Laplace’s operator with a structure reflecting the solution domain geometry and its use in the determination of the disturbing potential by a convergent series of successive approximations

The structure of the Laplace operator is relatively simple when expressed in terms of spherical or ellipsoidal coordinates. The physical surface of the Earth, however, substantially differs from a sphere or an oblate ellipsoid of revolution, even if optimally fitted. The same holds true for the solution domain and the exterior of a sphere or of an oblate ellipsoid of revolution. The situation is more convenient in a system of general curvilinear coordinates such that the physical surface of the Earth (smoothed to a certain degree) is embedded in the family of coordinate surfaces. Therefore, a transformation of coordinates is applied in treating the gravimetric boundary value problem. The transformation contains also an attenuation function. Tensor calculus and its rules are used and the Laplace operator is expressed in the new coordinates. Its structure becomes more complicated now. Nevertheless, in a sense it represents the topography of the physical surface of the Earth. Subsequently the Green’s function method is used together with the method of successive approximations in the solution of the gravimetric boundary value problem expressed in terms of the new coordinates.