Domain decomposition methods in multiscale geopotential determination from SST and SGG

Satellite-to-Satellite Tracking (SST) and Satellite Gravity Gradiometry (SGG) are two useful ways of reconstructing the earth's gravitational potential from satellite data, delivered by low-flying satellite missions such as CHAMP (launched 2000) or the future mission GOCE. In SST and SGG, discrete scalar data of the first and the second radial derivative of the earth's gravitational potential, respectively, are given on the satellite orbit. The determination of the gravitational potential on and outside the earth's surface from these data is an exponentially ill-posed problem. In this thesis, we investigate the reconstruction of the earth's gravitational potential from such data with two different approaches, using splines, wavelets, regularization techniques, and a domain decomposition method. These methods are based on space-localizing structures and allow global reconstructions, as well as local reconstructions of the potential from only locally given data. The numerical simulations in this thesis are carried out with data corresponding to a frequency band of EGM96 (Earth Gravitational Model 96). In the first approach, we compute a smoothing spline from the gravitational data. This smoothing spline is an approximation of the earth's gravitational potential. Here, spline smoothing compensates for the ill-posedness of the problem. Scaling functions and wavelets yield a multiresolution analysis of the potential. In the second approach, we approximate the (first or second order) radial derivative of the potential on the satellite orbit with the help of a spline. Then regularization scaling functions and regularization wavelets are applied to reconstruct the gravitational potential on and outside the earth's surface in a multiscale representation. A central feature in both approaches is the computation of an approximating spline from a large data set. This demands the solution of a linear system with a positive definite, symmetric $N\\\\times N$-matrix, in case of $N$ measurements. These systems are solved efficiently with a domain decomposition method, namely a multiplicative variant of the Schwarz alternating algorithm. This algorithm is an iterative scheme which splits the large matrix into smaller overlapping positive definite, symmetric submatrices and solves corresponding small linear systems in each iterative step. Application of the algorithm to spline interpolation and spline smoothing reduces the CPU-time and the memory requirement considerably and allows one to solve systems of a size that could not be handled before. The splines in this thesis are directly related to the set of bounded linear measurement functionals, and the spline interpolation operator is the orthogonal projector onto the spline space. In addition, parameter choice strategies for spline smoothing, based on a structural similarity between Tikhonov regularization and spline smoothing, are proposed and applied.