INTERPOLATION WITH 3 = 1.8 INTERDOLATION WITH B = 1,2 
  
P O SIMPLE POINT » 
  
  
  
  
Fig. 5. Interpolation with Practals, 
(Y) 2 
J = 15 de min (11) 
ox Y 
The results of the finite element method (11) are identical to the results 
of the Prediction method using variogramme (8) or the correlation function 
(5), as it was shown already in 1971 by Kimeldorf and Wabha (see also Dolph 
and Woodbury, 1952; Kubik, 1973). When interpolating with the prediction 
method (Wiener, 1949) 
Z(x) = K(X,xj)' KG x) *Z(X,) or 
= (12) 
Z(x) » Var(x-x.)* Var(x.-x.) b. 2(x.) 
i i. uj j 
(with Z(x,) vector of sample values) the numerical stability of the compu- 
) 
tations should be carefully controlled. When using variogrammes, as advo- 
cated by few, the smallest elements occur along the diagonal of the matri- 
ces involved; the matrices are not positive definite. The accumulation of 
rounding errors is very serious and meaningless results are obtained when 
using desk top computers without special precautions. The authors there- 
fore recommend the use of the correlation function (5). 
Figure 5 shows some examples of interpolation. For B = 1 we obtain linear 
interpolation, for B = 3 piecewise 3rd degree interpolation (Spline-inter- 
polation), and for B values between one and three we obtain interpolation 
forms, which properly model break lines in the terrain while preserving 
relative smoothness in the other profile sections (cf. Botman and Kubik, 
1984).