, Volume 86, Issue 5, pp 359367
First online:
On weighted total leastsquares for geodetic transformations
 Vahid MahboubAffiliated withDepartment of Surveying and Geomatics Engineering, Geodesy Division, Faculty of Engineering, University of Tehran Email author
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In this contribution, it is proved that the weighted total leastsquares (WTLS) approach preserves the structure of the coefficient matrix in errorsinvariables (EIV) model when based on the perfect description of the dispersion matrix. To achieve this goal, first a proper algorithm for WTLS is developed since the quite recent analytical solution for WTLS by Schaffrin and Wieser is restricted to the condition \({{P}_{\rm A} =\left({P_0 \otimes P_x}\right)}\) (where \({\otimes}\) is used to denote the Kronecker product) for the weight matrix of the coefficient matrix in the EIV model. This situation can be seen in the case of an affine transformation where the univariate approach can be an appropriate alternative to the multivariate WTLS approach, which has been applied to the affine transformation by Schaffrin and Felus, resp. Schaffrin and Wieser with restrictions similar to \({{P}_{\rm A} =\left( {P_0 \otimes P_x}\right)}\). In addition, this algorithm for WTLS can be interpreted well in the geodetic literature since it is based on the perfect description of the inverse dispersion matrix (or variance–covariance). By using the algorithm of WTLS, one obtains more realistic results in some applications of transformation where a high precision is needed. Some empirical examples, resp. simulation studies give insight into the efficiency of the procedure.
Keywords
EIV model Weighted total leastsquares principle Similarity transformation Affine transformation Title
 On weighted total leastsquares for geodetic transformations
 Journal

Journal of Geodesy
Volume 86, Issue 5 , pp 359367
 Cover Date
 201205
 DOI
 10.1007/s0019001105245
 Print ISSN
 09497714
 Online ISSN
 14321394
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 EIV model
 Weighted total leastsquares principle
 Similarity transformation
 Affine transformation
 Industry Sectors
 Authors

 Vahid Mahboub ^{(1)}
 Author Affiliations

 1. Department of Surveying and Geomatics Engineering, Geodesy Division, Faculty of Engineering, University of Tehran, North Kargar Ave., AmirAbad, Tehran, Iran