Comparison of grid-based algorithms for computing upslope contributing area

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1 Click Here for Full Article WATER RESOURCES RESEARCH, VOL. 42,, doi: /2005wr004648, 2006 Comparison of grid-based algorithms for computing upslope contributing area Robert H. Erskine, 1,2 Timothy R. Green, 1 Jorge A. Ramirez, 3 and Lee H. MacDonald 4 Received 7 October 2005; revised 13 May 2006; accepted 22 June 2006; published 28 September [1] Terrain attributes based on upslope contributing area, A, are used widely in distributed hydrologic models. Several grid-based algorithms are available for estimating A. In this study, five algorithms (D8, 8, MFD, DEMON, and D1) were compared quantitatively on two undulating agricultural fields (63 and 109 ha) in northeastern Colorado. Global positioning system (GPS) data (0.02-m accuracy) were used to generate grid digital elevation models (DEMs) at 5-, 10-, and 30-m cell sizes. Relative differences between A values estimated using single- and multiple-direction algorithms increased with decreasing grid cell size. Relative differences were greatest along ridges and side slopes, and differences decreased where the terrain became more convergent. Multiple-direction algorithms (MFD, DEMON, and D1), allowing for flow divergence, are recommended on these undulating terrains for 5- and 10-m grids where A is most sensitive to the algorithm selection. Citation: Erskine, R. H., T. R. Green, J. A. Ramirez, and L. H. MacDonald (2006), Comparison of grid-based algorithms for computing upslope contributing area, Water Resour. Res., 42,, doi: /2005wr Introduction [2] Upslope contributing area, A, estimated from digital elevation models (DEMs) is fundamental to a wide range of distributed hydrologic models. This attribute, also called upslope area, contributing area, and flow accumulation, represents the area that can potentially produce runoff to the location (i.e., a grid cell) of interest. Terrain attributes derived from A have been used to model channel networks [O Callaghan and Mark, 1984; Jenson and Domingue, 1988; Jenson, 1991; Tarboton et al., 1991; Martz and Garbrecht, 1992; Pilotti et al., 1996], soil moisture distributions and zones of surface saturation [Beven and Kirkby, 1979; Burt and Butcher, 1985; O Loughlin, 1986; Moore et al., 1988; Nyberg, 1996; Western et al., 1999; Florinsky et al., 2002; Green and Erskine, 2004; Güntner et al., 2004], soil erosion and landslides [Montgomery and Dietrich, 1994; Desmet and Govers, 1995; Pack et al., 2001], and pedologic variables [Moore et al., 1993; Odeh et al., 1994; Gessler et al., 2000; Florinsky et al., 2002]. Moore et al. [1991] summarized various terrain attributes of hydrologic significance. [3] Specific catchment area, a, and steady state wetness index, WI, are two of the most commonly used terrain attributes, and each of these is derived from A. Specific catchment area (a = A / L) is defined as the upslope 1 USDA-ARS Agricultural Systems Research Unit, Fort Collins, Colorado, USA. 2 Also at Department of Civil Engineering, Colorado State University, Fort Collins, Colorado, USA. 3 Department of Civil Engineering, Colorado State University, Fort Collins, Colorado, USA. 4 Department of Forest, Rangeland, and Watershed Stewardship, Colorado State University, Fort Collins, Colorado, USA. Copyright 2006 by the American Geophysical Union /06/2005WR004648$09.00 contributing area per unit length of contour, L [Moore et al., 1991]. The wetness index, WI = ln(a / S) [Beven and Kirkby, 1979], is a function of a and the local slope, S, where computed values of S are sensitive to DEM grid cell size [Chang and Tsai, 1991; Jenson, 1991; Panuska et al., 1991; Wolock and Price, 1994; Zhang and Montgomery, 1994; Thieken et al., 1999; Thompson et al., 2001; Erskine, 2005] and DEM accuracy [Sasowski et al., 1992; Bolstad and Stowe, 1994; Giles and Franklin, 1996; Hunter and Goodchild, 1997; Endreny et al., 2000; Holmes et al., 2000; Erskine, 2005]. We expect estimations of A to depend on both the characteristics of the DEM and the algorithm used for estimating A from a DEM. [4] Several algorithms exist for DEM estimation of A, and different algorithms are applied without a quantitative basis for comparison between methods. Advances in remote sensing and global positioning system (GPS) technology mean that fine-scale spatial data are becoming increasingly available. Evaluation of terrain analysis methods on highresolution DEMs (i.e., less than 30-m cell size) is now necessary as differences between methods are likely dependent on the resolution of the DEM. This paper examines five algorithms for estimating A with grid DEMs: D8 [O Callaghan and Mark, 1984], 8[Fairfield and Leymarie, 1991], MFD [Freeman, 1991], DEMON [Costa-Cabral and Burges, 1994], and D1 [Tarboton, 1997]. The aim of this study is to quantify the relative differences between A values estimated by these algorithms for a range of DEM grid cell sizes derived from high-resolution GPS data. Results from this study will provide improved guidance for the growing number of developers and users of high-resolution terrain models. 2. Background [5] The main distinction between different algorithms for estimating A is how the area potentially contributing flow is 1of9

2 partitioned from a center cell to its downslope neighbors. Algorithms generally have been categorized into two types: single-direction and multiple-direction. A single-direction algorithm transfers all flow from the center cell to one downslope neighbor, while multiple-direction algorithms can partition flow to multiple downslope neighbors. [6] The most commonly used single-direction algorithm is D8 [O Callaghan and Mark, 1984], which assigns all flow from the center cell to the neighbor that yields the steepest downslope gradient. The D8 method is computationally efficient, but it produces unrealistic straight and parallel flow paths. [7] The 8 algorithm [Fairfield and Leymarie, 1991] attempts to overcome the problem of straight flow paths, while maintaining the simplicity of a single-direction algorithm. This algorithm randomly assigns flow from the center grid cell to one of its downslope neighbors with the probability proportional to slope. This randomness creates more tortuous flow paths for each realization than D8, but the lack of a deterministic result is a drawback for most applications. [8] The MFD algorithm [Freeman, 1991] is a multipledirection method that partitions flow to all downslope neighbors by a function of slope to an exponent, 1.1. Increasing this exponent to infinity would be equivalent to the D8 method, while averaging many realizations of 8 would be equivalent to MFD with an exponent of unity [Quinn et al., 1991]. Optimal levels of flow partitioning have also been found with the exponent between 4 and 6 based on simulated flow distributions [Holmgren, 1994]. [9] The DEMON algorithm [Costa-Cabral and Burges, 1994] is an extension of the method of Lea [1992], where grid elevation values are taken as pixel corners and flow directions are based on the aspect of a plane surface fit to each pixel. Flow is assumed to be uniform over the pixel area and partitioned to two downslope neighbors in cardinal directions. Two-dimensional flow creates conceptual stream tubes, where the widths of these tubes increase over divergent terrain, decrease over convergent terrain, and remain constant over planar surfaces. Upslope contributing areas are estimated by the summation of influence matrices created for each grid cell. [10] The D1 algorithm [Tarboton, 1997] routes flow in the direction of steepest descent of the eight triangular facets formed in a 3 by 3 grid cell window. Flow is then partitioned to the two neighbors forming the steepest triangle based on the resulting downslope vector. Similar to DEMON, flow is partitioned to no more than two downslope neighbors and the dominant flow direction is determined continuously rather than discretely in 45-degree increments. However, D1 is more computationally efficient than DEMON. [11] Several studies have shown differences between single- and multiple-direction algorithms based on predicted channel networks [McMaster, 2002; Endreny and Wood, 2003], the location of ephemeral gullies [Desmet and Govers, 1996], modeled erosion and sedimentation rates [Schoorl et al., 2000], spatial patterns of saturated areas [Güntner et al., 2004], and the statistical distributions of terrain attributes [Quinn et al., 1991; Wolock and McCabe, 1995; Desmet and Govers, 1996; Tarboton, 1997]. Locations of ephemeral gullies and channel networks are better correlated with terrain attributes derived from the algorithms with limited flow divergence (DEMON and D1). Surface soil moisture (0 0.3 m) and winter wheat grain yields were better correlated (r ) with log(a) and WI derived from multiple-direction algorithms (MFD, DEMON, and D1) than the same attributes derived from single-direction algorithms (D8 and 8) [Erskine, 2005]. In another study, WI was more correlated with soil moisture when WI was derived from DEMON (r ) rather than D8 (r ) [Schmidt and Persson, 2003]. This paper builds upon these studies by quantifying the differences in A between selected algorithms for various DEM grid cell sizes up to 30 m. 3. Site Description [12] This study used data from two agricultural fields in northeastern Colorado: the North field (40.48 N, W) on the Lindstrom Farm and the Scott field (40.59 N, W) on the Drake Farm (Figure 1). The field data were collected when both fields were in a winter wheat/fallow crop rotation. These two fields are 155 km apart but have a similar undulating terrain formed by aeolian deposition. The 63-ha North field has 21 m of relief and a maximum slope of The 109-ha Scott field has 29 m of relief with a maximum slope of Because of tillage, neither site has clearly incised channels. 4. Methods 4.1. Elevation Data Collection and DEM Production [13] Real-time kinematic GPS provided position data to within ±0.02 m in each dimension relative to National Geodetic Survey control points. Data were collected along transects at approximately 5-m spacing within each field by mounting a GPS on an all-terrain vehicle. Areas outside the field boundary that contributed flow to areas within the field were surveyed at approximately 10-m spacing. The irregularly spaced elevation data were interpolated to 5-, 10-, and 30-m grid DEMs by ordinary kriging based on the eight nearest neighbors. Gaussian semivariogram models were fit to separation distances required to achieve eight neighboring elevation values. Cross validation, where each measured elevation point is removed individually and recomputed by the interpolation method, was used to evaluate interpolation errors. The distribution models and parameters were chosen to minimize these errors. Interpolation cross-validation errors were 0.04 and 0.03 m root-mean-square error (RMSE) for the North and Scott fields, respectively Upslope Contributing Area Calculations [14] Upslope contributing areas were estimated for each DEM by each of the five algorithms. No sinks were encountered in the Scott field, and the sinks in the North field were filled to ensure that all drainage area contributed to a field edge. This was accomplished by raising the elevation of a grid cell, or grid cells for sinks larger than one cell, to create a flat area equal in elevation to the minimum elevation surrounding the sink. All algorithms were applied to the same sink-filled DEM. The first four algorithms (D8, 8, MFD, and DEMON) were implemented using the terrain analysis software package TAPES-G [Gallant and Wilson, 1996]. TAPES-G modifies the DEMON algorithm so that the output is grid-centered, like 2of9

3 Figure 1. The location and topography of the two study sites in northeastern Colorado (contour interval = 3.0 m, source data = U.S. Geological Survey). The three-dimensional maps were created from the 5-m DEMs from this study. the output of the other algorithms, rather than edge-centered (i.e., on grid cell corners). This allowed for exact comparisons of A at each grid cell. The D1 algorithm was implemented using an ArcGIS TM extension called TauDEM (see [15] By comparing values of log(a) at each grid cell, relative differences between algorithms are quantified by a coefficient of relative difference, D: D ¼ X n i¼1 ð i Þ 2 Xn i¼1 X n i¼1 ð i i Þ 2 ð i Þ ; ð1þ where i is the log(a) value at location i for the base method of comparison, i is the log(a) value at the same spatial location for the method being compared to the base method, and is the arithmetic mean of i over n grid cells. This coefficient is equivalent to the coefficient of efficiency [Nash and Sutcliffe, 1970], except that the domain is over space rather than time and there are no observed values, only a base method of comparison. If D is equal to 1, then there is a 1:1 relationship between the values of log(a) derived from two different methods. If D is negative, then the variance between methods is greater than the variance of log(a) of the base method. Because the variance in log(a) can differ substantially between algorithms, the D values for each pairwise comparison can differ depending upon which algorithm is treated as the base method. For every paired comparison, each algorithm was used as the base method, yielding two D values per pair. [16] Pairwise plots indicated a pattern used to separate regions of low and high log(a) values. Paired comparisons were made separately for the two regions, log(a) above and below a defined threshold according to the base method. The percentage of the total drainage area classified into each region was computed for each algorithm at each grid cell size. Seven classes were created for D values to aid in the analysis of the results where category 1 rates methods as being very similar (0.9 D 1) and category 7 rates methods as being very dissimilar (D< 1). 5. Results and Discussion 5.1. Geomorphic Patterns [17] Maps of log(a) provide a qualitative, visual comparison between algorithms at each field site (Figures 2 and 3), and these are similar to visual comparisons in earlier studies 3of9

4 Figure 2. Maps of log(a) estimated by five different algorithms on the North field using 5-, 10-, and 30-m DEMs. [e.g., Tarboton, 1997, Figures 8 and 9]. Unrealistic drainage patterns (i.e., straight flow paths) resulting from the grid structure are present in all algorithms to some degree, but such grid artifacts are most pronounced in single-direction algorithms. The D8 method yields straight and parallel flow paths along hillslopes for all grid cell sizes from 5 to 30 m. The 8 method creates more tortuous flow paths, but these flow paths also tend to parallel each other without converging along the hillslopes. The drainage patterns along hillslopes are different for the multiple-direction algorithms (MFD, DEMON, and D1) due to a more gradual increase of log(a). DEMON and D1 produce flow pathways along the valley bottoms that are very similar to the singledirection algorithms, but the MFD algorithm produces wider flow pathways due to excessive flow divergence. Patterns of log(a) differ visually among algorithms at all grid cell sizes, but higher-resolution maps show distinctly different patterns on hillslope areas. [18] These visual comparisons highlight both the similarities and the differences between algorithms, but they do not 4 of 9

5 Figure 3. Maps of log(a) estimated by five different algorithms on the Scott field using 5-, 10-, and 30-m DEMs. Figure 4. A visual comparison of drainage patterns is provided for the northeast quadrant of the Scott field by an aerial photograph, shaded relief, and log(a) estimated by a multiple-flow direction (MFD) and a single-flow direction (D8) algorithm. The shaded relief and log(a) maps are derived from a 5-m DEM. 5of9

6 Figure 5. Cumulative frequency distributions for log(a) on the Scott and North fields for (a and b) 5-m, (c and d) 10-m, and (e and f) 30-m DEMs. provide any basis for ground-truthing. While there are no incised channels at these sites, patterns of drainage are apparent from the existing topography. Field observations are partially confirmed by comparing log(a) maps, a shaded relief map that was derived from the DEM, and an aerial photograph. For example, a shaded relief map of the 5-m DEM on the northeast quadrant of the Scott field reproduces the main drainage pathways present in an aerial photograph (Figure 4). The drainage patterns shown by the MFD algorithm are more consistent with field observations than the drainage patterns produced by the D8 algorithm (Figure 4). The drainage patterns generated by the multipledirection algorithms are more consistent with field observations at each study site than the drainage patterns developed with any of the single-direction algorithms Statistical Distributions of Log(A) [19] The cumulative frequency distributions of log(a) distinguish between the two single-direction algorithms (D8 6of9 and 8) and two of the multiple-direction algorithms (MFD and DEMON) (Figure 5). The distribution for D1 falls between the single-direction and the other multiple-direction algorithms, which is most apparent for the 5- and 10-m grid cell sizes (Figures 5a 5d). Differences between algorithms are less distinct as grid cell size increases up to the maximum value of 30 m (Figures 5e and 5f) due to a truncation of the lower tail of the distribution. Truncation occurs when increasing the grid cell size because the minimum value for A, the area of one grid cell, is increased. Thus highresolution DEMs (i.e., 5-m cells) may be needed to fully distinguish the differences among lumped statistical distributions of log(a) resulting from these five algorithms. [20] Single-direction methods have a greater proportion of grid cells with low log(a) values, decreasing the mean and increasing the variance of log(a) relative to multipledirection methods. This result can be attributed to the lack of divergence, which orphans cells away from and adjacent

7 Figure 6. Matrix plot of grid cell comparisons of log(a) as estimated by various algorithms on the Scott field 5-m DEM. The 1:1 line is included on all plots. to the concentrated flow paths. The interquartile ranges for all algorithms and grid resolutions reveal systematic decreases in the spread of the distribution of log(a) for both field sites going from the single- to multiple-flow directions algorithms. The decreasing trend in the spread parallels the increasing trend in the mean, where D1 falls between the other algorithms. Thus the spatial statistical distributions of log(a) vary among computational algorithms. DEMON and MFD are the most similar, yielding the highest mean and lowest spread about the mean Grid Cell to Grid Cell Comparisons [21] The pairwise plots for the Scott field 5-m DEM (Figure 6) indicate a low correlation between algorithms when values of log(a) are less than 4.5, or A 3 ha. Upslope contributing area values in this range correspond to the ridges and side slopes for these landscapes. The pairwise plots tend to become more linear and have a slope near one when the values of log(a) are greater than 4.5. This indicates a stronger agreement between algorithms in the convergent valleys. This same pattern was seen on the North field (not shown here) and at all grid cell sizes [Erskine, 2005]. [22] Table 1 presents D values as seven classes for upper (log(a) < 4.5) and lower (log(a) > 4.5) regions. In general, D values are higher (i.e., class numbers are lower) when comparing two multiple-direction algorithms or two singledirection algorithms. Also, D generally increases as grid cell size increases regardless of the algorithms being compared. In almost all cases, lower regions yield substantially higher values of D than upper regions. For example, D commonly exceeds 0.9 (class 1) in the lower convergent areas while it is often negative (classes 5, 6, 7) in upper regions. Exceptions to this pattern are found with MFD as the base method, when grid cells with high A values experience divergence. This divergence creates very large differences between MFD and other algorithms at these lower grid cells. Sink filling can also accentuate differences between algorithms in the lower regions [Erskine, 2005]. [23] As grid cell size increases, the percentage of the total drainage area classified in the lower region increases (Table 1). At 5-m grid cell sizes, less than 3% of the area is classified in the lower regions for each algorithm at each field. This percentage increases for each algorithm at each field when increasing the grid cell size to 30 m; up to 9.8% for MFD on the North field. Algorithms are most similar in the estimation of A along the main drainage pathways making up the lower regions. Therefore increasing the area of the lower region will lead to increased similarity in maps of log(a) between different algorithms. Increasing grid cell size also reduces the number of possible drainage pathways before reaching a convergent grid cell, and this reduces the opportunity for 7of9

8 Table 1. Classes of D (Coefficient of Relative Difference, Equation (1)) for Log(A) Estimated With Different Algorithms in Both Upper and Lower Regions a Base Method D8 8 MFD DEMON D1 Scott Field, 5-m D8... 5,1 6,3 6,1 5,2 8 5,2... 7,3 7,2 6,2 MFD 5,1 6,1... 2,1 2,1 DEMON 4,1 5,1 2,2... 2,1 D1 4,1 5,1 3,2 2,1... % Lower Scott Field, 10-m D8... 4,1 5,2 5,1 4,1 8 5,1... 7,2 7,1 5,1 MFD 4,1 5,1... 2,1 2,1 DEMON 4,1 5,1 2,2... 2,1 D1 3,1 4,1 2,2 2,1... % Lower Scott Field, 30-m D8... 4,2 4,2 5,2 2,1 8 4,2... 5,2 6,2 4,1 MFD 4,1 5,1... 2,1 3,1 DEMON 4,1 5,1 2,2... 3,1 D1 2,1 3,1 2,2 3,1... % Lower North Field, 5-m D8... 4,1 6,3 5,1 4,2 8 4,1... 7,3 7,2 5,2 MFD 6,1 6,1... 4,1 4,1 DEMON 4,1 5,1 3,3... 3,2 D1 4,2 4,2 4,3 4,2... % Lower North Field, 10-m D8... 4,1 5,3 5,1 4,2 8 4,1... 5,3 6,1 5,2 MFD 6,1 6,1... 4,1 5,1 DEMON 4,1 5,1 2,2... 3,2 D1 3,2 4,2 3,3 3,2... % Lower North Field, 30-m D8... 3,1 3,3 3,1 3,2 8 3,1... 4,3 4,1 3,2 MFD 5,1 6,1... 4,1 4,1 DEMON 3,1 4,1 2,2... 3,1 D1 3,2 3,2 2,2 3,2... % Lower a Upper regions are defined by log(a) <4.5orA < 3 ha using the base method. The percentage of the total area classified as lower region, log(a) > 4.5, is reported for the base method. Classification of D (Upper, Lower) is 0.9 D 1.0! 1; 0.6 D<0.9! 2; 0.3 D<0.6! 3; 0.0 D<0.3! 4; 0.5 D<0.0! 5; 1.0 D< 0.5! 6; 1< D< 1.0! 7. differences to occur between algorithms in the divergent upper regions. 6. Summary and Conclusions [24] We compared five algorithms (D8, 8, MFD, DEMON, and D1) for estimating the upslope contributing area, A, on two undulating agricultural fields. Our goal was to quantify relative differences between these algorithms for DEM grid cell sizes ranging from 5 to 30 m. The direct results are limited to geomorphic patterns in moderately undulating terrain, but the analytical methods and the results are believed to be more generally applicable. The main results are as follows: [25] 1. The drainage paths and geomorphic patterns developed with multiple flow direction algorithms (MFD, DEMON, and D1) are consistent with the visual patterns on a shaded relief map, derived from a 5-m DEM, and aerial photography independent of the DEM. [26] 2. The visual differences between algorithms based on maps of log(a) are apparent at all DEM resolutions, but are most pronounced at the highest DEM resolution (5-m grid cells). [27] 3. The cumulative frequency distributions of log(a) were less sensitive to the algorithm as grid cell size increased. Thus relative differences between single- and multiple-direction algorithms were amplified at higher grid resolutions. The cumulative frequency distributions of log(a) derived with MFD and DEMON were very similar, but they differed from the distributions derived with D8 and 8. The frequency distribution derived with D1 fell between the distributions derived with MFD and D8. [28] 4. The greatest relative differences between singleand multiple-direction algorithms occur in the divergent areas (i.e., ridges and side slopes). Increasing the grid cell size decreases the relative differences between algorithms, as the percentage of grid cells in convergent lower regions increases. [29] The observed geomorphic patterns, distributions of log(a), and quantitative comparisons indicate that multipledirection algorithms should be used for applications of upslope contributing area derived from higher-resolution DEMs (5- and 10-m grids). This recommendation is based on the observation that estimates of A were most sensitive to the selected algorithm as grid cell size decreased. The methods of analyses discussed here could be applied to terrain of different topology, geological origin, level of dissection or incision, land use, and watershed scale to assess the sensitivities of estimated terrain attributes in such landscapes. [30] Acknowledgments. 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