153 
(4) 
reach the values from -0.02 mm/km to -30.85 mm/km. 
Interpolated scale of the official network in the mm/km is 
shown on the Figure 8. Practical value of the computed scale 
factor is that the correctly measured distances should be 
corrected according to the scale factor deformation valid for the 
measured distance position in the net. Such distance could be 
then included into the existing coordinate system with the non- 
homogenous scale without difficulties. 
Figure 7: Scale deformation of the astrogeodetic network of 
Slovenia in mm/km 
It is obvious that also other types of transformation, like affine 
transformation, projection transformation, or adding additional 
parameters representing systemtic errors in the net, could be 
performed for the detailed investigation of the existing 
astrogeodetic network. But the transformation procedure should 
be done very carefully, since many transformation parametrs 
will fit the data better, i.e. produce smaller residuals, than the 
one with few transformation parameters. Here we mention one 
not very common transformation procedure using least squares 
collocation. 
TRANSFORMATION USING LEAST SQUARES 
COLLOCATION 
Mikhail (1976) describes collocation as "a general least squares 
techique combining classical adjustment with interpolation and 
filtering..". Moritz (1980) states that "least squares collocation 
is optimal in the sense that it gives the most accurate results that 
are obtainable on the basis of the available data." 
The collocation equation can be represented as follows: 
v + BA = f (3) 
s + n + BA = f 
The residuals v are decomposed into a correlated signal 
component s and a random noise n. The noise in the model is 
simply the random measuring errors and the signal can be 
desribed as that component of the model which reflects the 
inability of the selected parameters À to accurately describe the 
physical relationship (Deakin et ah, 1994). The signal vector s 
can be subdivided into signal s' at the observations (common) 
points-and the signal r at computation (new) points. Assuming 
that no correlation exists between signal and noise, the a-priori 
variance-covariance matrix between the random quantities must 
be computed. This enables covariance function and allows 
quantities, which are normally linked by mathematical 
relationships, to be described in a statistical manner. 
For the covariance function can be used the simple exponential 
function (Mikhail, 1976): 
C(r) = C„ e- (r,D)! 
The free parameters in the models are variance (C 0 ) and 
characteristic distance D. r is the spatial distance between two 
points. For the collocation the residuals are smaller then for 7- 
parameters transformation. Results form collocation are 
strongly influnced by the choice of free paramaters (variance 
and the characteristic distance) in covariance function. It is well 
known that C 0 is a scale factor for interpolation errors. 
Characteristic distance describes the behavior for distances on 
the order of D itself. 
In this article we used the collocation as a mean by which we 
tried to model the distorsions of the national coordinate system. 
Because of only having 27 observation points, covariance 
function was not determined empirically. Characteristic distance 
of the covariance function is chosen to be the minimum distance 
in the network. With the choice of the characteristic distance 
equal to the minimum distance in the net we tried to simulate 
the situation where the error in the position in certain point has 
no influence to the position error in the neighbouring points, 
which is certainly not very close to the reality. 
In the collocation model described above parameters in A are 
seven transformation parameters (f,a,p,y,dm). Signal s' in 
observation (common) points is calculated as follows: 
s' = C s yD‘‘(f- BA) (5) 
where: 
D = (C sV + C nn ) 
C sV is covariance matrix of the signal in observation points, C, in 
is a covariance matrix of the noise n. 
Residuals v=f-BA after the 7-parameters transformation of the 
national coordinate system into the ETRS 89 coordinate system 
were used for the computation of the signal r in computation 
points as follows: 
r = C rs D'(f-BA) (6) 
where C rs ' is the covariance matrix between the signals at the 
computation and observation points. The signal r computed at 
the computation points can be treated as the local distortion of 
the national coordinate system. 
The distortions of the national coordinate system of Slovenia 
computed using the collocation method are sketched in Figure 
8. 
Figure 8:The distortions of the national coordinate system of 
Slovenia computed using the collocation method