The scale free and scale - bound properties of land surfaces: fractal analysis and specific geomorphometry from digital terrain models

Abstract

The scale-bound view of landsurfaces, being an assemblage of certain landforms, occurring within limited scale ranges, has been challenged by the scale-free characteristics of fractal geometry. This thesis assesses the fractal model by examining the irregularity of landsurface form, for the self-affine behaviour present in fractional Brownian surfaces. Different methods for detecting self-affine behaviour in surfaces are considered and of these the variogram technique is shown to be the most effective. It produces the best results of two methods tested on simulated surfaces, with known fractal properties. The algorithm used has been adapted to consider log (altitude variance) over a sample of log (distances) for: complete surfaces; subareas within surfaces; separate directions within surfaces. Twenty seven digital elevation models of landsurfaces arc re-examined for self- affine behaviour. The variogram results for complete surfaces show that none of these are self-affine over the scale range considered. This is because of dominant slope lengths and regular valley, spacing within areas. For similar reasons subarea analysis produces the non-fractal behaviour of markedly different variograms for separate subareas. The linearity of landforms in many areas, is detected by the variograms for separate directions. This indicates that the roughness of landsurfaces is anisotropic, unlike that of fractal surfaces. Because of difficulties in extracting particular landforms from their landsurfaces, no clear links between fractal behaviour, and landform size distribution could be established. A comparative study shows the geomorphometric parameters of fractal surfaces to vary with fractal dimension, while the geomorphometry of landsurfaces varies with the landforms present. Fractal dimensions estimated from landsurfaces do not correlate with geomorphometric parameters. From the results of this study, real landsurfaces would not appear to be scale- free. Therefore, a scale-bound approach towards landsurfaces would seem to be more appropriate to geomorphology than the fractal alternative.