# Probabilistic Evaluation of the Integer Least-Squares and Integer Aperture Estimators

## Abstract

The carrier phase observations start to act as very precise pseudorange observations once the ambiguities are resolved as integers. However, the integer ambiguity estimates should only be used if the reliability of the integer solution is high. The question is then how to assess this reliability. A well-known a-priori reliability measure is the ambiguity success rate. But even with a high success rate, integer ambiguity validation remains indispensable in order to check whether or not a specific integer solution is sufficiently more likely than any other integer candidate. A solution to the integer validation problem is the use of integer aperture estimation. With this approach an aperture space is defined such that only float samples that fall into this space are fixed to the corresponding integer estimates, otherwise the float solution is maintained. The aperture space is built up of translationally invariant aperture pull-in regions centered at all integers. The size of these pull-in regions is determined by the condition of a fixed failure rate.

In this contribution, we will present the probabilistic measures that can be used to assess the reliability of the integer least-squares and the integer aperture ambiguity estimators, as well as the reliability of the corresponding baseline estimators. These probabilities will also be evaluated in the presence of a bias in order to study the bias-robustness of the integer ambiguity estimators. A case study is carried out with several GNSS models, which shows that the integer aperture estimator has some favorable probabilistic properties as compared to integer least-squares estimation, both in the unbiased and in the biased case.

## Keywords

Integer least-squares integer aperture estimation bias-robustness## Preview

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