The GNSS integer ambiguities: estimation and validation

three vc-matrices which are scaled versions of each other: the RTIA aperture pull-in
regions corresponding to situation b are all equal.
The fixed fail rate approach works much better than using a fixed critical value for the
ratio test, because the size of the aperture pull-in region is nicely adjusted in case of a
fixed fail rate, as can be seen by comparing the results corresponding to situation c for
the first three vc-matrices. In practice the choice of a fixed µ often works satisfactorily,
but that is because either the value is chosen very conservative, cf. (Han 1997), or it is
required that the ILS success rate is close to one before an attempt to fix the ambiguities
is made.
The aperture pull-in regions of the DTIA and OIA estimators become more and more
identical the higher the precision is, so that the relations in equation (5.67) become
valid, as can be seen by comparing these regions for the first three vc-matrices.
5.8.3 IAB and IALS estimation
From table 5.2 it follows that IALS estimation may perform close to optimal, but IAB
estimation does not perform so good. This can also be seen in the bottom panels of
figures 5.28 and 5.29. For a certain fail rate, the success rates with IAB estimation are
clearly lower than with IALS estimation.
The advantage of IAB estimation, however, is that exact evaluation of the fail rate and
success rate is possible. Moreover, the results in table 5.2 indicate that for a fixed fail
rate, the aperture parameters of both estimators do not differ that much. This can also
be seen in the top panels of figures 5.28 and 5.29. For all examples µB < µLS. This
gives rise to the idea to choose a maximum allowed fail rate, i.e. Pf < β, and compute
the aperture parameter µB such that Pf,IAB = β. Then apply IALS estimation with
µLS = µB. For this choice it follows from equations (5.29) and (5.35) that the IALS
success rate will always be larger than or equal to the IAB success rate. Based on the
examples it can be expected that the IALS fail rate will then be smaller than β. However,
this approach is not optimal since the corresponding success rate will also be lower than
the one obtained with the IALS fail rate equal to β. This is illustrated with two examples
in table 5.3. Moreover, it is not guaranteed that indeed Pf,IALS(µ) ≤ Pf,IAB(µ).
In section 5.7.1 an approximation of the OIA aperture parameter was given which makes
use of the IAB aperture parameter. For examples 06 01, 06 02, 10 01 and 10 03 the
results are shown in figures 5.26 and 5.27. The approximation does not work very well,
and the results are often too conservative: the aperture pull-in regions are smaller than
needed. Only for low fail rates the approximated aperture parameter is sometimes too
large, so that the actual OIA fail rate will be larger than allowed.
5.9 Performance of IA estimation
The examples in the preceding section aimed at showing the potentials of IA estimation
in general and comparing the different IA estimators. Based on the results it can be
Performance of IA estimation 131