The GNSS integer ambiguities: estimation and validation

Conclusions and recommendations 6
6.1 Integer estimation and validation
The goal of this thesis was to evaluate different approaches to the integer estimation
and validation problem. In this thesis three approaches were considered, based on three
classes of integer estimators. The relationships between the estimators are shown in the
Venn diagram of figure 6.1.
The smallest, most restrictive class is the class of admissible integer estimators. Integer
rounding, bootstrapping, and integer least-squares are the well-known admissible estimators.
From this class, integer least-squares is the optimal estimator, since it maximizes
the probability of correct integer estimation. Therefore, almost all ambiguity resolution
methods currently used in practice are based on this estimator. A parameter resolution
theory can, however, not be considered complete without the appropriate measures to
validate the solution. For that purpose, many integer validation tests have been proposed
in literature, but all of these tests lack a sound theoretical foundation and most tests are
based on the invalid assumption that the integer ambiguity candidates are deterministic.
The largest class of integer estimators is the class of best integer equivariant (BIE)
estimators. This BIE estimator is in some sense always superior to its float and fixed
counterparts: it is best in the sense that it minimizes the mean squared errors of the
estimators. This is 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 close to one, since then the fixed ambiguities may not be considered
deterministic and the integer validation procedures that are currently available should
not be used. So, in that case the ambiguities cannot be fixed reliably.
The BIE ambiguity estimator is equal to a weighted sum over all integer vectors, and the
weights depend on the probability density function of the float ambiguities. Therefore,
the BIE estimator can be considered as a compromise between the float and fixed solution.
These are approximated in the extreme cases of very bad or very good precision
respectively. The disadvantages of BIE estimation are that it does still not provide an exact
probabilistic evaluation of the final solution, and that it is computationally complex.
Moreover, the results in this chapter indicate that only in a limited number of cases the
BIE estimator performs obviously better than either the float or the fixed estimator.
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