The GNSS integer ambiguities: estimation and validation

5.1 Integer Aperture estimation
The problem described above is related to the definition of admissible integer estimators,
which only distinguishes two situations: success if the float ambiguity falls inside the
pull-in region Sa, and failure otherwise. In practice, a user will decide not to use the
fixed solution when the probability of failure is too high. This gives rise to the thought
that it might be interesting to consider not only the two above-mentioned situations,
but also a third, namely undecided. Actually, with combined integer estimation and
validation using a discrimination test, these three situations are also distinguished, but
the discrimination tests can only be used in the case of high precision. Therefore a
new class of integer estimators is defined that allows for the three situations mentioned
above. Then integer estimators within this class can be chosen which somehow regulate
the probability of each of those situations. This can be accomplished by considering
only conditions (ii) and (iii) of definition 3.1.1, stating that the pull-in regions must be
disjunct and translational invariant.
5.1.1 Class of IA estimators
The new class of ambiguity estimators was introduced in Teunissen (2003d;2003g), and
is called the class of Integer Aperture (IA) estimators. It is defined as:
(i)
z∈Z n
Ωz =
z∈Z n
(Ω ∩ Sz) = Ω ∩ (
z∈Z n
Sz) = Ω ∩ R n = Ω
(ii) Ωu ∩ Ωz = (Ω ∩ Ωu) ∩ (Ω ∩ Ωz) = Ω ∩ (Su ∩ Sz) = ∅, u, z ∈ Z n , u = z
(iii) Ω0 + z = (Ω ∩ S0) + z = (Ω + z) ∩ (S0 + z) = Ω ∩ Sz = Ωz, ∀z ∈ Z n
(5.1)
Ω ⊂ R n is called the aperture space. From (i) follows that this space is built up of
the Ωz, which will be referred to as aperture pull-in regions. Conditions (ii) and (iii)
state that these aperture pull-in regions must be disjunct and translational invariant.
Note that the definition (5.1) is very similar to the definition of the admissible integer
estimators (3.1.1), but that R n is replaced by Ω.
The integer aperture estimator, ā, is now given by:
ā =
zωz(â) + â(1 −
ωz(â)) (5.2)
z∈Z n
z∈Z n
with the indicator function ωz(x) defined as:
ωz(x) =
1 if x ∈ Ωz
0 otherwise
(5.3)
So, when â ∈ Ω the ambiguity will be fixed using one of the admissible integer estimators,
otherwise the float solution is maintained. It follows that indeed the following three cases
88 Integer Aperture estimation