Global Positioning System Reference
In-Depth Information
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km
2
km
Q
pi
2
Q
pi
x
i
x
i
x
i
˜
· ˜
−
−
˜
· ˜
x
m
+˜
x
m
=
0
(7.71)
This equation can be verified using
2
Λ
= ˜
x
m
· ˜
x
m
(7.72)
x
i
Q
pi
km
,
x
i
Q
pi
km
1
2
i
α
=
(7.73)
x
1
˜
y
1
˜
z
1
˜
B
=
x
2
˜
y
2
˜
z
2
˜
(7.74)
x
3
y
3
z
3
˜
˜
˜
[26
km
Q
p
1
km
Q
p
2
km
Q
p
3
T
τ
=
−
−
−
(7.75)
Lin
—
0.6
——
No
PgE
B
−
1
(
x
m
=
˜
Λ
τ
+
α
)
(7.76)
Su
bstituting (7.76) in (7.72) gives the quadratic equation for
Λ
,
B
−
1
τ
−
1
Λ
2
B
−
1
α
Λ +
B
−
1
α
=
,
B
−
1
2
,
B
−
1
,
B
−
1
τ
+
τ
α
0
(7.77)
Λ
Th
e two solutions for
are substituted in (7.76) to obtain two positions for station
m
. The ellipsoidal height can be used to decide which of the positions for
m
is the
co
rrect one.
The closed formulas can be generalized for more than four satellites. In this case
th
e number of rows in
B
equals the number of satellites or the number of double
di
fferences. We multiply (7.52) from the left with
B
T
[26
,
B
and set
α
=
B
T
α
=
B
T
B
,
B
T
and
τ
=
τ
. Equations (7.59) or (7.77) can then be rewrite in the bar-notation and
so
lved for
Λ
.
7.7.2 Double-Difference Float and Triple-Difference Solutions
R
receivers observing
S
satellites at
T
epochs generate at most
RST
carrier phase
ob
servations. In many cases, the data set might not be complete due to cycle slips and
sig
nal blockage. To see the symmetry of the expressions, we order the undifferenced
ph
ase observations
ψ
first by epoch, then by receiver, and then by satellite. For epoch
i
w
e have
ϕ
1
(i)
ϕ
R
(i)
]
T
ψ
i
=
[
ϕ
1
(i)
···
···
ϕ
R
(i)
···
(7.78)
ψ
1
.
ψ
T
ψ
=
(7.79)
