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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
  
3). The advantages are again tlie freedom in the selection of 
the theodolite stations, made independent from each other, as 
long as they observe a certain number of common points, like 
in photogrammetry no image is normally visible in any other 
image. The adjustment of a control network is similar then to 
a photogrammetric block bundle adjustment. 
2 THE OBSERVATION EQUATIONS 
Like in photogrammetry where the observations are the two 
image coordinates x and y per point, (that are indeed 
directions from the projection centre), the theodolite angular 
observations are the horizontal direction and the zenital 
angle. In analogy with the co-linearity equations, we can set 
up two equations for the two direction, the horizontal one / 
and the vertical one ÿ. One can add, if it is the case, the 
equations of the measured distances. The surveyor is then 
free to place the theodolite where he likes. He needs only one 
tripod and the instrument only, and no more the complete 
traversing equipment. The survey can be greatly speed up. 
The disadvantage is that this procedure is more sensitive to 
gross errors that are more difficult to find and eliminate. In 
addition it can be difficult to estimate the approximated value 
of the co-ordinates for the unknown points. For this reason it 
is recommended to use the co-planarity to estimate the 
approximate coordinates of the unknown points. For any 
station it is very useful to measure the magnetic bearing by 
means of a compass, in order to get an approximate value of 
the bearing. 
2.1 The Equation of the horizontal direction 
The relationship that links the co-ordinates of the points A 
and B and the bearing 0, is: 
0457arctg((Xg-X/(Yn-Y4)). (1) 
[n 
Figure 3 — Zero bearing 045 and horizontal reading /og 
The discrepancy is the difference between the observed 
distance 0, and its approximated value 0*,5. Normally the 
bearing is not measured, since the direction of the Y axis is 
unknown; instead of it we measure the direction /og that is 
linked to the bearing by : /og* 0,9 70,5 (2) 
Setting: X47X?,*dX4 Y,í7Y? 4 *dY, Z=L°,+dZ,, 
X47X? dX, Yg- Y?gtdYg Zy7Z^gtdZp, (3) 
the last equation of the bearing becomes (after its 
linearisation about the approximate value X° Y°Z°) then the 
so-called equation of direction: 
y? = Y? Y 0 y? y? y? x? x 0 
TF 4 , AB Th qu no. > Vini 1 , 
AAN + HA EN à HA— A dY, 
0:2 4 o + : () o 2 
d di, di 18 
se SE 9 dere (4) 
lo the traditional unknown co-ordinates another one is 
added, the so-called bearing of the origin 0,5. 
2.2 The zenital angle 
The height Zp of a point aimed [rom the station S of elevation 
Zs is given by: 
fa * 
Z,2 Zt decia td : 
SP 
DR. 7 (5) 
deriving the zenital angle psp , we get: 
(XFN 4 
VE 1 SR HEN 
(Z, -Z,)-h, - Am, RJ| 2R 
  
\ 
(6) 
with k refraction coefficient, Z,,=(Zp+Zs)/2 mean height, R 
radius of the local sphere, Ag instrumental height in S , Am, 
height of the target in P 
Its linearisation, neglecting the term Z,/R, brings to: 
  
| A ; © 9 yo. A 
Nyy rim) Ik ; a TL e (dY, -dX.) | 
dd Zn) 2R]L 4s de 
0 
ds, 
T HA 
ES (az?, —h, + An © 
  
oll, vl YE Dp Oo 
  
(7) 
with the position 
der 1-k 
Psp -acig| —; i Sas ds, 
AZ op = hy dm, 2R (8) 
(Mussio, 1982). The unknowns are the variations of the 
coordinates dX, dY, dZ for the station S and for the aimed 
point P (6 in total). With this equation we take into account 
the existing correlation between the three coordinates. The 
correlations are strong, and then it is useful to consider them, 
in the case of local networks and of technical networks, 
where the visual directions are often very inclinated. The 
estimate of the variance matrix of the unknowns is then more 
correct. 
Provided that all the angular observations have the same 
accuracy, the weight of the observations will inversely be 
proportional to the square distance to the observed point. 
2.3 The equation of the distance 
Between two points S and P on the X, Y, Z plane, we can 
measure the slope distance ds» . The equation of the distance, 
is. then dsp-((Xp-Xs *( Y- Y^ *( Z- Za)" (9) 
Its linearisation brings to: 
X,-X. y-Y Z-Z X.- X, Y-r Z-Z 
  
S IV 3 7 ^d Fl EL S JV E £5 I7 — 
4. dX, 4 dy 7 dX+ d aX, 3 d dy 3 d. dz - 
z(d,-d,)tv 
(10) 
The normal system so formed will be singular since it is 
necessary to define a system. 
To be able to determine the rank deficiency, it is necessary to 
keep in mind that in an altimetric problem it is equal to one, 
(one vertical translation), in a planimetric problem it is equal LN 
to three (two horizontal translations and one rotation in the x A 
y plane). In a spatial problem the rank deficiency is in general 
cqual to six, however in this case for the nature of the fi 
observations it reduces again to four (three translations and a l 
rotation in the plain xy). Therefore a vertex and the direction 
to a second vertex have to be kept fixed.