**Preferred Slope Algorithm**

Three slope methods appear to have the broadest support:

- Four closest neighbors (FCU):
- Eight neighbors unweighted (ENU): present preferred method in MICRODEM. Users must still be aware that this method produces peaks that are too flat.
- Steepest Adjacent Neighbor (SAN): requires adjustments to get reliable aspect distribution, and does not give a smooth slope distribution. Preferred method in MICRODEM until the summer of 2003, and there are still reasons to prefer it.

The Eight Neighbors Unweighted method became the default in the summer of 2003, primarily because it produces the smoothest, most realistic slope histograms, and reasonable aspect distributions It still has problems with smoothing of slopes in valleys, peaks, and ridges.

Guth (1995) reviewed all the independent slope and aspect algorithms he could find. He suggested that the steepest adjacent neighbor provided the best estimate of slope, and the 8 neighbors with even weights provided the best estimate of aspect. This is modestly called the Guth Hybrid slope algorithm.

More recent comparisons of slope algorithms include Hodgson (1998) and Jones (1998), but neither cited Guth (1995) and their dismissal of the steepest adjacent neighbor does not appear completely valid.

Justification to use largest slope of 8 to adjacent elevations in grid:

- Each point is surrounded by 8 neighbors. Four are to the N,S,E,& W, and 4 approximately NE,SE,SW,& NW. How close those are depend upon the type of DEM and latitude; it is exact at the equator where x and y spacing is equal or using a UTM based DEM. The worst cases will be in the vicinity of 50°, where just to the south the spacing is 60 and 90 m, and just to the north the spacing is 90 and 120 m. There the diagonal values are at 53 and 34° instead of 45° (arc tan (120/90) and arc tan (60/90)).
- The direction of maximum slope can thus occur no more than 28° away from the largest of the eight directional slopes to nearest neighbors in the DTED grid (half of 56, the complement of 34).
- If we assume a plane for the surface in the vicinity of the greatest slope, we can use vector arithmetic to project the slope in various directions. The projection will be a function of the cosine. For the worst case, the estimated slope to the DTED value in the closest direction will be 87.9% of the true slope (cos(28) = 0.879), and that occurs only when the true maximum lies directly between two grid values. At the equator, with maximum error in estimating, the estimated slope is 92.4% of its true value.
- Since maximum error is only about 10%, and given the overall accuracy of the data to start with, we do not feel it necessary to obtain the added rigor of fitting a surface to the data points in the vicinity and then computing the slope of the tangent surface.
- The other methods effectively compute the slope are a region twice the size of the data spacing.
- See Guth (1995) for additional details and a comparison on the effect of algorithm on calculated slope.

Maps showing where slope algorithms differ.

*Last revised 10/20/2011*