An optimality property of the integer least-squares estimator P. J. G. Teunissen Department of Mathematical Geodesy and Positioning, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Thijsseweg 11, 2629 JA Delft, The Netherlands e-mail: P.J.G.Teunissen@geo.tudelft.nl; Fax:+31 15 278 3711 Received: 11 January 1999 / Accepted: 9 July 1999 Abstract. A probabilistic justi®cation is given for using the integer least-squares (LS) estimator. The class of admissible integer estimators is introduced and classical adjustment theory is extended by proving that the integer LS estimator is best in the sense of maximizing the probability of correct integer estimation. For global positioning system ambiguity resolution, this implies that the success rate of any other integer estimator of the carrier phase ambiguities will be smaller than or at the most equal to the ambiguity success rate of the integer LS estimator. The success rates of any one of these estimators may therefore be used to provide lower bounds for the LS success rate. This is particularly useful in case of the bootstrapped estimator. Key words. Integer LS á GPS ambiguity resolution á Ambiguity success rate 1 Introduction Ambiguity resolution applies to a great variety of global positioning system (GPS) models currently in use. These range from single-baseline models used for kinematic positioning to multi-baseline models used as a tool for studying geodynamic phenomena. An overview of these and other GPS models, together with their application in surveying, navigation and geodesy, can be found in textbooks such as those of Leick (1995), Parkinson and Spilker (1996), Hofmann-Wellenhof et al. (1997), Strang and Borre (1997) and Teunissen and Kleusberg (1998). Despite the dierences in application of the various GPS models, it is important to understand that their ambi- guity resolution problems are intrinsically the same. That is, the GPS models on which ambiguity resolution is based can all be cast in the following conceptual frame of linear(ized) observation equations y Aa Bb e 1 where y is the given GPS data vector of order m, a and b are the unknown parameter vectors respectively of order n and o, and e is the noise vector. The matrices A and B are the corresponding design matrices. The data vector y will usually consist of the `observed minus computed' single- or dual-frequency double-dierence (DD) phase and/or pseudorange (code) observations accumulated over all observation epochs. The entries of vector a are then the DD carrier phase ambiguities, expressed in units of cycles rather than range. They are known to be integers, a 2 Z n . The entries of the vector b will consist of the remaining unknown parameters, such as for instance baseline components (coordinates) and possibly atmospheric delay parameters (tropo- sphere, ionosphere). They are known to be real-valued, b 2 R o . The procedure which is usually followed for solving the GPS model of Eq. (1) can be divided into three steps (for more details we refer to e.g. Teunissen 1993 or de Jonge and Tiberius 1996). In the ®rst step we simply disregard the integer constraints a 2 Z n on the ambigu- ities and perform a standard adjustment. As a result we obtain the (real-valued) estimates of a and b, together with their variance±covariance matrix ^ a ^ b 2 4 3 5 ; Q ^ a Q ^ a ^ b Q ^ b ^ a Q ^ b 2 4 3 5 2 This solution is referred to as the `¯oat' solution. In the second step the `¯oat' ambiguity estimate ^ a is used to compute the corresponding integer ambiguity estimate a. This implies that a mapping F : R n 7! Z n , from the n- dimensional space of real numbers to the n-dimensional space of integers, is introduced such that a F ^ a 3 Once the integer ambiguities are computed, they are used in the third step to ®nally correct the ¯oat estimate of b. As a result we obtain the `®xed' solution b ^ b Q ^ b^ a Q 1 ^ a ^ a a 4 Journal of Geodesy (1999) 73: 587±593