On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms

Burkhard Schaffrin, Yaron A. Felus

Research output: Contribution to journalArticlepeer-review

113 Scopus citations


The multivariate total least-squares (MTLS) approach aims at estimating a matrix of parameters, Ξ, from a linear model (Y - EY = (X - EX) Ξ) that includes an observation matrix, Y, another observation matrix, X, and matrices of randomly distributed errors, EY and EX. Two special cases of the MTLS approach include the standard multivariate least-squares approach where only the observation matrix, Y, is perturbed by random errors and, on the other hand, the data least-squares approach where only the coefficient matrix X is affected by random errors. In a previous contribution, the authors derived an iterative algorithm to solve the MTLS problem by using the nonlinear Euler-Lagrange conditions. In this contribution, new lemmas are developed to analyze the iterative algorithm, modify it, and compare it with a new 'closed form' solution that is based on the singular-value decomposition. For an application, the total least-squares approach is used to estimate the affine transformation parameters that convert cadastral data from the old to the new Israeli datum. Technical aspects of this approach, such as scaling the data and fixing the columns in the coefficient matrix are investigated. This case study illuminates the issue of "symmetry" in the treatment of two sets of coordinates for identical point fields, a topic that had already been emphasized by Teunissen (1989, Festschrift to Torben Krarup, Geodetic Institute Bull no. 58, Copenhagen, Denmark, pp 335-342). The differences between the standard least-squares and the TLS approach are analyzed in terms of the estimated variance component and a first-order approximation of the dispersion matrix of the estimated parameters.

Original languageEnglish
Pages (from-to)373-383
Number of pages11
JournalJournal of Geodesy
Issue number6
StatePublished - Jun 2008
Externally publishedYes


  • Empirical coordinate transformation
  • Errors-in-variables modeling
  • Multivariate total least-squares solution (MTLSS)
  • Quasi-linear models


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