Global Positioning System Reference
In-Depth Information
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TABLE 4.8
Magnification Factor for Standard Ellipses
Probability 1
− α
n
−
r
95%
98%
99%
1
20.00
50.00
100.00
2
6.16
9.90
14.10
3
4.37
6.14
7.85
4
3.73
4.93
6.00
5
3.40
4.35
5.15
6
3.21
4.01
4.67
8
2.99
3.64
4.16
10
2.86
3.44
3.89
12
2.79
3.32
3.72
15
2.71
3.20
3.57
[14
20
2.64
3.09
3.42
30
2.58
2.99
3.28
50
2.52
2.91
3.18
Lin
—
-
——
Lon
PgE
100
2.49
2.85
3.11
∞
2.45
2.80
3.03
The magnification factor,
2
F
2
,n
−
r,
α
, as a function of the probability and the de-
gree of freedom, is shown in Table 4.8. The table shows immediately that a small
degree of freedom requires a large magnification factor to obtain, e.g., 95% prob-
ability. It is seen that in the range of small degrees of freedom, an increase in the
degree of freedom rapidly decreases the magnification factor, whereas with a large
degree of freedom, any additional observations cause only a minor reduction of the
magnification factor. After a degree of freedom of about 8 or 10, the decrease in the
magnification factor slows down noticeably. Thus, based on the speed of decreas-
ing magnification factor, a degree of 10 appears optimal, considering the expense of
additional observations and the little gain derived from them in the statistical sense.
For a degree of freedom of 10, the magnification factor is about 3 to cover 95%
probability.
The hypothesis (4.290) can readily be generalized to three dimensions encompass-
ing the Cartesian coordinates of a three-dimensiona
l network
station. The magnifica-
[14
tion factor of the respective
standard ellipsoid
is
3
F
3
,n
−
r,
α
for it to contain
(
1
)
probability. Similarly, the standard deviation of an individual coordin
ate is co
nverted
to a
(
1
− α
)
probability confidence interval by multiplication with
F
1
,n
−
r,
α
. These
magnification factors are shown in Figure 4.4 for
− α
0
.
05. For higher degrees of
freedom, the magnification factors converge toward the respective chi-square values
because of the relationship
rF
r,
∞
= χ
α =
r
.
For drawing the confidence ellipse at station
P
i
, we need the rotation angle
ϕ
between the
(
x
i
)
and
(
z
i
)
coordinate systems as well as the semimajor and semiminor
axis of the ellipse. Let
(
y
i
)
denotes the translated
(
x
i
)
coordinate system through the
adjusted point
x
i
, then Equation (4.296) becomes
R
i
y
i
z
i
=
(4.299)
