ErrorAnalysisonGrid-Based Slope andAspectAlgorithms

Slope and aspect are the most frequently used surface geomorphic parameters in terrain analysis. While derived from grid DEM. the parameters often display noticeable errors due to errors {aJ in data. {bJ inherent in data structure. and {c} created by algorithms. It has been observed that some controversial results were reported in evaluating the results by various slope and aspect algorithms. largely because of the variety in assessment methodology and the difficulties in separating errors in data and those generated by the algorithms. This paper reports the study that assesses and compares the results from numerous grid-based slope and aspect algorithms using an analytical approach. Tests were made based on artificial polynomial surfaces which can be defined by mathematical formulae. with control/able "added" data errors. By this approach. different algorithms were quantitatively tested and their error components were analyzed. Thus. their suitability and tolerance related to DEM data characteristics can be described. Introduction Slope and aspect have been regarded as two of the most important geomorphic parameters. as they not only efficiently describe the relief and structure of the land surface. but are also widely applied as vital parameters in hydrological models (Band. 1986; Moore et al. 1988; Quinn et al. 1991; 1995; Jenson. 1994). landslide monitoring and analysis (Ouan and Grant. 2000), mass movement and soil erosion studies (Dietrich et al. 1993; Desmet and Govers. 1996a; Mitasova et al. 1996; Biesemans et al. 2000) and landuse planning (Desmet and Govers. 1996b; Stephen and Irvin. 2000). In most applications today. slope and aspect are typically derived from OEM that is based on raster data structure. To date. there have been numerous mathematical models and algorithms that calculate slope and aspect from elevation data (e.g. Sharpnack and Akin. 1969; Fleming and Hoffer. 1979; Horn. 1981; Unwin. 1981; O'Callaghan and Mark. 1984; Zevenbergen and Thorn. 1987; Wood, 1996). Although the mathematical definition of slope and aspect is quite clear. its implementation based on grid-based OEM may vary. since some assumptions must be made on how the continuous surface is approximated by discrete sample points (i.e., grid cells). The variation in such implementation would only present minor problems in applications such as surface visualization and classification. but its impact on terrain analysis that is based on quantitativp. models could be very significant (Zhou et al. 1998; Tang. 2000; Zhou and Liu, 2002). It was pointed out that selection of algorithms could be a critical Q. Zhou is with the Dp.partment of Geography. Hong Kong Baptist University. Kowloon Tong, Kowloon. Hong Kong. China (qiming@hkbll.edll.hk). X. Liu is with the School of Geographical Science. Nanjing Normal University. Nanjing 210097. China. I . PHOTOGRAMMETRIC ENGINEERING& REMOTESENSING factor that might create a grp.at impact on the analytical results (Moore. 1996; Burrough and McDonnell, 1998). Studies have been conducted to analyze the errors created by the slope and aspect algorithms with a variety of approaches and methodologies. One approach emphasizes errors in OEM itself while paying little attention on models. For example. Skidmore (1989) and Florinsky (1998) analyzed slope and aspect errors using the actual survey data. Another approach focused on mathematical models while ignoring OEM errors (e.g. Hodgson. 1995; Jones. 1998). The conclusions from these studies have been quite different. sometimes controversial. For example. Skidmore (1989) concluded that third-order finite difference method (Horn 1981) derived better results than the second-order finite difference method. His finding was also confirmed by Florinsky (1998). On the other hand. Hodgson (1995) and Jones (1998) stated that the latter performed better than the former in computing slope and aspect values. To make a fair comparison between different slope and aspect algorithms. three pre-conditions must be satisfied: 1. The source of error should be identified; 2. The kinds of errors must be independent. identifiable and controllable in the test so that their impact can be quantified: and 3. r\ reference (or true value) must be established to make the results comparable. Zhou and Liu (2002) reported a method that is based on mathematical surfaces to evaluate the errors generated by flow routing algorithms in digital terrain analysis. Using this method. errors created by an algorithm can be isolated from OEM errors so that the quantitative measures can be made to compare a derived. specific catchment area with the true value defined by the polynomial surfaces. Using an analytical approach. Florinsky (1998) developed test for the precision of four methods for computing partial derivatives of elevations. and produced formulae for Root Mean Square Error (RMSE)of four local topographic variables including slope and aspect as focused by this paper. Florinsky also used a real-world OEM to map the error distribution for visual analysis. but the contribution from the OEM data error largely remained unknown. This paper reports the study that attempts to extend the above methods into the test of slope and aspect calculation. In this study. we extended Florinsky's method to general terms and designed tests on artificial polynomial surfaces with controllable data errors. Two representative surfaces. namely. the inverse ellipsoid and Gauss synthetic surface. were used to test six commonly-employed slope and aspect algorithms Photogrammetric Engineering & Remote Sensing Vol. 70. No.8, August 2004. pp. 957-962. 0099-1112/04/7008-0957/$3.00/0 @ 2004 American Society for Photogrammetry and Remote Sensing