On the importance of intra-frame and inter-frame covariances in frame transformation theory

The coordinate frame transformation (CFT) problem in geodesy is typically solved by a stepwise approach which entails both inverse and forward treatment of the available data. The unknown transformation parameters are first estimated on the basis of common points given in both frames, and subsequently they are used for transforming the coordinates of other (new) points from their initial frame to the desired target frame. Such an approach, despite its rational reasoning, does not provide the optimal accuracy for the transformed coordinates as it overlooks the stochastic correlation (which often exists) between the common and the new points in the initial frame. In this paper we present a single-step least squares approach for the rigorous solution of the CFT problem that takes into account both the intra-frame and inter-frame coordinate covariances in the available data. The optimal estimators for the transformed coordinates are derived in closed form and they involve appropriate corrections to the standard estimators of the stepwise approach. Their practical significance is evaluated through numerical experiments with the 3D Helmert transformation model and real coordinate sets obtained from weekly combined solutions of the EUREF Permanent Network. Our results show that the difference between the standard approach and the optimal approach can become significant since the magnitude of the aforementioned corrections remains well above the statistical accuracy of the transformation results that are obtained by the standard (stepwise) solution.