Global Positioning System Reference
In-Depth Information
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concerns lies again in the linearized double-difference equations. Without loss of
generality, it is sufficient to investigate the difference between one satellite and two
ground stations. Scaled to distances, the relevant portion of the double-difference
equation is
P
pq
p
k
(t)
p
km
(t)
= ρ
− ρ
m
(t)
+···
(5.57)
Th
e linearized form is
d
x
m
+
e
k
e
m
·
dP
pq
km
e
k
e
m
·
d
x
p
=−
·
d
x
k
+
−
(5.58)
N
ext, we transform the coordinate corrections into their differences and sums. This
is
accomplished by
[18
d
x
k
−
d
x
m
=
d(
x
k
−
x
m
)
=
d
b
(5.59)
d
x
k
+
d
x
k
+
d
x
m
x
m
Lin
—
0.8
——
Lon
PgE
=
=
d
x
c
(5.60)
2
2
Th
e difference (5.59) represents the change in the baseline vector, i.e., the change in
le
ngth and orientation of the baseline, and (5.60) represents the change in the geo-
ce
ntric location of the baseline center. The latter can be interpreted as the translatory
un
certainty of the baseline, or the uncertainty of the fixed baseline station. Trans-
fo
rming (5.58) to the difference and sum gives
[18
2
e
k
e
m
·
−
e
k
e
m
·
d
x
c
+
e
k
e
m
·
1
dP
pq
km
d
x
p
=−
+
d
b
−
−
(5.61)
There is a characteristic difference in magnitude between the first bracket and the
others. Allowing an error of the order
O(b/
k
)
, the first bracket simplifies to 2
e
m
or
2
e
k
. The second and the third brackets are of opposite signs but the same magnitude.
It is readily verified that the terms in the latter two brackets are of the order
O(b/
p
ρ
p
k
)
.
ρ
When the baseline vector is defined by
p
k
p
m
b
≡
ρ
−
ρ
(5.62)
Equation (5.61) becomes, after neglecting the usual small terms,
b
ρ
b
ρ
dP
pq
km
e
m
·
d
x
p
=−
d
b
+
m
·
d
x
c
−
m
·
(5.63)
p
p
Th
e orders of magnitude for the coefficients in this equation will not change, even
if
double-difference expressions are fully considered. Equating the first two terms
in
(5.63), we get the relative impact of changes in the baseline and the translatory
po
sition of the baseline from
m
ρ
·
d
b
=
b
·
d
x
c
(5.64)
