The Uniform Tykhonov-Phillips Regularization (α-weighted S-homBLE) and its Application in GPS Rapid Static Positioning

  • J. Cai
  • E.W. Grafarend
  • C. Hu
  • J. Wang
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 132)


In high accuracy GPS positioning the conventional least-squares method is widely applied in processing of carrier phase observation. But it will not be always succeed in estimating of unknown parameters, in particular when the problem is ill-posed, for example, there is the weak multicollinear problem in the normal matrix with shorter period GPS phase observation. Here the newly developed method of determining the optimal regularization parameter α in uniform Tykhonov-Phillips regularization (α-weighted S-homBLE) by A-optimal design (minimizing the trace of the Mean Square Error matrix MSE) is reviewed. This new algorithm with A-optimal Regularization can be applied to overcome this kind problem in both GPS rapid static and real time kinematic positioning with single or dual frequency measurements, especially for the shorter period observation. In the case study, both the estimate methods are applied to process the two-epoch L1 data in single frequency GPS rapid static positioning. A detailed discuss about effects of the initial coordinate accuracy will also be presented. The results show that newly algorithm with optimal regularization can significantly improve the reliability the GPS ambiguity resolution in shorter observation period.


Integer least-squares ambiguity resolution regularization GPS rapid static positioning (α-weighted S-homBLE A-optimal design 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Cai J. (2004): Statistical inference of the eigenspace components of a symmetric random deformation tensor, Dissertation, Deutsche Geodätische Kommission (DGK) Reihe C, Heft Nr. 577, München, 2004.Google Scholar
  2. Cai J., E. Grafarend and B. Schaffrin (2004): The A-optimal regularization parameter in uniform Tykhonov-Phillips regularization – α-weighted BLE, IAG Symposia 127. In F. Sanso (ed.) V Hotine-Marussi Symposium on Mathematical Geodesy, pp. 309–324, Springer, Berlin, Heidelberg, New York.Google Scholar
  3. Chen D. and G. Lachapelle (1995): A comparison of the FASF and least-squares search algorithms for on-the-fly ambiguity resolution, Navigation: Journal of the Institute of Navigation, Vol. 42, No. 2, pp. 371–390.Google Scholar
  4. Euler H.-J. and H. Landau (1992): Fast GPS ambiguity resolution on-the-fly for real-time application. In Proceedings of Sixth International Geodetic Symposium on Satellite Positioning, Columbus, OH, pp. 650–659.Google Scholar
  5. Frei E., and G. Beulter (1990): Rapid Static Positioning Based on the Fast Ambiguity Resolution Approach ‘FARA’: Theory and First Results, Manuscripts Geodaetica, Vol.15, pp. 325–356.Google Scholar
  6. Grafarend E. (2000): Mixed integer-real valued adjustment (IRA) problems, GPS Solutions, Vol. 4, pp. 31–45.CrossRefGoogle Scholar
  7. Grafarend E. (2006): Linear and Nonlinear Models – Fixed Effects, Random Effects and Mixed Models, Walter de Gruyter, Berlin, New York.Google Scholar
  8. Han S. and C. Rizos (1996): Improving the computational efficiency of the ambiguity function algorithm, Journal of Geodesy, Vol. 70, No. 6, pp. 330–341.Google Scholar
  9. Hansen P. (1992): Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review, Vol. 34, pp. 561–580.CrossRefGoogle Scholar
  10. Hatch R. (1990): Instantaneous ambiguity resolution. In Proceedings of KIS’90, Banff, Canada, September 10–13, pp. 299–308.Google Scholar
  11. Hatch R. and H.-J. Euler (1994): Comparison of several AROF kinematic techniques. In Proceedings of ION GPS-94, Salt Lake City, Utah, September 20–23, pp. 363–370.Google Scholar
  12. Kim D and R.B. Langley (2000): GPS Ambiguity Resolution and Validation: Methodologies, Trends and Issues, the 7th GNSS Workshop, International Symposium on GPS/GNSS, Seoul, Korea.Google Scholar
  13. Lou L. and E. Grafarend (2003): GPS integer ambiguity resolution by various decorrelation methods, Zeitschrift für Vermessungswesen, Vol. 128, No. 3, pp. 203–210.Google Scholar
  14. Mallows C.L. (1973): Some comments on Cp, Technometrics, Vol. 15, pp. 661–675.CrossRefGoogle Scholar
  15. Ou J. and Z. Wang (2004): An improved regularization method to resolve integer ambiguity in rapid positioning using single frequency GPS receivers, Chinese Science Bulletin, Vol. 49, pp. 196–200.CrossRefGoogle Scholar
  16. Phillips D.L. (1962): A technique for the numerical solution of certain integral equations of the first kind, Journal of the Association for Computational Machinery, Vol. 9,pp. 84–96.Google Scholar
  17. Remondi B.W. (1984). Using the Global Positioning System (GPS) phase observable for relative geodesy: Modeling, processing and results, Ph.D. Dissertation, Center for Space Research, University of Texas at Austin.Google Scholar
  18. Shen Y. and B. Li (2005): A new approach of regularization based GPS fast ambiguity resolution in rapid GPS positioning, GNSS Hong Kong 2005 Conference, Hong Kong, PR China; 08.12.2005 – 10.12.2005; in: “GNSS Hong Kong 2005”.Google Scholar
  19. Teunissen P.J.G. (1993): Least-squares estimation of the Integer GPS ambiguities, Invited lecture, Section IV: Theory and Methodology, IAG General Meeting, Beijing, China.Google Scholar
  20. Tykhonov A.N. (1963): The regularization of incorrectly posed problem, Soviet Mathametical Doklady 4,1624–1627.Google Scholar
  21. Wang Z., C. Rizos and S. Lim (2006): Single epoch algorithm based on TIKHONOV regularization for dam monitoring using single frequency receivers, Survey Review, Vol. 38, pp. 682–688.Google Scholar
  22. Weisberg S. (1985): Applied Linear Regression, 2nd ed, Wiley, New York.Google Scholar
  23. Xu P. (1998): Truncated SVD methods for discrete linear ill-posed problems, Geophysical Journal International, Vol. 135, pp. 505–514.CrossRefGoogle Scholar
  24. Xu P. (2001): Random simulation and GPS decorrelation, Journal of Geodesy, Vol. 75, pp. 408–423.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • J. Cai
    • 1
  • E.W. Grafarend
    • 1
  • C. Hu
    • 2
  • J. Wang
    • 2
  1. 1.Department of Geodesy and GeoInformaticsUniversity of StuttgartD-70174 StuttgartGermany
  2. 2.Department of Surveying and Geo-informaticsTongji University200092 ShanghaiP.R. China

Personalised recommendations