The GNSS integer ambiguities: estimation and validation

This means that the higher the precision, the more identical the aperture pull-in regions
of the difference test and the optimal test become. It follows from equation (5.66) that
then
µO ≈ exp{− 1
2 µD} + 1 or µD ≈ −2 ln(µO − 1) (5.67)
In figures 5.26 and 5.27 the aperture parameter µO is shown as function of the fail rate for
different examples. The black solid lines show the µO that follow from the simulations,
the dashed grey line show the µ ′ O as approximated with (5.67) as function of the DTIA
fail rate. For example 10 01, the precision is high and the results in table 5.2 already
showed that the DTIA estimator performs almost identical to the OIA estimator. It
could therefore be expected that µO ≈ exp{− 1
2 µD} + 1, which is indeed shown in the
figure. On the other hand, for the other examples the DTIA estimator performs worse
than the OIA estimator. Because of the poorer precision the terms Ri, i > 3, and thus
the last term on the right-hand side of (5.63), cannot be ignored. Not surprisingly µO
is clearly different from µ ′ O .
In the bottom panels of the figures, the OIA success rates as function of the ’fixed’ fail
rate are shown in the case µO would be used, and in the case µ ′ O would be used. So,
in the latter case the success rate is shown as function of the DTIA fail rate which was
used to determine µ ′ O . As expected, only for example 10 01 approximately the same
results are obtained.
From equation (5.66) follows that it is not possible to rewrite the OIA estimator in the
same form as the ratio test (3.78), but both tests depend on R1 and R2, which is the
reason that similar results can be obtained with RTIA and OIA estimation if it would be
possible to choose the critical value of the ratio test in an appropriate way. In practice
µR is often chosen equal to a fixed value and the ratio test should only be applied if the
precision is high. Figure 5.24 shows on the left the aperture parameter µR as function
of the fail rate for the different examples. It can be seen that for the examples where
n = 10 a small fail rate will be obtained when µR = 0.3; in the case of poorer precisions
on the other hand, it is not clear how the appropriate µR can be chosen such that the fail
rate will be low enough, but at the same time such that the test is not too conservative.
The F -ratio test
ê2 + R1 Qy
ê2 ≤ µR
+ R2 Qy
also looks at R1 and R2, but as explained this IA estimator is not really comparable
with all others because it also depends on the float residuals ê. In practice the aperture
parameter is often chosen equal to 1
2 or as 1/Fα(m − p, m − p, 0), see section 3.5. The
panels on the right of figure 5.24 show the aperture parameter as function of the fail
rate and as function of α in the case the F -distribution would be used to determine the
aperture parameter. As explained, the latter choice is actually not correct. It follows
that it is not obvious how α must be chosen, which confirms that it is better not to
incorrectly use the F -distribution. With µR = 1
2 the fail rate will generally be quite low.
The results of this section give an explanation for the fact that it is difficult to find
a good criterion to choose the critical value for the ratio test and the difference test:
126 Integer Aperture estimation