On the Foundation of the Popular Ratio Test for GNSS Ambiguity Resolution Peter J.G. Teunissen and Sandra Verhagen Delft Institute of Earth Observation and Space Systems Delft University of Technology The Netherlands BIOGRAPHY Peter Teunissen is full professor at the Delft Institute of Earth Observation and Space Systems. His research focuses on GNSS data processing strategies for medium scaled net- works with an emphasis on ambiguity resolution. Sandra Verhagen is a PhD student at the Delft Institute of Earth Observation and Space Systems. She is working on quality control and integer ambiguity resolution for high precision GNSS and pseudolite applications. ABSTRACT 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 in- teger 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 valida- tion plays a crucial role in the process of ambiguity resolu- tion. 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 under- lying 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 solu- tion. That is, we will also show that the procedure under- lying 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 ingre- dients 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 illus- trated. 1 INTRODUCTION Integer carrier phase ambiguity resolution is the key to fast and high-precision global navigation satellite system (GNSS) positioning and navigation. It applies to a great variety of current and future models of GPS, modernized GPS and Galileo, with applications in surveying, naviga- tion, geodesy and geophysics. These models may dif- fer greatly in complexity and diversity. They range from single-baseline models used for kinematic positioning to multi-baseline models used as a tool for studying geody- namic phenomena. The models may or may not have the relative receiver-satellite geometry included. They may also be discriminated as to whether the slave receiver(s) is stationary or in motion, or whether or not the differen- tial atmospheric delays (ionosphere and troposphere) are in- cluded as unknowns. An overview of these models can be found in textbooks like (Hofmann-Wellenhof et al., 2001; Leick, 2003; Parkinson and Spilker, 1996; Strang and Borre, 1997; Teunissen and Kleusberg, 1998). Any linear(ized) GNSS model can be cast in the following system of linearized observation equations: E{y} = Aa + Bb , a ∈ Z n ,b ∈ R p (1) with E{.} the mathematical expectation operator, y the m- vector of observables, a the n-vector of unknown integer parameters and b the p-vector of unknown real-valued pa- rameters. The data vector y will then usually consist of the ’observed minus computed’ single- or multi-frequency double-difference (DD) phase and/or pseudorange (code) observations accumulated over all observation epochs. The entries of vector a are then the DD carrier phase ambigu- ities, expressed in units of cycles rather than range, while the entries of the vector b will consist of the remaining unknown parameters, such as for instance baseline compo- nents (coordinates) and possibly atmospheric delay param- eters (troposphere, ionosphere). The procedure for solving the above GNSS model can be divided conceptually into three steps. In the first step one simply discards the integer constraints a ∈ Z n and per- 2529 ION GNSS 17th International Technical Meeting of the Satellite Division, 21-24 Sept. 2004, Long Beach, CA