A robust weighted total least-squares solution with Lagrange multipliers

Weighted total least-squares (WTLS) is becoming popular for parameter estimations in geodesy and surveying. However, it does not take into consideration the possible gross errors in observations, which may lead to a reduction in the robustness and reliability of parameter estimations. In order to solve this problem, in this study, Lagrange multipliers (LM) are employed to make WTLS solution rigorous and the IGG (Institute of Geodesy and Geophysics) weight function is employed to make WTLS solution more robust and reliable, resulting in a new robust WTLS solution (RWTLS-LM-IGG). A comparison with existing WTLS and robust WTLS solutions is conducted for linear regression and coordinate transformation, through experimental evaluation with simulation data sets (with different numbers and magnitudes of gross errors) and two sets of real-life data. The results of simulation experiments show that the variance component and the mean square error of estimated parameter vector obtained by using the existing methods increase almost linearly with an increase in the numbers and magnitudes of gross errors, but these values obtained by using the proposed method are almost stable, which means an effective reduction of the influence of the gross errors by the proposed method as compared with the existing methods. It is also found that the larger the numbers and magnitudes of gross errors, the more obvious such a reduction. Furthermore, the experiment results with two sets of real-life data are consistent with the results of simulation experiments.