2. ORIENTATION MODELS 
2.1 Physical model 
The model used for the orientation of linear array 
imagery from satellite sensors is described in Neto 
[1992, 1993] but is summarised here. It differs slightly 
for across and along track stereopairs. 
A geocentric co-ordinate systems is used and the satellite 
orbit is described using Eulerian parameters ( a, e, i, Q, 
Q, F ) to fix the position of the satellite in space where a 
is the semi-major axis, e the eccentricity, i the 
inclination, €2 the longitude, w the argument of perigee 
and F is the true anomaly of the orbit. 
The major components of dynamic motion are the 
Earth's rotation and the satellite movements along the 
orbit path. These motions have been modelled as linear 
angular changes of F and Q with time. 
For across track stereopairs, the collinearity equations are 
used for the orientation of a single image and the 
rotations are described by the orbital elements and the 
attitude of the sensor. 
For alongtrack stereopairs an additional parameter is 
required, namely a variable to represent the time 
displacement, At. Hence if the first image is arbitrarly 
chosen, At sets the position of the second image relative 
to the first, as shown in figures 1 and 2. 
The two images of an alongtrack stereopair are taken 
during a common orbit and have the same values for 
semi-major axis, inclination, longitude of the ascending 
node and argument of perigee. If the origin of the first 
image is taken as origin of the second image, the values 
of the other orientation parameters are also common to 
both images, as they are set for the origin. However, the 
points are identified in the second image by their line 
and sample values, and the line number being a 
measurement of time on the second image is not related 
to the origin of the first image. The time displacement At 
acts as the translation in time suffered by the second 
image relative to the first, so that the orientation 
parameters for each line of the second image are affected 
by the line position x (measurement of time) plus At. The 
orientation parameters for each position become then 
dependent on At and correlation occurs. 
An attitude model can be initially formed using the 
attitude data file provided. The attitude parameters used 
in the iterative process adjust this attitude model to the 
ground control. 
An across track stereopair can be oriented using 6 control 
points per image, providing 12 observation equations, for 
the 10 parameter solution. An along track stereopair can 
be oriented using as few as 3 control points per image, 
562 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
   
providing 12 observation equations, for an 11 parameter 
solution. The number of control points may be decreased 
if a poorer orientation is accepted, for example when 2 
control points per image are used, then a 7 parameter 
orientation is possible with some redundancy. 
The model can accept information on the position and 
attitude of the sensor if it is available, but this is not a 
prerequisite for a solution. 
image 1 image 2 
e.g. e.g. 
forward looking ^ backward looking 
  
  
  
yx 
ie X ) (direction of flight) 
t 
  
  
  
  
  
  
  
  
  
  
  
  
  
Figure 1. Time displacement At between two images 
taken during the same orbit (represented on the image). 
origin of 
origin of 
image 1 At 
image 2 
      
   
direction 
of flight 
Figure 2. Time displacement At between two images 
taken during the same orbit (orbit representation). 
2.2 Polynomial model 
The polynomial model varies from the physical model in 
the way that the position of the sensor is modelled. In 
this case, first order polynomials were adopted to 
describe the position of the sensor in space, instead of 
using the orbital parameters. Higher order polynomials 
were tested, with no improvement in the final accuracy 
of the models. Due to the smooth characteristics of 
satellite orbits, this model can be applied to short arcs of 
orbits. Tests with simulated data for long arcs of orbits 
show that the accuracy worsens and the model is not as 
stable. 
For the orientation of along track stereoscopic pairs, the 
use of the time displacement parameter is still used but 
the algorithm needs more iterations for convergence. The 
physical model is clearly more stable than the 
polynomial approach. 
In this case, an across track stereo pair can be oriented 
using as few as 7 control points per image, providing 14 
    
   
  
   
    
  
    
     
  
   
   
    
     
    
   
    
     
   
    
  
  
   
    
  
  
  
  
  
   
   
    
    
   
  
   
    
   
   
    
   
   
  
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