Internal Reliability of Planar Coordinate Transformations

Coordinate transformations are essential in geodesy, surveying engineering, and many other disciplines working with coordinate systems. The task is to estimate transformation parameters from given coordinates of control points by least-squares adjustment. We focus on the internal reliability of the underlying adjustment model. It measures the ability of an adjustment model to detect discrepancies (biases) in the control points. The best established metric for the internal reliability is the minimum detectable bias (MDB). We derive explicit formulas for the MDB of the most important planar coordinate transformations: the planar similarity transformation and the planar rototranslation transformation. They are worked out for a bias in one coordinate, in both coordinates of a control point, and in all four coordinates of two control points of the target system. We investigate situations, where the MDB is infinite, such that biases are undetectable. The results are formulated in nine theorems.