An attempt to solve the Geodetic Boundary Value Problem, using a double layer approach for Malkin’s function Stokes’ accuracy accepted

The Boundary Value Problem of Physical Geodesy is usually treated as the problem of finding reliable corrections to be applied to an accepted ellipsoidal model of the Earth in order to get the true values of the gravitational force and the shape of its level surfaces in external space. The accuracy in determining these corrections corresponds, as a rule, to neglecting errors of the order of the flattening of the Earth (3‰, of obtained values). This first approximation identifies the Earth’s topography with the so-called Telluroid (see Fig. 1 and below), and uses as only boundary values s.c. true free air anomalies (see not:s).

To study the solvability of this problem, integral equations are well suited. For numerical work, they are now-a-days not difficult to handle, because of the availability of powerful high-speed computers. As to my opinion, there is not yet any fully satisfactory solution obtained, which fulfills all boundary conditions. If it should turn out, that no available harmonic models can satisfy the surface-condition and the infinity-conditions (see below) with sufficient accuracy (see above), we have to admit, that we need additional information along the Earth’s surface to be able to solve this important problem, by reducing it to the level case of Stokes (see Appendix 1).

The paper describes a direct use of a double-layer approach for the s.c. Malkin’s function Hy (see below), from which all wanted corrections in space may be deduced. This approach is suggested, because the solvability condition of the resulting singular Fredholm-equation can be used for defining, beforehand, the correct model gravity y. The solution, by definition of the double layer potential, satisfies the weaker infinity condition.

lim_r→∞ rH_y = 0

If it also satisfies the stronger condition

lim_r→∞ r²H_y = 0

accurately enough, ought to be investigated further.

Appendix 1 deals with the simpler problem of a purely oceanic Earth. It was mainly written to explain theoretically the procedure, when solving the general case (arbitrary topography) with double layer approach.

Appendix 2 develops several formulas, which could be used in numerical computations, when dealing with the general case.

Appendix 3 gives an additional explanation of the relationship between model gravity and the measured gravity along the Earth’s surface.