The GNSS integer ambiguities: estimation and validation

Since H3 ⊂ H1 ⊂ H2, starting point is to test H1 against H2. This will answer the
question whether or not the model on the basis of which the float solution is computed
may be considered valid or not. This is important, since the data may be contaminated
by undetected errors and/or the model may not represent reality because some physical
or geometrical effects are not captured correctly (e.g. multipath, atmospheric delays).
The test statistic that allows to test H1 against H2 is given by ˆσ2
σ 2 , and H1 is accepted
when
ˆσ 2
σ 2 < Fα(m − n − p, ∞, 0) (3.73)
where Fα(m − n − p, ∞, 0) is the critical value that corresponds to the central F -
distribution, F (m − n − p, ∞, 0), and the chosen level of significance α. This test is
identical to the overall model test in section 2.5.2.
If the value of the test statistic has passed the test in expression (3.73), the next step
is to verify whether or not hypothesis H3 may be considered valid. This means testing
H3 against H1. Alternatively, one could test H3 against H2 in the case one is not sure
if the first hypothesis is true. This would result in the following test:
ˇσ 2
σ 2 < Fα(m − p, ∞, 0) (3.74)
However, testing H3 against H1 is more powerful and focuses on the question whether
the fixed ambiguity vector ǎ may be considered valid. The test statistic is given by
(â−ǎ) T G −1
â (â−ǎ)
nσ2 , which is distributed under H3 as F (n, ∞, 0). Hence, the fixed ambiguities
are considered valid if:
(â − ǎ) T G −1
â (â − ǎ)
nσ2 = R1
nσ2 < Fα(n, ∞, 0) (3.75)
Note that from equation (3.14), with the last term equal to zero, it follows that:
R1 = (â − ǎ) T G −1
â (â − ǎ) = ěT G −1
y ě − ê T G −1
y ê = ˇ Ω − ˆ Ω
This shows that test (3.75) can also be expressed in terms of the test statistics of (3.73)
and (3.74).
The test in (3.75) is based on the distance - as measured by the metric Qâ - between the
float and fixed solution. This is reasonable, since one would not have much confidence
in the fixed solution if the distance to the float solution is large. In that case one should
not use the fixed solution and instead stick to the float solution and/or gather more data
so that the estimation and validation process can be repeated.
The value of the variance factor is not always known. In that case σ 2 can be replaced
by ˆσ 2 , so that the test becomes:
R1
nˆσ 2 < Fα(n, m − n − p, 0) (3.76)
The means of the test statistics of (3.73), (3.74) and (3.75) are all equal to one, whereas
the mean of the test statistic of (3.76) equals (m − n − p)/(m − n − p − 2), which is
Validation of the fixed solution 59