Global Positioning System Reference
In-Depth Information
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Note that the caret identifies the estimated parameters. Consequently, the estimate of
the quadratic form is
R
z
2
=
z
2
(4.253)
Be
cause there are
r<u
equations for the
u
parameters in (4.252), there always
ex
ists a solution for
x
. The simplest approach is to equate
u
−
r
parameters to zero.
Th
is would be identical to having these
u
r
parameters treated as constants in
th
e adjustment. They could be left out when setting up the design matrix and thus,
th
e singularity problem would be avoided altogether. Equation (4.252) can be solved
su
bject to
u
−
r
general conditions between the parameters. The resulting solution
is
a minimal constraint solution. If the particular condition (4.201) is applied, one
ob
tains the inner constraint solution. If
A
has no rank defect, then the system (4.252)
co
nsists of
u
equations for
u
unknowns.
The estimate for the quadratic form (4.253) does not depend on the parameters
x
and, thus, is invariant with respect to the selection of the minimal constraints
fo
r finding the least-square estimate of
x
. Moreover, the residuals themselves are
in
dependent of the minimal constraints. Substituting the solution (4.252) into (4.247)
gives
−
[13
Lin
—
1.9
——
Lon
PgE
F
T
G
T
v
o
G
T
=
(4.254)
¯
Si
nce the matrix [
FG
] is orthonormal, the expression for the residuals becomes
[
FG
]
o
G
T
[13
¯
v
GG
T
=
=
(4.255)
¯
Thus, the residuals are independent of the specific solution for
x
. The matrix
G
depends only on the structure of the design matrix
A
. By applying the law of variance-
covariance propagation to (4.255), we clearly see that the covariance matrix of the
adjusted residuals, and thus the covariance matrix of the adjusted observations, does
not depend on the specific set of minimal constraints. Note that the transformation
(4.235) does not invalidate these statements, since the
D
matrix is not related to the
parameters.
Returning to the derivation of the distribution of
v
T
Pv
, we find from (4.248) that
F
T
Ax
o
−
E
(
z
)
=
(4.256)
us
ing (4.245) and the fact that
E(
¯
Ax
according to (4.236). Making use of
)
=−
(4
.240) the covariance matrix is
F
T
G
T
I
[
FG
]
F
T
FF
T
G
G
T
FG
T
G
IO
OI
2
0
2
0
2
0
Σ
z
= σ
= σ
= σ
(4.257)
