Numerical solutions to altimetry-gravimetry boundary value problems in coastal regions
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AbstractThis thesis presents novel solutions of altimetry-gravimetry boundary value problems (AGBVPs) with compatibility (smoothness) conditions along the coastline for geoid determination. After an analysis of the state of the art for AGBVPs it was found that a lot of work has already been done in terms of theoretical problem formulation and solution investigation of AGBVPs. The conditions under which the solutions of different AGBVPs exist and are unique have been provided. It has been shown that without additional compatibility conditions along the coastline a solution with higher level of regularity does not exist. This was the starting point of this work and, after three preliminary experiments, it was found that the effect of smoothness along the coast line is significant for cm-geoid determination. Further theoretical investigations by the author resulted in the following achievements and contributions to local and regional determination of the geoid. The theory of spherical pseudo-differential operators (PDOs), spherical wavelets and spherical harmonics was combined for local and regional geoid determination in coastal areas. Using the theory of PDOs, it has been proven that the fixed AGBVP II has a unique solution and different methods applied to solve this problem yield the same solution. It has been shown that the compatibility conditions given by Svensson (1988) are equivalent to the condition that the data and their first and second order gradients coincide with each other along the coastline. It has been proven that PDOs are uniformly Lipschitz α in this case, they can be combined with wavelets that are locally Lipschitz α to increase the regularity (smoothness) of the solution along the coastline. A modified algorithm for reconstruction of a signal using wavelet modulus maxima points is suggested to detect and to smooth existing discrepancies and irregularities along the coastline. Any kind of functionals of disturbing potential is presented in a discrete form, which allows spherical PDOs and wavelets to be applied numerically; even the compatibility conditions are expressed in an explicit form in discrete form as a sum of two functionals described above. Final numerical solution with spherical HShannon wavelets was proposed and conclusions are drawn about its advantages from an application point of view. Finally, as a result of the proposed procedure a smooth transition of the geoid from land to sea can be achieved, which will result in an improved geoid model in coastal areas.
Bibliography: p. 202-209