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...DEMs include terrain mapping and visualization, location analysis, estimation critical slopes and landslides, forecasting flooding and its impact, optimal routing, visibility analysis, and many more. Digital elevation data are one of the most, if not the most, critical layer in what has been called the National Framework data (FGDC 2006). Within the digital representations of terrain, which Longley et al. (2005) claim can be stored in six different ways, by far the majority of data worldwide are stored in the simple grid or raster GIS data structure. The dominance of this type of data was recently further reinforced by the global release of the Shuttle Radar Topographic Mapping Mission data (Rodriguez et al. 2005).
The value of DEMs for computing terrain surface structure and hydrographic features dates back some time (Douglas 1986; Wood et al. 1988; Band 1986; O'Callaghan and Mark 1984). Once algorithms were generated to compute the basic terrain parameters (Sharpnack and Akin 1969; Evans 1980; Moore et al. 1993; Quinn et al. 1991), attention focused on hydrographic feature delineation, extraction, and use in modeling (CostaCabral and Burgess 1994; Gallant and Wilson 1996; Jenson and Domingue 1988; Lea 1992; Ritter 1987; Tarboton et al. 1993; Zevenbergen and Throne 1987). More recent research has been able to comprehensively review developments in the field (Evans 1998; Wood 1996; Jenson 1991) and produce educational material based on software tools (Wilson and Gallant 2000).
The role of error in the computation of surface parameters and their derivatives has been a subject of DEM research ever since the publication of some of the earliest literature in this area (Carter 1992; Chu and Tsai 1995; Hunter and Goodchild 1997; Lee 1996; Lee et al. 1992; Zhou and Liu, 2002). There has also been considerable attention paid to the quantitative differences among results computed by different algorithms (Jones 1988; Ryder and Voyadgis 1996; Wolock and McCabe 1995). This attention to sensitivity tests and error, particularly as a function of DEM resolution (Wolock and Price 1994; Wood et al. 1988; Endreny et al. 2000) has been important as the research algorithms found their way into various specialist hydrological packages and GISs.
Algorithms for Slope and Aspect
An aspect of the work comparing algorithms and investigating the impact of DEM error on hydrographic structure has been the critical impact of slope and aspect, and how their means of computation propagates errors forward into more sophisticated derivatives. For example, Quinn et al. (1995) investigated their impact in the TOPMODEL topographic index [ln(a/tan [beta]), where a is the upslope contributing area and tan [beta] is local slope]. Computing this index in various ways led to the conclusion that, "The spatial pattern and statistical distribution of the index is shown to be substantially different for different calculation procedures and differing pixel resolutions" (Quinn et al. 1995, p. 161). This led to the observation, echoed several times in research scholarship, that the algorithms for computing DEM slope and aspect account for many of the differences in their terrain--parameter derivatives, including downslope flow measures.
Some of the more frequently employed slope and aspect algorithms are defined by the equations included in Figure 1. Topographic slope is the first derivative of elevation, given as the rise in elevation over the run in distance, and it is obviously impacted by direction and range. Assuming angles clockwise from north in degrees, and Zx representing east-west, and Zy representing north-south gradients, then:
Slope = arctan ([square root of [Z.sup.2] x + [Z.sup.2] y) (1)
Aspect = 180[degrees] -arctan(Zy/Zx) + 90[degrees](Zx/|Zx|) (2)
[FIGURE 1 OMITTED]
Wilson and Gallant (2000) summarized the available slope and aspect algorithms and listed them using the TAPES-G labels (Figure 1). Algorithm 2FD is the Second-order Finite Difference method (Fleming and Hoffer 1979; Zevenbergen and Thorne 1987; Ritter 1987), which deals only with the principal compass directions and normalizes the slope vector between them. The 3FD algorithm is similar, named the Third-order Finite Difference (Sharpnack and Akin 1969; Horn 1981; Wood 1996). This algorithm averages the three vertical and horizontal cross vectors, without the center grid cell, and it normalizes the vector for X and Y. The 3FD creates smoother slopes than the 2FD, because it averages over more samples.
A modification of 3FD is 3FDWRD or Third-order Finite Difference Weighted by Reciprocal of Distance (Unwin 1981). Although almost identical to 3FD, it double-counts the vertical and horizontal profiles running through the center cell, simulating distance-decay. A further extension uses inverse-squared distance weighting, computing the third-order Finite Difference Weighted by Reciprocal of Squared Distance (3FDWRSD) (Horn 1981). This method weights the center by twice root 2, rather than the square.
The much simpler FFD (Frame Finite Difference) method (Chu and Tsai 1995) uses the cell corners rather than the center cell profiles, but otherwise it is similar to 2FD. Lastly, the SIMPLE Difference (SIMPLE D or SIMPLD) method (Jones 1988) uses a 4-cell neighborhood, looking only west and south and using only a single cell distance. SIMPLD is most sensitive to local variance but suffers a direction bias because the north and east cells slopes are not represented. In all cases, the slope and aspect solutions are yielded from Equations (1) and (2).
In prior research, we showed that these various algorithms have consequences for the manner in which slope and aspect are computed, and that these consequences propagate as errors into derivative terrain surface values, such as slope curvature and other fundamental terrain metrics, including downslope flow (Lee and Clarke 2005). In this current work, we focus specifically on downslope flow and flow accumulation--the two values used to generate stream positions on a DEM.
Our first goal is to examine the consequences of algorithmic or "method-produced" error in derived DEM digital maps, especially those showing the locations and directions of surface water flow. The values derived from DEMs at three different spatial resolutions were compared with the vector representations of streams on topographic maps for a test area in Santa Barbara County, California. The second goal is to investigate, and if possible determine, the impact of the spatial resolution of the DEM data on the derived downslope flow values and the resulting stream network.
Downslope Flow: Direction and Accumulation
The choice of the algorithm is central...
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