Journal of Geodesy

, Volume 85, Issue 4, pp 229–238 | Cite as

An iterative solution of weighted total least-squares adjustment

  • Yunzhong ShenEmail author
  • Bofeng Li
  • Yi Chen
Original Article


Total least-squares (TLS) adjustment is used to estimate the parameters in the errors-in-variables (EIV) model. However, its exact solution is rather complicated, and the accuracies of estimated parameters are too difficult to analytically compute. Since the EIV model is essentially a non-linear model, it can be solved according to the theory of non-linear least-squares adjustment. In this contribution, we will propose an iterative method of weighted TLS (WTLS) adjustment to solve EIV model based on Newton–Gauss approach of non-linear weighted least-squares (WLS) adjustment. Then the WLS solution to linearly approximated EIV model is derived and its discrepancy is investigated by comparing with WTLS solution. In addition, a numerical method is developed to compute the unbiased variance component estimate and the covariance matrix of the WTLS estimates. Finally, the real and simulation experiments are implemented to demonstrate the performance and efficiency of the presented iterative method and its linearly approximated version as well as the numerical method. The results show that the proposed iterative method can obtain such good solution as WTLS solution of Schaffrin and Wieser (J Geod 82:415–421, 2008) and the presented numerical method can be reasonably applied to evaluate the accuracy of WTLS solution.


Errors-in-variables model Total least-squares adjustment Iterative least-squares adjustment Unbiased variance component estimate 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Dermanis A (1987) Geodetic applications of interpolation and prediction. Lecture presented at the international school of geodesy “A. Marussi”, 4th course: applied and basic geodesy: present and future trends. Ettore Majorana Centre for Scientific Culture, Erice-Sicily, 15–25 June 1987Google Scholar
  2. Felus Y, Schaffrin B (2005) Performing similarity transformations using the error-in-variables model. American society for photogrammetry and remote sensing (ASPRS) annual meeting on CD, Baltimore, MarylandGoogle Scholar
  3. Golub H, Van Loan Ch (1980) An analysis of the total least squares problem. SIAM J Numer Anal 17(6): 883–893CrossRefGoogle Scholar
  4. Golub G, van Loan Ch (1996) Matrix computations, 3rd edn. Johns Hopkins University Press, BaltimoreGoogle Scholar
  5. Golub G, Hansen P, O’Leary D (1999) Tikhonov regularization and total least squares. SIAM J Matrix Anal 21: 185–194CrossRefGoogle Scholar
  6. Koch K (1999) Parameter estimation and hypothesis testing in linear models, 2nd edn. Springer, BerlinGoogle Scholar
  7. Markovsky I, Rastello M, Premoli A, Kukush A, Van Huffel S (2006) The element-wise weighted total least-squares problem. Comput Stat Data Anal 50: 181–209CrossRefGoogle Scholar
  8. Mikhail E (1976) Observations & least squares. Harper&Row, New YorkGoogle Scholar
  9. Neri F, Saitta G, Chiofalo S (1989) An accurate and straightforward approach to line regression analysis of error-affacted experimental data. J Phys Ser E Sci Instr 22: 215–217CrossRefGoogle Scholar
  10. Pope A (1972) Some pitfalls to be avoided in the iterative adjustment of nonlinear problems. In: Proceedings of 38th annual meeting of the American society photogrammetry, Washington, DC, pp 449–473Google Scholar
  11. Pope A (1974) Two approaches to nonlinear least squares adjustments. Can Surv 28(5): 663–669Google Scholar
  12. Schaffrin B, Felus Y (2005) On total least-squares adjustment with constraints. In: A window on the future of Geodesy, IAG-Symposium, vol 128. Springer, Berlin, pp 417–421Google Scholar
  13. Schaffrin B (2006) A note on constrained total least-squares estimation. Linear Algebra Appl 417(1): 245–258CrossRefGoogle Scholar
  14. Schaffrin B, Felus Y (2008) Multivariate total-least squares adjustment for empirical affine transformations. In: Xu P, Liu J, Sermanis A (eds) International association of geodesy symposia: VI Hotine–Marussi symposium on theoretical and computational Geodesy, 29 May–2 June 2006, Wuhan, China, vol 132. Springer, Berlin, pp 238–242Google Scholar
  15. Schaffrin B, Lee I, Felus Y, Choi Y (2006) Total least-squares for geodetic straight-line and plane adjustment. Boll Geod Sci Aff 65: 141–168Google Scholar
  16. Schaffrin B, Felus Y (2008) On the multivariate total least-squares approach to empirical coordinate transformations. three algorithms. J Geod 82: 373–383CrossRefGoogle Scholar
  17. Schaffrin B, Wieser A (2008) On weighted total least-squares adjustment for linear regression. J Geod 82: 415–421CrossRefGoogle Scholar
  18. Schaffrin B, Neitzel F, Uzum S (2009) Empirical similarity transformation via TLS-adjustment: exact solution vs. Cadzow’s approximation. International Geomatics Forum, 28–30 May 2009, Qingdao, People’s Republic of ChinaGoogle Scholar
  19. Teunissen PJG (1990) Nonlinear least squares. Manuscr Geod 15: 137–150Google Scholar
  20. Van Huffel S (ed) (1997) Recent advances in total least squares techniques and errors-in-variables modeling. SIAM proceedings series, SIAM, PhiladelphiaGoogle Scholar
  21. Van Huffel S, Cheng C, Mastronardi N, Paige C, Kukush A (2007) Total least squares and errors-in-variables modeling. Comput Stat Data Anal 52: 1076–1079CrossRefGoogle Scholar
  22. Van Huffel, S, Lemmerling, P (eds) (2002) Total least squares and errors-in-variables modeling: analysis, algorithms and applications. Kluwer Academic Publishers, DordrechtGoogle Scholar
  23. Van Huffel S, Vandewalle J (1989) Analysis and properties of the generalized total least-squares problem AX ≈ B when some or all columns of A are subject to error. SIAM J Matrix Anal Appl 10: 294–315CrossRefGoogle Scholar
  24. Van Huffel S, Vandewalle J (1991) The total least Squares problem: computational aspects and analysis. SIAM, PhiladelphiaGoogle Scholar
  25. Xu P (1986) Variance-covariance propagation for a non-linear function. J Wuhan Techn Uni Surv Mapp 11(2): 92–99 (in Chinese with an English abstract)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Surveying and Geo-informaticsTongji UniversityShanghaiPeople’s Republic of China
  2. 2.Key Laboratory of Advanced Surveying Engineering of State Bureau of Surveying and MappingShanghaiPeople’s Republic of China

Personalised recommendations