The GNSS integer ambiguities: estimation and validation

Integer Aperture estimation 5
In section 3.2 the success rate was introduced. This probability is a valuable measure
for deciding whether one can have enough confidence in the fixed solution. As soon
as the success rate is close enough to one, a user can fix the ambiguities, so that the
precise carrier phase measurements start to contribute to the solution. However, when
the success rate is considered too low with respect to a given threshold, it will not yet be
possible to benefit from the precise phase observations. This also means that there is a
probability that the user unnecessarily sticks to the float solution just because the fail rate
is too high. In practice, not only the success rate is considered to decide on acceptance
or rejection of the integer solution. The integer solution is also validated using one of
the methods described in section 3.5. Since no sound validation criterion exists yet, in
this chapter a new class of integer estimators is defined, the class of integer aperture
estimators. Within this class estimators can be defined such that the acceptance region
of the fixed solution is determined by the probabilities of success and failure.
The new class of integer estimators is defined in section 5.1. Then different examples
of integer aperture estimators are presented. First, the ellipsoidal aperture estimator in
section 5.2, for which the aperture space is built up of ellipsoidal regions. Then in section
5.3 it is shown that integer estimation in combination with the ratio test, difference test,
or the projector test belongs to the class of integer aperture estimators. When the
aperture space is chosen as a scaled pull-in region, the integer aperture bootstrapping
and least-squares estimators can be defined. This will be shown in section 5.4. Finally,
the penalized integer aperture estimator and optimal integer aperture estimator are
presented in sections 5.5 and 5.6. Section 5.7 deals with implementation aspects. A
comparison of all the different estimators is given in section 5.8, and examples in order
to show the benefits of the new approach are given in section 5.9.
In order to illustrate the principle of IA estimation, throughout this chapter a twodimensional
(2-D) example will be used. Simulations were carried out to generate
500,000 samples of the float range and ambiguities, see appendix B, for the following
vc-matrix:
Q =
Qˆ b
Q â ˆ b
Qˆ bâ
Qâ
Note that Qâ = Qˆz 02 01.
⎛
0.1800 −0.1106
⎞
0.0983
= ⎝−0.1106
0.0865 −0.0364⎠
0.0983 −0.0364 0.0847
87