pae. NEU
4d
— Mv Mv va iv Lv ve XP Mv Yusha
Mv ça -
where
F'z(F-0)-90'
il’ =90°-i
Q’=Q- 180°
The major components of dynamical 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 which the second degree zonal component J,
of the Earth gravitational potential is the principal component.
The first order perturbations caused by J, and the Earth's
rotation is given by equations 5, 6 and 7 (Kaula, 1966).
(003 LanRe
Q a EM (5)
(3 knReé
WwW -- 7 : ad : [1-5.cos(i)] (6)
3.J2.n-Re?- (3 cos?i - 1)
4-23.(1-e2)32 (7)
where Re is the Earth's semi-major axis, v is the Earth rotation
and n is the mean motion of the satellite. M is the mean anomaly
and M' is its variation with time, from which the value of F can
be calculated.
Since the satellite is not pointing precisely towards the centre of
the Earth, additional attitude rotations are introduced by an
orthogonal matrix RA, which characterizes the effects of pitch,
roll and yaw, and their variations with time. The viewing angle
also varies for each image and the effect of its geometry is added
to the formulae as another orthogonal matrix R,,.
The collinearity equations (8a and 8b) are used for the
orientation of a single image, where uij are the elements of
matrix U (Equation 9), Xa, YA, Z4 are the geocentric
coordinates of each control point and s is a scale factor.
U11(XA-Xg)*u;2(Y A-Y s)*uis(ZA-Zs)
= 0 8
u31(XA-X5)+U32(Ÿ A-Y 5)+U33(ZA-Zs) Ba)
| u21(XA-X5)+u72(YA-Ys)+u73(ZA-Z5) | 3 (8b)
7 ug(XA-Xg)tus(YA-Ys)tus(ZA-Zg) 7
1
U- - R, RA Ro ©)
Since along-track stereopairs are taken during the same orbit,
with a short time interval in between, the method described
above may be advantageously modified. However, an additional
parameter is required, namely a variable to represent the time
displacement, At. Hence if the first image is arbitrary chosen, At
sets the position of the second image relative to the first, as
shown in figures 4a and 4b.
317
image 1 image 2
eg V. e.g.
forwardlooking ^ backward looking
x
K X MI (direction of flight)
Figure 4a - Time displacement At between two images taken
during the same orbit (represented on the image).
origin of
origin of
image 1 At s
image 2
direction
of flight
Figure 4b - Time displacement At between two images taken
during the same orbit (orbit representation).
The two images of an along-track stereopair are taken during a
common orbit, being the values for semi-major axis, inclination,
longitude of the ascending node and argument of perigee the
same for the two images. 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.
In all 10+n parameters are used, 10 to orientate the strip, and n
time displacement parameters for the n+1 images. The 10
parameters used to orientate the strip are the semi-major axis,
the true anomaly, the longitude of ascending node, the orbit
inclination, and six more parameters defining the attitude of the
sensor with time. The argument of perigee was considered nil in
the computations because the orbit eccentricity is very small in
the cases considered (<0.002) and its effect on the model
orientation can be expressed by the effect of the true anomaly.
An attitude model is initially formed using the attitude data file
provided and tue attitude parameters used in the iterative process
adjust this attitude model to the ground control.
