PC, = pC; + Vi; 
The usual linearization gives the following expression : 
  
  
  
  
  
  
1 o PCy 9 PC; o PC; 
Vi = F; OR > X.) - pe; + DOR] dOR, + JOR, dOR, = ese TOR dOR 
OPC; e APC; 
Usi ; ; bas : 
sing a numerical differentiation the element 5 OR, OR is replaced by AOR. don; 
where 
' 1 ' 1 ' 1 1 ! 
APC; ; F; (OR, , OR, P oni +AOR,, vos OR) -F (OR, : OR, ++; OR: > +. OR 
AOR; ^ OR, 
In program-writing the procedure is this : 
With approximated a priori values for all the unknown orientation parametres, picture 
coordinates for every point in the object are calculated through the transformations TRANS, 
TRANS2, ... These picture coordinates are compared to the measured observations and the 
difference is the numerical constant. By returning to the approximated orientation elements and 
giving the first element a small increment (AX,) the computing procedure through TRANS 1, 
TRANS 2, ... is repeated. This time the picture coordinate is compared to the first computed 
picture coordinate, and the difference Ax'(and Ay!) is divided with the increment AX, giving a 
1 
numerical value for = , the numerical coefficient. By repeating this procedure for every 
o 
unknown parameter, all the coefficients for the linearized observation equations can be computed. 
Establishing the normal equations and solving these are then done in the usual manner. 
In fig. 5 a picture coordinate is shown as a function of an orientation element. By the 
normal partial differentiation, the curve is replaced by a tangent to the curve at the point of the 
approximated value of the unknown. By numerical differentiation the curve is replaced by a 
chord. None of these replacements are correct, and both replacements will only work for small 
corrections of the unknown. The numerical differentiation works best if the increment AX has a 
value comparable to the a posteriori correction to the approximated unknown. 
Introducing new transformations or new unknowns in the adjustment is rather simple 
with this method. 
PC, 4 
AOR; 
OR; OR, 
Fig. 5. The fundamental equation and the numerical coefficients. 
-- 
  
Problems arising from the correlation of the unknowns 
  
By letting the unknowns of the adjustment be geometrical quantities that logically 
describe the physical system, it is impossible to avoid correlation between the unknowns, some 
being even to a great degree correlated. At the same time the a priori knowledge of the unknown, 
especially the exterior elements can be so unaccurate, that the approximated values are very far 
from the real values of the unknown. Anybody who has worked with systems of this kind will know, 
that it is just about impossible, or at best, only with luck possible to get iterative adjustments to 
converge. 
192