Global Positioning System Reference
In-Depth Information
Figure 8.13 extends the GPS interferometer to two satellites. For
q
, the addi-
tional SV, a second SD metric can be formed:
q
q
q
q
SD
=
φ
+
N
+
S
+
f
τ
(8.11)
km
km
km
km
km
As with (8.6), the expected cancellation of SV transmitted signal phase and clock
bias occurs, and a short baseline will be assumed such that ionospheric and tropo-
spheric propagation delays cancel as well.
The interferometric DD is now formed using the two SDs. Involved in this met-
ric are two separate satellites and the two receivers, one at either end of the baseline,
b
. Differencing (8.10) and (8.11) yields the following:
pq
pq
pq
pq
(8.12)
DD
=
φ
+
N
+
S
km
km
km
km
where the superscripts
p
and
q
refer to the individual satellites, and
k
and
m
are the
individual antennas. With the formation of the DD, the receiver clock-bias terms
now cancel. Remaining is a phase term representing the combined carrier-phase
measurements made at
k
and
m
by the receivers using SVs
p
and
q
, an integer term
made up of the combined unknown integer ambiguities, and a system phase-noise
term consisting primarily of combined multipath and receiver effects [27]. It now
remains to relate the DD to the unknown baseline
b
, which exists between the two
receiver antennas.
Referring again to Figure 8.13, it is evident that the projection of
b
onto the
LOS between
p
and
m
can be written as the inner (dot) product of
b
with a unit vec-
tor
e
p
in the direction of SV
p
. This projection of
b
(if converted to wavelengths by
dividing by
)is
SD
k
p
. Similarly, the dot product of
b
with a unit vector
e
q
in the
direction of SV
q
would result in
SD
k
q
λ
. Rewriting SD (8.10) and (8.11) with this
substitution yields:
(
)
p
p
p
p
p
SD
=⋅
be
λ
−
1
=
φ
+
N
+ +
S
f
τ
km
km
km
km
km
(8.13)
(
)
q
q
−
1
q
q
q
SD
=⋅
be
λ
=
φ
+
NS
+
+
f
τ
km
km
km
km
km
SV q
e
q
SV p
SD
km
p
e
p
m
SD
km
q
b
k
Figure 8.13
GPS interferometer—two satellites.
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