Global Positioning System Reference
In-Depth Information
or, in the case of multipath, some mean value. Equation (8.33) is rewritten next in
light of these ideas:
~
~
[
]
[
]
qDD
−
n
λ
+
q DD
−
n
λ
1
s
1
1
2
s
2
2
(8.34)
~
~
[
]
[
]
+
qDD
−
n
λ
+
qDD
−
n
λ
=
γ
3
s
3
3
4
s
4
4
1-2m,
depending on the multipath environment. With this in mind, the values for
n
in
(8.34) can be adjusted such that the result is near to zero—at least within some pre-
determined threshold (
Once again it is noted that the smoothed-code DDs are bounded within
±
γ
). Assuming that the receiver noise and interchannel biases
can be kept to below
/2 (which is generally the case), it becomes possible to use
(8.34) to resolve the carrier-cycle ambiguities. Putting (8.34) into matrix form:
λ
[
]
qDD N
s
−
λγ
(8.35)
[
~~ ~~
]
where
N
12 34
and represents a set of integer values that, when substituted
into the equation, satisfy the threshold constraint (i.e., ). The question now
becomes one of how to find the
N
vectors that produce such a result.
Since there are only four multiplication operations and three additions required
to examine each case, one answer to such a question is to use an exhaustive search.
With a
=
nn nn
1-2m bound on the accuracy using the smoothed-code DDs, such a search
requires that components of
N
contain iterations covering
±
at L1, where the
wavelength is 19.03 cm. There are 23
4
, slightly less than 300,000, possible candi-
dates for the first epoch, which is not an unreasonable number. If necessary, more
efficient search strategies could be implemented; however, when the wide-lane
wavelength is examined at the end of this chapter, the number of candidates will
drop to less than 3,000, which then makes the exhaustive search almost trivial. In
any event, as the integer values are cycled from [-11 -11 -11 -11] to [
±
11
λ
+
11
+
11
+
11
+
11], those integer sets that are within the threshold are retained and become candi-
dates for the fixed baseline solution.
8.4.1.7 Final Baseline Determination (Fixed Solution)
For each epoch, the various
N
sets that meet the threshold constraint of (8.35) are
stored, or, if stored previously, a counter (
j
) is incremented to indicate persistence of
the particular ambiguity set. For those sets that persist, a sample mean (
η
avg
) is calcu-
lated based on the first 10 values of the residual. The variance (
η
σ
2
) about the sample
mean is determined as well. These calculations are as follows:
[
]
(
)
η
j
−+
1
η
avg
j
η
=
j
−
1
j
≤
10
and
η
=
0
(8.36)
avg
avg
j
j
0
(
)
2
η
=
η
+
η
−
η
j
=
12
,
2
2
avg
j
σ
σ
j
j
j
−
1
[
]
(8.37)
2
(
)
(
)
j
−
2
η
+
η
−
η
avg
j
2
σ
j
η
=
j
−
1
j
>
2
(
)
2
σ
j
−
1
j
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