International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
APPENDIX A. EXAMPLE CALCULATION 
Let’s go through an example where four control points are and 
the point No. 11 has a real gross-error in Y: 
  
  
  
  
Ck=75.00 
Number of points: 4 
No X Y X Y Z 
11/.714.99085 11.57 0.200 1401.000 0.200 
12 40.44218 71.30058 550.000 1400.000 3.000 
27 14.343652 66 94081 0.200 0.200 0.200 
28 40.35546 | -68.87416 550.000 0.200 6.000 
Wrong point: 11 
Ey = +1.0 M (a real gross-error) 
  
  
  
  
Table 1 Dataset of control points 
From the equations of (3)-(6) we get: 
11-12-27 11-12-28 11-27-28 12-27-28 
44.513645613 .063867943 0.9985977178 0.938647821 
-19.6426097 0.68254725 -0.8692 19 0.39 340 
1.2413 4 0.02 . 854 
55691 .049113251 .00 402 
7210.7 141.2 
-10.649 148.910 
141.5 560. 135 
149.141 560.191 
140 
1 
1.06375 
1.06375 
111.28460 1.06249 «0 7 
.998044 
.008492 
0.008492 
111.284608 1.061676 
0.999264 2 :999076 
0.999264 2.503617 239 
Table 2. Solutions from every combination 
where 
X1,x2,x3,x4  : roots of the equation, gained by the equation 
(3) 
Xp, Yp,Zp : projection center 
n,m : scalar factors 
After this we chose the common solutions: 
  
  
Incremental value: 0.005 mm 
i 
0.0130 0.0742 .-0.0136 .-0.0640 0.0130 06.0742 
0.1460 0.0026 -0.1237 0.0710 0.1460 0.0026 
0.0024 0.0133 0.0025 0.0121 0.0024 0.0133 
2 
-0.0370 0.0635 0.0367 .—-0.0926 | -0.0370 0.0635 
0.1396 0.0293- -0.1372 0.0505 0.1396 0.0293 
-0.0203 0.0347 0.0067. -0.0166 '-0.0203 0.0347 
0.0501 0.0104 0.0133. -0.0765 0.0501 0.0104 
06.0226. —0.0232 -0.1478 0.0018 0.0226 0.0232 
-0.0046 0.0253 0.0023. —-0.0138  —0.0046 070253 
4 
0.0502 .-0.0288 -0.0372 '-0.0659 0.0502 ..-Q.0288 
0.0029 —0.0297 -0.1413 0.0294 0.0029 070297 
0.0141 0.0183 -0.0205 -0.0362 0.0141 0.0183 
  
  
Table 4. The F matrices in each group 
To each solution we can calculate now the weight matrices by 
the equation (10): 
  
  
0.0020 0.0001 -0.0048 
0.0001 0.0004 -0.0012 
-0.0048 -0.0012 0.0629 
2 
0.0068 0.0006 -0.0138 
0.0006 0.0005 -0.0005 
~0.0138 -0.0005 0.0362 
3 
0.0024 -0.0001 -0.0020 
20.0001 0.0011 0.0014 
-0.0020 0.0014 0.0196 
4 
0.0035 -0.0007 -0.0042 
-0.0007 0.0012 0.0005 
-0.0042 0.0005 0.0140 
  
  
  
1 xp= 141.509 yp= 700.036 zp= 750.271 
2 xp= 141.295 yp= 700.581 zp= 750.711 
3 xp= 140.207 yp- 699.525 zp= 750.515 
4 xp= 140.000 yp= 700.000 zp= 750.000 
  
  
Table 3. Common solutions 
Now, let's determine the F dispersion matrix for each solution 
incrementing each x,y image coordinates with a small value 
(see Table 4.). 
682 
Table 5. Weight matrices for each group 
By the equation (11) the adjusted coordinates of the projection 
center are: 
xp- 140.549 m 
yp- 700.226 m 
Zp= 750.301 m 
By the equation (14) we got mo = 0.026. Applying the 
equations (15) the RMS values for each coordinate are: 
mx= 0.262 m 
my= 0.465 m 
mz= 0.087 m 
We gain the estimated RMS values before the adjustment by the 
formulas (16) as follows: 
mx~= 0.025 m 
my~= 0.037 m 
mz-= 0.008 m 
Setting up the null-hypothesis by the formula (18) we got the 
following statistical value: Ft=9.28. 
Also we calculate the F values by (19) for each coordinate and 
compare them with Ft: 
Fx= 9.52> Ft 
Fy= 13.79>Ft 
Fz= 10.08 > Ft 
The null-hypothesis is not fulfilled, so we can declare that the 
space resection has a gross-error. 
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