Slope is defined by a plane tangent to a topographic surface, as modelled by the DEM at a point (Burrough, 1986). Slope is classified as a vector; as such it has a quantity (gradient) and a direction (aspect). Slope gradient is defined as the maximum rate of change in altitude (tan Q), aspect (y) as the compass direction of this maximum rate of change (Cf. Fig. 1). More analytically, slope gradient at a point is the first derivative of elevation (Z) with respect of the slope (S), where S is in the aspect direction (y). At the same time the first derivative of a function (i.e. S stands for slope) at a point can be defined as the slope (angular coefficient or trigonometric tangent) of the tangent to the function on that particular point, hence:
tan Q = rise/run = ¶Z/¶S
Figure 1. Slope components,
note that slope gradient can be express in percent or in degrees (Modified
The technique/algorithm adopted to calculate slope gradient and aspect varies according to the type of DEM selected to model the topography. Models for structuring elevation database can be square-grid, TIN (triangular irregular network), and contour-based ones (Check here for more details on DEMs).
After conducting a literature review (though not a comprehensive one), it appears that a greater number of algorithms have been developed for gridded DEMs (Cf. Table 1).
Table 1. Techniques for calculating gradient on DEMs.
Type of DEM
Technique for calculating gradient
1. Average triangle value (Tajchman, 1981)
2. TAPES C (Moore, 1988)
3. Neighbourhood method (CERL, 1988)
4. Quadratic surface method (Evans, 1980; Zevenbergen and Thorne, 1987)
5. Best fit plane method (Beasley and Huggins, 1982)
6. Maximum slope method (Travis et al., 1975)
A study conducted by Srinivasan and Engel (1991) compared methods 3, 4, 5 and 6 (Cf. Table 1) against field measurements. According to their results the neighbourhood and quadratic surface methods would follow more faithfully ground measurements, the maximum slope method being the least reliable.
N.B. DEM derivatives are also highly dependent on the type of interpolation adopted during the process of DEM creation (e.g. kriging, spline, etc.).
Among a number of possible applications, I’d like to stress in a Pacific Northwest context, how important is to automatically derive slope gradient and aspect in the prospective to compile more sophisticated thematic maps such as:
· Terrain Stability Assessment
· Snow Avalanche Risk Mapping
· Debris Flow Risk Mapping
Beasley, D.B. and Huggins, L.F. 1982. ANSWERS: User’s manual. U.S. EPA-905/9-82-001, Chicago, IL. 54pp.
Burrough, P.A. 1986. Principles of Geographical Information Systems for Land Resources Assesment. Oxford: Clarendon Press.
CERL, 1988. GRASS reference manual, version 3.0. Champaign, IL: U.S. Army, Corps of Engineers, Construction Engineering Research Laboratory.
Evans, I.S. 1980. An integrated system of terrain analysis for slope mapping. Zeitschrift fur Geomorphologie. 36: 274-295.
Moore, I.D., R.B. Grayson and A.R. Ladson. 1991. Digital Terrain Modelling: a review of hydrological, geomorphological, and biological applications. Hydrological Processes. 5: 3-30.
Srinivasan, R. and B.A. Engel. 1991. Effect of slope prediction methods on slope and erosion estimates. Applied Engineering in Agriculture. 7: 779-783.
Tajchman, 1981. On computing topographic characteristics of a mountain catchment. Canadian J. Forest Res., 11: 768-774.
Travis, M.R., Elsner, G.H., Iverson, W.D., and Johnson, C.G. 1975. VIEWIT: computation of seen areas, slope, and aspect for land-use planning. USDA F.S. Gen. Tech. Rep. PSW-11/1975, 70p. Berkeley, California, U.S.A.
Zevenbergen, L.W. and C.R. Thorne. 1987. Quantitative analysis of land surface topography. Earth Surface Processes and Landforms, 12: 47-56.
Evans, I.S. 1972. General geomorphometry, derivatives of altitude and descriptive statistics. In: Chorley, R.J. (Ed.), Spatial Analysis in Geomorphology, pp. 17-90. Methuen.
Jenson, S. K. 1985. Automated derivation of hydrologic basin characteristics from digital elevation data. Proc. Auto-Carto 7:Digital Representation of Spatial Knowledge, Washington D.C., 301-310.
Kwamme, K.L. 1990. GIS algorithms and their effects on regional archaeological analysis. In: K.M. Allen, S.W. Green and E.B. Zubrow (Eds.), Interpreting space: GIS and archaeology, pp. 112 – 124. Taylor & Francis.
Skidmore, A.K. 1989. A comparison of techniques for calculating gradient and aspect from a gridded digital elevation model. International Journal of Geographical Information Systems. 4: 323-334.
A case study:
A gridded DEM perspective:
Slope of 3-dimentional equations:
Planar and profile curvature in a TIN environment: