Spacetime Modeling of the Earth’s Gravity Field by Ellipsoidal Harmonics

All planetary bodies like the Earth rotate causing centrifugal effect! The result is an equilibrium figure of ellipsoidal type. A natural representation of the planetary bodies and their gravity fields is in terms of ellipsoidal harmonics and ellipsoidal wavelets. Here, various effects are treated when considering the Earth to be “ellipsoidal”: Sections 2 and 3 start the chapter with the celebrated ellipsoidal Dirichlet and ellipsoidal Stokes (to first order) boundary-value problems. Section 4 is devoted to the definition and representation of the ellipsoidal vertical deflections in gravity space, extended in Sect. 5 to the representation in geometry space. The potential theory of horizontal and vertical components of the gravity field, namely, in terms of ellipsoidal vector fields, is the target of Sect. 6. Section 7 is concentrated on the reference potential of type Somigliana-Pizzetti field and its tensor-valued derivatives. Section 8 illustrates an ellipsoidal harmonic gravity field for the Earth called SEGEN (Gravity Earth Model), a setup in ellipsoidal harmonics up to degree/order 360/360. Five plates are shown for the West-East/North-South components of type vertical deflections as well as gravity disturbances referring to the International Reference Ellipsoid 2000. The final topic starts with a review of the curvilinear datum problem referring to ellipsoidal harmonics. Such a datum transformation from one ellipsoidal representation to another one in Sect. 9 is a seven-parameter transformation of type (i) translation (three parameters), (ii) rotation (three parameters) by Cardan angles, and (iii) dilatation (one parameter) as an action of the conformal group in a three-dimensional Weitzenbäck space W(3) with seven parameters. Here, the chapter is begun with an example, namely, with a datum transformation in terms of spherical harmonics in Sect. 10. The hard work begins with Sect. 11 to formulate the datum transformation in ellipsoidal coordinates/ellipsoidal harmonics! The highlight is Sect. 12 with the characteristic example in terms of ellipsoidal harmonics for an ellipsoid of revolution transformed to another one, for instance, polar motion or gravitation from one ellipsoid to another ellipsoid of reference. Section 13 reviews various approximations given in the previous three sections.