The GNSS integer ambiguities: estimation and validation

Table 3.5: Overview of all test statistics to validate the integer ambiguity solution as
proposed in literature. The fixed solution is accepted if both the integer test and the
discrimination test are passed.
integer test discrimination test equation
R1
nˆσ 2 < Fα(n, m − n − p)
R1
nˆσ 2 < Fα(n, m − n − p)
R1
nˆσ 2 < Fα(n, m − n − p)
ˇΩ2
ˇΩ
> Fα ′(m − p, m − p) (3.77)
R2 > k (3.78)
R1
R2
nˆσ 2 > Fα ′(n, m − n − p) (3.83)
ˇΩ < χ 2 β (m − p, δ) ˇ Ω2 > χ 2 α(m − p) (3.84)
R1
nˆσ 2 < Fα(n, m − n − p) R2 − R1 > k (3.86)
R1
nˆσ 2 < Fα(n, m − n − p)
R2−R1
2 √ ˆσ 2 δ
> tα ′(m − n − p) (3.87)
An overview of all test statistics described in this section is given in table 3.5. It can
be seen that there are three pairs of tests which have a similar ’structure’: the two
ratio tests of (3.77) and (3.78); two tests (3.83) and (3.84) that consider the best
and second-best integer solutions separately; and the two tests (3.86) and (3.87) that
look at the difference d = R2 − R1. Note, however, that the discrimination tests of
(3.78) and (3.86) have an important conceptual difference as compared to the other
tests. These two tests take only the float and fixed ambiguity solutions into account
for discrimination, whereas the other tests take the complete float solution into account
together with the fixed ambiguities.
It would be interesting to know which of the approaches works best in all possible situations.
For this purpose in Verhagen (2004) simulations were used for a two-dimensional
example (geometry-free model with two unknown ambiguities). It followed that the ratio
test (3.78) and the difference test (3.86) perform well, provided that an appropriate
critical value is chosen. However, that is exactly one of the major problems in practice.
It should be noted that in practice the ratio tests are often applied without the integer test
(3.76), as it is thought that if one can have enough confidence in the float solution, the
discrimination test is sufficient to judge whether or not the likelihood of the corresponding
fixed ambiguity solution is large enough.
In chapter 5 it will be shown that it is possible to give a theoretical foundation for some
of the discrimination tests. The performance of the tests will also be evaluated, and it
will be shown that one can do better by applying the optimal integer aperture estimator
to be presented there.
3.6 The Bayesian approach
The Bayesian approach to ambiguity resolution is fundamentally different from the approach
using integer least-squares, which is generally used in practice but lacks sound
64 Integer ambiguity resolution