Global Positioning System Reference
In-Depth Information
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7.7.4 Ambiguity Function
Th
e least-squares techniques discussed above require partial derivatives and the min-
im
ization of
v
T
Pv
, with
v
and
P
being the double-difference residuals and double-
di
fference weight matrix. The derivatives and the discrepancy terms depend on the
as
sumed approximate coordinates of the stations. The least-squares solution is iter-
ate
d until the solution converges. In the case of the ambiguity function technique,
we
search for station coordinates that maximize the cosine of the residuals. Consider
ag
ain the double-difference observation equation
f
c
ρ
v
pq
km
ϕ
pq
km,a
ϕ
pq
km,b
pq
km,a
N
pq
km,a
ϕ
pq
km,b
=
−
=
+
−
(7.103)
In usual adjustment notation, the subscripts
a
and
b
denote the adjusted and the
observed values, respectively. In (7.103) we have neglected again the residual double-
difference ionospheric and tropospheric terms, as well as the signal multipath term.
The residuals in units of radians are
[26
Lin
—
5
——
Lon
PgE
pq
km
v
pq
km
ψ
=
2
π
(7.104)
The key idea of the ambiguity function technique is to realize that a change in the
integer
N
pq
pq
km
by a multiple 2
km
changes the function
ψ
π
and that the cosine of this
function is not affected by such a change because
cos
ψ
km,L
=
cos
2
km,L
=
cos
2
π
v
pq
km,L
(7.105)
pq
v
pq
N
pq
π
km,L
+ ∆
[26
N
pq
where
km,L
denotes the arbitrary integer. The subscript
L
, denoting the frequency
id
entifier, has been added for the purpose of generality.
There are 2
(R
∆
1
)
double differences available for dual-frequency ob-
se
rvations . If we further assume that all observations are equally weighted, then the
su
m of the squared residuals becomes, with the help of (7.104),
−
1
)(S
−
2
R
−
1
S
−
1
2
R
−
1
S
−
1
v
T
Pv
x
m
,N
pq
km,L
=
v
pq
km,L
2
ψ
km,L
2
1
pq
=
(7.106)
4
π
2
L
=
1
m
=
1
q
=
1
L
=
1
m
=
1
q
=
1
If the station coordinates
x
k
are known, the function could be minimized by varying
the coordinates
x
m
and the ambiguities using least-squares estimation. The ambiguity
function is defined as
2
R
−
1
S
−
1
cos
ψ
km,L
pq
AF (
x
m
)
≡
L
=
1
m
=
1
q
=
1
cos
2
f
L
c
2
R
−
1
S
−
1
pq
km,a
N
pq
km,L,a
ϕ
pq
km,L,b
=
π
ρ
+
−
L
=
1
m
=
1
q
=
1
