Transforming ellipsoidal heights and geoid undulations between different geodetic reference frames

Abstract

Transforming height information that refers to an ellipsoidal Earth reference model, such as the geometric heights determined from GPS measurements or the geoid undulations obtained by a gravimetric geoid solution, from one geodetic reference frame (GRF) to another is an important task whose proper implementation is crucial for many geodetic, surveying and mapping applications. This paper presents the required methodology to deal with the above problem when we are given the Helmert transformation parameters that link the underlying Cartesian coordinate systems to which an Earth reference ellipsoid is attached. The main emphasis is on the effect of GRF spatial scale differences in coordinate transformations involving reference ellipsoids, for the particular case of heights. Since every three-dimensional Cartesian coordinate system ‘gauges’ an attached ellipsoid according to its own accessible scale, there will exist a supplementary contribution from the scale variation between the involved GRFs on the relative size of their attached reference ellipsoids. Neglecting such a scale-induced indirect effect corrupts the values for the curvilinear geodetic coordinates obtained from a similarity transformation model, and meter-level apparent offsets can be introduced in the transformed heights. The paper explains the above issues in detail and presents the necessary mathematical framework for their treatment.