A Numeric-Symbolic Solution of GNSS Phase Ambiguity

  • Béla Paláncz

    Department of Photogrammetry and Geoinformatics, Faculty of Civil Engineering, Budapest University of Technology and Economics, H-1521 Budapest, P.O.B. 91, Hungary

  • Lajos Völgyesi

    Department of Geodesy and Surveying, Faculty of Civil Engineering, Budapest University of Technology and Economics, H-1521 Budapest, P.O.B. 91, Hungary


Solution of the Global Navigation Satellite Systems (GNSS) phase ambiguity is considered as a global quadratic mixed integer programming task, which can be transformed into a pure integer problem with a given digit of accuracy. In this paper, three alter-native algorithms are suggested. Two of them are based on local and global linearization via McCormic Envelopes, respectively. These algorithms can be effective in case of simple configuration and relatively modest number of satellites. The third method is a locally nonlinear, iterative algorithm handling the problem as {-1, 0, 1} programming and also lets compute the next best integer solution easily. However, it should keep in mind that the algorithm is a heuristic one, which does not guarantee to find the global integer optimum always exactly. The procedure is very powerful utilizing the ability of the numeric-symbolic abilities of a computer algebraic system, like Wolfram Mathematica and it is properly fast for minimum 4 satellites with normal configuration, which means the Geometric Dilution of Precision (GDOP) should be between 1 and 8. Wolfram Alpha and Wolfram Clouds Apps give possibility to run the suggested code even via cell phones. All of these algorithms are illustrated with numerical examples. The result of the third one was successfully compared with the LAMBDA method, in case of ten satellites sending signals on two carrier frequencies (L1 and L2) with weighting matrix used to weight the GNSS observation and computed as the inverse of the corresponding covariance matrix.

Keywords: GNSS, GPS cycle ambiguities, computer algebra, local and global linearization, successive nonlinear method, mixed integer programming
Published online
How to Cite
Paláncz, B., Völgyesi, L. “A Numeric-Symbolic Solution of GNSS Phase Ambiguity”, Periodica Polytechnica Civil Engineering, 64(1), pp. 223-230, 2020. https://doi.org/10.3311/PPci.15092
Research Article