Inversion of satellite gravity data using spherical radial base functions

摘要

The regional gravity field modelling based on satellite data in spherical radial base functions (SRBF) is investigated in this thesis. The SRBF have been recently used for the regional representation of the Earth's gravity field as an alternative to the global spherical harmonic analysis. The use of SRBF for the regional gravity field modelling requires several choices to be made. The shape of the SRBF and their positions, the treatment of boundary effects and the use of a prior gravity field model to reduce the long wavelengths can be mentioned as different choices which have to be made. There is a wide variety of options for these choices which make it almost impossible to define a unique and standard way for regional gravity field modelling. Moreover, no matter how these choices are made, the resulting observation equations are strictly inconsistent and the associated design and normal matrices are strongly ill-posed. The solution must be then obtained by means of a proper regularization method. The main objective of this thesis is the development of new methods for the regularization of regional gravity field solutions based on satellite data. In the first three chapters of the thesis, the basic concepts of the satellite gravimetry and gradiometry are given briefly. Several mathematical models for the functional link between satellite observations and the gravitational potential are addressed. Furthermore, the mathematical expressions of the global gravity field modelling using spherical harmonics and SRBF are given. It will be numerically shown that these two groups of base functions provide the same accuracy for the representation of gravity field on the global scale. The spatial pattern of the estimated scaling coefficients on the global scale (for the SRBF) gives a perspective about the expected 'geometry' of the coefficients (as unknown parameters) in the regional modelling. Such perspective can be then used as prior knowledge about the unknown parameters. In Chapter 4, the mathematical description of global gravity field modelling using SRBF, is extended to regional solutions. We classify different choices for the model setup to seven groups. These groups are investigated in detail and our approach to define the choices is proposed. In Chapter 5, the issue of regularization of discrete ill-posed problems will be investigated generally. We describe the mathematical description of the ill-posed problems in general. In addition, some 'diagnostic' tools to determine the extent of the 'ill-posedness' will be introduced. The Tikhonov regularization and the singular value decomposition, as powerful tools for the treatment of ill-posed problems, are explained further. Several techniques for the choice of (Tikhonov) regularization parameter such as the variance component estimation (VCE), the generalized cross validation (GCV) and the L-curve method are considered. Based on the space-localization properties of the SRBF, we introduce our proposed method, called the parameter-signal correlation (PSC), for the choice of regularization parameter. Since the regularization parameter should be chosen from an extremely large set of numbers, we also propose two methods to obtain an initial and realistic value for the regularization parameter. These methods significantly reduce the computation costs and lead to a very fast convergence. The connection between the regularization and the shape of SRBF will also be explained which simplifies the choice of the shape functions. Finally, the regional gravity field modelling will be numerically investigated in several test areas. We chose three test regions according to their geographical locations as well as their signal contents. These regions are: Scandinavia, Central Africa and South America along the Andes. In addition, we considered two types of satellite observations: simulated GRACE-type data corrupted with coloured noise and real GOCE gravity gradients. Therefore the numerical considerations are divided into two steps. In the first step, the simulated GRACE data are considered. A global gravity field solution using spherical harmonics up to degree and order 120, as well as several regional solutions, are determined based on the same simulated satellite data. For the regional solutions, the VCE, GCV, L-curve and the proposed PSC methods have been used as the regularization parameter choice methods. The results are then compared to the input model to quantify the quality of different solutions. In all solutions, our PSC method gives the most promising results with the least geoid RMS on the Earth's surface. It also gives better results when the regional solutions are compared to the global spherical harmonic solution in the corresponding regions. The north-south GRACE stripes are remarkably reduced as the result of PSC regularization. In the second step, we employed real GOCE observations for regional modelling. Two months of calibrated V_(zz) components are used. The solutions are compared to the global gravity field model EGM2008 as well as the recent combined model GOCO03s. Again, the performance of four different regularization approaches are compared in the test areas. The PSC method gives the least geoid RMS compared to other approaches which shows the success of the proposed method. Moreover, the solutions show a considerable improvement compared to the global model EGM2008. The deviations between the regional solutions and the model GOCO03s, are also in the range of accumulated geoid errors of the recent global models. This indicates that even a short period of GOCE observations can provide promising results in medium and short wavelengths of the Earth's gravity field. In addition, this provides evidence that the regional gravity field determination based on satellite data provides satisfactory results if the solution is properly regularized.