Browsing by Author "Austen, Gerrit"

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On the treatment of the geodetic boundary value problem by means of regular gravity space formulations
(2009) Austen, Gerrit; Keller, Wolfgang (Prof. Dr. sc. techn.)
The aim of this thesis is to present an alternative for the solution of a fundamental problem of geodesy. This problem, the so-called classical geodetic boundary value problem, comprises the determination of the figure of the Earth as well as the recovery of the Earth's gravity field in the exterior of the terrestrial masses. Already in 1849, G.G. Stokes addressed the problem of finding the Earth's gravity potential together with the physical shape of the Earth, i.e. the geoid. Later on in 1962, M.S. Molodensky proposed his famous theory for the direct gravimetric determination of the Earth's topographical surface along with the external gravity potential. Both approaches solve the initially nonlinear free boundary value problem, which implies considerable mathematical difficulties in the investigation of its existence and uniqueness properties, by means of a twofold linearization strategy. For this purpose, adequate approximations for the solution of the physical problem component, i.e. the determination of the gravity field, and for the geometrical part, i.e. the determination of the shape of the Earth's body, must be assumed. In detail, a normal potential to approximate the true potential as well as a reference surface for the geoid or the topography is required. In 1977, F. Sansò found an elegant approach to solve the geodetic boundary value problem by transforming it from the ordinary or geometry space into a dual space. This auxiliary space is generally referred to as gravity space. F. Sansò's break-through idea is based on the application of Legendre's transformation, a member of the family of contact transformations, to obtain the corresponding boundary value problem in the newly introduced gravity space. In contrast to the conventionally treated problem, the boundary value problem in gravity space relies on a fixed boundary. Naturally, such a situation is preferable from the mathematical point of view. Remaining only is the necessity to find a suitable linearization procedure for the gravity potential determination. Nevertheless, F. Sansò's transformed problem still suffered from a distinct singularity at the origin. Due to this reason, W. Keller encouraged the use of a modified contact transformation in 1987, which provided a fixed boundary value problem in gravity space free of any singularities. Moreover, W. Keller's revised theory succeeded to also overcome several other shortcomings of F. Sansò's gravity space transformation. Thus, in the framework of this work the terminology regular gravity space formulations is applied for the newly elaborated class of gravity space approaches related to the methodology pioneered by W. Keller. Indeed, W. Keller's concept additionally benefits from the fact that in its linearized version the resulting boundary value problem in dual space is analogous to the one of the simple Molodensky problem, which results within the scope of the classical solution procedure. This agreement clearly allows for making further use of all computing tools currently available for the well-established procedure of solving the classical Molodensky problem. It remains an open question why the regular gravity space concept has not yet been implemented numerically despite its obvious conceptual advantages. Hence, after setting the classical theory, F. Sanò's approach and the two new regular formulations in contrast with each other, thereby introducing the basic theoretical principles and discussing the benefits and drawbacks of the particular methods, this work aims for the first time at the systematic numerical implementation still outstanding for the latest two regular approaches. In particular, the dissertation aims to examine the suitability of using a more sophisticated linearization process. Following the familiar example of the Somigliana-Pizzetti type of normal potential, the application of an ellipsoidal normal potential or, more precisely, spheroidal normal potential is intended for the linearization procedure. The key questions to be answered are whether the overall formulae work and, in particular, the mathematical structure of a simple Molodensky problem can be preserved and whether introducing a spheroidal linearization point is also numerically advantageous. In summary it can be stated that the present work not only attends at length to the central issues of solving the well-known geodetic boundary value problem analytically but also provides a set of applicable numerical methods. The conducted numerical experiments document the successful accomplishment of the intended proof of concept for the approaches devised in regular gravity space. All the same, it should be pointed out that the focus of the thesis has been rather on the expansion of the theory of the geodetic boundary value problem than on the refinement of the already existing computational tools.