Center strategies for universal geodetic transformations: modified iteration policy and two alternative models

Although centralized coordinates are applied in geodetic coordinate transformations implicitly or explicitly, the centering strategy has not been comprehensively investigated from the theoretical perspective. We rigorously model and extend the empirically used three center strategies based on different models:

• Original model: Based on the partition representations of the solution, we propose a modified iteration policy, which reduces the parameter number and improves numerical stability during iteration. Also, its simplified version is analyzed when the cofactor matrix has the Kronecker product structures. It can be regarded as the extension of the work of Teunissen, since we essentially follow the same idea of partitioning the transformation parameters and the translation parameters, but more general covariance matrix structures are investigated in our consideration.
• Shifting model: With the partitioned solution forms, we prove the estimated transformation matrix and the residual vector are translational invariant. For iteration, with the classical iteration policy, the shifts should be chosen properly; with the modified iteration policy, there is no restriction since it is numerically equivalent to the original model. In addition, this model shows the feasibility of conducting the adjustment with the centralized coordinates and the original stochastic model.
• Translation elimination model: By multiplying the transformation relation with a specific matrix from both sides, we formulate the translation elimination model with the coordinates centralized and the translation parameters eliminated. With this model reduction, the covariance matrix has also been transformed since the observation equations are comprised of coordinate combinations. In addition, Leick’s model reduction strategy is a special case of this model, which is conducted by subtracting one particular observation equation from the remaining equations. 

Test computations with different weight structures show the validity of these strategies.