more efficient real-time realisation, but extended with one more 
parameter for improved precision. This model assumes a simplified 
orbital, which could be approximated to a plane circle with adequate 
accuracy during the imaging period of a scene. The orbit radii 
determined from the ephemeris are fitted to a third-degree polynomial 
to represent the deviations of the orbit from this circular shape. The 
constant term of the polynomial is allowed to change in the adjustment, 
but the linear, quadratic and third degree terms are kept fixed. The 
orbital parameters used are thus reduced to four: 
Inclination (I), Right Ascension (£2), Time at Ascending node (t = t 0), 
Orbital radius at Ascending node (ro). 
But the central travel angle is derived by v = (t - t?) 217P, where P is the 
orbit period which for SPOT is 101.46 hours. A ‘leader’ (*.lea) file is 
delivered with SPOT imagery. This leader file contains a predicted 
ephemeris and measured angular velocities. Initial orbital parameters 
could be derived from a block adjustment of ephemeris data. The 
angular velocities measured about every 82 lines (125 ps) are assumed 
to be of high, but not sufficient, accuracy to model attitude variations 
of the imaging platform. Relative attitude angles in kappa, phi and 
omega are computed by integration, but are interpolated into an 80- 
lines spacing (76 values) to ensure a fast look-up access. This is crucial 
to the real-time implementation for the Leica mapping terminal (LMT). 
With these relative attitude angles, only the constant offsets in kappa, 
phi, and omega are left to be determined. A linear rate of change for 
phi is added for a total of 8 parameters. Linear rates for kappa and 
omega were found to be insignificant, and the same was found for 
quadratic rates of change for the three attitude parameters. Simulations 
showed that the Westin model does not sufficiently model SPOT 
without at least a linear phi parameter. The orientation model is usually 
set-up for a SPOT-panchromatic image. A SPOT-XS image set-up 
would only require minor refinements for its doubled size of imaging 
elements and reduction by half in number of pixels across- and along- 
track. If unchanged, results will still be similar. 
3.0 SPOT FUNCTIONAL MODEL 
The adjustment is done in the earth centred inertial geocentric co- 
ordinate system (ECI), but transformations are required between these 
other systems: 
° The earth centred, earth fixed geocentric system (ECEF). 
° A local geodetic system with a known relationship to 
geographical co-ordinates. 
° À sensor co-ordinate system as described in section 3.21. 
° À local orbital system which coincides with the attitude reference 
system when all three attitude values are zero. 
Control information would normally be available in a local geodetic 
coordinate system. These must be tranformed to geographical 
coordinates, then the ECEF, and finally to the ECI system before they 
could be used in this model. 
The link between a ground control point and its image co-ordinate is: 
Xg = Xs + ARiRbRsxP 
where 
° xP = image co-ordinates vector. 
° Xg = ground co-ordinates vector in ECI 
° Xs = satellite position vector in ECI 
° Au. -= scale factor 
° Ri = Rotation from the orbital reference system to the ECI 
° Rb = Rotation between the attitude reference system and the 
orbital system. The interpolated attitude changes from the 
measurement system are used here. Only the values of kappa, phi, 
and omega offsets at the beginning of the scene are calculated in 
the adjustment. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
e Rs = Rotation between the sensor and the attitude reference 
system; the primary rotation in omega is the mirror inclination; it 
would also take care of the CCD-sensor off-sets, if any. SPOT is 
no longer publishing angular CCD off-sets, usually derived from 
calibration. 
Rs is optional because the system can accomodate for the absence of 
this rotation in the model. Accordingly, the mirror inclination angle can 
be used as an initial value for omega in Rb instead of a zero value. 
The reverse equation is: 
xb E 1/A .R .(Xg- X0) 
where R z(Rs . Rb . Ri) 
In matrix form, this becomes: 
0 Ta ea os Xm Xo 
mne ,. 4 
v, EAE me 5n e yo y 
zc 3 LES 133 Z5 Zo 
e 
yp is the y-image co-ordinate. and c is the focal length of the SPOT 
camera. The SPOT header file provides a scene-center-time from which 
imaging times for any lines could be calculated using the CCD 
integration time which is 1.5004 us for SPOT. Radial data from the 
ephemeris data is fitted to a third degree polynomial with respect to 
time. Ephemeris data is also used to calculate the time at the acending 
node, enabling the dynamic calculation of the travel angle at any image 
point. The radius at any image point could be used to generate the 
camera position co-ordinates by the inverse transformation with the 
orthogonal matrix of keplerian rotation angles (Ri); see below. Of the 
keplerian rotations parameters, Right ascension (Q) and Inclintion (I) 
remain fairly constant; only the travel angle (v) is changing rapidly 
with time. 
AX, - XQ) + 1, (X,- X0) + 1, (X, - X0) 
  
  
F = 0 = —C® : 
nX,-X,) +1, (X,-X,) t n, (X, - X,) 
F. (Xe) + D, (X, - X4) + La (X, - X,) 
= ) = —C® E - = 
s " : Bi (Xe - XQ) + Ba (X 7 Xe) + I3 (X. X) 
The adjustment could correct five observations y-image co-ordinate, 
time of imaging GCP (t, equivalent to x-pixel coordinate), Longitude 
(9). Latitude(A) and height (h).: 
[yp, t, 9, A, h]. 
The 8 parameters of orientation to be corrected are: 
[lo, Qo, to, ro, wo, po, Ko, pto]. 
pto is the linear component of the phi rotational parameter. A Taylor's 
series expansion of the collinearity equations are done but only the first 
order terms are taken. 
8k on 3r Om oF 3 
Bo Op 81. 3 
OF, OF, OF, OF, OF 
> 
2t 
Il 
« 
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