Geography Reference
In-Depth Information
Using the approach detailed by Horn (1981), the gradient of the central cell is com-
puted using Equations 10.1 and 10.2:
∂
z
x
(44
+¥+-+¥+
(2
39)
34)
(45
(2
43)
38)
=
=
0.1625
∂
(8
¥
10)
∂
z
y
(44
+¥+-+¥+
(2
44)
45)
(34
(2
32)
38)
=
=
0.5125
∂
(8
¥
10)
Using these i gures, the gradient is calculated from:
2
2
∂
z
ʈ
∂
z
ʈ
g
=
+
∂˯
(10.3)
Á˜
Á˜
˯
∂
x
y
For this example, this leads to:
2
2
g
=
(0.1625)
+
(0.5125)
=
0.5376 m
h is can be converted to gradient in degrees (
gd
):
(10.4)
gd
=
atan(
g
)
¥
57.29578
where atan (also given by tan
-1
) is the inverse tangent (see Appendix C). Gradient is
ot en also expressed as percentages. Here gradient in degrees is:
gd
=
0.4933
¥
57.29578
=
28.264
∞
Figure 10.8 shows gradient (in degrees) derived from a DEM of Northern Ireland.
Note that the spatial resolution of the DEM from which gradient was derived is 740.224 m.
h e gradients are never as large as in the computed example above because any dra-
matic gradients are, in ef ect, averaged out because of the coarse spatial resolution.
Note that the gradient map is visually similar to the standard deviation map given
earlier in the chapter (Figure 10.7). h is is not surprising as the standard deviation
picks out the edges of features and gradient does the same. Maps of gradient are used
widely for modelling erosion. Mitásová
et al.
(1996) computed gradient and aspect as a
part of a set of procedures for modelling erosion and deposition. Li
et al.
(2004) provide
a detailed account of gradient and aspect derivation as well as other terrain parameters.
Other products derived from surfaces
10.6
Besides gradient and other derivatives of altitude, many other products are derived
from DEMs. h ese include maps indicating the direction of steepest downhill descent
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