PDF Evaluation of the Integer Ambiguity Residuals

  • S. Verhagen
  • P. J. G. Teunissen
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 127)


A parameter estimation theory is incomplete if no rigorous measures are available for validating the parameter solution. Since the classical theory of linear estimation does not apply to the integer GPS model, rigorous validation is not possible when use is made of the classical results. As with the classical theory, a first step for being able to validate the integer GPS model is to make use of the residuals and their probabilistic properties. The residuals quantify the inconsistency between data and model, while their probabilistic properties can be used to measure the significance of the inconsistency.

In this contribution we will present and evaluate the joint probability density function (PDF) of the multivariate integer GPS carrier phase ambiguity residuals, which are defined as the difference between the real-valued float ambiguity estimates and the integer-valued fixed ambiguity estimates. Since the residuals and their properties depend on the integer estimation principle used, we will present the PDF of the ambiguity residuals for the whole class of admissible integer estimators. This includes the estimation principles of integer rounding, integer bootstrapping and integer least-squares. In order to get a better understanding of the various features of the joint PDF of the ambiguity residuals we will use a step-by-step construction aided by graphical means. Although the results apply for any dimension, the one-dimensional case and the two-dimensional case are highlighted.


GPS integer ambiguity residuals parameter distributions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Euler, H.J. and B. Schaffrin (1990). On a measure of discernibility between different ambiguity solutions in the static-kinematic GPS-mode. In: K.P. Schwarz and G. Lachapelle (eds.), IAG-symp., vol. 107, Kinematic Systems in Geodesy, Surveying, and Remote Sensing, Springer, New York, pp. 285–295.Google Scholar
  2. Hoffmann-Wellenhof, B., H. Lichtenegger, and J. Collins (1997). Global Positioning System: Theory and Practice. 4th edn., Springer, Berlin Heidelberg New York.Google Scholar
  3. Jonkman, N.F. (1998). Integer Ambiguity Estimation without the Receiver-Satellite Geometry. LGR Series, no. 18, Delft Geodetic Computing Centre, Delft.Google Scholar
  4. Leick, A. (1995). GPS Satellite Surveying. 2nd edn., John Wiley, New York.Google Scholar
  5. Parkinson, B. and J.J. Spilker (eds) (1996). GPS: Theory and Applications. Vols. 1 and 2, AIAA, Washington, DC.Google Scholar
  6. Strang, G. and K. Bone (1997). Linear Algebra, Geodesy, and GPS. Wellesley-Cambridge Press.Google Scholar
  7. Teunissen, P.J.G. (1993). Least-squares Estimation of the Integer GPS Ambiguities. Invited lecture, Section IV Theory and Methodology, IAG General Meeting. Beijing, August. Also in: LGR Series, No. 6, Delft Geodetic Computing Centre, Delft, pp. 59–74.Google Scholar
  8. Teunissen, P.J.G. (1998). On the Integer Normal Distribution of the GPS Ambiguities. Artiftc Sat 33 (2): 49–64.Google Scholar
  9. Teunissen, P.J.G. (1999). An Optimality Property of the Integer Least-squares Estimator. J Geod 73: 587–593.CrossRefGoogle Scholar
  10. Teunissen, P.J.G. (2001). Integer Estimation in the Presence of Biases. J Geod 75: 399–407.CrossRefGoogle Scholar
  11. Teunissen, P.J.G. (2002a). The Parameter Distributions of the Integer GPS Model. J Geod 76: 41–48.CrossRefGoogle Scholar
  12. Teunissen, P.J.G. (2002b). Integer Least-Squares. In: Proc. V Hotine-Marussi Symposium. Matera, June 17–21 (this volume).Google Scholar
  13. Teunissen, P.J.G. and A. Kleusberg (eds) (1998). GPS for Geodesy. 2’d edn., Springer, Berlin Heidelberg New York.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • S. Verhagen
    • 1
  • P. J. G. Teunissen
    • 1
  1. 1.Department of Mathematical Geodesy and PositioningDelft University of TechnologyDelftThe Netherlands

Personalised recommendations