Surface-fitting by partitioned multiple regression equations in oblique territories and its use in coordinate transformations

A bivariate polynomial is an important type of explicit function that can approximate a physical quantity that varies continuously and smoothly over a surface. Such functions can be derived by successive least-squares optimisations which determine which terms are statistically significant, the results being designated as multiple regression equations (MREs). One application is coordinate shifts from one geodetic positioning system (datum) to another. Recent research has found accuracy benefits from partitioned MREs in which two polynomials are joined together smoothly, reducing the need for high-power terms. Case studies from that research suggested that partitioning should be across whichever of the north–south and east–west extents was the greater. The paper investigates whether diagonal partitioning is best for data in oblique territories (where the line of greatest extent is diagonal). The case study selected was coordinate transformations in Slovenia which is relatively oblique. Oblique partitioning was not in itself an improvement on east–west partitioning but became so when contributing to two hybrid models. More research is needed but results so far suggest that the best course is to test all partitioning options and apply the one that gives the closest fit to the particular data.