Determining the fractal dimension of terrain surfaces is a contentious process. In practice such analysis involves obtaining an estimate that is highly dependent on the way in which the data have been captured and stored, and the modelling choices made by the user, rather than simply being a reflection of the complexity of original surface. For many datasets the fractal dimension as computed in one program will differ from that computed by other programs, and may only apply across a limited scale range.

Figure 6‑42 shows the result of fractal analysis of one of our test surfaces, OS tile TQ81NE, using the Landserf package. The fractal dimension in this example is computed by selecting a fixed interval lag and computing the standard variogram for the surface based on this lag, assuming an isotropic process, and then plotting the log of the lag against the log of the variogram. The best fit line y=ax+b is then used to compute the fractal or capacity dimension, D_C, using the expression:

D_C=3‑a/2

Hence in this example D_C=2.2. The Pentland Hills test dataset (tile OS NT04) has a higher value of 2.4, although interpretation of the computed differences is not straightforward. This approach yields a single measure of fractal dimension for the entire surface.

Figure 6‑42 Fractal analysis of TQ81NE

Idrisi computes fractal values for an entire grid using a 3x3 window or kernel, based on the slope values computed for that window. If s_ij is the slope value in degrees from 0 at location (i,j) then the fractal dimension, D_C,_ij , is computed as:

where the logs can be in any base. This formula results in a linear (profile-like) fractal measure, providing an index of surface texture in the range [1,2].