The GNSS integer ambiguities: estimation and validation

The test is then defined by the constraint that P (γ < 0.5 | Ha) > β. However, the
problem is again that the test statistic does actually not have a Student’s t-distribution
for the same reason as described above for test (3.87). Moreover, it can be easily shown
that ˆγ is always smaller than or equal to 0.5 so that the definition of the test is not
rigorous:
ˆγ = (ǎ2 − ǎ) T G −1
â (â − ǎ)
(ǎ − ǎ2) T G −1
â (ǎ − ǎ2)
= ǎT2 G −1
â â − ǎT G −1
â â − ǎT2 G −1
â ǎ − ǎT G −1
â ǎ
ǎT 2 G−1
â ǎ2 − 2ǎT 2 G−1
â ǎ + ǎT G −1
â ǎ
R1 + Γ
=
R1 + R2 + 2Γ
≤ R1 + Γ 1
=
2(R1 + Γ) 2
with Γ = ǎ T 2 G −1
â â + ǎT G −1
â â − ǎT 2 G −1
â ǎ − âT G −1
â â.
(3.90)
It should be noted that in Wang et al. (1998) another test statistic was presented with
a similar form as that of Han (1997).
3.5.3 Evaluation of the test statistics
A problem with all of the above-mentioned tests is the choice of the critical values.
Either an empirically determined value has to be used, or they are based on the incorrect
assumption that the fixed ambiguity estimator ǎ may be considered deterministic. In
principle, this is not true. Firstly, because the entries of the vector ǎ depend on the same
vector y as used in the formulation of the hypothesis H3 in (3.71). So, if y changes,
also ǎ will change. Secondly, since the vector y is assumed to be random, also the fixed
ambiguities obtained with integer least-squares estimation will be stochastic.
Integer validation based on for example the ratio tests in (3.77) and (3.78) often work
satisfactorily. The reason is that the stochasticity of ǎ may indeed be neglected if there
is sufficient probability mass located at one integer grid point of Z n , i.e if the success
rate is very close to one. So, one could use the additional constraint that the success rate
should exceed a certain limit. But this immediately shows that this will not be the case in
many practical situations. In those cases, the complete distribution function of the fixed
ambiguities should be used for proper validation. It would thus be better to define an
integer test that takes into account the randomness of the fixed ambiguities. Starting
point is to replace the third hypothesis in (3.71) with the following, see (Teunissen
1998b):
H3 : y = Aa + Bb + e, a ∈ Z n , b ∈ R p , e ∈ R m
(3.91)
This hypothesis is more relaxed than the one used in (3.71), but makes it possible to
find a theoretically sound criterion to validate the integer solution.
Another point of criticism is that the combined integer estimation and validation solution
lacks an overall probabilistic evaluation.
Validation of the fixed solution 63