Global Positioning System Reference
In-Depth Information
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A different form for the solution of the augmented system (4.106) is obtained by
using alternative relations of the matrix partitioning inverse Expressions (A.82) to
(A.89). It follows readily that
=−
N
1
+
N
2
−
1
A
1
M
−
1
w
2
A
2
M
−
1
x
w
1
+
1
2
=−
N
1
+
N
2
−
1
−
w
2
A
2
M
−
1
(4.118)
N
1
x
∗
+
2
N
2
)
−
1
N
2
x
∗
+
w
2
A
2
M
−
1
x
∗
−
=
(
N
1
+
2
Th
e procedure implied by the first line in (4.118) is called the method of adding
no
rmal equations. The contributions of the new observations are simply added ap-
pr
opriately.
The cofactor matrix
Q
x
of the parameters can be written in sequential form as
[11
Q
x
∗
A
2
M
2
+
A
2
Q
x
∗
A
2
−
1
A
2
Q
x
∗
Q
x
=
Q
x
∗
−
(4.119)
Lin
—
0.5
——
No
PgE
=
Q
x
∗
+ ∆
Q
x
is the cofactor matrix of the first group of observations and equals
N
−
1
1
Q
x
∗
. The
contribution of the second group of observations to the cofactor matrix is
Q
x
∗
A
2
M
2
+
A
2
Q
x
∗
A
2
−
1
A
2
Q
x
∗
∆
Q
x
=−
(4.120)
Th
e change
Q
x
can be computed without having the actual observations of the
se
cond group. This is relevant in simulation studies.
The computation of
v
T
Pv
proceeds as usual
∆
[11
v
T
Pv
v
1
P
1
v
1
+
v
2
P
2
v
2
=
(4.121)
k
1
w
1
−
k
2
w
2
=−
The second part of (4.121) follows from (4.95) to (4.99). Using (4.102) for
k
1
, (4.114)
for
x
, (4.116) for
∆
x
, and (4.117) for
k
2
, then the sequential solution becomes
v
T
Pv
v
T
Pv
∗
+ ∆
v
T
Pv
=
(4.122)
v
T
Pv
∗
+
A
2
x
∗
+
w
2
T
M
2
+
A
2
−
1
A
2
x
∗
+
w
2
A
2
N
−
1
=
1
with
v
T
Pv
∗
being obtained from (4.60) for the first group only.
The a posteriori variance of unit weight is computed in the usual way:
v
T
Pv
r
1
+
2
0
=
σ
(4.123)
r
2
−
u
where
r
1
and
r
2
are the number of equations in (4.86) and (4.87), respectively. The
letter
u
denotes, again, the number of parameters.
