sample spacing can be derived. Jacobi (1980), Frederiksen et al (1978) and 
Tempfli (1982) proved that the accuracy 0 of interpolation is largely in- 
dependent of the method of interpolation; it depends mainly on the spacing 
D between the sample points and the surface characteristics a: 0° prop parie 
4. TERRAIN MODELS FROM RANDOM FUNCTIONS 
" This conversation is going on a Little 
too fast; Lets go back to the last remark 
but one". 
Another class of terrain models is based on random functions. We shall de- 
fine a random function as a function Z(x), the value of which for any value 
x of its argument is a random variable. The argument x is considered a non- 
random argument. This concept may be used to describe the elevations of a 
terrain profile as a function of the profile length x. 
In order to characterize the random function, the knowledge of the density 
distribution pr(Z) of the values Z for different arguments x is necessary. 
The random function can be considered to be defined if all multi-dimensio- 
nal density distributions are given for any values X, ;/X.......z +. Although 
this often can be done, this method is not always convenient. Hence, in the 
majority of cases we limit ourselves to specify selected parameters of these 
density distributions. One can choose various quantities as such parame- 
ters, however the most convenient are: 
- The expectation or mean 
+ co 
m(x) = E(Z(x)) 5 f z-pr(zlx)az (4a) 
- co 
- The variance 
c?(x) » Var(Z(x)) » E((Z(x) - m(x))?] (4b) 
and 
- The correlation(covariance) function 
K (x (X2) zm E((Z(Xx 
1 ) - m(x,)) (Z(x,) = m(x„))} (4c) 
1 
where E{.} denotes the mathematical expectation of the argument. The corre- 
lation function describes the correlation between the two random variables 
Z(x,) and Z(x,) . 
A very important property of a random function is the dependence or inde- 
pendence of its distribution function on the origin of x. In accordance 
with this, stationary and non-stationary random functions are distinguished. 
For stationary random functions, the mathematical expectation and variance 
are constant and the covariance function depends only on the difference of 
the coordinates x5-x,, which is also called lag d. A second attribute which 
is also used as the basis of a classification of random functions is the 
  
  
    
  
    
  
   
  
  
   
  
   
  
  
   
  
  
  
   
  
  
   
   
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