Global Positioning System Reference
In-Depth Information
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LAMBDA is also applicable to estimating a subset of ambiguities. For example,
in the case of dual-frequency ambiguities one might parameterize in terms of the
widelane and the L1 ambiguities. LAMBDA could operate initially on the wide-
lane covariance submatrix and fix the wide-lane ambiguities immediately, and then
attempt to fix the L1 ambiguities as sufficient geometry becomes available. Teunissen
(1997) shows that the
Z
transformation always includes the widelane but goes far
beyond that to achieve an even better decorrelation.
In order to judge the expected performance of the ambiguity resolution, one can
compute the success rate, i.e., the probability of correct integer estimation. The suc-
cess rate depends on the covariance matrix and as such on the geometry embedded
in the functional and stochastic model (Teunissen, 1998).
[29
7.
8.3 Discernibility
Lin
—
*
3
——
Nor
*PgE
The ambiguity testing outlined above is a repeated application of null hypotheses
testing for each ambiguity set. The procedure tests the changes
v
T
Pv
due to the
constraints. The decision to accept or to reject the null hypothesis is based on the
probability of the type-I error, which is usually taken to be
∆
0
.
05. In many cases,
se
veral of the null hypotheses will pass, thus identifying several qualifying ambiguity
se
ts. This happens if there is not enough information in the observations to determine
th
e integers uniquely and reliably. Additional observations might help resolve the
sit
uation. The ambiguity set that generates the smallest
α =
v
T
Pv
fits the float solution
be
st and, consequently, is considered the most favored fixed solution. The goal of
ad
ditional statistical considerations is to provide conditions that make it possible to
di
scard all but one of the ambiguity sets that passed the null hypotheses test.
The alternative hypothesis
H
a
is always relative to the null hypothesis
H
0
. The
fo
rmalism for the null hypothesis is given in Section 4.9.4. In general, the null and
al
ternative hypotheses are
∆
[29
A
2
x
∗
+
2
=
H
0
:
o
(7.194)
A
2
x
∗
+
2
+
H
a
:
w
2
=
o
(7.195)
U
nder the null hypothesis the expected value of the constraint is zero. See also
Eq
uation (4.270). Thus,
E
z
H
0
≡
E
A
2
x
∗
+
2
=
o
(7.196)
Because
w
2
is a constant, it follows that
E
z
H
a
≡
E
A
2
x
∗
+
2
+
w
2
=
w
2
(7.197)
The random variable
z
H
a
is multivariate normal distributed with mean
w
2
, i.e.,
