Global Positioning System Reference
In-Depth Information
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Substituting Expression (4.178) for
S
into (4.172) gives the nonsymmetric matrix
B
T
EB
T
−
1
E
A
T
PAQ
B
=
T
B
≡
I
−
(4.182)
This expression is modified with the help of (4.174), (4.176), and (4.182):
A
T
PA
B
T
B
Q
B
+
E
T
BE
T
−
1
EB
T
−
1
E
+
=
I
(4.183)
It can be solved for
Q
B
:
Q
B
=
A
T
PA
B
T
B
−
1
E
T
EB
T
BE
T
−
1
E
+
−
(4.184)
The least-squares solution of
x
B
subject to condition (4.169) is, according to (4.170),
(4.171), and (4.181),
[12
Q
B
A
T
P
x
B
=−
(4.185)
Lin
—
1.5
——
Nor
PgE
The cofactor matrix of the parameters follows from the law of variance-covariance
propagation
Q
B
A
T
PAQ
B
=
Q
x
B
=
Q
B
(4.186)
The latter part of (4.186) follows from (4.182) upon multiplying from the left by
Q
B
and using (4.174). Multiplying (4.182) from the right by
A
T
PA
and using (4.177)
gives
[12
A
T
PA
A
T
PAQ
B
A
T
PA
=
(4.187)
The relation implied in (4.186) is
Q
B
A
T
PAQ
B
=
Q
B
(4.188)
r
conditions are necessary to solve the least-squares problem; i.e., the minimal
number of conditions is equal to the rank defect of the design (or normal) matrix. Any
solution derived in this manner is called a minimal constraint solution. There are ob-
viously many different sets of minimal constraints possible for the same adjustment.
The only prerequisite on the
B
matrix is that it have full row rank and that its rows be
linearly independent of
A
. Assume that
u
−
Cx
C
=
o
(4.189)
is an alternative set of conditions. The solution
x
C
follows from the expressions given
by simply replacing the matrix
B
by
C
. The pertinent expressions are
Q
C
A
T
P
x
C
=−
(4.190)
