Coordinate transformation by minimizing correlations between parameters
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Date
1972-07
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Publisher
Ohio State University. Division of Geodetic Science
Abstract
The subject of this investigation is to determine the transformation parameters (three rotations, three translations and a scale factor) between two Cartesian coordinate systems from sets of coordinates given in both systems. The objective is the determination of well separated transformation parameters with reduced correlations between each other, a problem especially relevant when the sets of coordinates are not well distributed. The above objective is achieved by preliminarily determining the three rotational parameters and the scale factor from the respective direction cosines and chord distances (these being independent of the translation parameters) between the common points, and then computing all the seven parameters from a solution in which the rotations and the scale factor are entered as weighted constraints according to their variances and covariances obtained in the preliminary solutions. Numerical tests involving two geodetic reference systems were performed to evaluate the effectiveness of this approach as follows: (a) A non-constrained solution for general transformation for the seven parameters (including the three translations and scale factor). (b) A constrained solution for general transformation for the seven parameters utilizing the three rotations with their statistics as constraints. (c) A constrained solution for general transformation for the seven parameters using the three rotations and scale factor with their statistics as constraints. The above schemes were then separately repeated for each of the following three cases: (i) Using the full variance-covariance matrix between coordinates of the geodetic reference systems. (ii) Using only a (3 x 3) banded diagonal variance-covariance matrix, thus assuming no correlation between coordinates of any two points within the system. (iii) Using only variances for the coordinates, thereby further omitting the correlation between the three coordinates of any one point in the system. In the case of seven parameter general transformation, the best estimates were obtained using full variance-covariance matrix and constraining three rotations and the scale factor, case (c) and (iii) above. The improvement in correlation between translations and rotations was more significant compared to between translation and scale factor.
Description
Prepared for National Aeronautics and Space Administration, Washington, D.C.: Contract No. NGR 36-008-093, OSURF Project No. 2514