its 
The solution for X is only a unique one when the inverse of (A!PA) is unique, or, 
which is the same, when the normal matrix is regular. 
An adjustment of type Il involves a direct solution of unknown elements. In 
geodesy and photogrammetry these unknown elements are coordinates and 
orientations expressed with respect to a certain coordinate system. Without 
chosing a coordinate system equation (4) cannot fulfil the condition of uniqueness. 
There are many solutions of (4) all depending on the location and orientation of an 
arbitrary chosen coordinate system. Hence atpa) is a singular" normal matrix if 
no coordinate system is chosen. 
For two dimensional networks the rank-deficiency is four. For three dimensional 
networks the rank-deficiency is seven. 
Therefore the introduction of an S-base (Baarda, ref. 1, Molenaar, ref. 2 and 
Teunissen, ref. 3 and 4) involves the choice of four (2-D) or seven (3-D) non- 
stochastic quantities. 
What happens if no S-base is chosen? 
In that case (A*PA) is singular, which means that some of the A-columns are a 
linear combination of the remaining columns. 
Assume the last p-columns of A to be a linear combination of the first (m-p)- 
columns. The rank-deficiency of A then becomes p, and A can be split up into two 
parts: 
(5) A = (A, | A,) 
nxm  nx(m-p) nxp 
As is a linear combination of A, denoted as: 
(6) (A,) = (A,).(L94) 
nxp | nx(m-p) (m-p)xp 
* : : ; JA: 
Here it is assumed that the network configuration will not lead to singularities 
in the nomal equations. 
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