New Algorithm of GPS Rapid Positioning Based on Double- -Type Ridge Estimation
Publication: Journal of Surveying Engineering
Volume 133, Issue 4
Abstract
In this paper, a new algorithm is employed in global positioning system (GPS) rapid positioning using several-epoch single-frequency phase data. First, we define the double- -type ridge estimator (DKRE) based on the structure characteristics of multicollinearities of the normal equations matrix in the double-difference (DD) model, and prove when the DKRE is not worse than the least-squares estimator (LSE) in the sense of a reduced MSEM. Taking into account how the ridge parameter in the ordinary ridge estimator is confirmed based on the generalized ridge estimator (GRE), we propose a method of estimating two ridge parameters for the DKRE. Second, we improve the LAMBDA method through replacing the cofactor matrix computed by the LSE with the cofactor matrix computed by the DKRE. The ambiguities-fixed solution is found by the sequential LSE. A theorem stating that the success rate of the improved LAMBDA method is bigger than the original LAMBDA method is given. Finally, through the GPS positioning tests, it is shown that the present method is highly efficient and reliable, and some very valuable conclusions are obtained by analyzing the computation results.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The writers wish to thank the editor and referee for helpful suggestions which improved the quality of the paper. This work is supported jointly by National Science Foundation of China (Grant No. NSFC40474007), National Splendid Youth Science Foundation of China (Grant Nos. UNSPECIFIED40125013, UNSPECIFIED49825107), “Basic Geographic Information and Digital Technique” Task of Important Laboratory of Shandong Province (Grant No. UNSPECIFIEDSD040202), and Henan Province Nature Science Foundation of China (Grant No. NSFC0511010100).
References
Frei, E., and Beutler, G. (1990). “Rapid static positioning based on the fast integer ambiguity resolution approach ‘FARA’: Theory and first results.” Manuscr. Geod., 15(3), 172–179.
Hoerl, A. E., and Kennard, R. W. (1970). “Ridge regression: Biased estimation for non-orthogonal problems.” Technometrics, 12, 55–67.
Hofmann-Wellenhof, B., Lichtenegger, H., and Collins, J. (1997). GPS: Theory and practice, 4th Ed., Springer, Berlin.
Horemuz, M., and Sjöberg, L. E. (2002). “Rapid GPS ambiguity resolution for short and long baselines.” J. Geodesy, Berlin, 76, 381–391.
Kibria, B. W. G. (2003). “Performance of some new ridge regression estimators.” Commun. Stat.-Simul. Comput., 32(2), 419–435.
Kim, D., and Langley, R. B. (1999). “An optimized, least-squares technique for improving ambiguity resolution and computational efficiency.” Proc., ION GPS-1999, 1579–1588.
Koch, K. R. (1999). Parameter estimation and hypothesis testing in linear models, 2nd Ed., Springer, Berlin.
Löwner., K. (1934). “Über monotone Matrixfunktionen.” Math. Z., 38, 177–216.
Massy, W. F. (1965). “Principal component regression in exploratory statistical research.” J. Acoust. Soc. Am., 60, 234–266.
Ou, J. K., and Wang, Z. J. (2003). “An improved regularization method to resolve integer ambiguity in rapid positioning using single frequency GPS receivers.” Chin. Sci. Bull., 49(2), 196–200.
Qingming, G., and Jinsan, L. (1999). “Generalized shrunken-type robust estimation.” J. Surv. Eng., 125(4), 177–184.
Qingming, G., and Jinsan, L. (2000). “Biased estimation in Gauss–Markov model.” Allgemeine Vermessungs-Nachrichten, 107, 104–108.
Qingming, G., and Zhang, J. J. (1998). “Robust biased estimation and its applications in geodetic adjustments.” J. Geodesy, Berlin, 72, 430–435.
Strang, G., and Borre, K. (1997). Linear algebra, geodesy and GPS, Wellesley-Cambridge Press, Cambridge, Mass.
Teunissen, P. J. G. (1993). “Least-squares estimation of the integer GPS ambiguities.” LGR Series. No. 11, Delft Geodetic Computing Centre, Delft, The Netherlands, 59–74.
Teunissen, P. J. G. (1995). “The least-square integer ambiguity decorrelation adjustment: A method for fast GPS integer ambiguity estimation.” J. Geodesy, Berlin, 70(1–2), 65–82.
Teunissen, P. J. G. (1996). “The LAMBDA method for integer ambiguity estimation: Implantation aspects.” LGR Series No. 12, Delft Geodetic Computing Centre, Delft, The Netherlands.
Teunissen, P. J. G. (1999). “An optimality property of the integer least-squares estimator.” J. Geodesy, Berlin, 73, 587–593.
Teunissen, P. J. G., and Kleusberg, A. (1998). GPS for geodesy, 2nd Ed., Springer, Berlin.
Trenkler, G. (1985). “Mean square error matrix comparisons of estimators in linear regression.” Commun. Stat: Theory Meth., 14, 2495–2509.
Information & Authors
Information
Published In
Copyright
© 2007 ASCE.
History
Received: May 26, 2006
Accepted: Jan 23, 2007
Published online: Nov 1, 2007
Published in print: Nov 2007
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.