The rigorous model for similarity transformation under intra-frame and inter-frame covariance

The coordinates are obtained from observations by using least-squares method, so their precision should be contaminated by observation errors and the covariance also exists between common points and non-common points. The coordinate errors don't only exist in the initial frame but also in the target frame. But the classical stepwise approach for coordinate frame transformation usually takes the coordinate errors of the initial frame into account and overlooks the stochastic correlation between common points and non-common points. A new rigorous unified model is proposed for coordinate frame transformation that takes into account both the errors of all coordinates in both fame and inter-frame coordinate covariance between common points and non-common points, and the corresponding estimator for the transformed coordinates are derived and involve appropriate corrections to the standard approach, in which the transformation parameters and the transformed coordinates for all points are computed in a single-step least squares approach. The inter frame coordinate covariance should be consistent to their uncertainties, but in practice their uncertainties are not consistent. To balance the covariance matrices of both frames, a new adaptive estimator for the unified model is thus derived in which the corresponding adaptive factor is constructed by the ratio computed by Helmert covariance component estimation, reasonable and consistent covariance matrices are arrived through the adjustment of the adaptive factor. Finally, an actual experiments with 2000 points from the crustal movement observation network of China (abbreviated CMONOC) is carried out to verify the implement of the new model, the results show that the proposed model can significantly improve the precision of the coordinate transformation. © 2017, Surveying and Mapping Press. All right reserved.