## A Comparison of Algorithms used to Compute Hill Slopes and Aspects as a Property of the DEM.

Kevin H. Jones
Macaulay Land Use Research Institute, Craigiebuckler, Aberdeen , AB15 8QH, United Kingdom

Abstract
A comparison has been made of algorithms used to compute slope and aspect from the lattice/grid type Digital Elevation Model (DEM). The slope and aspect algorithms under test were applied to synthetic DEMs derived from the 49-term trigonometric surface of Morrison (1971; 1974). The values of slope and aspect produced by the algorithms being tested were compared with the true slope and aspect values obtained by analytic partial differentiation of Morrison's trigonometric surface. Morrison's 49-term equation for the test surface represents a least-squares fit to 121 data points read from a square lattice on Hsu and Robinson's (1970) Surface III, which is a real topographic surface.

Following the example of Jones (1996), in his study of slope algorithms using Morrison's surface, ten cell sizes of between 1/100 and 1/10 (1/100, 1/90, 1/80 etc. to 1/10) of the x and y dimensions of the area of definition of the surface were used to represent different sampling intensities in the test DEMs.

Morrison's surface was rotated through steps of twenty degrees in azimuth to produce eighteen ensembles of test DEMs, of the required sampling intensities. This was done to avoid any bias in the slope or aspect results due to the predominant direction of ridges and valleys on Morrison's surface.

To demonstrate the generality of our slope and aspect algorithm comparison results, further tests were carried out on various smaller areas of Morrison's surface, each representing 1/4 of the area of definition of the whole surface. The results of these tests gave broadly the same results as those using the entire area of definition of Morrison's surface.

Finally a relative comparison of slope and aspect algorithms was made using a DEM of a real topographic surface. A comparison was made between the "best" performing algorithm on Morrison's surface and the other slope and aspect algorithms tested. Using a real DEM in this way enabled us to evaluate the validity of our ranking of slope and aspect algorithms using Morrison's surface.

During this study a number of different parameters were calculated to determine how the various slope and aspect algorithms were performing. The difference between true and algorithm derived slope and aspect grids were taken, giving residual grids. The root-mean-square (RMS) average value of each residual slope and residual aspect grid was calculated to give an estimate of the error in the numerical slope and aspect values for each algorithm at each cell size.

In addition, the variance and mean of the residual slope and residual aspect grids were also calculated. Other parameters calculated included the "percentage of total sum of squares" of the true and algorithm derived slope and aspect grids (Harbaugh, 1964), the differences between maximum theoretical and maximum calculated values of slope, and differences between mean theoretical and mean calculated values of slope.

These parameters were used to rank the slope and aspect algorithms from "best" to "worst"; special attention being paid when use of different parameters resulted in a different ranking of the algorithms. A general synthesis of these results resulted in a procedure for evaluating the performance of slope and aspect generating algorithms and recommendations for the "best" slope and aspect algorithms for use on the selected terrain type.

Acknowledgements
Funding for this work was provided by the Scottish Office, Agriculture, Environment and Fisheries Department.

References
Harbaugh, J.W. 1964. "A Computer Method for Four-Variable Trend Analysis illustrated by a Study of Oil-Gravity Variations in Southeastern Kansas", State Geological Survey of Kansas Bulletin 171, p.32.

Hsu, M., and Robinson, A.H. 1970. The Fidelity of Isopleth Maps. Minneapolis: University of Minnasota Press, p.17.

Jones, K.H. 1996. "A Comparison of Eight Algorithms used to Compute Slopes as a Local Property of the DEM", Proceedings of the GIS Research UK 1996, 7-12.

Morrison, J.L. 1971. Method-Produced Error in Isarithmic Mapping, Technical Monograph No. CA-5, American Congress on Surveying and Mapping, Washington, D.C., p.24.

Morrison, J.L. 1974. "Observed Statistical Trends in Various Interpolation Algorithms Useful for First Stage Interpolation", The Canadian Cartographer, 11(2), 142-159.