Abstract
A weighted least-squares (WLS) solution to a 3-D non-linear symmetrical similarity transformation within a Gauss-Helmert (GH) model, and/or an errors-in-variables (EIV) model is developed, which does not require linearization. The geodetic weight matrix is the inverse of the observation dispersion matrix (second-order moment). We suppose that the dispersion matrices are non-singular. This is in contrast to Procrustes algorithm within a Gauss-Markov (GM) model, or even its generalized algorithms within the GH and/or EIV models, which cannot accept geodetic weights. It is shown that the errors-invariables in the source system do not affect the estimation of the rotation matrix with arbitrary rotational angles and also the geodetic weights do not participate in the estimation of the rotation matrix. This results in a fundamental correction to the previous algorithm used for this problem since in that algorithm, the rotation matrix is calculated after the multiplication by row-wise weights. An empirical example and a simulation study give insight into the efficiency of the proposed procedure.
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Mahboub, V. A weighted least-squares solution to a 3-D symmetrical similarity transformation without linearization. Stud Geophys Geod 60, 195–209 (2016). https://doi.org/10.1007/s11200-015-1109-1
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DOI: https://doi.org/10.1007/s11200-015-1109-1