Ellipsoidal geoidal undulations (ellipsoidal Bruns formula): case studies

The Bruns formula is presented in its nonlinear form for any type of refere nce fields and reference equipotential surfaces. The classical Bruns formul a is derived from the reference field of the first term of spherical harmon ic expansion (w = gm/r) based on the reference equipotential sphere S-r0(2) of radius r(0) = gm/W-0. W-0 is the potential value of the Gauss-Listing g eoid. Four reference fields of ellipsoidal type, namely (1) the first term of ellipsoidal (spheroidal) harmonic expansion of the external gravitationa l field of the Earth, (2) the first term of ellipsoidal harmonic expansion of the external gravitational potential field of the Earth plus the centrif ugal field, (3) the Somigliana-Pizzetti gravity potential field, and (4) th e ellipsoidal harmonic expansion of the external gravitational field of the Earth to degree/order 50/50 plus the centrifugal field are considered. Bas ed on each of the aforementioned ellipsoidal fields, the corresponding Brun s formula up to second-order nonlinear terms is computed. The ellipsoidal B runs formula has been applied for a geoid computation in the state of Baden -Wurttemberg (Germany). More specifically, the geoidal undulations computed based on the first three reference ellipsoidal fields of choice are compar ed with the geoidal undulation computed based on the fourth reference field of ellipsoidal harmonic expansion up to degree/order 50/50 plus the centri fugal field. The results indicate that the Bruns formula computed based on the first two reference ellipsoidal fields does not provide centimetre accu racy unless the nonlinear terms of the Bruns formula up to degree 3 in term s of disturbing potential values are incorporated. However, the Bruns formu la computed based on the Somigliana-Pizzetti reference field offers millime tre accuracy even for the linear part of the ellipsoidal Bruns formula. Owi ng to the fact that the ellipsoidal Bruns formula based on the Somigliana-P izzetti field has the advantage of fulfilling the Gauss criterion of zero v alue for the global mean of geoidal undulations over the reference equipote ntial surface, the ellipsoidal Bruns formula based on the Somigliana-Pizzet ti reference field is recommended for up-to-date geoid computations with su b-millimetre accuracies.