Basin level statistical properties of topographic index for North
America
Praveen Kumar
a,*, Kristine L. Verdin
b, Susan K. Greenlee
b aEnvironmental Hydrology and Hydraulic Engineering, Department of Civil and Environmental Engineering, University of Illinois, Urbana, IL 61801, USA
b
Earth Resources Observation Systems (EROS) Data Center, Sioux Falls, SD 57198, USA
Received 17 July 1999; accepted 14 October 1999
Abstract
For land±atmosphere interaction studies severalTopmodel based land-surface schemes have been proposed. For the imple-mentation of such models over the continental (and global) scales, statistical properties of the topographic indices are derived using GTOPO30 (30-arc-second; 1 km resolution) DEM data for North America. River basins and drainage network extracted using this dataset are overlaid on computed topographic indices for the continent and statistics are extracted for each basin. A total of 5020 basins are used to cover the entire continent with an average basin size of 3640 km2. Typically, the ®rst three statistical moments of
the distribution of the topographic indices for each basin are required for modeling. Departures of these statistical moments to those obtained using high resolution data have important implications for the prediction of soil-moisture states in the hydrologic models and consequently on the dynamics of the land±atmosphere interaction. It is found that a simple relationship between the statistics obtained at the 1 km and 90 m resolutions can be developed. The mean, standard deviation, skewness, L-scale and L-skewness all show approximate linear relationships between the two resolutions making it possible to use the moment estimates from the GTOPO30 data for hydrologic studies by applying a simple linear downscaling scheme. This signi®cantly increases the utility value of the GTOPO30 datasets for hydrologic modeling studies. Ó 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction
The treatment of land-surface heterogeneity in land± atmosphere coupled model studies has emerged as a pressing research issue during the last several years. The concept of topographic index (also called wetness in-dex), originally proposed in theTopmodel [1], for char-acterizing the distribution of moisture states in a basin has gained considerable attention and successful models based on this concept have been developed for land± atmosphere interaction studies [3,4,11,15]. For this purpose basins, and not typically used rectangular grids conforming to the atmospheric models, are more ap-propriate units for modeling the terrestrial hydrologic processes as they are better suited to capture the heter-ogeneity arising from topographic controls over surface and sub-surface ¯ow. Regions of ¯ow convergence and low vertical soil moisture de®cit are identi®ed as large values for the topographic index, and low values cor-respond to uphill areas of ¯ow divergence and/or high
vertical soil-moisture de®cit. In Topmodela hydrologic similarity assumption is invoked. This states that all locations in a basin with the same topographic index will have the same hydrologic response. Models for land± atmosphere studies that utilize the probability distribu-tion of the topographic index over the basin to capture this behavior for land±atmosphere interaction studies have been tested and validated for individual basins or over limited areas [4,15], but have not been implemented in GCMs, among other reasons, for lack of basin level topographic characteristics with continental scale coverage.
Recently the United States Geological Survey (USGS) has developed a digital elevation model (DEM) at 30-arc-second resolution with global coverage [5]. This enables the ecient estimation of derivative infor-mation such as slope, aspect, hydrologic ¯ow paths, ¯ow accumulation and basin boundaries [17]. The basins are represented hierarchically at ®ve levels of subdivisions with the average basin size ranging from 2;209;207 km2
at Level 1 to 3640 km2at Level 5. These developments
pave the way for the implementation of basin level hydrologic models with continental and global coverage.
*
Corresponding author.
E-mail address:kumar1@uiuc.edu (P. Kumar).
The objective of this paper is to describe the char-acteristics of the topographic indices at the basin level for North America, extracted from the 30-arc-second DEM. Modeling studies using the data are not dis-cussed. Usually, the ®rst three statistical moments of the distribution of the topographic indices for each basin is required for modeling. These moments enable the parameter estimation of the probability distribu-tion. Typically a 3-parameter gamma (or Pearson-III) distribution is used [14]. The properties of the ®tted distribution are sensitive to the DEM resolution and this impacts the performance of the hydrologic model [19]. In order to address this problem in the use of 30-arc-second DEM, we adopt the following procedure: (i) estimate topographic index using the single ¯ow algorithm from the 30-arc-second DEM data; (ii) overlay basin boundaries and estimate statistical mo-ments; (iii) analyze topographic indices obtained from high resolution 3-arc-second data for selected regions representing a range of topographic features to quan-tify the impact of resolution on the estimates of the statistical moments; (iv) establish a functional rela-tionship (downscaling function) to convert statistical moments from the 1 km data to equivalent estimates at the higher resolution. Once the downscaling func-tion is identi®ed, we can convert the estimates from the 30-arc-second data to estimates that would be representative of topographic index if it was estimated from high resolution data. This method is useful because using high resolution data directly for the es-timation of the distributional properties of the topo-graphic index is very dicult due to the enormous volume of the data, especially if global applications are considered.
In a similar study, Wolock and McCabe [20] studied the dierences in topographic attributes obtained at the two dierent resolutions to identify if the dierences were due to terrain-discretization eects or smoothing. Our work is complimentary in that it is geared toward studying the properties at the basin scale for GCM ap-plications. Analysis regions for the higher resolution data are chosen to coincide with that of Wolock and McCabe [20] so that the two results can be used together for further studies.
2. Data description
US Geological Survey's EROS (Earth Resources Observation Systems) Data Center in Sioux Falls, SD has developed a global digital elevation model called GTOPO30. The resolution of this dataset is 30-arc-sec-ond (81
310ÿ
3 degrees). The vertical resolution is 1 m,
and the elevation values for the globe range fromÿ407 to 8752 m above mean sea level. Additional details about the dataset are available in [5].
A subset of this dataset corresponding to the North American Continent is used for this study. The dataset was ®rst projected from geographic coordinates to Lambert Azimuthal Equal Area coordinate system at a resolution of 1 km. This renders each cell, regardless of the latitude, to represent the same ground dimensions (length and area) as every other cell. Consequently, de-rivative estimates such as drainage areas, slope, etc. are easier, consistent, and reliable. The extraction of these hydrographic features from the 30-arc-second dataset are based on the drainage analysis algorithm of Jenson and Domingue [9]. This algorithm ®rst identi®es and ®lls spurious sinks (or pits). Eort is made to preserve nat-ural sinks such as lakes in the landscape. Then for each cell, direction of steepest descent from among its eight neighbors is computed. This information is then used to compute the ¯ow accumulation for each cell. A 1000 km2 threshold is then applied to the ¯ow accumulation
values to obtain a drainage network in raster format and then vectorized [17]. The drainage network is then used to identify basins and sub-basins. In order to represent the basins hierarchically, a system developed by Pfaf-stetter [12] is used which utilizes an ecient coding scheme [17] (the reader is encouraged to refer to the web publication [16]). Basins at ®ve levels of subdivision are developed. For this study, Level 5 description was chosen with the mean basin size of 3640 km2. Fig. 1
il-lustrates a typical layout of basin patterns at Level 5 for a region in the north-eastern United States. It is found that at this level the basins provide a sucient subgrid resolution for GCM applications [3,10,11].
In order to assess the impact of the resolution on the estimates of topographic indices, a higher resolution dataset (3-arc-second) available from USGS is also used for selected regions (see Fig. 2) These datasets are available in blocks of 1°1°latitude±longitude cover-age for the entire United States. The ground dimensions are 92.6 m along the latitude and 92:6cos ph=180m along the longitude where h is the latitude in degrees. Consequently the ground resolution along the longitude ranges from roughly 80 m in the southern US to 60 m near the Canadian border. This dataset is also referred to as the 90 m resolution data in this paper. Thirty-®ve 1°1°latitude±longitude grids, as shown in the Fig. 2, were selected for comparative analysis at the high and low resolutions.
3. Analysis
the selected 1°1° latitude±longitude regions (Fig. 2) using the 3-arc-second dataset. At these resolutions the multiple ¯ow algorithm [13] does not provide a better estimate and consequently were not used. Appropriate measures are taken to account for the distortions in the
estimates of contributing area and slopes due to the curvilinear latitude±longitude coordinates of the 3-arc-second data.
Spatial statistical moments of topographic indices over each of the 5020 basins were computed. Figs. 3 Fig. 1. Level 5 subdivision of a region of the eastern United States overlaid on the GTOPO30 DEM. Black and white lines represent the streams and the basin boundaries, respectively.
and 4 show the spatial distribution of these statistical moments. We observe that the mean is generally larger in ¯at areas and the standard deviation is generally larger in mountainous regions. This is intuitive since in ¯at regions we tend to have larger contributing areas and smaller slopes giving rise to a larger topographic index. The variability of the topographic relief and local slopes in mountainous regions gives rise to more spread in the distribution function and consequently to larger values of the standard deviation. Fig. 5 shows the spatial dis-tribution of the skew. As expected the skew is generally larger in areas where there are large ¯ow accumulations. For a very small fraction of the continental area (2.2%) the skew was found to be negative. The basins corre-sponding to these areas were typically very small. Con-sequently very few values were used in computing the skew and therefore the estimates have large estimation errors. We believe that the presence of the negative value is a consequence of the limitation of statistical estima-tion due to small sample size rather than due to the estimation of topographic index from a low resolution dataset.
For each of the thirty-®ve 1°1°latitude±longitude regions selected for high resolution analysis, the spatial statistical moments were computed from estimates obtained from both the 3-arc-second and 30-arc-second DEM data. Fig. 6 shows the plots of the three mo-ments and skewness obtained at the two resolutions. The regression lines (solid lines) are obtained using the least trimmed square robust regression which is based on a genetic algorithm [2]. This algorithm provides
better linear ®ts by excluding outliers as compared to the usual least-squares regression (plotted as dotted lines). On the downside, however, no coecient of correlation can be estimated. As seen from the ®gures, there are nice linear relationships for the ®rst three moments. However, there are noticeably large devia-tions from linearity in the skewness plots (no system-atic dependence of these deviations on topographic features could be established). This can have signi®cant in¯uence on the estimate of the probability distribution and subsequently on the predicted hydrologic response. For example, the parameter a of the gamma distribu-tion (see Eq.(A.1)), often used to model the probability distribution of the topographic index [14], is completely determined by the skewness (see Eq. (A.10)). Thus er-ror in this parameter quickly propagates to the model response.
The estimates of second and higher order statistical moments suer from the limitation that a few very large values can distort the estimate, particularly if the sample size is small. The parameter estimates for a probability distribution obtained from these moments can be severely aected. In order to overcome this limitation, L-moments based on probability weighted moments [6±8], can be used. A probability weighted moment of order r, for a random variable X with cu-mulative distribution function F x, is given as ar EfX1ÿF xrg. The L-moments of the ®rst four or-ders are given as
k1a0; k2a0ÿ2a1;
k3a0ÿ6a16a2; k4a0ÿ12a130a2ÿ20a3:
Notice that k1 is the usual mean. The moment k2 is
called the L-scale as it measures the spread of the dis-tribution. The L-moment ratio of orderris de®ned as
srkr=k2; r3;4;. . .:
The quantities s3 and s4 are called L-skewness and
L-kurtosis, respectively. Note that all moments are estimated using linear combination of X thereby Fig. 5. Spatial distribution of basin skew of the topographic index over North America. Negetive skew values for approximately 2.2% of the total continental area are displayed as 0 (dark blue).
overcoming the problem associated with raising a few large values to higher powers. Parameters for several distributions including gamma, normal, log-normal, etc. can be estimated using the L-moments [8].
Fig. 7 (top) shows the plots of L-moments at the two resolutions. Since the ®rst moment is the same as the usual ®rst moment, i.e., the mean, only the second (k2:
L-scale) and the third (s3: L-skewness) moments are
tested for linearity at the two resolutions. We see that L-skewness shows signi®cantly improved linear relation-ship as compared to the usual skewness estimates. No noticeable improvement in the estimate of L-scale as compared to skewness is gained. Linear approximation for the L-kurtosis between the two resolutions is very weak and hence is not shown. These results suggest that: (i) simple linear relationship between spatial moments of topographic indices obtained at the two dierent reso-lutions (1 km and 90 m) exists; and (ii) L-moments provide a better regression equation to downscale the moment estimates from the low resolution of 1 km to higher resolution of 90 m.
A sense for the appropriate distribution to describe the topographic indices can be obtained from the moment ratio diagram which is a plot between L-skewness and L-kurtosis [8]. Fig. 7 (bottom) shows this diagram obtained from the estimates at the two resolu-tions. Comparing these with the theoretical values of
gamma distribution (solid line) we see that in the ma-jority of the cases it provides a better approximation at the higher resolution. The theoretical curves for other commonly used 3-parameter distributions such as log-normal lie above the gamma distribution curve and are not plotted. In Appendix A we summarize the parameter estimation equations for the gamma distribution using the L-moments, for completeness. Additional details can be found in [8].
4. Discussion and conclusions
The topographic index obtained using the GTOPO30 DEM data captures the general spatial distribution over the North American continent. The existence of simple linear downscaling functions to obtain the ®rst three statistical moments (usual and L-moments) from the 30-arc-second DEM data with global coverage to a reso-lution that is a factor of 10 higher signi®cantly increases its utility for hydrologic applications. The analysis also suggests that the use of L-moments for downscaling will provide better estimates than the usual moments. For hydrologic applications, this is particularly signi®cant since the models utilizing the data are quite sensitive to the tail of the probability distribution. The strong linear relationship of L-skewness suggests that information Fig. 6. Downscaling functions for obtaining 90 m equivalent of usual statistical moments of topographic index from estimates using the 1 km DEM data for the North American Continent. Solid lines represent linear ®t using the least trimmed square robust regression and the dotted lines represent the usual least squares regression ®t. The equations correspond to the solid lines.
about the tail of the distribution is simply scaled but not lost. The L-moment ratio diagram suggests that gamma distribution provides a reasonable approximation to the probability distribution function at the higher resolution of 90 m. Its parameters can be estimated from the ®rst three L-moments.
Several issues can be raised about the use of these results for Topmodel application where typically DEM data at 30 m or higher resolution are recommended. However, the empirically observed linear relationships suggest that in the absence of high resolution data, a course resolution data as used in this paper, along with the linear downscaling scheme, provide a viable means for the application of Topmodel concepts over signi®-cantly larger areas. Obviously, care has to be taken that other model assumptions should be valid in the region of interest. In addition, sensitivity studies should be performed to assess the magnitude of the model sponse error associated with the approximations re-sulting from the linear downscaling scheme.
The theoretical reason of the empirically observed linear relationship is not clear but we speculate that it is a result of similarity in topographic features at the dif-ferent resolutions. Similar conclusions were also reached by Wolock and McCabe [20] where they also present a more detailed account of the eects of discretization and smoothing on the estimates of the slopes and contrib-uting area. Whether fractal or multifractal
characteris-tics of topographic features can give rise to such a behavior is an intriguing hypothesis and will be pursued in a separate study.
Acknowledgements
This research was partially funded by NASA grants NAG5-3661, NAGW-5247, and NAG5-7170, and NSF grant EAR 97-06121. Thanks are also due to Dave Wolock for providing the code to estimate the topo-graphic index from the GTOPO30 dataset using ARC/ INFO and to Margie Caisley for performing a lot of the data analysis work.
Appendix A. Estimation of gamma distribution using L-moments
Parameter estimation for the 3-parameter gamma distribution using the L-moments is described here for completeness (see [8] for details). The probability dis-tribution for the gamma disdis-tributions is given as
f x xÿn
aÿ1
eÿ xÿn=b
baC a ; A:1
wherea,n, andbare the parameters of the distribution. The L-moments are given as
k1nab; A:2
HereIx p;qis the incomplete beta function ratio
Ix p;q
ments k1 and k2, respectively. Then the estimates of
parametersnandb are given by the equations
^
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