in which Sij may be chosen as 
a 7? or.1 t d.. 
4 
$3 J 
and 
+, 
ad, = V (x - X 3? + (y - y y? 
1j 
  
while all methods applying interpolation in a stochastic field require that the data are statio- 
nary random functions with unchanged probability distributions from point to point, and that 
egodicity applies (each data point observation is representative for the stochastic nature of the 
field), the weighted arithmetic mean further requires an isotropic field, so that the distance 
weight is uninfluenced by the direction. It is therefore only applicable for a preprocessed field. 
The moving average expresses the coordinate discrepancies as polynomials, which permits to drop 
the condition of isotropy: 
2 2 
à, + a,x + 3 Y * (a xy + a,x + acy ) 
Bb +b 
O 
2 2 
x" b.y ) 
x * by * (bxy * b 
1 4 
For each point to be interpolated the coefficies of the polynomial must be soved in a time con- 
suming manner, however. Equations (13b) and (13c) are valid for the weights. 
The most elaborate method, which consequently offers itself only as a post - treatment to test the 
validity of an applied deterministic model at a number of check values is the method of linear pre- 
diction (linear least squares interpolation) without or with filtering, as introduced into geodesy 
by Krarup and Moritz, as introduced into photogrammetry by Kraus, later Leberl /59/ and Mikhail, 
and as applied by Bähr /4, 5/ 
Cx? ' 
3; 
c Li t 
Yj Y4 
Cross - covariance matrix for the 
p. and for the n control points i 
J discrepancies 
at n control 
points 
in which c is the autocovariance matrix for the n control points 
The filtering operation allows the vector of discrepancies Ax! to be consisting of correlated 
and uncorrelated positions Ax! and Ax;