Regularized Solution to Fast GPS Ambiguity Resolution
Publication: Journal of Surveying Engineering
Volume 133, Issue 4
Abstract
In rapid global positioning systems (GPS) positioning one of the key problems is to quickly determine the ambiguities of GPS carrier phase observables. Since carrier phase observations are generally collected only for a few minutes in the mode of rapid GPS positioning, the least squares floating solution of the ambiguities will be highly correlated and the decorrelation approach has often been used in order to reduce the search space of integer ambiguities. In this paper we propose a regularized algorithm as an alternative approach to decorrelation, and compute the regularization parameter by minimizing the trace of mean squared errors. Since regularization has been essential to solve inverse ill-posed problems and shown to be very significant in reducing the condition number of normal matrices, we will explore possible applications of regularization for improving the high correlation of the estimated float ambiguities. Numerical experiments with 50 epochs of single frequency observations show that the condition number after regularization reduces to half of that of the floating solution if the ambiguities could be known to . If better knowledge about the ambiguities could be obtained to within , further improvement can be achieved. The results indicate that regularization could be used for fast GPS ambiguity resolution. Our experiments also demonstrate that a scale factor of about 8 is needed to multiply the estimated variance of unit weight for obtaining a reasonable estimator for the accuracy of float ambiguities.
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Acknowledgments
This paper is substantially supported by the National Natural Science Foundation of China (Project No. NNSCF40674003). The anonymous reviewer is very much appreciated for his constructive comments and clarifying this manuscript.
References
Frei, E., and Beulter, G. (1990). “Rapid static positioning based on the fast ambiguity resolution approach ‘FARA’: Theory and first results.” Manuscr. Geod., 15, 326–356.
Grafarend, E. W. (2000). “Mixed integer/real valued adjustment (IRA) problems: GPS initial cycle ambiguity resolution by means of LLL algorithm.” GPS Solutions, 4, 31–44.
Hadamard, J. (1932). Lecture on Cauchy’s problem in linear partial differential equations, Yale University Press, reprinted by Dover, New York, 1952.
Hansen, P. C. (1992). “Analysis of discrete ill-posed problems by means of the L-curve.” SIAM Rev., 34, 561–580.
Hoerl, A. E., and Kennard, R. W. (1970a). “Ridge regression: Applications to nonorthogonal problems.” Technometrics, 12, 69–82.
Hoerl, A. E., and Kennard, R. W. (1970b). “Ridge regression: Biased estimation for nonorthogonal problems.” Technometrics, 12, 55–67.
Liu, L. T., Hsu, H. T., Zhu, Y. Z., and Ou, J. K. (1999). “A. new approach to GPS ambiguity decorrelation.” J. Geodesy, Berlin, 73, 478–490.
Lou, L. Z., and Grafarend, E. W. (2002). “GPS integer ambiguity resolution by various decorrelation methods.” ZFV. Z. Vermessungswes., 127, 1–8.
Tarantola, P. (2005). Inverse problem theory, SIAM, Philadelphia
Teunissen, P. J. G. (1994). “A new method for fast carrier phase ambiguity estimation.” Proc., IEEE PLANS’94, Las Vegas, 562–573.
Teunissen, P. J. G. (1995). “The least-squares ambiguity decorrelation adjustment: A method for fast GPS integer ambiguity estimation.” J. Geodesy, Berlin, 70, 65–82.
Teunissen, P. J. G. (1996). “An analytical study of ambiguity decorrelation using dual frequency code and carrier phase.”J. Geodesy, Berlin, 70, 515–528.
Teunissen, P. J. G., de Jonge, P. J., and Tiberius, C. C. (1997). “The least-squares ambiguity decorrelation adjustment: Its performance on short GPS baselines and short observation spans.” J. Geodesy, Berlin, 71, 589–602.
Tikhonov, A. N. (1963a). “Regularization of ill-posed problems.” Dokl. Akad. Nauk SSSR, 151 (1), 49–52.
Tikhonov, A. N. (1963b). “Solution of incorrectly formulated problems and the regularization method.” Dokl. Akad. Nauk SSSR, 151 (3), 501–504.
Xu, P. L. (1992). “Determination of surface gravity anomalies using gradiometric observables.” Geophys. J. Int., 110, 321–332.
Xu, P. L. (1998a). “Mixed integer observation models and integer programming with applications to GPS ambiguity resolution.” J. Geod. Soc. Jpn., 44, 169–87.
Xu, P. L. (1998b). “Truncated SVD methods for discrete linear ill-posed problems.” Geophys. J. Int., 135, 505–514.
Xu, P. L. (2001). “Random simulation, and GPS decorrelation.” J. Geodesy, Berlin, 75, 408–423.
Xu, P. L. (2006). “Voronoi cells, probabilistic bouds, and hypothesis testing in mixed integer linear models.” IEEE Trans. Inf. Theory, 52, 3122–3138.
Xu, P. L., Fukuda, Y., and Liu, Y. (2006). “Multiple parameter regularization: Numerical solution and application to the determination of geopotential from precise satellite orbits.” J. Geodesy, Berlin, 80, 17–27.
Xu, P. L., and Rummel, R. (1994a). “A simulation study of smoothness methods in recovery of regional gravity fields.” Geophys. J. Int., 117, 472–486.
Xu, P. L., and Rummel, R. (1994b). “Generalized ridge regression with applications in determination of potential fields.” Manuscr. Geod., 20, 8–20.
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Information
Published In
Journal of Surveying Engineering
Volume 133 • Issue 4 • November 2007
Pages: 168 - 172
Copyright
© 2007 ASCE.
History
Received: Jul 12, 2006
Accepted: Nov 13, 2006
Published online: Nov 1, 2007
Published in print: Nov 2007
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