Global Positioning System Reference
In-Depth Information
=
R
3
(
p
)
x
p
(t
p
)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
p
k
ρ
−θ
k
)
x
k
−
R
3
(
−θ
(5.43)
If
τ
denotes the travel time for the signal, then the earth rotates during that time by
= Ω
e
t
k
−
t
p
= Ω
e
τ
p
θ = θ
k
− θ
(5.44)
with
Ω
e
being the earth rotation rate and
τ
the travel time of the signal. The topocentric
distance becomes
R
3
(
)
x
p
(t
p
)
p
k
ρ
=
−θ
k
)
x
k
−
R
3
(
−θ
k
+ θ
=
R
3
(
p
)
R
3
(
x
p
(t
p
)
−θ
−θ
)
x
k
−
(5.45)
=
R
3
(
x
p
(t
p
)
−θ
)
x
k
−
[17
x
k
−
)
x
p
(t
p
)
=
R
3
(
θ
Lin
—
-
——
No
PgE
In modifying (5.45), we used the facts that a distance is invariant with respect to the
rotation of the coordinate system and that the rotation matrix
R
3
is orthonormal.
Because
θ
is a function of
τ
, Equation (5.45) must be iterated. A good initial
estimate is
τ
0
=
0
.
075 sec. Computing
θ
1
from (5.44) and using this value in (5.45)
gives the initial value
ρ
1
for the distance. The second estimate of the travel time
follows from
τ
2
= ρ
1
/c
. This value is used in (5.44) to continue the iteration loop.
5.
3.3 Cycle Slips
[17
A cycle slip is a sudden jump in the carrier phase observable by an integer number
of cycles. The fractional portion of the phase is not affected by this discontinuity
in the observation sequence. Cycle slips are caused by the loss of lock of the phase
lock loops. Loss of lock may occur briefly between two epochs or may last several
minutes or more, if the satellite signals cannot reach the antenna. If receiver software
would not attempt to correct for cycle slips, it would be a characteristic of a cycle
slip that all observations after the cycle slip would be shifted by the same integer.
This is demonstrated in Table 5.1, where a cycle slip is assumed to have occurred for
TA
BLE 5.1
Effect of Cycle Slips on Carrier Phase Differences
Double
Triple
Carrier Phase
Difference
Difference
ϕ
k
(i
−
2
)
ϕ
m
(i
−
2
)
ϕ
k
(i
−
2
)
ϕ
m
(i
−
2
)
ϕ
pq
∆
ϕ
pq
km
(i
−
2
)
km
(i
−
1
,i
−
2
)
ϕ
k
(i
−
1
)
ϕ
m
(i
−
1
)
ϕ
k
(i
−
1
)
ϕ
m
(i
−
1
)
ϕ
pq
∆
ϕ
pq
km
(i, i
−
1
)
−
∆
km
(i
−
1
)
ϕ
k
(i)
ϕ
m
(i)
ϕ
k
(i)
+
∆
ϕ
m
(i)
ϕ
pq
∆
ϕ
pq
km
(i)
−
∆
km
(i
+
1
,i)
ϕ
k
(i
ϕ
m
(i
ϕ
k
(i
ϕ
m
(i
ϕ
pq
ϕ
pq
+
∆
−
∆
∆
+
1
)
+
1
)
+
1
)
+
1
)
km
(i
+
1
)
km
(i
+
2
,i
+
1
)
ϕ
k
(i
ϕ
m
(i
ϕ
k
(i
ϕ
m
(i
ϕ
pq
+
∆
−
∆
+
2
)
+
2
)
+
2
)
+
2
)
km
(i
+
2
)
