Global Positioning System Reference
In-Depth Information
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The rows of
E
are linearly independent of
A
because of (4.177). Thus, replacing the
matrix
C
by
E
in Equations (4.190) through (4.198) gives this special solution:
Q
P
A
T
P
x
P
=−
(4.202)
=
A
T
PA
E
T
E
−
1
E
T
EE
T
EE
T
−
1
E
+
−
Q
P
(4.203)
E
T
EE
T
−
1
E
A
T
PAQ
P
T
P
≡
=
I
−
(4.204)
A
T
PAQ
P
A
T
PA
A
T
PA
=
(4.205)
Q
P
A
T
PAQ
P
=
Q
P
(4.206)
[12
T
B
x
P
x
B
=
(4.207)
T
B
Q
P
T
B
=
Q
B
(4.208)
Lin
—
*
1
——
Nor
*PgE
T
P
x
B
x
P
=
(4.209)
T
P
Q
B
T
P
Q
P
=
(4.210)
The solution (4.202) is called the inner constraint solution. The matrix
T
P
in (4.204)
is symmetric. The matrix
Q
P
is a generalized inverse, called the pseudoinverse of the
normal matrix; the following notation is used:
[12
N
+
=
A
T
PA
+
Q
P
=
(4.211)
The pseudoinverse of the normal matrix is computed from available algorithms of
generalized matrix inverses or, equivalently, by finding the
E
matrix and using Equa-
tio
n (4.203). For typical applications in surveying, the matrix
E
can be readily iden-
tifi
ed. Because of (4.177), the solution (4.202) can also be written as
=−
A
T
PA
E
T
E
−
1
A
T
P
x
P
+
(4.212)
N
ote that the covariance matrix of the adjusted parameters is
2
0
Q
B,C,P
Σ
x
=σ
(4.213)
de
pending on whether constraint (4.169), (4.189), or (4.201) is used.
The inner constraint solution is yet another minimal constraint solution, although it
ha
s some special features. It can be shown that among all possible minimal constraint
solutions, the inner constraint solution also minimizes the sum of the squares of the
parameters, i.e.,
