Abstract: Multiple regression equations (MREs) provide an empirical direct method of transforming coordinates between geodetic datums. Since they offer a means of modelling distortions, they are capable of a more accurate fit to datum-shift datasets than more basic direct methods. MRE models of datum shifts traditionally consist of polynomials based on relative latitude and longitude. However, the limited availability of low-power terms often leads to high-power terms being included, and these are a potential cause of instability. This paper introduces three variations based on simple partitions and 2 or 4 smoothly conjoined polynomials. The new types are North/South, East/West and Four-Quadrant. They increase the availability of low-order terms, enabling distortions to be modelled with fewer side effects. Case studies in Great Britain, Slovenia and Western Australia provide examples of partitioned MREs that are more accurate than conventional MREs with the same number of terms.

Abstract: Standard Molodensky is a recognised method of transforming coordinates between geodetic datums. Although less accurate than some other methods, it has the merit of being direct. That is to say it can be applied to geodetic coordinates, without involving Cartesian coordinates that give rise to difficulties in computing latitude. This paper considers the use of Standard Molodensky when at least one of the datums is 2D+1D in nature, meaning that that horizontal and vertical positions are obtained by different methods. This was generally the case before 3D positioning by satellites and is a widespread characteristic of local datums that are still used. The 2D+1D property weakens the argument for 3D conformality, and invites the possibility that different translation parameters might be used for horizontal and vertical shifts. The possibility of including a Z-rotation as a 7th parameter is also considered. Besides being ideal for those who favour the simplicity of Standard Molodensky, the variations introduced in this paper offer significant improvements in accuracy such as error reductions of 75%, 69% and 99% in the three selected case studies.