Global Positioning System Reference
In-Depth Information
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P
pq
km,
1
P
pq
km,
2
Φ
pq
km
T
pq
km
d
pq
km,
1
,P
d
pq
km,
2
,P
d
pq
km,
1
,
Φ
d
pq
km,
2
,
Φ
ε
pq
km,
1
,P
ε
pq
km,
2
,P
ε
pq
km,
1
,
Φ
ε
pq
km,
2
,
Φ
ρ
+
11 00
1
I
pq
km,
1
,P
N
pq
km,
1
N
pq
km,
2
α
f
00
=
+
+
pq
km,
1
1
−
1
λ
1
0
pq
km,
2
1
−α
f
0
λ
2
Φ
(7.37)
In
single differencing the satellite clock error and the interfrequency bias
T
GD
cancel.
In
double differencing the receiver clock error cancels. The fast changing terms
p
km
ρ
pq
km
in the expressions above can be eliminated by subtracting the respective first
eq
uation. For example, for the case of single differencing we get
and
ρ
=
+
P
km,
2
−
P
km,
1
I
km,
1
,P
N
km,
1
N
km,
2
p
km,
2
,P
p
km,
1
,P
α
f
−
100
δ
− δ
[24
p
km,
1
P
km,
1
p
km,
1
,
p
km,
1
,P
−
2
λ
1
0
Φ
−
δ
Φ
− δ
p
km,
2
P
km,
1
−α
f
−
10
λ
2
p
km,
2
,
p
km,
1
,P
Φ
−
δ
Φ
− δ
Lin
—
2.8
——
Lon
PgE
ε
km,
2
,P
−
ε
km,
1
,P
ε
km,
1
,
Φ
−
ε
km,
1
,P
+
(7.38)
ε
km,
2
,
Φ
−
ε
km,
1
,P
This expression is especially useful for discovering and repairing cycle slips in single-
difference ambiguities. The expression can be further simplified by recognizing that
I
km,
1
,P
[24
0 for short baselines. For longer baselines, the ionospheric term can
be modeled by a first-order polynomial in time (ionospheric bias and drift). The
ambiguities can readily be transformed to
N
km,w
and
N
km,
1
≈
or
N
pq
km,w
and
N
pq
km,
1
,
respectively, by using the
Z
matrix and thus providing the possibility of fixing the
wide-lane ambiguities early. Expressing the carrier phases in cycles, the following
expression can be readily verified from (7.37),
f
d
pq
km,
2
,ϕ
f
2
N
pq
km,w
+
f
1
f
1
−
f
1
+
f
2
N
pq
km,
1
ϕ
pq
km,
1
ϕ
pq
I
pq
km,
1
,ϕ
km,
1
,ϕ
,d
pq
=
+
−
+
km,w
f
2
4
.
5
N
pq
km,w
km,w
ϕ
pq
km,
1
ϕ
pq
≈
+
−
+···
(7.39)
Th
is is the extra-wide-lane equation. If the wide-lane ambiguities are known from
pr
ior analysis, the double-differenced L1 ambiguity can be computed from the carrier
phases only. Fortunately, Expression (7.39) does not depend on the large pseudorange
multipath terms, but on the smaller carrier phase multipath terms. If the wide-lane
ambiguity happens to be incorrectly identified by one, a situation that might occur for
satellites at low elevation angles, the computed L1 ambiguity changes by 4.5 cycles.
The first decimal of the computed L1 ambiguity would be close to 5. Because of prior
knowledge that the L1 ambiguity is an integer, we can use that fact to decide between
two candidate wide-lane ambiguities. This procedure is known as extra widelaning
(Wübbena, 1990).
