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ANALYTICAL AERIAL TRIANGULATION, SCHUT 15 
7 In the case of redundant observations, for proper weighting and correlating of 
equations they must be differentiated also to the observations, that is, to the photograph 
coordinates. This results in additional terms in the linear correction equation. Putting 
these on the right hand side, this side reads 
  
0 : 0 
i 
0X, 0X, 
he ( 0 >) UNS duoc Br Ô s) X% 1 1dy; 
in which v, and y, are the photograph coordinates with respect to an origin in the 
principal point. 
The four derivatives are column vectors in the orientation matrices of photographs 
i and 1, that is, unit vectors in the direction of the %; 4175 Ysry> C7 and y-axis 
respectively. - 
If we assume the measurements to be correlation free the equations are also cor- 
relation free and the weight of each equation is inversely proportional to the sum of the 
squares of the coefficients of dæ;+ 1» dy; + 1» dx, and dy, each square being first multi- 
plied by the weight of the corresponding observation. 
Assuming further equal weights of the observations and assuming the photographs 
to be approximately vertical and flown approximately in a straight line with the x-axes 
approximately in the direction of this line, the weight to be applied to each linear 
equation is approximately equal to 1/2b2f2 where b is the length of the vector B and fis 
the focal length. In this case all equations may be given equal weight. 
8. The four methods use the linear equations in an iteration procedure. Approximate 
values of the elements are chosen and are substituted in the linear equations. The equa- 
tions are then solved and the solution is used to improve the orientation. Following the 
procedure and the notation employed in paragraph 6, we have 
Dew X4, 7 A07 X61 With X. m AS 1 N 
These two equations can be replaced by one equation 
14-1 
new X^ 11 — (Ac: A0, +1)" X; +1 
and multiplication of the two dyadics gives a new dyadic which is an improved ap- 
proximation of Ayıı- The multiplication amounts to multiplication of the orthogonal 
matrices of the two dyadies. This multiplication is economical if after completing the 
orientation vectors will be transformed which have not been used in establishing the 
orientation. Examples are strip triangulation by the first and second procedures and 
computation of points that are not used in the orientation. 
The improved approximations are in turn substituted in the linear equations and 
new improved values are computed. This process is repeated until a criterion for con- 
vergence is satisfied. 
An evaluation of four triangulation methods. 
9. In the method developed by the author at N.R.C. the first triangulation procedure is 
used. The condition equation B X X,- X;,1 = 0 is differentiated with respect to the 
orientation elements. This gives the linear equation in paragraph 6. The matrix multi- 
plieation in paragraph 8 is employed. Assuming equal weights for the observations and 
freedom from correlation the linear equations are assumed to be of equal weight and cor- 
relation free. Evidently the method follows the preceding specifications and on this basis 
it can be pronounced to be perfectly sound. 
Initial approximations of the orientation elements are the matrix of the preceding 
photograph and the base components of the preceding model. If the difference in rotation