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rectified using few GCPs and a second order 
polynomial fit between image and map coordinates. 
In order to retain the parallax effect in these 
pseudo-rectified (intermediate level) products, 
GCP locations must be specified in elevation above 
an arbitrary datum plane as well as 
projected position. 
The correction to intermediate level imagery 
also requires knowledge of incidence angle 0^ 
for each GCP. An orbit model derived from the 
initial ephemeris data for each path was 
established in order to provide the incidence 
angle estimates for any point (see Figure 1). 
Consistency between relative positions as measured 
from the orbital model and GCP coordinates and 
those from ground/slant range values obtained from 
the imagery were also used to refine the 
positioning of the orbital flight. Those new 
ephemeris values were subsequently used for the 
three-dimensional terrain modeling. 
2.1 Orbit model 
The shuttle position in a Cartesian coordinate 
system can be specified completely either by a set 
of six orbit elements (p,Q,I,W,e x ,ey) (see 
Figure 2) or by six vector componpnts - a position 
vector It plus a velocity vector £. As these 
parameters evolve with time, variation resulting 
from non-spherical and non-homogeneous earth make 
it easier to handle these parameters in their 
orbit element form. 
X-Y: the equatorial plane; 
S: the shuttle position; 
0: the center of the earth; 
p: the Euclidean distance between S and 0; 
N: the ascending node; 
P: the perigee of the orbit; 
I: the inclination of the orbit relative tc 
ascending node N; 
Q: right ascension of the ascending node; 
oj: the argument of the perigee; 
W: the argument of the shuttle (ui + true 
anomaly); 
Figure 2. Geometry of the elliptic orbit 
movement. 
Path and position of the shuttle can be 
extrapolated using linear development of orbit 
equations, for any time t^=t + At, providing we 
know at least one set of parameters. Considering 
that ephemeris data were available approximately 
every ten seconds, it has been sufficiently 
accurate to keep only the linear variation part of 
the orbital parameters. Assuming also that e x 
and ey (which are the components of eccentricity 
vector and are expressed as follows: e x =e cosw; 
ey=e sinw, for a quasi-circular but 
non-equatorial orbit) are relatively constant over 
a short period of time, the development of the 
osculatory parameters as a function of time reduces 
to the following: 
p = p 0 + 6p t 
Q = Qq + 6Q t 
I = I 0 + 61 t 
W = W Q + 6W t 
where linear variations 6p,6I3,6l,6W are known by 
celestial mechanic laws. 
Taking the image centre as a reference point, 
linearization of the osculatory parameters for any 
time tj**t allows for an accurate position S of 
the shuttle for each corresponding image point 
M^. Considering that parameter linearization and 
shuttle position make it possible to know the time 
at which any ground point was imaged and that 
shuttle apparent heading (taking into account for 
earth rotation effect) is orthogonal to the line of 
sight, (Figure 3), one can write the line of sight 
equation developed as a function of time which 
makes the scalar product of the two following 
vectors equal to zero: 
[X( ti ) - t ± t e - XiJ.foti) -7 e ]=0 
where • is the scalar product operator; 
X( ti ) is the shuttle position vector at time 
Jrl* 
X(t^) is the shuttle velocity vector at time 
" e is the earth velocity vector; 
Hfj. is the geocentric coordinates of point M^. 
Figure 3. Line of sight representation in 
non-inertial (terrestrial) reference system. 
Developing this equation, we obtain this 
non-linear expression for the time t-^: 
t = 1 
i [arcsin(C) - y - W ] 
6W o 
where C and y are a function of V e , and of 
the osculatory parameters also a function of t^. 
If an approximate value of the viewing time 
t^ 0 (obtained from image measurement) is known 
for each point M^, one can use an iterative 
computation process to refine the value of t^ at 
each step: