Slope Discussion

Slope identifies the maximum rate of change in the terrain surface. Slope values can be calculated as percent slope or degree of slope.

Percent slope is defined as the ratio of rise (y) over run (x) times 100 (i.e., (y/x)*100). Percent slopes can range from 0 to infinity as the surface approaches an undefined state of vertical. Degrees slope is defined as the arc tangent of the ratio of rise over run (i.e., ATAN(y/x)). Degree slope values range from 0 to 90 degrees.

The algorithm used to compute slope depends on the surface method used to produce the elevation surface. A TIN uses the inclination of the normal vector of the terrain with the xy plane to compute slope (Berg 2000). The algorithm used to compute slope for the IDW method is the average maximum technique (ArcView™ 3.2 Help System, Burrough, 1998):

run = (dz/dx) = ((a + 2d + g) - (c + 2f + i)) / (8 * x_cellsize)
rise = (dz/dy) = ((a + 2b + c) - (g + 2h + i)) / (8 * y_cellsize)

where:

a	b	c
d	e	f
g	h	i

Example of a Slope Map

Map of Slope (High Slopes (Red) to Low Slopes (Green)) blended with hillshade

References:

ArcView™ 3.2 Help System

Berg, Mark de, Marc van Kreveld, Mark Overmars, Otfried Schwarzkopf. Computational Geometry. Springer, 2nd Edition, 2000.

Burrough, Peter A., Rachael A. McDonnell. Principles of Geographic Information Systems. Chapter 8, "Spatial Analysis Using Continuous Fields". Pages 200-202. Oxford. 1998.