Slope Algorithms

Znw

Zn

Zne

Zw

Z

Ze

Zsw

Zs

Zse

The slope is taken as a dZ value divided by a horizontal distance.  The horizontal distance is the data spacing (east-west, north-south, or diagonal) for the one point and nine point methods, and twice that distance for the four and eight point methods.  For DEMs like the USGS 10 m and 30, the NS and EW spacings are the same; they differ for data like the USGS NED, USGS 1:250K DEM, SRTM, and DTED with geographic spacing.

Aspect, the downhill direction, can be computed as a byproduct of the slope computation.

Slope and aspects, the magnitude and direction of the vector tangent to the topographic surface pointing downhill at a point, have been computed with a multitude of methods; Eyton (1991) and Carter (1992) list a number of references. The following discussion centers on the methods implemented in MICRODEM; the references point to clear expositions of the method, and not necessarily the first use of the method, many of which are in hard to location publications in the gray literature. The algorithms fall into several categories, depending on the number of neighboring points considered to compute the slope and aspect. 

Preferred slope algorithm,  In general the choice of slope algorithm does not make much difference, and you should just stick with the default.


Two neighbors:

Three neighbors:

Four neighbors to N, S, E, and W (excluding point itself) (rook's case):

Four neighbors to NW, SE,N E, and SW (excluding point itself) (bishop's case):

Eight neighbors (excluding point itself) (Queen's case):

Nine points:


Table. Selected Slope and Aspect Algorithms adapted from Guth (1995).

Neighbors 
Points Used 
Selected Reference
 2 
z,ze 
Batson & others (1975); No aspect
 3 
z,zn,zne 
O'Neill & Mark (1987)
 4 
zn,zs,ze,zw 
Eyton (1991); Carter (1992)
 8  
zn,zne,ze,zse,zs,zsw,zw,znw
Differential weights (Horn, 1981); 
Sobel operator (Richards, 1986)
 8 
zn,zne,ze,zse,zs,zsw,zw,znw
Identical weights (Sharpnack & Akin, 1969)
 8 
zn,zne,ze,zse,zs,zsw,zw,znw 
Heerdegen & Beran (1982)
 8
zn,zne,ze,zse,zs,zsw,zw,znw
Local trend surface (Davis, 1973)
 9 
z,zn,zne,ze,zse,zs,zsw,zw,znw
Steepest Neighbor (Collins, 1975); Only 8 aspects
 9 
z,zn,zne,ze,zse,zs,zsw,zw,znw
Steepest Downhill Neighbor; Only 8 aspects
 9
z,zn,zne,ze,zse,zs,zsw,zw,znw
Average Neighbor; Only 8 aspects

Maps showing where slope algorithms differ.

Slope references

Slope units, in percent.


Last revised 5/23/2014