Theoretical Upper Bound and Lower Bound for Integer Aperture Estimation Fail-Rate and Practical Implications Tao Li and Jinling Wang (School of Surveying and Geospatial Engineering, The University of New South Wales, Australia) (E-mail: jinling.wang@unsw.edu.au) Integer ambiguity validation is pivotal in precise positioning with Global Navigation Satellite Systems (GNSS). Recent research has shown traditionally used ambiguity validation methods can be classified as members of the Integer Aperture (IA) estimators, and by the virtue of the IA estimation, a user controllable IA fail-rate is preferred. However, an appropriately chosen fail-rate is essential for ambiguity validation. In this paper, the upper bound and the lower bound for the IA fail-rate, which are extremely useful even at the designing stage of a GNSS positioning system, have been analysed, and numerical results imply that a meaningful IA fail-rate should be within this range. KEY WORDS 1. GNSS. 2. Integer least-squares estimation. 3. Integer Aperture (IA) estimation. 4. Ambiguity validation. Submitted: 29 June 2012. Accepted: 27 September 2012. First published online: 20 November 2012. 1. INTRODUCTION. Global Navigation Satellite Systems (GNSS) can employ two types of measurements to locate the position of a receiver, namely the carrier phase measurements and the code measurements. The carrier phase measurements are more accurate than the code measurements so that for precise GNSS positioning, carrier phase measurements are indispensable. However, the unknown integer cycles of wavelength are difficult to determine and consequently the problem of integer ambiguity resolution and validation arises. In general, there are three steps to resolve the double differenced integer ambiguity vector. The first step is to estimate the float solution and its variance-covariance matrix by the least-squares or Kalman filter, regardless of the constraints of the integer THE JOURNAL OF NAVIGATION (2013), 66, 321–333. © The Royal Institute of Navigation 2012 doi:10.1017/S0373463312000513