Global Positioning System Reference
In-Depth Information
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If two receivers
k
and
m
observe two satellites
p
and
q
at the same nominal time,
the double-difference phase observable (5.2) is
f
c
ρ
f
ϕ
pq
pq
N
pq
I
pq
c
T
pq
d
pq
ε
pq
km,
1
,ϕ
(5.25)
km
(t
p
)
km,
1
(t)
=
+
km,
1
(
1
)
+
km,
1
,ϕ
(t)
+
km
(t)
+
km,
1
,ϕ
(t)
+
where
pq
p
q
km
(t
p
)
km
(t
p
)
km
(t
p
)
ρ
= ρ
− ρ
(5.26)
N
pq
N
km
(
1
)
N
km
(
1
)
km
(
1
)
=
−
(5.27)
I
pq
I
km,ϕ
(t)
I
km,ϕ
(t)
km,ϕ
(t)
=
−
(5.28)
[17
T
pq
T
km
(t)
T
km
(t)
km
(t)
=
−
(5.29)
Lin
—
1.3
——
Lon
*PgE
d
pq
d
km,ϕ
(t)
d
km,ϕ
(t)
km,ϕ
(t)
=
−
(5.30)
ε
pq
ε
km,ϕ
−
ε
km,ϕ
km,ϕ
=
(5.31)
Th
e most important feature of the double-difference observation is the cancellation
of
the large receiver clock errors
d
t
k
and
d
t
m
(in addition to the cancellation of
th
e satellite clock errors and the satellite hardware delays). The receiver hardware
de
lays at a given receiver also cancel, as long as they are the same for every satellite
ob
served. Because multipath is a function of the geometry between receiver, satellite,
an
d reflector, the term (5.30) does not cancel in the double-difference observable.
The double-difference integer ambiguity
N
pq
[17
km
plays an important role in accurate
re
lative positioning using double differences. Estimating the ambiguity together with
th
e other parameters as a real number, one gets the so-called
float solution.
If the
es
timated ambiguities
N
pq
km
can be successfully constrained to integer, one gets the
am
biguity fixed solution.
Because of residual model errors the estimated ambiguities
w
ill, at best, be close to integers. Imposing integer constraints adds strength to the
so
lution, because the number of parameters is reduced and the correlations between
pa
rameters reduce as well. Much effort has gone into extending the baseline length
ov
er which ambiguities can be fixed. At the same time, much research has been
ca
rried out to develop algorithms that allow the ambiguities to be fixed from short
ob
servation spans over short baselines. Having the possibility of imposing the integer
co
nstraint on the estimated ambiguity is a major strength of the double-differencing
ap
proach. For details on this topic, see Chapter 7.
The triple difference (5.3) is the difference of two double differences over time
km
(t
p
, t
q
)
f
1
c
ϕ
pq
pq
∆
km,
1
(t
2
,t
1
)
=
∆ρ
+
(5.32)
f
c
∆
I
pq
T
pq
d
pq
ε
pq
km,
1
,ϕ
+ ∆
km,
1
,ϕ
(t
2
,t
1
)
+
km
(t
2
,t
1
)
+ ∆
km,
1
,ϕ
(t
2
,t
1
)
+ ∆
