An Iterative Solution of the Scalar Free Boundary Value Problem and the Choice of the Reference Surface
In the formulation of the scalar free boundary value problem we solve for the gravity potential in the external space outside the earth’s surface and for the vertical position of the boundary surface. After linearization the reduced boundary condition refers to the Telluroid s ∋ p, and the new difference quantity δω, the disturbing potential, is introduced. To represent the unknown disturbing potential in the global basis of spherical harmonics a harmonic analysis has to be applied to the given boundary data. In this context the boundary data have to be (downward) continued from s to a reference surface which shows a rotational symmetry with respect to the earth’s mean rotational axis.
In general a sphere K or the surface E of an ellipsoid of revolution is selected.
After the analytical continuation of the evaluation operator E s the boundary condition can be split in two parts. The main component is covered by the isotropic term which corresponds to the Stokes-problem. The second part consists of the ellipsoidal and topographical component which are functionals of δω. Therefore an iterative solution stategy is appropriate. In numerical investigations the different behaviour in convergence—whether K or E is used as reference surface—is object of this contribution.