Published by Maney Publishing (c) Survey Review Ltd Iteratively reweighted total least squares: a robust estimation in errors-in-variables models V. Mahboub* 1 , A. R. Amiri-Simkooei 2,3 and M. A. Sharifi 4 In this contribution, the iteratively reweighted total least squares (IRTLS) method is introduced as a robust estimation in errors-in-variables (EIV) models. The method is a follow-up to the iteratively reweighted least squares (IRLS) that is applied to the Gauss–Markov and/or Gauss–Helmert models, when the observations are corrupted by gross errors (outliers). In a relatively new class of models, namely EIV models, IRLS or other known robust estimation methods introduced in geodetic literature cannot be directly applied. This is because the vector of observations or the coefficient matrix of the EIV model may be falsified by gross errors. The IRTLS can then be a good alternative as a robust estimation method in the EIV models. This method is based on the algorithm of weighted total least squares problem according to the traditional Lagrange approach to optimise the target function of this problem. Also a new weight function is introduced for IRTLS approach in order to obtain better results. A simulation study and an empirical example give insight into the robustness and the efficiency of the procedure proposed. Keywords: Errors-in-variables model, Weighted total least squares, Robust estimation, Iteratively Reweighted total least squares Introduction If the observations are distorted by gross errors in addition to random ones, one can use robust estimation techniques. The concept of robustness has been already introduced in the last years. The term ‘robust’ was coined in statistics by [4]. Various definitions of greater or lesser mathematical rigor are possible for the term, but in general, referring to a statistical estimator, it means ‘insensitive to small departures from the idealised assumptions for which the estimator is optimized’ [9], [13]. In the last two decades, several publications concern- ing robust estimation and outlier detection methods have been published in geodetic literature. We may for instance refer to [18], [2], [11], [27], and [3]. None of them, however, has been applied to the relatively new class of models named errors-in-variables (EIV) models, which can be solved using the total least squares (TLS) method developed by [7], although [19] and [20] have proposed a method for outlier detection in EIV model based on the traditional statistical test. The method is, however, applicable when only one outlier appears, either in the observation vector or in the coefficient matrix. In the last decade, there has been a growing demand to use the TLS method, generally in engineering sciences and particularly in geodetic surveying applica- tions. Therefore, a robust estimation method for EIV models needs to be established. Although some researchers such as [28] and [5] have investigated this problem in the statistical literature, the methods proposed can only be applied to a linear regression model. In addition, the methods are difficult to use. We aim to adapt the methods for many engineering applications. This came out of the real and simulated examples that show, in order to achieve more accurate results, one has to further develop the method and adopt it for engineering applications. In the present contribution, one approximate approach is introduced for robust estimation in EIV models which is coming from the fields of optimal control and filtering rather than the field of mathematical statistics. It is the iteratively reweighted total least squares (IRTLS) which is a follow- up to the iteratively reweighted least squares (IRLS) that was originally introduced by [12] into the geodetic applications. To develop the IRTLS algorithm, we select one algorithm among the several existing algorithms that solve the TLS problem. This algorithm was developed by [15] for the weighted total least squares (WTLS) problem. It is based on the traditional Lagrange approach of which 1 Department of Surveying and Geomatics Engineering, Geodesy Division, Faculty of Engineering, University of Tehran, North Kargar Ave., Amir- Abad, Tehran, Iran 2 Department of Surveying Engineering, Faculty of Engineering, University of Isfahan, 81746-73441, Isfahan, Iran 3 Acoustic Remote Sensing Group, Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS, Delft, The Netherlands 4 Department of Surveying and Geomatics Engineering, Geodesy Division, Faculty of Engineering, University of Tehran, North Kargar Ave., Amir- Abad, Tehran, Iran *Corresponding author, email vahid_mahboobk@yahoo.com ß 2013 Survey Review Ltd. Received 29 November 2011; accepted 8 May 2012 92 Survey Review 2013 VOL 45 NO 329 DOI 10.1179/1752270612Y.0000000017