On the problem of geoid height transformation between different geodetic reference frames

Abstract

Transforming geoid heights between different geodetic reference frames (GRFs) is an essential task in gravity field modeling and its proper implementation is crucial for many applications involving the use of the geoid. In this paper an attempt is made to investigate the problem of geoid height transformation between different GRFs, without considering other variations in the auxiliary geophysical or geodynamical hypotheses that may be a-priori specified for the desired geoid type (e.g. variations in the values of M and/or $W_o$, variations in terms of treatment of permanent tidal effects). The aim here is to present the required methodology to deal with the problem: “how should we transform geoid heights, referring to a fixed equipotential surface (W = $W_o$), from a given GRF to another GRF when we know the seven similarity transformation parameters linking the two frames?”. Special emphasis is given on the effect of GRF scale variations in coordinate transformations involving reference ellipsoids, for the particular case of geoid heights. Since every Cartesian coordinate system “gauges” an attached ellipsoid according to its own particular scale, there will exist a small contribution from the scale variation between the involved GRFs on the relative size of their adopted reference ellipsoids. Neglecting such a scale-induced indirect effect corrupts the values for the curvilinear geodetic coordinates obtained from a similarity transformation model, and significant errors can be introduced in the transformed geoid heights. The paper explains the above issues in detail and presents the necessary mathematical framework for their solution.