Ellipsoidal geoidal undulations (ellipsoidal Bruns formula): case studies: Fid-Bau Portal

Ellipsoidal geoidal undulations (ellipsoidal Bruns formula): case studies

Ardalan, A. A. / Grafarend, E. W.

Abstract. The Bruns formula is presented in its nonlinear form for any type of reference fields and reference equipotential surfaces. The classical Bruns formula is derived from the reference field of the first term of spherical harmonic expansion (w=gm/r) based on the reference equipotential sphere Sr02 of radius r0=gm/W0. W0 is the potential value of the Gauss–Listing geoid. Four reference fields of ellipsoidal type, namely (1) the first term of ellipsoidal (spheroidal) harmonic expansion of the external gravitational field of the Earth, (2) the first term of ellipsoidal harmonic expansion of the external gravitational potential field of the Earth plus the centrifugal field, (3) the Somigliana–Pizzetti gravity potential field, and (4) the ellipsoidal harmonic expansion of the external gravitational field of the Earth to degree/order 50/50 plus the centrifugal field are considered. Based on each of the aforementioned ellipsoidal fields, the corresponding Bruns formula up to second-order nonlinear terms is computed. The ellipsoidal Bruns formula has been applied for a geoid computation in the state of Baden–Württemberg (Germany). More specifically, the geoidal undulations computed based on the first three reference ellipsoidal fields of choice are compared with the geoidal undulation computed based on the fourth reference field of ellipsoidal harmonic expansion up to degree/order 50/50 plus the centrifugal field. The results indicate that the Bruns formula computed based on the first two reference ellipsoidal fields does not provide centimetre accuracy unless the nonlinear terms of the Bruns formula up to degree 3 in terms of disturbing potential values are incorporated. However, the Bruns formula computed based on the Somigliana–Pizzetti reference field offers millimetre accuracy even for the linear part of the ellipsoidal Bruns formula. Owing to the fact that the ellipsoidal Bruns formula based on the Somigliana–Pizzetti field has the advantage of fulfilling the Gauss criterion of zero value for the global mean of geoidal undulations over the reference equipotential surface, the ellipsoidal Bruns formula based on the Somigliana–Pizzetti reference field is recommended for up-to-date geoid computations with sub-millimetre accuracies.