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Journal of Geodetic Science

Editor-in-Chief: Eshagh, Mehdi

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On weighted total least squares adjustment for solving the nonlinear problems

C. Hu
  • Corresponding author
  • College of Surveying and Geo- Informatics, Tongji University, Shanghai, People’s Republic of China
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Y. Chen
  • College of Surveying and Geo-Informatics, Tongji University, Shanghai, People’s Republic of China / Key Laboratory of Advanced Surveying Engineering of State Bureau of Surveying and Mapping, Shanghai, People’s Republic of China
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Y. Peng
Published Online: 2014-06-06 | DOI: https://doi.org/10.2478/jogs-2014-0007


In the classical geodetic data processing, a non- linear problem always can be converted to a linear least squares adjustment. However, the errors in Jacob matrix are often not being considered when using the least square method to estimate the optimal parameters from a system of equations. Furthermore, the identity weight matrix may not suitable for each element in Jacob matrix. The weighted total least squares method has been frequently applied in geodetic data processing for the case that the observation vector and the coefficient matrix are perturbed by random errors, which are zero mean and statistically in- dependent with inequality variance. In this contribution, we suggested an approach that employ the weighted total least squares to solve the nonlinear problems and to mitigate the affection of noise in Jacob matrix. The weight matrix of the vector from Jacob matrix is derived by the law of nonlinear error propagation. Two numerical examples, one is the triangulation adjustment and another is a simulation experiment, are given at last to validate the feasibility of the developed method.

Keywords: nonlinear adjustment; nonlinear error propagation; weighted total least squares


  • Acar M., Ozludemir M., Akyilmaz O., Celik R. and Ayan T., 2006, Deformation analysis with Total Least Squares, Nat. Hazard. Earth. Sys, 6,4, 663-669.CrossrefGoogle Scholar

  • Amiri-Simkooei A., and Jazaeri S., 2012, Weighted total least squares formulated by standard least squares theory, J Geod. Sci., 2,2, 113-124.Google Scholar

  • Amiri-Simkooei A., and Jazaeri S., 2013, Data-snooping procedure applied to errors-in-variables models, Stud. Geophys. Geod, 1-16.Web of ScienceGoogle Scholar

  • Davis T., 1999, Total least-squares spiral curve fitting, J Surv. Eng., 125,4, 159-176.Google Scholar

  • Felus Y., 2004, Application of total least squares for spatial point process analysis, J Surv. Eng., 130,3, 126-133.Google Scholar

  • Felus Y. and Scha_rin B., 2005, A total least-squares approach in two stages for semivariogram modeling of aeromagnetic data, 215-220. Google Scholar

  • Golub G. H. and Van Loan C., 1980, An analysis of the total least squares problem, SIAM J Num. Ana., 17,6, 883-893.Web of ScienceGoogle Scholar

  • Jazaeri S., Amiri-Simkooei A. and Shari_ M., 2013, Iterative algorithm for weighted total least squares adjustment, Surv Rev, in press.Web of ScienceGoogle Scholar

  • Mahboub V., 2011, On weighted total least-squares for geodetic transformations, J Geod., 1-9.Web of ScienceGoogle Scholar

  • Schaffrin B. and Wieser A., 2008, On weighted total least-squares adjustment for linear regression, J Geod, 82,7, 415-421.Google Scholar

  • Schaffrin B. and Wieser A., 2009, Empirical afine reference frame transformations by weighted multivariate TLS adjustment, Geod.c Ref. Fram., 134, 213-218.Google Scholar

  • Schaffrin B. and Felus Y., 2009, An algorithmic approach to the total least-squares problem with linear and quadratic constraints, Stud Geophys Geod, 53, 01.Web of ScienceGoogle Scholar

  • Schaffrin B. and Wieser A., 2011, Total least-squares adjustment of condition equations, Stud. Geophys. Geod., 55,3, 529-536.Web of ScienceGoogle Scholar

  • Shen Y., Li B. and Chen Y., 2011, An iterative solution of weighted total least-squares adjustment, J Geod, 85,4, 229-238.Google Scholar

  • Tong X., Jin Y. and Li L., 2011, An improved weighted total least squares method with applications in linear fitting and coordinate transformation, J Surv. Eng., 137,4, 120-128. Web of ScienceGoogle Scholar

About the article

Received: 2013-09-25

Accepted: 2014-03-24

Published Online: 2014-06-06

Published in Print: 2014-06-01

Citation Information: Journal of Geodetic Science, Volume 4, Issue 1, ISSN (Online) 2081-9943, DOI: https://doi.org/10.2478/jogs-2014-0007.

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© 2014 by C. Hu et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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