LEAST-SQUARES ADJUSTMENTS - GPS Satellite Surveying - page 153

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given in Table 4.1 and the specification (4.342) for the redundancy number r i as the
diagonal element of Q v P , it follows that if the redundancy number is close to 1,
then the variance of the residuals is close to the variance of the observations, and
the variance of the adjusted observations is close to zero. If the redundancy number
is close to zero, then the variance of the residuals is close to zero, and the variance of
the adjusted observations is close to the variance of the observations.
Intuitively, it is expected that the variance of the residuals and the variance of
the observations are close; for this case, the noise in the residuals equals that of the
observations, and the adjusted observations are determined with high precision. Thus
the case of r i close to 1 is preferred, and it is said that the gain of the adjustment is high.
If r i is close to zero, one expects the noise in the residuals to be small. Thus, small
residuals as compared to the expected noise of the observations are not necessarily
desirable. Because the inequality (4.344) is a result of the geometry as represented
by the design matrix A , small residuals can be an indication of a weak part of the
network.
Because the weight matrix P is considered diagonal, i.e.,

[15

Lin
—
3.7
——
No
PgE

2
0

= σ

p i (4.346)

i

σ

it follows that

r i

=σ 0 √ q i =σ 0 r i

i

σ σ 0

σ 0 σ i √ r i

σ v i

p i =σ 0

= (4.347)

2
0

σ

[15

From (4.341) it follows that the average redundancy number is

n − R( A )

n

r av = (4.348)

Th e higher the degree of freedom, the closer the average redundancy number is to 1.
Ho wever, as seen from Table 4.8, the gain, in terms of probability enclosed by the
sta ndard ellipses, reduces noticeably after a certain degree of freedom.

4. 10.2 Controlling Type-II Error for a Single Blunder

Baarda's (1967) development of the concept of reliability of networks is based on un-
Studentized hypothesis tests, which means that the a priori variance of unit weight is
assumed to be known. Consequently, the a priori variance of unit weight (not the a
posteriori variance of unit weight) is used in this section. The alternative hypothesis
H a specifies that the observations contain one blunder, that the blunder be located at
observation i , and that its magnitude be ∇ i . Thus the adjusted residuals for the case

of the alternative hypothesis are

v | H a = v − Q v Pe i ∇ i (4.349)