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Figure 3: Location of control, check and tie points in the 
THR image 
3.4 Bundle Adjustment 
In order to compensate for possible systematic errors in 
either the exterior orientation or/and the interior orientation 
data (look angles) of SPOT-5 a new functional model is 
implemented into ICC's adjustment software GeoTeX 
(Colomina et al., 1992), which is described in the following. 
The idea behind is to estimate global correction terms of the 
given position and attitude of one common trajectory for all 
cameras and also of the given look angles for each single 
camera. As correction functions serve 3 order polynomials. 
3.4.1 Functional Model 
The functional model is based on equation [1], which relates 
the look direction vector u, in the navigation reference frame 
to the look direction in the terrestrial coordinate frame, 
defined by the subtraction of the projection centre vector [Xo, 
No Za. from the point vector [X, Y, Zr 
- tan(y, ),, des A 
U, T tan(y )o = A. ; Re ; Y RT Y, [1] 
24 5 zu. 
where 
u, defines the look direction vector in the navigation 
reference frame. It depends on the look direction angles 
(Wx), and (¥y),, which are given for each pixel p of the 
sensor line). 
R,, transforms the navigation reference frame into the orbital 
frame. It depends on the interpolated attitude angles a(t), 
a,(t), a,(t) around the roll, pitch and yaw axes at time t, 
which are given as a time series at 8 Hz frequency. 
R5; transforms the orbital frame into the terrestrial frame. It 
depends on the centre of mass position P(t) of the 
satellite and the velocity vector V(t), which are given as a 
time series at a 30 second time interval. 
Eliminating the scale factor u the applied pseudo-observation 
equations [2] are obtained: 
s nC - X n Q -h)t*tn(Z - Z) 
nO - XornmQ-Y)enmz-2z) 
Oe EX > An) th -Y)rm(Z-2) 
X A) (Ya) (Z Ze) 
—tan(y, ), 
+tan(w,), 
with A 
R - R5 a, (0.4, (0.2, ()] R2 [PG (0]- 
Fi fü. "s 
mm FP I5 [3] 
Fa foo fus 
The coefficients of the correction polynomials are the actual 
unknowns of the adjustment, which are applied to the 
parameters of the external orientation E, (equation [4]), i.e. to 
the position vector P(t)=[Xo, Yo, zr and to the attitude 
angles a(t), ay(t), a,(t) as well as to the parameters of the 
internal orientation I. (equation[5]), i.e. to the look direction 
angles (Wx), and (Fy), of sensor line S and pixel p. 
E-ZE -AÀ,*B,(t-t)- C,(t - ty 9 Dj(r- ny 
[4] 
Pelle Bey EEE Ley DEP: 
p / ;( 1000 ) 1 1000 ) i 1000 
[5] 
The external orientation parameters E (equation [4]) enter in 
the equations [2] and [3]. They are derived from the 
interpolated parameter E, at time t of the corresponding 
image line and the 3" order correction polynomial with its 
unknown coefficients Ag, Bp, Cg and Dg. tc here is defined as 
time of the central line (212001) of the THR image. The 
internal orientation parameters I? (equation [5]) also enter in 
equation [2]. They are derived from the parameter LS of the 
interpolated sensor position p and the 35 order correction 
polynomial with its unknown coefficients AS, BS, CP and 
Dj. pc here is defined as the centre pixel of the respective 
sensor line S. 
This model involves 48 unknowns, if 3 viewing directions 
are involved (like in this evaluation: HRS1, HRS2 and THR), 
ie. 4 unknowns for each of the 6 external orientation 
parameters and 2 x 4 unknowns for each of the 3 sensor lines. 
In practice only a subset of these 48 unknowns will be 
significantly determinable and the rest of the parameters need 
to be fixed in order to avoid over-parametrization problems. 
3.4.2 Input 
The following observations are introduced into the 
adjustment: 
— Image coordinates of 19 control and 17 check points 
(020.5 pixel), measured in the HRSI, HRS2 and the 
THR images (see figure 3), 
—