in a specified area of given extension. In order to eliminate the depen-
dence of the definition on this reference area, the concept of Relative
Relief is introduced, by dividing the Local Relief by the extend of the
reference area (its diameter or perimeter). Relative Relief is a dimen-
sionless quantity; its dependence on the extend of the reference area
and its frequency distribution allows classification of terrain accor-
ding to roughness and genesis.
Slope js possibly the most important parameter of terrain forms, because
it controls the gravitational forces available for geomorphic work (Evans,
1972). The slope is the first derivative of the elevation in arbitrary
direction, or in the direction of steepest descent. On a macroscopic
scale, slope is defineable at any point of the terrain, except at break
lines. Slope SL is either measured as angle or in percent of inclination
of triangular facets in a regular grid, or it is measured by derived quan-
tities like the roughness factor RF - 100(1-cos SI), Mark (1975).
Wavelength or extend of the terrain form is the third important parameter
here mentioned. The wavelength in a terrain profile is defined as the
average distance between successive (local) maxima or minima. This wave-
length is measured in length units (meters) and may be studied in various
characteristic directions. Its relationship to the magnitude of the ter-
rain forms can be ideally studied by Fourier Transformation (see section 3).
The terrain forms may be classified with respect to these three parameters -
relief, slope and wavelength, to their magnitude and their fluctuations and
with respect to their mutual relationships. These parameters will be our
basic reference when studying other models of the terrain.
3. TERRAIN MODELS FROM FOURIER SPECTRA
"If any of them can explain it", said Alice,
"I'LL give him sixpence. I don't believe
there is an atom of meaning in it". "If
there is no meaning in it", said the King,
"that saves a world of trouble, you know, as
we needn't try to find any”.
The terrain may be described in form of a Fourier series, and many of the
above parameters may be directly related to the coefficients of the series
(Ayeni, 1976; Tempfli and Makarovic, 1979; Frederiksen et al, 1978; Frede-
riksen, 1981; Tempfli, 1982). Any continuous and continuously differentiable
function Z(x), O « x « L, can be expanded into a uniformly convergent Fourier
series
217 211
= + . — 5 . . 1 — 08 .
Z (x) à, Y ac cos Cr x*f)^4 $ De sint xf) (1)
where a, and b, are parameters and f is a relative frequency related to the
length L of the profile. :
This model may be used to describe terrain profiles in function of their
sine - and cosine components. Equally, twodimensional terrain may be trans-
formed, using twodimensional Fourier series expansions.
