«6- 
According to the theory, this interpolation can be expected to give 
better results than any other method, if the data has indeed the character of a 
stationary random function and if the proper correlation function is used. 
To qualify as a stationary random function, one requirement is that the 
systematic trend in the data must be eliminated by reducing the heights to a 
suitable reference surface. This surface is accordingly called a trend surface. 
Such a surface can be computed in advance. Kraus [45], [47] suggests to use for 
this a low-degree polynomial surface. Alternatively, a trend surface can be 
computed simultaneously with the interpolation. Hardy [32] does this by adding a 
polynomial of low degree to the left side of eq. (3). Lauer [52] and Chiles and 
Deiner [23] also add a linear sum of functions but do not specify the individual 
unctions. 
These analytic functions will seldom be flexible enough to eliminate the 
systematic trend in extensive areas. A better trend surface can be obtained by a 
preliminary interpolation which produces an appreciable amount of smoothing. A 
moving surface method has been used for this purpose by Koch [42] and by Schut [9]. 
Arthur [13] was the first to publish a method of this type. Having 
initially used a distance function 
f3l-2 (4a) 
which proved in many cases to give a singular matrix B, he later [14] changed to a 
Gaussian curve 
f * exp(-ar?) (4b) 
in which 222.5 and, as before, » is the ratio between the distance under consider- 
ation and a fixed distance for which he chose the average distance between 
reference points. 
Kraus, first in 1970 [44] and in several following publications, also 
makes use of a Gaussian curve. The coefficient a is here computed to make this 
curve agree as well as possible with the values of the correlation function 
computed according to the theory from the reference heights. Smoothing becomes 
possible by replacing the value of the curve at r-0 by a larger value. This 
formulation is used in the Stuttgart Contour Program, described by Stanger [68]. 
Assmus [15] documents an extension of this program in which breaklines are dealt 
with very satisfactorily by specifying that the value of the correlation function 
is zero for any two points separated by a breakline. 
Lauer [52] describes the use in his dissertation of a distance function 
which can be written in the form 
= by. 
f = exp(-ar’); a0 (4c) 
and concludes from experiments that »=1.2 is an optimum value. Koch [43] and 
Fuchs [10-3] make use of the summation of surfaces as well as of a moving surface 
method; filtering is possible also. Koch [42] prefers the distance function, due 
to Hirvonen, 
f = 1/(1+2) (4d) 
Lauer, in [53], lists as satisfactory this function and also a weighted average of 
this function and 1/(1+r). 
Finally, Hardy, first in 1971 [32], makes use of the distance function 
f = (d?+C)P (4e)