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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: