rm of the density distribution function of Z. The distribution law most ‘ 
equently assumed is the normal law, although this. in many. cases is not 
proper model for elevations. 
For a stationary random function, the Fourier Transform of the correla- 
tion function K(d) is equal to the power spectrum S(F), (2)(cf. Champeny, 
73). For the model (3) for terrain spectra the correlation function E) 
llows from the inverse Fourier Transform approximately to 
K(d) 2 c? - ca”, B = (4-1), (5) 
This model is a suitable model for terrain profiles, in particular if o? 
is chosen sufficiently large. 
Modelling the terrain by the correlation function is in principle possible, 
but estimation of the variance and the correlation is difficult in praxis. 
This has two reasons: Firstly, the elevations are not normally distributed; 
and individual extreme elevations or local flats have undue influence on 
the estimates. Robust estimation of the variance and correlation is neces- 
sary. The fluctuation of the elevations is estimated from their median 
(Crüger et al, 1984), correlation is estimated by rank correlation methods 
(Kendall, 1948). 
Secondly, long trends or semi-systematic fluctuations seriously distort 
the estimates. A proper manner to tackle this problem is to estimate the 
correlation function from visual inspection of the terrain profiles, and 
to describe large regional terrain forms by an appropriately large corre- 
lation length (which may equal or exceed the profile length) and by a cor- 
respondingly large variance (cf. Kubik, 1975). Once these parameters are 
chosen large enough, the deductions in the model become independent of the 
exact numerical values of these parameters. 
The above pitfalls in estimating the correlation function may: also be over- 
come by relaxing either the condition of staticnarity or the condition of 
normal distribution. The notorious long tailedness of most empirical histo- 
grammes may in many cases best be interpreted by accepting the possibility 
of an infinite variance *), This means in everyday language, that the vari- 
ance of height fluctuations in the terrain profile increases with the pro- 
file length. 
In order to avoid this somewhat unmanageable value e, we may model the 
differences of elevations instead of the elevations themselves. De Wijs 
(1972), Matheron (1971) and Mandelbrot (1982) all proposed the variance 
of the difference 8 023) for modelling terrain forms, var (z,-2,); (6) 
This quantity is also called (difference-)variance function, and it de- 
pends, for stationary differences, only on (x;-x;) = d. For a stationar, 
function Z(x), the variance function may be related to the correlati 
function by 
Var(d) s 2(90^ - K(d)). 
  
   
   
wer Chan