Global Positioning System Reference
In-Depth Information
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Substituting Equation (4.358) into (4.362) gives the limit for the marginally de-
tectable blunder, given the probability levels
α
and
β
0
:
δ
0
√
r
i
σ
i
|∇
0
i
| ≥
(4.363)
Eq
uations (4.361) and (4.363) state that in 100
(
1
− β
0
)
% of the cases, blunders
gr
eater than those given in (4.363) are detected. In 100
β
0
% of the cases, blunders
gr
eater than those given in (4.363) remain undetected. The larger the redundancy
nu
mber, the smaller is the marginally detectable blunder (for the same
σ
i
). It
is
important to recognize that the marginally detectable blunders (4.363) are based
on
adopted probabilities of type-I and type-II errors for the normal distribution.
Th
e probability levels
δ
0
and
β
0
refer to the one-dimensional test (4.357) of the
in
dividual residual
v
i
, with the noncentrality being
α
and
δ
0
. The assumption is that only
on
e blunder at a time is present. The geometry is shown in Figure 4.10. It is readily
cl
ear that there is a simple functional relationship
[15
β
0
)
between two normal
di
stributions. Table 4.9 contains selected probability levels and the respective
δ
0
= δ
n
(
α
,
δ
0
Lin
—
6.9
——
Sho
PgE
values.
The chi-square test (4.263) of the a posteriori variance of unit weight
2
0
is also
σ
se
nsitive to the blunder
δ
i
fo
r the chi-square distribution of the alternative hypothesis. One can choose the
pr
obabilities
∇
i
. In fact, the blunder will cause a noncentrality of
α
chi
and
β
chi
for this multidimensional chi-square test such that
δ
0
=
δ
c
hi
α
chi
,
u
. The factor
δ
0
depends on the degree of freedom because the
ch
i-square distribution depends on it. Baarda's B method suggests equal traceability
of
errors through one-dimensional tests of individual residuals,
v
i
, and the multi-
di
mensional test of the a posteriori variance of unit weight
β
chi
,n
−
[15
2
0
. This is achieved
by
requiring that the one-dimensional test and the multidimensional test have the
sa
me type-II error, i.e.,
σ
β
0
= β
chi
. Under this condition there exists a relationship
be
tween the probability of type-II error, the significance levels, and the degree of
fre
edom expressed symbolically by
r
. The
B
method assures equal traceability but implies different significance levels for the
= δ
n
α
β
0
= δ
chi
α
chi
,
δ
0
,
β
0
,n
−
TABLE 4.9
Selected Probability Levels
in Reliability
α
β
0
δ
0
0.05
0.20
2.80
0.025
0.20
3.1
0.001
0.20
4.12
0.05
0.10
3.24
0.025
0.10
3.52
0.001
0.10
4.57
