Spheroidal integral formulas for computing the disturbing gravitational potential and its first-, second- and third-order directional derivatives from disturbing gravitational potential and its vertical and horizontal derivatives

Integral transformations derived from boundary-value problems (BVPs) of potential theory constitute the core mathematical apparatus of physical geodesy and gravity-field modelling. Classical Green’s function solutions of the Dirichlet, Neumann, and Stokes problems generate a complete family of integral transformations that relate the disturbing gravitational potential and its directional derivatives. In the spherical approximation, this framework—summarised by the Meissl scheme—has reached a high level of completeness and currently provides mutual transformations among all components of the gravitational-gradient tensors up to the third order. These tools underpin the processing of heterogeneous gravity observations acquired by terrestrial, airborne, and satellite sensors.

Increasing accuracy requirements and the geometric proximity of the Earth to a rotational ellipsoid, however, necessitate a transition from spherical to spheroidal formulations. Although analytical solutions of the three fundamental BVPs on an oblate spheroid have been derived and several corresponding integral equations have been proposed, the spheroidal analogue of the Meissl scheme remains incomplete.

In this contribution, we derive spheroidal integral formulas for computing the disturbing gravitational potential and its first-, second-, and third-order directional derivatives from the disturbing potential and its vertical and horizontal derivatives. The correctness of the newly derived integral formulas is verified by closed-loop tests using data from a global geopotential model.