Utilizing non-iterative linear transformations between non-uniformly dilated 3D frames

Linear transformations have been routinely applied in various field applications since they permit a straightforward geometric relationship between different coordinate systems. The most general form of linear transformations which is also referred to as the affine model allows coordinate transformations between non-uniformly dilated systems. This particular model is intrinsically useful for datum applications in deforming areas. In this study, a twelve-parameter 3D affine transformation is proposed to model the physical properties between the positional coordinates from different sets of reference systems. Its transformation parameters can be estimated using a novel Non-Iterative Solution for Linear Transformations (NISLT) algorithm. In order to illustrate the capability of the proposed approach, numerical analyses were performed for the coordinate transformations in two real cases: 1) a mild non-uniform dilatation case transforming coordinates between two continental networks, an International Terrestrial Reference Frame 2000 (ITRF2000) solution and its corresponding International GNSS Service (IGS) solution, and 2) a strong non-uniform dilatation case transforming coordinates between two regional networks around Taiwan based on IGS solutions at two different epochs. Results reveal that a significant improvement on the transformation quality can be achieved when the proposed procedure is implemented in modern-day datum applications, especially for those in a deforming area.