Global Positioning System Reference
In-Depth Information
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one-dimensional and multidimensional tests. For details see Baarda (1968, p. 25). In
practical applications one chooses the factor
δ
0
on the basis of a reasonable value for
α
and
β
0
from Table 4.9.
4.10.3 Internal Reliability
Ev
en though the one-dimensional test is based on the assumption that only one
bl
under exists in a set of observations, the limit (4.363) is usually computed for
all
observations. The marginally detectable errors, computed for all observations,
ar
e viewed as a measure of the capability of the network to detect blunders with
pr
obability
(
1
− β
0
)
. They constitute the internal reliability of the network. Because
th
e marginally detectable errors (4.363) do not depend on the observations or on
th
e residuals, they can be computed as soon as the configuration of the network
an
d the stochastic model are known. If the limits (4.363) are of about the same
siz
e, the observations are equally well checked, and the internal reliability is said
to
be consistent. The emphasis is then on the variability of the marginally detectable
bl
unders rather on their magnitude. A typical value is
[15
Lin
—
1
——
Sho
*PgE
δ
0
=
4.
4.10.4 Absorption
Ac
cording to (4.336) the residuals in the presence of one blunder are:
v
=
Q
v
P
(
−
e
i
∇
i
)
(4.364)
[15
Th
e impact on the residual of observation
i
is
∇
v
i
=−
r
i
∇
i
(4.365)
Equation (4.365) is used to estimate the blunders that might cause large residuals.
Solving for
∇
i
gives
v
i
=−
∇
v
i
r
i
+∇
v
i
v
i
r
i
∇
≈−
≈−
(4.366)
i
r
i
because
v
i
v
i
, where
v
i
denotes the residual without the effect of the blunder.
The computation (4.366) provides only estimates of possible blunders. Because the
matrix
Q
v
P
is not a diagonal matrix, a specific blunder has an impact on all residuals.
If several blunders are present, their effects overlap and one blunder can mask others;
a blunder may cause rejection of a good observation.
Equation (4.365) demonstrates that the residuals in least-squares adjustments are
not robust with respect to blunders in the sense that the effect of a blunder on the
residuals is smaller than the blunder itself, because
r
varies between 0 and 1. The ab-
sorption, i.e., the portion of the blunder that propagates into the estimated parameters
and falsifies the solution, is
∇
