25. 
0.3. This same procedure, with a small value of s, is needed for all weight 
functions which approach infinity when » approaches zero. 
The behaviour of these weight functions is shown in Table 1, where they 
are listed in sequence of increasingly sharp drop-off at small values of ». Those 
which become infinite at »=0 have been normalized to w=1 at r=0.02. 
Two further details require attention. The maximum distance within 
: which reference points are used must be chosen large enough to provide some over- 
determination throughout the interpolation area. An increase too far beyond this 
size will also produce smoothing. Breaklines in the terrain must be taken into 
account. This is done, especially in the German programs, by measuring or comput- 
ing a denser than elsewhere number of points on these lines and by not using points 
beyond them. 
Table 1: Values of the weight functions in the equations (2a) to (2f) 
  
(2c) (2f) (2e) (2e) (2a) 
a=14 a=20 
  
edi 
. 
1.00 1.00 
e 
-— 
.00 
97 
57 
57 
28 
il 
.03 
.95 
82 
.45 
.16 
.04 
.00 
„99 
.96 
32 
.86 
78 
.68 
.56 
.42 
27 
.10 
. 
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4. Summation of surfaces 
  
The methods of this group employ an algebraic formulation that is 
identical with one used in the correlation theory of stationary random functions. 
There, the interpolation is called linear least-squares interpolation or predic- 
tion. Smoothing of the data is possible also and is called filtering. A distance 
function, called the correlation function or covariance function, is either 
defined or computed. From this function follow the elements of a matrix B: the 
element in row Z and column j is the value of the distance function for the 
distance between the z-th and the j-th reference point. The numbering of those 
points is arbitrary. Further, for each interpolated point the i-th component of a 
vector b is the value of the distance function for the distance from the inter- 
polated point to the Z-th reference point. If z is the vector whose components 
are the heights of the reference points in the sequence of the numbering, the 
height of the interpolated point is found as: 
h - blB^!z (3) 
Hardy [32] has given this formulation its geometrical interpretation as a 
summation of as many surfaces as there are reference points. Each surface is a 
surface of revolution, centred at one of the reference points. It js obtained by 
multiplying the distance function by the component of the vector B “z which has 
the same sequence number as the reference point and by rotating the graph of the 
resulting function about the ordinate axis.