Weakly Multicollinear Datum Transformations
Publication: Journal of Surveying Engineering
Volume 138, Issue 4
Abstract
Geodetic network design and optimization is a very well-known concept in geodesy. However, in many cases, the available geodetic network configuration with respect to the estimation model is insufficient because of the physical and financial limitations. For the case of estimating the datum transformation parameters between two datums, the colocated points are only an unevenly and inhomogenously distributed subset of the available national/regional networks. Because the transformation parameters are defined with respect to an earth-centered, earth-fixed (ECEF) frame, very limited geographic coverage of the national/regional networks often leads to a weakly multicollinear estimation problem. Such limited geographical coverage is often coupled with the intrinsic geometrical distortions as well as the relatively lower precision of the observations, in particular, when transforming a terrestrial network into a space-based network. In such cases, the individual parameters become highly correlated and oversensitive to the network configuration, and the individual transformation parameters cannot be estimated reliably. In this study, the concept of an idealized three-dimensional (3D) regional network geometry is introduced, its inverse cofactor matrix is analytically derived, and a regularized estimation method based on the inverse cofactor matrix of an ideal network distribution is presented to deal with the weakly multicollinear datum transformation problem. The efficiency of the proposed method is shown in three realistically simulated networks. The proposed method outperforms the standard least squares in terms of mean square error (MSE) and reduces the correlations among the parameters.
Get full access to this article
View all available purchase options and get full access to this article.
References
Aktuğ, B. (2009). “Inverse and compound datum/frame transformations.” J. Surv. Eng., 135(2), 46–55.
Aktuğ, B., Kaypak, B., and Çelik, R. N. (2010). “Source parameters for the , 03 February 2002, Çay Earthquake (Turkey) and aftershocks from GPS, Southwestern Turkey.” J. Seismol., 14(3), 445–456.
Baarda, W. (1968). “A testing procedure for use in geodetic networks.” Publications on Geodesy, Vol. 2, Netherlands Geodetic Commission, Delft, Netherlands.
Burša, M. (1966). Fundamentals of the theory of geometric satellite geodesy, Trav. Inst. Géophys. Acad. Tchéc. Sci., Československá Akademie Věd., Prague, Czech Republic.
Bürgmann, R., et al. (2002). “Deformation during the 12 November 1999 Düzce, Turkey, Earthquake, from GPS and InSAR Data.” Bull. Seismol. Soc. Am., 92(1), 161–171.
Cai, J. (2000). “The systematic analysis of the transformation between the German geodetic reference system (DHDN, DHHN) and the ETRF system (DREF91).” Earth Planets Space, 52(11), 947–952.
Cai, J., Grafarend, E., and Schaffrin, B. (2004). “The A-optimal regularization parameter in uniform Tykhonov-Phillips regularization α-weighted BLE.” V Hotine-Marussi Symposium on Mathematical Geodesy, F. Sansi, ed., Vol. 127. Int. Association of Geodesy Symp., Matera, Italy, 309–324.
Collilieux, X., Altamimi, Z., Ray, J., van Dam, T., and Wu, X. (2009). “Effect of the satellite laser ranging network distribution on geocenter motion estimation.” J. Geophys. Res., 114(B04402), 1–17.
Cooper, M. A. R. (1987). Control surveys in civil engineering, Collins, London.
Cross, P. A. (1985). “Numerical methods in network design.” Optimization and design of geodetic networks, E. W. Grafarend and F. Sanso, eds., Springer, Berlin, 429–435.
Even-Tzur, G. (1999). “Sensitivity design for monitoring deformation networks.” Boll. Geod. Sci. Affini, 58(4), 313–324.
Even-Tzur, G. (2002). “GPS vector configuration design for monitoring deformation networks.” J. Geod., 76(8), 455–461.
Flores, A., Ruffini, G., and Rius, A. (2000). “4D tropospheric tomography using GPS slant wet delays.” Ann. Geophys., 18(2), 223–234.
Golub, G., and von Matt, U.(1997). “Tikhonov regularization for large scale problems.” Workshop on scientific computing, G. H. Golub, S. H. Lui, F. Luk, and R. Plemmons, eds., Springer, New York, 3–26.
Grafarend, E. W. (1970). “Optimization of geodetic networks.” Bolletino di Geodesia a Science Affini, 33(4), 351–406.
Hansen, P. (1992). “Analysis of discrete ill-posed problems by means of the L-curve.” SIAM Rev., 34(4), 561–580.
Hoerl, A. E., and Kennard, R. W. (1970). “Ridge regression: Biased estimation for nonorthogonal problems.” Technometrics, 12(1), 55–67.
Howe, B. M., Runciman, K., and Secan, J. A. (1998). “Tomography of ionosphere: Four dimensional simulations.” Radio Sci., 33(1), 109–128.
Kuang, S. L. (1996). Geodetic network analysis and optimal design: Concepts and applications, Ann Arbor Press, Chelsea, MI.
Kumar, M. (1972). “Coordinate transformation by minimizing correlations between parameters.” Rep. No. 184, Dept. of Geodetic Science, Ohio State Univ., Columbus, OH.
Kutoğlu, H. S. (2004). “Figure condition in datum transformation.” J. Surv. Eng., 130(3), 138–141.
Kwon, J. H., Bae, T.-S., Choi, Y.-S., Lee, D.-C., Lee, Y.-W. (2005). “Geodetic datum transformation to the global geocentric datum for seas and islands around Korea.” Geosciences J., 9(4), 353–361.
Leick, A., and van Gelder, B. H. W.(1975). “On similarity transformations and geodetic network distortions based on Doppler satellite” Rep. No. 235, Dept. of Geodetic Science, Ohio State Univ., Columbus, OH.
Magnus, J. R., and Neudecker, H. (1999). Matrix differential calculus with applications in statistics and econometrics, Wiley, New York.
Mallows, C. L. (1973). “Some comments on Cp.” Technometrics, 15(4), 661–675.
Malys, S. (1988). “Dispersion and correlation among transformation parameters relating two satellite reference frames.” M.S. thesis, Ohio State Univ., Columbus, OH.
Phillips, D. L. (1962). “A technique for the numerical solution of certain integral equations of the first kind.” J. ACM, 9(1), 84–96.
Schaffrin, B. (1985). “Aspects on network design.” Optimization and design of geodetic networks, E. W. Grafarend and F. Sanso, eds., Springer, Berlin, 548–597.
Schaffrin, B. (2008). “Minimum mean squared error (MSE) adjustment and the optimal Tykhonov-Phillips regularization parameter via reproducing best invariant quadratic uniformly unbiased estimates (repro-BIQUUE).” J. Geod., 82(2), 113–121.
Schmitt, G. (1985). “Review of network design: Criteria, risk functions, design ordering.” Optimization and design of geodetic networks, E. W. Grafarend and F. Sanso, eds., Springer, Berlin, 6–10.
Soler, T. (1976). “On differential transformations between Cartesian and curvilinear (geodetic) coordinates.” Rep. No. 236, Dept. of Geodetic Science, Ohio State Univ., Columbus, OH.
Tikhonov, A. N. (1963). “The regularization of incorrectly posed problem.” Soviet Math. Dokl., 4(6), 1624–1627.
Vaníček, P., and Steeves, R. R. (1996). “Transformation of coordinates between two horizontal geodetic datums.” J. Geod., 70(11), 740–745.
Vaníček, P., Novák, P., Craymer, M. R., and Pagiatakis, S. (2002). “On the correct determination of transformation parameters of a horizontal geodetic datum.” Geomatica, 56(4), 329–340.
Wang, J., Chen, Y. (1994). “On the reliability measure of observations.” Acta Geodaet. Cartograph Sin., English Edition (1994), 42–51.
Wells, D. E., and Vaníček, P. (1975). “Alignment of geodetic and satellite coordinate systems to the average terrestrial system.” Bull. Geod., 117(1), 241–257.
Wright, T. J., Lu, Z., and Wicks, C. (2003). “Source model for the Mw 6.7, 23 October 2002, Nenana Mountain Earthquake (Alaska) from InSAR.” Geophys. Res. Lett., 30(18), 1974–1978.
Xu, P., and Rummel, R. (1994). “Generalized ridge regression with applications in determination of potential fields.” Manuscr. Geod., 20(1), 8–20.
Information & Authors
Information
Published In
Journal of Surveying Engineering
Volume 138 • Issue 4 • November 2012
Pages: 184 - 192
Copyright
© 2012 American Society of Civil Engineers.
History
Received: Oct 7, 2011
Accepted: Mar 7, 2012
Published online: Mar 10, 2012
Published in print: Nov 1, 2012
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.