Integer estimation in the presence of biases P. J. G. Teunissen Department of Mathematical Geodesy and Positioning, Delft University of Technology, Thijsseweg 11, 2629 JA Delft, The Netherlands e-mail: p.j.g.teunissen@geo.tudelft.nl; Tel.: +31-15-278-2558; Fax: +31-15-278-3711 Received: 28 September 2000 / Accepted: 29 March 2001 Abstract. Carrier phase ambiguity resolution is the key to fast and high-precision GNSS Global Navigation Satellite System) kinematic positioning. Critical in the application of ambiguity resolution is the quality of the computed integer ambiguities. Unsuccessful ambiguity resolution, when passed unnoticed, will too often lead to unacceptable errors in the positioning results. Very high success rates are therefore required for ambiguity reso- lution to be reliable. Biases which are unaccounted for will lower the success rate and thus increase the chance of unsuccessful ambiguity resolution. The performance of integer ambiguity estimation in the presence of such biases is studied. Particular attention is given to integer rounding, integer bootstrapping and integer least squar- es. Lower and upper bounds, as well as an exact and easy-to-compute formula for the bias-aected success rate, are presented. These results will enable the evalu- ation of the bias robustness of ambiguity resolution. Keywords: GNSS ± Ambiguity Resolution ± Success Rate ± Bias Robustness 1 Introduction Ambiguity resolution applies to a great variety of GNSS models currently in use. They 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 models, together with their applications in surveying, navigation and geodesy, can be found in textbooks such as those of Hofmann-Wellenhof et al. 1997), Leick 1995), Parkin- son and Spilker 1996), Strang and Borre 1997) and Teunissen and Kleusberg 1998). Despite the dierences in application of the various GNSS models, their ambiguity resolution problems are intrinsically the same. Hence, any rigorous method of ambiguity resolution should be applicable to each of these models. Any such method should be able to e- ciently obtain integer ambiguity estimates from the `¯oat' solution, as well as provide the user or analyst with tools to evaluate the quality of the integer solution so obtained. Unfortunately the availability of proper indicators for the qualitative aspects of the integer am- biguity estimators is still lacking in most of the present- day GNSS positioning systems. It is of importance to be able to evaluate the quality of the integer solution, since unsuccessful ambiguity resolution, when passed unnoticed, will all too often lead to unacceptable errors in the positioning results. We therefore need to have a way of knowing how often we can expect the computed ambiguity solution to coincide with the correct, but unknown, solution. Is this nine out of 10 times, 99 out of 100, or a higher percentage? It will certainly never equal 100%. After all, the integer am- biguities are computed from the data. They are therefore subject to uncertainty just like the data are. In order to describe the quality of the integer ambi- guity solution, we require the probability distribution of the integer estimator. This distribution will be a proba- bility mass function, due to the integer nature of the ambiguities. Of this probability mass function, the probability of correct integer estimation ± also referred to as the success rate ± is of particular interest. This probability depends on three contributing factors: the functional model, the stochastic model and the chosen method of integer estimation. Changes in any of these will aect the ambiguity success rate. In this contribution we address the probabilistic as- pects of integer ambiguity estimation in the presence of biases. We will refrain however, from discussing the computational intricacies of integer estimation. For a discussion of these aspects, we refer to, for example, Teunissen 1993) and de Jonge and Tiberius 1996a), or to the textbooks of Hofmann-Wellenhof et al. 1997), Strang and Borre 1997) and Teunissen and Kleusberg 1998). A very ecient way of solving the integer esti- mation problem is provided by the LAMBDA method. A description of the LAMBDA method can be found in the aforementioned publications, while practical results Journal of Geodesy 2001) 75: 399±407