Global Positioning System Reference
In-Depth Information
cos
2
f
L
c
1
2
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2
R
−
1
S
−
1
pq
km,a
ϕ
pq
km,L,b
=
π
ρ
−
(7.107)
L
=
1
m
=
1
q
=
1
Th
e small double-difference ionospheric, tropospheric, and multipath terms are not
lis
ted explicitly in this equation, although they are present and will affect the ambigu-
ity
function technique just as they do the other solution methods. Nevertheless, if we
as
sume for a moment that these terms are negligible, and that the receiver positions
ar
e perfectly known, then Equation (7.107) shows that the maximum value of the
am
biguity function is 2
(R
1
)
because the cosine of each term could be
1.
Observational noise will cause the value of the ambiguity function to be slightly
be
low the theoretical maximum. Since the ambiguity function does not depend on
th
e ambiguities, it is also independent of cycle slips. This invariant property is the
m
ost outstanding feature of the ambiguity function and is unique among all the other
so
lution methods.
Because the function
−
1
)(S
−
[26
pq
km,L
in (7.104) is small when good approximate coordi-
na
tes are available (typically corresponding to several hundredths of a cycle), we can
ex
pand the cosine function in a series and neglect higher-order terms. Thus,
ψ
Lin
—
0.7
——
Nor
PgE
1
ψ
km,L
2
2!
2
R
−
1
S
−
1
2
R
−
1
S
−
1
pq
pq
km,L
AF (
x
m
)
=
cos
ψ
=
−
+···
L
=
1
m
=
1
q
=
1
L
=
1
m
=
1
q
=
1
2
R
−
1
S
−
1
(7.108)
ψ
km,L
2
1
2
pq
=
2
(R
−
1
)(S
−
1
)
−
[26
L
=
1
m
=
1
q
=
1
2
v
T
Pv
=
2
(R
−
1
)(S
−
1
)
−
2
π
The last part of this equation follows from (7.106). The ambiguity function and the
least-squares solution are equivalent in the sense that the ambiguity function reaches
maximum and
v
T
Pv
minimum at the point of convergence, the correct
x
m
(Lachapelle
et al., 1992).
There are several ways to initialize an ambiguity function solution. The simplest
procedure is to use a search volume centered at some initial estimate of the station
coordinates
x
m
. Such an estimate could be computed from point positioning with
pseudoranges; the size of the search volume would be a function of the accuracy of
the estimate. This physical search volume is subdivided into a narrow grid of points
with equal spacing. Each grid point is considered a candidate for the solution and
used to compute the ambiguity function (7.107). The double-difference ranges
pq
km,a
,
which are required in (7.107), are evaluated for the trial position. As the ambiguity
function is computed by adding the individual cosine terms one double difference at a
time, early exit strategies can be implemented to reduce the computational effort. For
example, if the trial position differs significantly from the true position, the residuals
are likely to be bigger than one would expect due to measurement noise, unmodeled
ionospheric and tropospheric effects, and the multipath. An appropriate strategy could
ρ
