Modification of ellipsoidal coordinates and successive approximations in the solution of the linear gravimetric boundary value problem
20/09/2016 | 17:30 | Session 3: Local/regional geoid determination methods and models
Author(s): Petr Holota and Otakar Nesvadba
Petr Holota and Otakar Nesvadba
Investigations of the external gravity field of the Earth are essentially connected with the theory of boundary value problems of mathematical physics. The aim of this paper is to discuss the solution of the linearized gravimetric boundary value problem by means of the method of successive approximations. We start with the relation between the geometry of the solution domain and the structure of Laplace’s operator. Similarly as in other branches of engineering and mathematical physics a transformation of coordinates is used that offers a possibility to solve an alternative between the boundary complexity and the complexity of the coefficients of the partial differential equation governing the solution. For instance Laplace’s operator has a relatively simple structure in terms of ellipsoidal coordinates which are frequently used in geodesy. However, the physical surface of the Earth substantially differs from an oblate ellipsoid of revolution, even if it is optimally fitted. Therefore, an alternative is discussed. A system of general curvilinear coordinates such that the physical surface of the Earth is imbedded in the family of coordinate surfaces is used. Clearly, the structure of Laplace’s operator is more complicated in this case. In a sense it represents the topography of the physical surface of the Earth. Nevertheless, the construction of the respective Green’s function is more simple, if the solution domain is transformed. In this connection Green’s function method together with the method of successive approximations is used for the solution of the linear gravimetric boundary value problem expressed in terms of new coordinates. The structure of iteration steps is analyzed and if useful, it is modified by means of the integration by parts. The individual steps are discussed and interpreted.