Global Positioning System Reference
In-Depth Information
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422211
242121
224112
211422
121242
112224
Q
∆
=
(7.90)
Each epoch adds a block to the diagonal of
Q
∆
. The triple-difference cofactor matrix
is in the case of
R
=
3
,S
=
4, and
T
=
3,
2
Q
Q
−
Q
∇
=
∆
∆
(7.91)
Q
∆
Q
∆
[26
−
2
The triple-difference cofactor matrix is band-diagonal for
T>
3. The triple-difference
observations between consecutive (adjacent) epochs are correlated. The inverse of the
triple-difference cofactor matrix, which is required in the least-squares solution, is a
full matrix. Eren (1987) gives an algorithm for computing the elements of the cofactor
matrices (7.88) and (7.89), requiring no explicit matrix multiplication. The subscripts
and superscripts of the undifferenced phase observations are used to compute the
elements of the cofactor matrices directly.
The relevant terms of the double-difference carrier phase equation (5.25) are
Lin
—
2.1
——
Nor
*PgE
c
x
p
x
k
−
x
p
x
m
−
x
q
x
k
+
x
q
x
m
f
[26
ϕ
pq
km
=
−
−
−
−
N
pq
km
d
pq
km,ϕ
ε
pq
km,ϕ
+
+
+
(7.92)
f
c
ρ
pq
km
N
pq
km
d
pq
km,ϕ
ε
pq
km,ϕ
=
+
+
+
The residual ionospheric and tropospheric terms are not explicitly listed in (7.92)
since they are expected to cancel over short baselines. Notice the addition of the am-
biguity term
N
pq
km
in (7.92) as compared to the expression (7.64) for pseudoranges.
Assuming again that the station coordinates
x
k
are known, the parameters to be esti-
mated are
x
m
and the double-difference ambiguities. Since the carrier phase multipath
d
pq
km,ϕ
is not known it is typically treated as a model error and ignored. A row of the
design matrix consists of
∂ϕ
pq
km
c
e
m
−
e
q
m
f
∂
x
m
=
(7.93)
and containsa1inthecolumn of the respective double-difference ambiguity param-
eter, and zero elsewhere. The least-squares solution or Kalman filter solution that
estimates the parameters
