Global Positioning System Reference
In-Depth Information
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of rows 1 and 2, which correspond to the respective positions of
x
1
and
x
2
in
x
. With
these specifications it follows that
q
x
1
q
x
1
,x
2
A
2
N
−
1
A
2
Q
i
=
=
(4.292)
q
x
2
,x
1
q
x
2
Q
i
contains the respective elements of the inverse of the normal matrix. With these
specifications
T
Q
−
1
i
=
and Expression (4.280) becomes
x
i
−
x
i,T
T
Q
−
i
x
i
−
x
i,T
∼
1
2
F
2
,n
−
r
(4.293)
2
0
σ
Given the significance level
, the hypothesis test can be carried out. The two-
dimensional confidence region is
P
(
x
i
−
α
[14
F
2
,n
−
r,
α
x
i
)
T
Q
−
1
i
(
x
i
−
x
i
)
≤
F
2
,n
−
r,
α
=
F
2
,n
−
r
dF
=
1
− α
(4.294)
2
0
2
σ
0
Lin
—
1
——
Lon
PgE
The size of the confidence ellipses defined by (4.294) depends on the degree of
freedom of the adjustment and the significance level. The ellipses are centered at
the adjusted position and delimit the
(
1
)
probability area for the true position.
The principal axis form of (4.294) is obtained through orthogonal transformation. Let
R
i
denote the matrix whose rows are the orthonormal eigenvectors of
Q
i
, then
− α
R
i
Q
−
1
R
i
=
Λ
−
1
(4.295)
i
i
[14
Q
i
according to (A.48). The matrix
Λ
i
is diagonal and contains the eigenvalues
λ
and
Q
2
of
Q
1
. With
λ
R
i
x
i
−
x
i
z
i
=
(4.296)
Expression (4.294) becomes
/
1
,
-
≤
z
1
z
2
P
2
+
1
2
σ
0
σ
0
2
.
Q
1
2
F
2
,n
−
r,
α
Q
2
2
F
2
,n
−
r,
α
λ
λ
(4.297)
F
2
,n
−
r,
α
=
F
2
,n
−
r
dF
=
1
− α
0
Fo
r
F
2
,n
−
r,
α
=
1
/
2, the ellipse is called the
standard ellipse
or the
error ellipse.
Thus,
th
e probability enclosed by the standard ellipse is a function of the degree of freedom
n
−
r
and is computed as follows:
1
/
2
P(standard ellipse)
=
F
2
,n
−
r
dF
(4.298)
0
