Scale and structure in landscapes

This thesis deals with the treatment of scale in the representation and analysis of topographic surfaces. It is now well accepted that terrain parameters computed at different scales cannot be directly compared and that process models calibrated at one scale must be recalibrated when used at a different scale. If the behaviour of the parameters and models as a function of scale could be characterised, these problems would be to some extent overcome. While scale poses difficult problems, there are signs that these problems are not intractable. Studies of the way terrain variables change with scale suggest that the changes are not unpredictable but systematic and amenable to mathematical characterisation. If a mathematical model describing the scaling properties of topographic surfaces could be developed many of the problems associated with scale and resolution in digital representations of terrain would be solved. The fractal model proposed by Mandelbrot and since supported by many researchers was the first such mathematical model used to study the scaling properties of natural landscapes. Its simplicity and mathematical depth make it very attractive, but on close examination it is found to be inadequate for several reasons. The two most important of these are its inability to represent distinctive characteristics of landscapes, in particular drainage networks, and the clear evidence from digital elevation models that a simple fractal scaling law does not apply at all scales. Fourier spectral analysis provides an alternative framework for investigating scaling that avoids imposing any particular scaling relationship. Spectral analysis permits identification of power-law scaling relationships over restricted ranges of scales, and the characteristic scales at which the scaling relationships change. Analysis of DEMs derived at several different resolutions from two different scales of source data leads to a model of the filtering effects associated with interpolation to a regular grid. Removal of these filtering effects demonstrates that the frequently observed increased smoothness of topography at short wavelengths (typically less than 200 m) is a property of the landscape itself, with some additional smoothness imparted by cartographic and interpolation effects. The wavelength at which the increased smoothness appears is shown to be related to hillslope length. Motivated by the desire to analyse the surface uszng mathematical forms more akin to real landform elements) a new method of representing and analysing landscapes is described which attempts to directly model real surface features . The contention is that an adequate model of the scaling properties of topographic surfaces requires a representation of the surface that explicitly captures topographic features at different scales. The new decomposition based on positive wavelets provides a method of isolating features of a pre-defined shape from the surface. Three study sites are analysed in terms of the amplitude) shape and orientation of component features at different scales) and in terms of the spatial distribution of features. The use of the representation for generalising and refining surfaces is demonstrated