions for the spatial scale considered and the data set used. Finally, these models permit a better understanding of the non-linearity of the relationships between snowpack and predictor variables.
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