Article | Open | Published:
An improved regularization method for estimating near real-time systematic errors suitable for medium-long GPS baseline solutions
Earth, Planets and Spacevolume 60, pages793–800 (2008)
It is well known that the key problem associated with network-based real-time kinematic (RTK) positioning is the estimation of systematic errors of GPS observations, such as residual ionospheric delays, tropospheric delays, and orbit errors, particularly for medium-long baselines. Existing methods dealing with these systematic errors are either not applicable for making estimations in real-time or require additional observations in the computation. In both cases, the result is a difficulty in performing rapid positioning. We have developed a new strategy for estimating the systematic errors for near real-time applications. In this approach, only two epochs of observations are used each time to estimate the parameters. In order to overcome severe ill-conditioned problems of the normal equation, the Tikhonov regularization method is used. We suggest that the regularized matrix be constructed by combining the a priori information of the known coordinates of the reference stations, followed by the determination of the corresponding regularized parameter. A series of systematic errors estimation can be obtained using a session of GPS observations, and the new process can assist in resolving the integer ambiguities of medium-long baselines and in constructing the virtual observations for the virtual reference station. A number of tests using three medium- to long-range baselines (from tens of kilometers to longer than 1000 kilometers) are used to validate the new approach. Test results indicate that the coordinates of three baseline lengths derived are in the order of several centimeters after the systematical errors are successfully removed. Our results demonstrate that the proposed method can effectively estimate systematic errors in the near real-time for medium-long GPS baseline solutions.
Alves, P., Development of Two Novel Carriers Phase-Based Methods for Multiple Reference Station Positioning, Ph.D thesis, University of Calgary, Canada, 2004.
Cai, J., C. Hu, and E. W. Grafarend, The Optimal Regularization Method and its Application in GNSS Rapid Static Positioning, Proc of the 20th Int Tech Meeting Satellite Division US Inst Navigation, September, Fort Worth, USA, 2007.
Chen, X. M., S. W. Han, C. Rizos, and P. C. Goh, Improving real-time positioning efficiency using the Singapore integrated multiple reference station network, Proc of the 13th Int Tech Meeting Satellite Division US Inst Navigation, Salt Lake City, UT, 19–22 September, 9–16, 2000.
Chen, H. Y., C. Rizos, and S. W. Han, From simulation to implementation: low-cost densification of permanent GPS networks in support of geodetic applications, J. Geod., 75, 515–526, 2001.
Chen, H. Y., C. Rizos, and S. W. Han, An instantaneous ambiguity resolution procedure suitable for medium-scale GPS reference station networks, Surv. Rev., 37, 396–410, 2004.
Collins, P. and R. Langley, Limiting factors in tropospheric propagation delay error modelling for GPS airborne navigation, Proc. of the U. S. Institute of Navigation Annual Meeting, 19–21 June, Cambridge, MA, USA, 519–528, 1996.
Dai, L., J. L. Wang, C. Rizos, and S. W. Han, Predicting atmospheric biases for real-time ambiguity resolution in GPS/GLONASS reference station networks, J. Geod., 76, 617–628, 2003.
Fotopoulos, G. and M. E. Cannon, Spatial and temporal characteristics of DGPS carrier phase errors over a regional network, Proc. U.S. Institute of Navigation Annual Meeting, San Diego, California, 26–28, June, 54–64, 2000.
Gao, Y., Z. Li, and J. F. McLellan, Carrier phase based regional area differential GPS for decimeter-level positioning and navigation, Proc 10th Int Tech Meeting Satellite Division Inst Navigation, Kansas City, Mo, 16–19 September, 1305–1313, 1997.
Han, S. and C. Rizos, GPS network design and error mitigation for realtime continuous array monitoring system, Proc. of 9th Int. Tech. Meeting of the Satellite Division of the U.S. Inst. of Navigation, Kansas City, Missouri, 17–20 September, 1827–1836, 1996.
Hansen, P. C., The L-Curve and its Use in the Numerical Treatment of Inverse Problems, Computational Inverse Problems in Electrocardiology, 5, 119–142, WIT Press, 2001.
Hoerl, A. E. and R. W. Kennard, Ridge regression: Biased estimation for non-orthogonal problems, Technometrics, 12, 55–67, 1970a.
Hoerl, A. E. and R. W. Kennard, Ridge regression: Applications to non-orthogonal problems, Technometrics, 12, 69–82, 1970b, correction, 12, 123.
Hofmann, B. W., H. Lichtenegger, and J. Collins, Global positioning system: theory and practice, 5th Edition, Springer, Berlin Heidelberg New York, 2001.
Horemuž, M. and L. E. Sjöberg, Rapid GPS ambiguity resolution for short and long baselines, J. Geod., 76, 381–391, 2002.
Hu, G. R., H. S. Khoo, P. C. Goh, and C. L. Law, Development and assessment of GPS virtual reference stations for RTK positioning, J. Geod., 77, 292–302, 2003.
Lachapelle, G., P. Alves, L. P. Fortes, M. E. Cannon, and B. Townsend, DGPS RTK positioning using a reference network, Proc of the 13th Int Tech Meeting Satellite Division US Inst Navigation, Salt Lake City, UT, 19–22 September, 1165–1171, 2000.
Landau, H., U. Vollath, and X. M. Chen, Virtual Reference Station Systems, J. GPS, 1.1, 137–144, 2003.
Leick, A., GPS satellite surveying, 3rd Edition, Wiley, New York, 2004.
Odijk, D., H. van der Marel, and I. Song, Precise GPS positioning by applying ionospheric corrections from an active control network, GPS Solution, 3, 49–57, 2000.
Ou, J. K., Uniform Expression of Solutions of Ill-posed Problems in Surveying Adjustment and the Fitting Method by a Selection of the Parameter Weights, ACTA Geod. Cartogr. Sin., 33, 284–288, 2004 (in Chinese).
Ou, J. K. and Z. J. Wang, An improved regularization method to resolve integer ambiguity in rapid positioning using single frequency GPS receivers, Chinese Sci. Bull., 49, 196–200, 2004.
Raquet, J. and G. Lachapelle, RTK positioning with multiple reference stations, GPS World, 12, 48–53, 2001.
Rizos, C. and S. Han, Reference station network based RTK systems—Concepts & progress, Wuhan Univ. J. Nat. Sci., 8, 566–574, 2003.
Roulston, A., N. Talbot, and K. Zhang, Evaluation of various GPS satellite ephemerides, Proc. 13th Int. Tech. Meeting Satellite Division US Inst. Navigation, 19–22 September, Salt Lake City, USA, 45–54, 2000.
Sun, H., M. E. Cannon, and T. E. Melgard, Real-time GPS reference network carrier phase ambiguity resolution, Proc. of Institute of Navigation National Technical Meeting, 25–27 January, San Diego, CA, USA, 193–199, 1999.
Teunissen, P. J., The geometry-free GPS ambiguity search space with a weighted ionosphere, J. Geod., 71, 370–383, 1997.
Tikhonov, A. N. and V. Y. Arsenin, Solutions of Ill-posed Problems, New York: Wiley, 10–21, 1977.
Vollath, U., A. Buerchl, H. Landau, C. Pagels, and B. Wagner, Long-Range RTK Positioning Using Virtual Reference Stations, Proc. of the 13th Int. Tech. Meeting Satellite Division US Inst Navigation, September, Salt Lake City, USA, 1143–1147, 2000.
Vollath, U., H. Landau, X. Chen, K. Doucet, and C. Pagels, Network RTK versus Single Base RTK—Understanding the Error Characteristics, Proc. of the 15th Int. Tech. Meeting Satellite Division US Inst. Navigation, 24–27 September, Portland USA, 2774–2781, 2002.
Wang, Z., C. Rizos, and S. Lim, Single epoch algorithm based on Tikhonov regularization for deformation monitoring using single frequency GPS receivers, Surv. Rev., 38, 682–688, 2006.
Xu, P. L., The Value of Minimum Norm Estimation of Geopotential Fields, Geophys. J. Int., 111, 170–178, 1992.
Xu, P. L., Truncated SVD Methods for Linear Discrete Ill-posed Problems, Geophys. J. Int., 135, 505–514, 1998.
Xu, P. L. and R. Rummel, A Generalized Ridge Regression Method with Applications in Determination of Potential Fields, Manuscr. Geod., 20, 8–20, 1994.
Xu, P. L., Y. Fukuda, and Y. M. Liu, Multiple Parameter Regularization: Numerical Solutions and Applications to the Determination of Geopotential from Precise Satellite Orbits, J. Geod., 80, 17–27, 2006.
Yang, Y., Robust estimation of geodetic datum transformation, J. Geod., 73, 68–274, 1999.
Zhang, K., F. Wu, S. Wu, C. Rizos, C. Roberts, L. Ge, T. Yan, C. Gordini, A. Kealy, M. Hale, P. Ramm, H. Asmussen, D. Kinlyside, and P. Harcombe, Sparse or Dense: Challenges of Australian Network RTK, Proceedings of IGNSS Symposium 2006, Holiday Inn Surfers Paradise, Australia, 17–21, July 2006.
Zhou, J. W., J. K. Ou, and Y. X. Yang, Research on the theory of observation data and errors, 29–35, Seismic Publishing House, Beijing, 1999 (in Chinese).
About this article
- Medium-long baselines
- systematic errors
- GPS network RTK positioning
- ill-conditioned equation
- Tikhonov regularization