Global Positioning System Reference
In-Depth Information
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By substituting (4.371) and (4.363), we can write this equation as
=
∇
0
i
e
i
PAN
−
1
A
T
Pe
i
∇
0
i
σ
2
0
i
e
i
P
(
I
2
=
∇
−
Q
v
P
)
e
i
=
∇
0
i
p
i
(
1
−
r
i
)
2
0
i
λ
(4.373)
0
0
0
σ
σ
or
1
−
r
i
0
i
0
λ
=
δ
(4.374)
r
i
The values
λ
0
i
are a measure of global external reliability. There is one such value
for each observation. If the
λ
0
i
are the same order of magnitude, the network is ho-
mogeneous with respect to external reliability. If
r
i
is small, the external reliability
factor becomes large and the global falsification caused by a blunder can be signif-
ica
nt. It follows that very small redundancy numbers are not desirable. The global
ex
ternal reliability number (4.374) and the absorption number (4.368) have the same
de
pendency on the redundancy numbers.
[15
Lin
—
9.3
——
No
PgE
4.10.6 Correlated Cases
Th
e derivations for detectable blunders, internal reliability, absorption, and external
re
liability assumes uncorrelated observations for which the covariance matrix
Σ
b
is
di
agonal. Correlated observations are decorrelated by the transformation (4.235). It
ca
n be readily verified that the redundancy numbers for the decorrelated observations
¯
are
[15
r
i
=
Q
v
P
ii
=
I
D
T
AN
−
1
A
T
D
ii
¯
−
(4.375)
In
many applications, the covariance matrix
Σ
b
is of block-diagonal form. For ex-
am
ple, for GPS vector observations, this matrix consists of 3
3 full block-diagonal
m
atrices if the correlations between the vectors are neglected. In this case, the matrix
D
is also block-diagonal and the redundancy numbers can be computed vector by
ve
ctor from (4.375). The sum of the redundancy numbers for the three vector compo-
ne
nts varies between 0 and 3. Since, in general, the matrix
D
has a full rank, the degree
of
freedom
(n
×
−
r)
of the adjustment does not change. Once the redundancy numbers
r
i
are available, the marginal detectable blunders
∇
0
i
, the absorption numbers
A
i
an
d other reliability values can be computed for the decorrelated observations. These
qu
antities, in turn, can be transformed back into the physical observation space by
pr
emultiplication with the matrix
(
D
T
)
−
1
.
¯
4.11 BLUNDER DETECTION
Errors (blunders) made during the recording of field observations, data transfer, the
computation, etc., can be costly and time-consuming to find and eliminate. Blunder
