Global Positioning System Reference
In-Depth Information
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The symbol
τ
n
−
r
denotes the
τ
distribution with
n
−
r
degrees of freedom. It is related
to Student's
t
by
√
n
−
rt
n
−
r
−
1
τ
n
−
r
=
n
(4.377)
t
n
−
r
−
1
−
r
−
1
+
Fo
r an infinite degree of freedom the
distribution converges toward the Student
di
stribution or the standardized normal distribution, i.e.,
τ
τ
∞
=
t
∞
=
n(
0
,
1
)
.
σ
0
)
. The
hy
pothesis is rejected, i.e., the observation is flagged for further investigation and
po
ssibly rejection, if
Pope's blunder rejection procedure tests the hypothesis
v
i
∼
n(
0
,
σ
v
i
/
|τ
i
| ≥
c
(4.378)
[16
Th
e critical value
c
is based on a preselected significance level. For large systems, the
re
dundancy numbers are often replaced by the average value according to Equation
(4
.348), in order to reduce computation time; thus
τ
i
=
σ
0
σ
0
Lin
—
*
2
——
Sho
PgE
v
i
σ
i
√
(n
(4.379)
−
r)/n
could be used instead of (4.376).
[16
4.
11.2 Data Snooping
Baarda's data snooping applies to the testing of individual residuals as well. The
theory assumes that only one blunder be present in the set of observations. Applying
a series of one-dimensional tests, i.e., testing consecutively all residuals, is called
a data snooping strategy. Baarda's test belongs to the group of un-Studentized tests
which assume that the a priori variance of unit weight is known. The zero hypothesis
(4.357) is written as
v
i
σ
0
√
q
i
∼
n
i
=
n(
0
,
1
)
(4.380)
At
a significant level of 5%, the critical value is 1.96. The critical value for this test is
no
t a function of the number of observations in the adjustment. The statistic (4.380)
us
es the a priori value
σ
0
and not the a posteriori estimate
σ
0
.
and the data snooping procedures work best for iterative solutions.
At each iteration step, the observation with the largest blunder should be removed.
Since least-squares attempts to distribute blunders, several correct observations might
receive large residuals and might be flagged mistakenly.
Both the
τ
