Regularization methods for the solution of the inverse Stokes problem
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AbstractSeveral approaches with similar concepts for the regularization of improperly posed problems have been developed in different areas of science, especially in geophysics and mathematics. In mathematics, the solution approaches are based on Tikhonov's regularization [TIKHONOV, 1963] which a-priori defines some parameters of the unknown function. In geophysics, a regularization method based on classical logic has been proposed where probability statements are employed as a-priori information [TARANTOLA, 1987]. In surveying engineering, collocation is widely used to incorporate additional knowledge into the least squares solution [MORITZ, 1980; RAPP, 1985]. A theoretical comparison of some regularization methods is given in LOUIS  and SANSO . However, these methods are seldom applied to actual data due to the enormous computational effort involved. This research compares several regularization methods for the solution of the inverse Stokes problem for gravity field determination from satellite altimeter data. The comparison is done from a theoretical point of view as well as by practical application. The theoretical comparison discusses equivalences and differences of the approaches. The application of the methods compares the numerical results obtained with the algorithms on two different data sets. The comparison of the regularization methods shows that the inversion of Stokes problem is only moderately unstable. Thus, a simple inversion routine works under favourable circumstances but provides no safeguard against instability. Tikhonov regularization with cross-validation is a user-independent methods and works well on all data sets. The truncated singular value decomposition gives good results only when the accuracy estimates of the singular values are adequate. Both these methods are very time-consuming due to the singular value decomposition routine. The probabilistic method reduces to a combined least squares adjustment for Gaussian statistics. It produces good results as long as the a-priori error estimates are correct. Collocation performs poorly on these small data sets. However, it is by far the fastest method when FFT algorithms are used. A method, which is fast and user-independent at the same time, would be ideal for gravity field determination from satellite altimeter data.
Bibliography: p. 203-210.