The GNSS integer ambiguities: estimation and validation

in practice, a user has to choose between two undesirable situations:
• Integer validation is based on wrong assumptions, since the randomness of the
fixed ambiguities is incorrectly ignored;
• No attempt is made to fix the ambiguities because of the low success rate, although
there is still a probability that the ambiguities can be fixed correctly.
Some integer validations tests are defined such that the invalid assumption of a deterministic
fixed ambiguity estimator is avoided. However, these tests lack a sound theoretical
foundation. Moreover, often fixed critical values are used based on experience. But that
implies that in many situations the tests will either be too conservative or too optimistic.
1.2 Objectives and contribution of this work
Obviously, validation of the integer ambiguity solution is still an open problem. In order
to deal with the above two situations, two approaches are investigated:
• A new class of ambiguity estimators is used, which results in estimators that in
some sense are always superior to their float and fixed counterparts.
This estimator is the Best Integer Equivariant (BIE) estimator. It is best in the
sense that it minimizes the mean squared errors of the estimators. This can be
considered a weaker performance criterion than that of the integer least-squares
estimator, the maximization of the success rate. This might be an advantage
if the success rate is not high, since then a user will generally not have enough
confidence in the fixed solution, and stick to the float solution. It is investigated
how BIE estimation can be implemented. The performance of this estimator is
then compared with the fixed and float estimators.
• A new ambiguity estimator is used by defining an a priori acceptance region, or
aperture space.
This new ambiguity resolution method is the so-called integer aperture estimation.
The problem of ambiguity validation is incorporated in the estimation procedure
with this approach. Instead of distinguishing only the probability of success and
of failure, also the probability of not fixing (undecided rate) is considered a priori.
An integer acceptance region is then defined by putting constraints on the
three probabilities; only if the float ambiguities fall in the acceptance region, the
ambiguities will be fixed using one of the well-known integer estimators.
The theoretical aspects of all approaches are considered, as well as implementation
aspects. Thereby it is avoided to go into too much detail with respect to proofs and
derivations; these are only included if they contribute to the understanding of the theory.
The methods are implemented in Matlab R○.
Furthermore, it is investigated how the various methods will perform. For that purpose,
Monte-Carlo simulations are used, since then the ’true’ situation is known and it is
possible to compare the performance of different estimators and validators.
2 Introduction