# GNSS Ambiguity Resolution: When and How to Fix or not to Fix?

## Abstract

In order to facilitate rapid and precise GNSS positioning, the integer carrier phase ambiguities need to be resolved. Since wrong integer ambiguity estimates may result in fixed position estimates which are worse than their float counterparts, very high success rates (i.e. high probabilities of correct integer estimation) or very low failure rates are required when performing ambiguity resolution. We discuss two different approaches of ambiguity resolution, a model-driven approach and a data-driven approach. The first is linked to the theory of integer estimation and the second is linked to the theory of integer aperture estimation. In the first approach, the user chooses an integer estimator and computes on the basis of his/her model the corresponding failure rate. The decision whether or not to use the integer ambiguity solution is then based on the thus computed value of the failure rate. This approach is termed model-driven, since the decision is solely based on the strength of the underlying model and not dependent on the actual ambiguity float estimate. This approach is simple and provides a priori information on the outcome of the decision process. A disadvantage of the model-driven approach is that it does not provide the user of any control over the failure rate. This disadvantage is absent when one uses the more elaborate data-driven approach of integer aperture estimation. With this approach the user sets his/her own failure rate (irrespective of the strength of the underlying model), thus generating an aperture space which forms the basis of the decision process: the integer solution is chosen as output if the float solution resides inside the aperture space, otherwise the float solution is maintained. Although more elaborate, the data-driven approach is more flexible than the model-driven approach and can provide a guaranteed failure rate as set by the user.

In this contribution we compare the model-driven and data-driven approaches, describe the decision making process of when and how to fix (or not to fix) and also give the optimal data-driven approach. We also show how the so-called ‘discrimination tests’, in particular the popular ‘ratio test’, fit into this framework. We point out that the common rationales for using these ‘tests’ are often incorrectly motivated in the literature and we show how they should be modified in order to reach an overall guaranteed failure rate for ambiguity resolution.

## Keywords

Integer least-squares integer aperture estimation## Preview

Unable to display preview. Download preview PDF.

## References

- Euler, H. J. and Schaffrin, B. (1991). On a measure for the discernibility between different ambiguity solutions in the statickinematic GPS-mode.
*IAG Symposia no. 107, Kinematic Systems in Geodesy, Surveying, and Remote Sensing*. Springer-Verlag, New York, pages 285–295.Google Scholar - Han, S. and Rizos, C. (1996). Integrated methods for instantaneous ambiguity resolution using new-generation GPS receivers.
*Proceedings of IEEE PLANS’ 96, Atlanta GA*, pages 254–261.Google Scholar - Hofmann-Wellenhoff, B. and Lichtenegger, H. (2001).
*Global Positioning System: Theory and Practice*. Springer-Verlag, Berlin, 5 edition.Google Scholar - Leick, A. (2003).
*GPS Satellite Surveying*. John Wiley and Sons, New York, 3rd edition.Google Scholar - Misra, P. and Enge, P. (2001).
*Global Positioning System: Signals, Measurements, and Performance*. Ganga-Jamuna Press, Lincoln MA.Google Scholar - Strang, G. and Borre, K. (1997).
*Linear Algebra, Geodesy, and GPS*. Wellesley-Cambridge Press, Wellesley MA.Google Scholar - Teunissen, P. J. G. (1993). Least squares estimation of the integer GPS ambiguities.
*Invited lecture, Section IV Theory and Methodology, IAG General Meeting, Beijing*.Google Scholar - Teunissen, P. J. G. (1998).
*GPS carrier phase ambiguity fixing concepts*. In: PJG Teunissen and Kleusberg A,*GPS for Geodesy*, Springer-Verlag, Berlin.Google Scholar - Teunissen, P. J. G. (1999a). An optimality property of the integer least-squares estimator.
*Journal of Geodesy*,**73**(11), 587–593.CrossRefGoogle Scholar - Teunissen, P. J. G. (1999b). The probability distribution of the GPS baseline for a class of integer ambiguity estimators.
*Journal of Geodesy*,**73**, 275–284.CrossRefGoogle Scholar - Teunissen, P. J. G. (2002). The parameter distributions of the integer GPS model.
*Journal of Geodesy*,**76**(1), 41–48.CrossRefGoogle Scholar - Teunissen, P. J. G. (2003a). Integer aperture GNSS ambiguity resolution.
*Artificial Satellites*,**38**(3), 79–88.Google Scholar - Teunissen, P. J. G. (2003b).
*Theory of integer aperture estimation with application to GNSS*. MGP report, Delft University of Technology.Google Scholar - Teunissen, P. J. G. (2003c). Towards a unified theory of GNSS ambiguity resolution.
*Journal of Global Positioning Systems*,**2**(1), 1–12.Google Scholar - Teunissen, P. J. G. (2004). Penalized GNSS ambiguity resolution.
*Journal of Geodesy*,**78**(4–5), 235–244.CrossRefGoogle Scholar - Teunissen, P. J. G. and Verhagen, S. (2004). On the foundation of the popular ratio test for GNSS ambiguity resolution.
*Proceedings of ION GNSS-2004, Long Beach CA*, pages 2529–2540.Google Scholar - Verhagen, S. (2004). Integer ambiguity validation: an open problem?
*GPS Solutions*,**8**(1), 36–43.CrossRefGoogle Scholar - Verhagen, S. and Teunissen, P. J. G. (2004).
*PDF evaluation of the ambiguity residuals*. In: F Sansó (Ed.),*V. Hotine-Marussi Symposium on Mathematical Geodesy*, International Association of Geodesy Symposia, Vol. 127, Springer-Verlag.Google Scholar - Wei, M. and Schwarz, K. P. (1995). Fast ambiguity resolution using an integer nonlinear programming method.
*Proceedings of ION GPS-1995, Palm Springs CA*, pages 1101–1110.Google Scholar