A New Regularized Solution to Ill-Posed Problem in Coordinate Transformation

ABSTRACT

Coordinates transformation is generally required in GPS applications. If the transformation parameters are solved with the known coordinates in a small area using the Bursa model, the precision of transformed coordinates is generally very poor. Since the translation parameters and rotation parameters are highly correlated in this case, a very large condition number of the coefficient matrix A exists in the linear system of equations. Regularization is required to reduce the effects caused by the intrinsic ill-conditioning of the problem and noises in the data, and to stabilize the solution. Based on advanced regularized methods, we propose a new regularized solution to the ill-posed coordinate transformation problem. Simulation numerical experiments of coordinate transformation are given to shed light on the relationship among different regularization approaches. The results indicate that the proposed new method can obtain a more reasonable resolution with higher precision and/or robustness.

Coordinates transformation is generally required in GPS applications. If the transformation parameters are solved with the known coordinates in a small area using the Bursa model, the precision of transformed coordinates is generally very poor. Since the translation parameters and rotation parameters are highly correlated in this case, a very large condition number of the coefficient matrix A exists in the linear system of equations. Regularization is required to reduce the effects caused by the intrinsic ill-conditioning of the problem and noises in the data, and to stabilize the solution. Based on advanced regularized methods, we propose a new regularized solution to the ill-posed coordinate transformation problem. Simulation numerical experiments of coordinate transformation are given to shed light on the relationship among different regularization approaches. The results indicate that the proposed new method can obtain a more reasonable resolution with higher precision and/or robustness.

Cite this paper

X. Ge and J. Wu, "A New Regularized Solution to Ill-Posed Problem in Coordinate Transformation,"*International Journal of Geosciences*, Vol. 3 No. 1, 2012, pp. 14-20. doi: 10.4236/ijg.2012.31002.

X. Ge and J. Wu, "A New Regularized Solution to Ill-Posed Problem in Coordinate Transformation,"

References

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[5] E. W. Grafarend and L. J. Awange, “Nonlinear Analysis of the Transformational Datum Transformation,” Journal of Geodesy, Vol. 77, 2003, pp. 66-76. doi:10.1007/s00190-002-0299-9

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[8] G. H. Golub and C. Reinsch, “Singular Value Decomposition and Least Squares Solutions,” Numerical Mathematics, Vol. 14, 1970, pp. 403-420. doi:10.1007/BF02163027

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[10] P. L. Xu and R. Rummel, “Generalized Ridge Regression with Applications in Determination of Potential Fields,” Manuscripta Geodaetica, Vol. 20, No. 1, 1994, pp. 8-20.

[11] P. L. Xu, Y. Fukuda and Y. Liu, “Multiple Parameter Regularization: Numerical Solution and Application to the Determination of Geopotential from Precise Satellite Orbits,” Journal of Geodesy, Vol. 80, No. 1, 2006, pp. 17-27. doi:10.1007/s00190-006-0025-0

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[15] P. C. Hansen, “The Discrete Picard Condition for Discrete Ill-Posed Problems,” BIT, Vol. 30, No. 4, 1990, pp. 658-672. doi:10.1007/BF01933214

[16] P. C. Hansen, “Truncated SVD Solutions to Discrete Ill- Posed Problems with Ill-Determined Numerical Rank,” Journal on Scientific and Statistical Computing, Vol. 11, 1990, pp. 503-518. doi:10.1137/0911028

[17] P. C. Hansen, “Analysis of Discrete Ill-Posed Problems by Means of the L-Curve,” SIAM Review, Vol. 34, No. 4, 1992, pp. 561-580. doi:10.1137/1034115

[18] M. Hanke, “Limitations of the L-Curve Method in Ill- Posed Problems,” BIT, Vol. 36, No. 2, 1996, pp. 287-301. doi:10.1007/BF01731984

[19] P. L. Xu, “Truncated SVD Methods for Discrete Linear Ill-Posed Problems,” Geophysical Journal International, Vol. 135, No. 2, 1998, pp. 505-514. doi:10.1046/j.1365-246X.1998.00652.x

[20] Y. Z. Shen and B. F. Li, “Regularized Solution to Fast GPS Ambiguity Resolution,” Journal of Surveying Engineering, Vol. 133, No. 4, 2007, pp. 168-172. doi:10.1061/(ASCE)0733-9453(2007)133:4(168)

[21] T. Regińska, “Regularization of Discrete Ill-Posed Problems,” BIT, Vol. 44, No. 3, 2004, pp. 119-133.

[22] G. H. Golub, M. T. Heath and G. Wahba, “Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter,” Technometrics, Vol. 21, No. 2, 1979, pp. 215- 223. doi:10.2307/1268518

[1] E. W. Grafarend, F. Krumm and F. Okeke, “Curvilinear Geodetic Datum Transformations,” Z Vermessungswesen, Vol. 120, No. 4, 1995, pp. 334-350.

[2] P. Vanicek and R. R. Steeves, “Transformation of Coordinates between Two Horizontal Geodetic Datum,” Journal of Geodesy, Vol. 70, No. 11, 1996, pp. 740-745.

[3] P. Vanicek, P. Novak and R. Craymerm, “On the Correct Determination of Transformation Parameters of Horizontal Geodetic Datum,” Geomatica, Vol. 56, No. 4, 2002, pp. 329-340.

[4] Y. Yang, “Robust Estimation of Geodetic Datum Transformation,” Journal of Geodesy, Vol. 73, No. 9, 1999, pp. 268-274. doi:10.1007/s001900050243

[5] E. W. Grafarend and L. J. Awange, “Nonlinear Analysis of the Transformational Datum Transformation,” Journal of Geodesy, Vol. 77, 2003, pp. 66-76. doi:10.1007/s00190-002-0299-9

[6] A. N. Tikhonov, “Regularization of Ill-Posed Problems,” Doklady Akademi Nauk, Vol. 151, No. 1, 1963, pp. 49- 52.

[7] A. N. Tikhonov, “Solution of Incurrectly Formulated Pro- blems and the Regularization Method,” Doklady Akade- mi Nauk, Vol. 151, No. 3, 1963, pp. 501-504.

[8] G. H. Golub and C. Reinsch, “Singular Value Decomposition and Least Squares Solutions,” Numerical Mathematics, Vol. 14, 1970, pp. 403-420. doi:10.1007/BF02163027

[9] P. C. Hansen, “The Truncated SVD as a Method for Re- gularization,” BIT, Vol. 27, 1987, pp. 534-553. doi:10.1007/BF01937276

[10] P. L. Xu and R. Rummel, “Generalized Ridge Regression with Applications in Determination of Potential Fields,” Manuscripta Geodaetica, Vol. 20, No. 1, 1994, pp. 8-20.

[11] P. L. Xu, Y. Fukuda and Y. Liu, “Multiple Parameter Regularization: Numerical Solution and Application to the Determination of Geopotential from Precise Satellite Orbits,” Journal of Geodesy, Vol. 80, No. 1, 2006, pp. 17-27. doi:10.1007/s00190-006-0025-0

[12] G. H. Golub and C. F. Van Loan, “Matrix Computation,” 3rd Edition, The Johns Hopkins University Press, Baltimore, 1996.

[13] P. C. Hansen, “Rank-Deficient and Discrete Ill-Posed Pro- blems,” SIAM, Philadelphia, 1998. doi:10.1137/1.9780898719697

[14] P. Tarantola, “Inverse Problem Theory,” SIAM, Philadelphia, 2005.

[15] P. C. Hansen, “The Discrete Picard Condition for Discrete Ill-Posed Problems,” BIT, Vol. 30, No. 4, 1990, pp. 658-672. doi:10.1007/BF01933214

[16] P. C. Hansen, “Truncated SVD Solutions to Discrete Ill- Posed Problems with Ill-Determined Numerical Rank,” Journal on Scientific and Statistical Computing, Vol. 11, 1990, pp. 503-518. doi:10.1137/0911028

[17] P. C. Hansen, “Analysis of Discrete Ill-Posed Problems by Means of the L-Curve,” SIAM Review, Vol. 34, No. 4, 1992, pp. 561-580. doi:10.1137/1034115

[18] M. Hanke, “Limitations of the L-Curve Method in Ill- Posed Problems,” BIT, Vol. 36, No. 2, 1996, pp. 287-301. doi:10.1007/BF01731984

[19] P. L. Xu, “Truncated SVD Methods for Discrete Linear Ill-Posed Problems,” Geophysical Journal International, Vol. 135, No. 2, 1998, pp. 505-514. doi:10.1046/j.1365-246X.1998.00652.x

[20] Y. Z. Shen and B. F. Li, “Regularized Solution to Fast GPS Ambiguity Resolution,” Journal of Surveying Engineering, Vol. 133, No. 4, 2007, pp. 168-172. doi:10.1061/(ASCE)0733-9453(2007)133:4(168)

[21] T. Regińska, “Regularization of Discrete Ill-Posed Problems,” BIT, Vol. 44, No. 3, 2004, pp. 119-133.

[22] G. H. Golub, M. T. Heath and G. Wahba, “Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter,” Technometrics, Vol. 21, No. 2, 1979, pp. 215- 223. doi:10.2307/1268518