Towards a unified theory of GNSS ambiguity resolution

In this invited contribution a brief review will be presented of the integer estimation theory as developed by the author over the last decade and which started with the introduction of the LAMBDA method in 1993. The re- view discusses three different, but closely related classes of ambiguity estimators. They are the integer estimators, the integer aperture estimators and the integer equivariant estimators. Integer estimators are integer aperture estima- tors and integer aperture estimators are integer equivari- ant estimators. The reverse is not necessarily true how- ever. Thus of the three types of estimators the integer es- timators are the most restrictive. Their pull-in regions are translational invariant, disjunct and they cover the ambi- guity space completely. Well-known examples are integer rounding, integer bootstrapping and integer least-squares. A less restrictive class of estimators is the class of inte- ger aperture estimators. Their pull-in regions only obey two of the three conditions. They are still translational invariant and disjunct, but they do not need to cover the ambiguity space completely. As a consequence the inte- ger aperture estimators are of a hybrid nature having either integer or non-integer outcomes. Examples of integer aper- ture estimators are the ratio-testimator and the difference- testimator. The class of integer equivariant estimators is the less restrictive of the three classes. These estimators only obey one of the three conditions, namely the condi- tion of being translational invariant. As a consequence the outcomes of integer equivariant estimators are always real- valued. For each of the three classes of estimators we also present the optimal estimator. Although the Gaussian case is usually assumed, the results are presented for an arbi- trary probability density function of the float solution. The optimal integer estimator in the Gaussian case is the inte- ger least-squares estimator. The optimality criterion used is that of maximizing the probability of correct integer es- timation, the so-called success rate. The optimal integer aperture estimator in the Gaussian case is the one which only returns the integer least-squares solution when the in- teger least-squares residual resides in the optimal aperture pull-in region. This region is governed by the probability density function of the float solution and by the probabil- ity density function of the integer least-squares residual. The aperture of the pull-in region is governed by a user- defined aperture parameter. The optimality criterion used is that of maximizing the probability of correct integer esti- mation given a fixed, user-defined, probability of incorrect integer estimation. The optimal integer aperture estimator becomes identical to the optimal integer estimator in case the success rate and the fail rate sum up to one. The best integer equivariant estimator is an infinite weighted sum of all integers. The weights are determined as ratios of the probability density function of the float so- lution with its train of integer shifted copies. The optimal- ity criterion used is that of minimizing the mean squared error. The best integer equivariant estimator therefore al- ways outperforms the float solution in terms of precision.