The GNSS integer ambiguities: estimation and validation
The GNSS integer ambiguities: estimation and validation
The GNSS integer ambiguities: estimation and validation
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Table 5.4: Overview of IA estimators. <strong>The</strong> last column gives the methods that can be used<br />
for the determination of the aperture parameter µ in the case of fixed fail rate IA <strong>estimation</strong>.<br />
definition aperture space: determination µ<br />
EIA ellipsoid â − ǎ 2 Qâ ≤ µ2 root finding method<br />
RTIA ratio test<br />
F -RTIA F -ratio test<br />
â−ǎ 2<br />
Qâ â−ǎ22 Qâ ê 2<br />
Qy +â−ǎ2 Qâ ê2 Qy +â−ǎ22 Qâ DTIA difference test â − ǎ2 2 Qâ − â − ǎ2 Qâ<br />
PTIA projector test<br />
IAB bootstrapping 1<br />
µ ˇɛB ∈ S0,B<br />
IALS least-squares 1<br />
µ ˇɛLS ∈ S0,LS<br />
OIA optimal<br />
≤ µ empirical/simulation<br />
≤ µ empirical/simulation<br />
≥ µ empirical/simulation<br />
(ǎ2−ǎ) T Q −1<br />
â (â−ǎ)<br />
≤ µ empirical/simulation<br />
ǎ2−ǎQâ root finding method<br />
= µB/simulation<br />
fˇɛ(ˇɛLS)<br />
fâ(ˇɛLS+a) ≤ µ simulation<br />
estimator is also not surprising, since it makes use of the scaled ILS pull-in region as<br />
aperture space, <strong>and</strong> the ILS estimator is known to be optimal in the class of admissible<br />
<strong>integer</strong> estimators.<br />
Which IA estimator to use?<br />
<strong>The</strong> RTIA <strong>and</strong> DTIA estimator are easy to apply <strong>and</strong> computationally simple, since the<br />
squared norms of the ambiguity residuals are already available after computing the ILS<br />
solution. However, these estimators are only close to optimal if an appropriate aperture<br />
parameter is used based on a fixed fail rate; <strong>and</strong> that is only possible by using simulations.<br />
<strong>The</strong> IALS estimator might be a good alternative: close to optimal performance, <strong>and</strong> easy<br />
to apply. <strong>The</strong> aperture parameter could be chosen equal to that of the IAB estimator<br />
for a given fail rate. Unfortunately, this will make the IALS estimator less optimal. If<br />
simulations are used for the determination of the aperture parameter, the computational<br />
efficiency of the IALS estimator will be much less than that of the RTIA <strong>and</strong> DTIA<br />
estimators, since ILS <strong>estimation</strong> has to be applied twice for each sample.<br />
In theory, the OIA estimator is of course to be preferred since it maximizes the success<br />
rate. However, computationally it is the most complex estimator because the probability<br />
density of the ambiguity residual must be computed. Furthermore, it is only possible to<br />
determine the aperture parameter for a given fail rate using simulations. If the aperture<br />
parameter is approximated using the IAB estimator, the resulting estimator will not be<br />
optimal.<br />
Summary 137