On the Foundation of the Popular Ratio Test for GNSS Ambiguity Resolution

Integer carrier phase ambiguity resolution is the key to fast and high-precision global navigation satellite system (GNSS) positioning and navigation. It is the process of resolving the unknown cycle ambiguities of the double-differenced carrier phase data as integers. For the problem of estimating the ambiguities as integers a rigorous theory is available. The user can choose from a whole class of integer estimators, from which integer least-squares is known to perform best in the sense that no other integer estimator exists which will have a higher success rate. Next to the integer estimation step, also the integer validation plays a crucial role in the process of ambiguity resolution. Various validation procedures have been proposed in the literature. One of the earliest and most popular ways of validating the integer ambiguity solution is to make use of the so-called Ratio Test. In this contribution we will study the properties and underlying concept of the popular Ratio Test. This will be done in two parts. First we will criticize some of the properties and underlying principles which have been assigned in the literature to the Ratio Test. Despite this criticism however, we will show that the Ratio Test itself is still an important, albeit not optimal, candidate for validating the integer solution. That is, we will also show that the procedure underlying the Ratio Test can indeed be given a firm theoretical footing. This is made possible by the recently introduced theory of Integer Aperture Inference. The necessary ingredients of this theory will be briefly described. It will also be shown that one can do better than the Ratio Test. The optimal test will be given and the difference between the optimal test and the Ratio Test will be discussed and illustrated.