Rotation- and Translation-Free Estimations of Symmetric, Rank-Two Tensors with a Case Study in LIDAR Surveying
Publication: Journal of Surveying Engineering
Volume 136, Issue 1
Abstract
An eigenparameter analysis plays an important role in varied fields where a symmetric tensor is involved. This technique allows one to investigate the principal behaviors (i.e., magnitudes and orientations) of a physical phenomenon that can be represented as a rank-two symmetric tensor. In this study, an analytical approach that enables rotation- and translation-free estimations of the eigenparameters from a symmetric tensor is developed, with a goal to remove the errors associated with a neglect and/or miscalculation of reference frame variations during a dynamical process. Two numerical examples, one with simulated data and the other with real light detection and ranging (LIDAR) surveying data, have been carried out to demonstrate the capability of the proposed approach in estimating the principal strains from a symmetric strain tensor. The results reveal that the proposed approach is capable of giving a direct estimate for the strain tensor without being affected by the rotation and translation of the reference frame and thus produces a principal strain estimate of a higher quality.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The writers thank Dr. Tomás Soler, Dr. Peiliang Xu, and the anonymous reviewers for their constructive comments which significantly improved the quality of the original manuscript. The funding support by the National Science Council in Taiwan (under Contract No. NSCTNSC 96-2218-E-002-027) is also gratefully acknowledged.
References
Altamimi, Z., Sillard, P., and Boucher, C. (2002). “ITRF2000: A new release of the international terrestrial reference frame for earth science applications.” J. Geophys. Res., 107(B10), 2214.
Angelier, J., Tarantola, A., Vallete, B., and Manoussis, S. (1982). “Inversion of field data in fault tectonics to obtain the regional stress. I: Single phase fault populations—A new method of computing the stress tensor.” Geophys. J. R. Astron. Soc., 69(3), 607–621.
Bodewig, E. (1959). Matrix calculus, 2nd Ed., North-Holland, Amsterdam.
Boucher, C., and Altamimi, Z. (2001). “ITRS, PZ90 and WGS84: Current realizations and the related transformation parameters.” J. Geod., 75(11), 613–619.
Bressan, G., Bragato, P. L., and Venturini, C. (2003). “Stress and strain tensors based on focal mechanisms in the seismotectonic framework of the Friuli-Venezia Giulia Region (northeastern Italy).” Bull. Seismol. Soc. Am., 93(3), 1280–1297.
Cai, J. (2004). “The statistical inference of eigenspace components of a symmetric random deformation tensor.” Ph.D. thesis, Deutsche Geodätische Kommission (DGK), Reihe C, München.
Cai, J., and Grafarend, E. W. (2007). “Statistical analysis of the eigenspace components of the two-dimensional, symmetric rank-two strain rate tensor derived from the space geodetic measurements (ITRF92-ITRF2000 data sets) in central Mediterranean and Western Europe.” Geophys. J. Int., 168(2), 449–472.
Cai, J., Grafarend, E. W., and Schaffrin, B. (2005). “Statistical inference of the eigenspace components of a two-dimensional, symmetric rank-two random tensor.” J. Geod., 78(7–8), 425–436.
DeMets, C., Gordon, R., Argus, D., and Stein, S. (1990). “Current plate motions.” Geophys. J. Int., 101(2), 425–478.
Grafarend, E. W., and Schaffrin, B. (1993). Ausgleichungsrechnung in linearen modellen, BI Wissenschaftsverlag, Mannheim.
Han, J. Y., and van Gelder, B. H. W. (2006). “Stepwise parameter estimations for a time-variant similarity transformation.” J. Surv. Eng., 132(4), 141–148.
Han, J. Y., van Gelder, B. H. W., and Soler, T. (2007). “On covariance propagation of eigenparameters of symmetric n-D tensors.” Geophys. J. Int., 170(2), 503–510.
Han, J. Y., van Gelder, B. H. W., Soler, T., and Snay, R. A. (2008). “Geometric combination of multiple terrestrial network solutions.” J. Surv. Eng., 134(4), 126–131.
Jaw, J. J., and Chuang, T. Y. (2008). “Registration of ground-based LIDAR point clouds by means of 3D line features.” J. Chinese Inst. Engr., 31(6), 1031–1045.
Owens, W. H. (2000a). “Error estimates in the measurement of anisotropic magnetic susceptibility.” Geophys. J. Int., 142(2), 516–526.
Owens, W. H. (2000b). “Statistical applications to second-rank tensors in magnetic fabric analysis.” Geophys. J. Int., 142(2), 527–538.
Rao, C. R., and Mitra, S. K. (1971). Generalized inverse of matrices and its applications, Wiley, New York.
Scheffé, H. (1959). The analysis of variance, Wiley, New York.
Snay, R. A. (1999). “Using the HTDP software to transform spatial coordinates across time and between reference frames.” Surv. Land Inf. Sys., 59(1), 15–25.
Soler, T. (1998). “A compendium of transformation formulas useful in GPS work.” J. Geod., 72(7–8), 482–490.
Soler, T., and van Gelder, B. H. W. (1991). “On covariances of eigenvalues and eigenvectors of second-rank symmetric tensors.” Geophys. J. Int., 105(2), 537–546.
Soler, T., and van Gelder, B. H. W. (2006). “Corrigendum: On covariances of eigenvalues and eigenvectors of second-rank symmetric tensors.” Geophys. J. Int., 165(1), 382.
Wyss, M., Liang, B., Tanigawa, W. R., and Wu, X. (1992). “Comparison of orientations of stress and strain tensors based on fault plane solutions in Kaoiki, Hawaii.” J. Geophys. Res., 97(B4), 4769–4790.
Xu, P. L. (1999). “Spectral theory of constrained second-rank symmetric random tensors.” Geophys. J. Int., 138(1), 1–24.
Xu, P. L. (2004). “Determination of regional stress tensors from fault-slip data.” Geophys. J. Int., 157(3), 1316–1330.
Xu, P. L., and Grafarend, E. (1996a). “Statistics and geometry of the eigenspectra of three-dimensional second-rank symmetric random tensors.” Geophys. J. Int., 127(3), 744–756.
Xu, P. L., and Grafarend, E. (1996b). “Probability distribution of eigenspectra and eigendirections of a two-dimensional, symmetric rank-two random tensor.” J. Geod., 70(7), 419–430.
Xu, P. L., Shimada, S., Fujii, Y., and Tanaka, T. (2000). “Invariant geodynamical information in geometric geodetic measurements.” Geophys. J. Int., 142(2), 586–602.
Information & Authors
Information
Published In
Journal of Surveying Engineering
Volume 136 • Issue 1 • February 2010
Pages: 23 - 28
Copyright
© 2010 ASCE.
History
Received: Apr 7, 2009
Accepted: Jun 4, 2009
Published online: Jun 6, 2009
Published in print: Feb 2010
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.