0
0
1 cm^{2 }
0
0
0
0
0
0
or
0
4 cm^{2 }0
0
0
0
2^{2}*0.01 m^{2 }
0
0
0
0
(24)
0
4 cm^{2 }
0
2 cm^{2 }
0
0 0
0
2 cm^{2 }0
The time separation from the central observation is
inversely proportional to the weight.
considered
between
observations.
The
No correlation was P (weight) matrix is
a
diagonal matrix with the following elements when a 1 Hertz sample
rate is used:
= 1/(2
2
= 1/(2
1
= 1/(2
0
= 1/(2
1
= 1/(2
2
Sigma
2
1
Sigma
2
2
Sigma
2
3
Sigma
2
4
Sigma
2
5
P
11
P
22
P
33
P
44
P
55
= 1.0 /
2 * 0.01 m )
(25)
= 1.0 /
2 * 0.01 m )
(26)
= 1.0 /
2 * 0.01 m )
(27)
= 1.0 /
2 * 0.01 m )
(28)
= 1.0 /
2 * 0.01 m )
(29)
The a priori variance of unit weight is 1.0.
The validity
of of
this the a
weight system can be proven using a Chi Square test posteriori variance of unit weight against the a priori
variance of in 43 of 45 be tempered
unit weight. The test selected interpolation
by
the
knowledge
that
indicated equal variances
tests.
The
validity
must
the
degrees
of
freedom
for
the 5
test is only 3 = 2).
2
(number
of
observations

number
of
unknowns,
The interpolated values for the antenna phase center at the time of the exposure are expressed as follows:
X_{exp }
= K_{x}(1) + K K_{x}(3)*(time
x exp
(2)*(time  time_{3})
exp 2
Y_{exp }
= K_{y}(1) + K K_{y}(3)*(time
y exp
(2)*(time  time_{3})
exp 2
Z_{exp }
= K_{z}(1) + K K_{z}(3)*(time
z exp
(2)*(time  time_{3})
exp 2

time_{3}) +
(30)

time_{3}) +
(31)

time_{3}) +
(32)
As a final note, the model used in this paper only
approximates mathematical
the actual trajectory of the aircraft.
The
modeling
process
acts
as
a
filter,
smoothing
the
aircraft
trajectory
using
a
second
order
polynomial.
It
would
be interesting to examine the extent of smoothing which actually
takes place.
The model
is
validated
by
the
results
illustrated
in Table 1. This same model James Lucas (Lucas 1989) and National Geodetic Survey.
was independently is currently used
developed by within the
ANTENNA PHASE CENTER TRANSFORMATION TO EXPOSURE STATION
The interpolated geocentric position of the antenna phase center at the exposure time can now be transformed through the previously determined system of spatial offsets to the position of the exposure station.