Scale and structure in landscapes
Abstract
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.