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Wavelet transform originated in 1980's for the analysis of seismic signals has seen an explosion of applications in geophysics. However, almost all of the material is based on wavelets over Euclidean spaces. This paper deals with the generalization of the theory and algorithmic aspects of wavelets to a spherical earth's model and geophysically relevant vector fields such as the gravitational, magnetic, elastic field of the earth.A scale discrete wavelet approach is considered on the sphere thereby avoiding any type of tensor-valued 'basis (kernel) function'. The generators of the vector wavelets used for the fast evaluation are assumed to have compact supports. Thus the scale and detail spaces are finite-dimensional. As an important consequence, detail information of the vector field under consideration can be obtained only by a finite number of wavelet coefficients for each scale. Using integration formulas that are exact up to a prescribed polynomial degree, wavelet decomposition and reconstruction are investigated for bandlimited vector fields. A pyramid scheme for the recursive computation of the wavelet coefficients from level to level is described in detail. Finally, data compression is discussed for the EGM96 model of the earth's gravitational field.

The following three papers present recent developments in multiscale gravitational field modeling by the use of CHAMP or CHAMP-related data. Part A - The Model SWITCH-03: Observed orbit perturbations of the near-Earth orbiting satellite CHAMP are analyzed to recover the long-wavelength features of the Earth's gravitational potential. More precisely, by tracking the low-flying satellite CHAMP by the high-flying satellites of the Global Positioning System (GPS) a kinematic orbit of CHAMP is obtainable from GPS tracking observations, i.e. the ephemeris in cartesian coordinates in an Earth-fixed coordinate frame (WGS84) becomes available. In this study we are concerned with two tasks: First we present new methods for preprocessing, modelling and analyzing the emerging tracking data. Then, in a first step we demonstrate the strength of our approach by applying it to simulated CHAMP orbit data. In a second step we present results obtained by operating on a data set derived from real CHAMP data. The modelling is mainly based on a connection between non-bandlimited spherical splines and least square adjustment techniques to take into account the non-sphericity of the trajectory. Furthermore, harmonic regularization wavelets for solving the underlying Satellite-to-Satellite Tracking (SST) problem are used within the framework of multiscale recovery of the Earth's gravitational potential leading to SWITCH-03 (Spline and Wavelet Inverse Tikhonov regularized CHamp data). Further it is shown how regularization parameters can be adapted adequately to a specific region improving a globally resolved model. Finally we give a comparison of the developed model to the EGM96 model, the model UCPH2002_02_0.5 from the University of Copenhagen and the GFZ models EIGEN-1s and EIGEN-2. Part B - Multiscale Solutions from CHAMP: CHAMP orbits and accelerometer data are used to recover the long- to medium- wavelength features of the Earth's gravitational potential. In this study we are concerned with analyzing preprocessed data in a framework of multiscale recovery of the Earth's gravitational potential, allowing both global and regional solutions. The energy conservation approach has been used to convert orbits and accelerometer data into in-situ potential. Our modelling is spacewise, based on (1) non-bandlimited least square adjustment splines to take into account the true (non-spherical) shape of the trajectory (2) harmonic regularization wavelets for solving the underlying inverse problem of downward continuation. Furthermore we can show that by adapting regularization parameters to specific regions local solutions can improve considerably on global ones. We apply this concept to kinematic CHAMP orbits, and, for test purposes, to dynamic orbits. Finally we compare our recovered model to the EGM96 model, and the GFZ models EIGEN-2 and EIGEN-GRACE01s. Part C - Multiscale Modeling from EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, UCPH2002_0.5, EGM96: Spherical wavelets have been developed by the Geomathematics Group Kaiserslautern for several years and have been successfully applied to georelevant problems. Wavelets can be considered as consecutive band-pass filters and allow local approximations. The wavelet transform can also be applied to spherical harmonic models of the Earth's gravitational field like the most up-to-date EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, UCPH2002_0.5, and the well-known EGM96. Thereby, wavelet coefficients arise and these shall be made available to other interested groups. These wavelet coefficients allow the reconstruction of the wavelet approximations. Different types of wavelets are considered: bandlimited wavelets (here: Shannon and Cubic Polynomial (CP)) as well as non-bandlimited ones (in our case: Abel-Poisson). For these types wavelet coefficients are computed and wavelet variances are given. The data format of the wavelet coefficients is also included.

A concept of generalized discrepancy, which involves pseudodifferential operators to give a criterion of equidistributed pointsets, is developed on the sphere. A simply structured formula in terms of elementary functions is established for the computation of the generalized discrepancy. With the help of this formula five kinds of point systems on the sphere, namely lattices in polar coordinates, transformed 2-dimensional sequences, rotations on the sphere, triangulation, and sum of three squares sequence, are investigated. Quantitative tests are done, and the results are compared with each other. Our calculations exhibit different orders of convergence of the generalized discrepancy for different types of point systems.

Spline functions that interpolate data given on the sphere are developed in a weighted Sobolev space setting. The flexibility of the weights makes possible the choice of the approximating function in a way which emphasizes attributes desirable for the particular application area. Examples show that certain choices of the weight sequences yield known methods. A pointwise convergence theorem containing explicit constants yields a useable error bound.

The paper discusses the approximation of scattered data on the sphere which is one of the major tasks in geomathematics. Starting from the discretization of singular integrals on the sphere the authors devise a simple approximation method that employs locally supported spherical polynomials and does not require equidistributed grids. It is the basis for a hierarchical approximation algorithm using differently scaled basis functions, adaptivity and error control. The method is applied to two examples one of which is a digital terrain model of Australia.

In this paper, we deal with the problem of spherical interpolation of discretely given data of tensorial type. To this end, spherical tensor fields are investigated and a decomposition formula is described. Tensor spherical harmonics are introduced as eigenfunctions of a tensorial analogon to the Beltrami operator and discussed in detail. Based on these preliminaries, a spline interpolation process is described and error estimates are presented. Furthermore, some relations between the spline basis functions and the theory of radial basis functions are developed.

In modern approximation methods linear combinations in terms of (space localizing) radial basis functions play an essential role. Areas of application are numerical integration formulas on the uni sphere omega corresponding to prescribed nodes, spherical spline interpolation, and spherical wavelet approximation. the evaluation of such a linear combination is a time consuming task, since a certain number of summations, multiplications and the calculation of scalar products are required. This paper presents a generalization of the panel clustering method in a spherical setup. The economy and efficiency of panel clustering is demonstrated for three fields of interest, namely upward continuation of the earth's gravitational potential, geoid computation by spherical splines and wavelet reconstruction of the gravitational potential.

The static deformation of the surface of the earth caused by surface pressure like the water load of an ocean or an artificial lake is discussed. First a brief mention is made on the solution of the Boussenesq problem for an infinite halfspace with the elastic medium to be assumed as homogeneous and isotropic. Then the elastic response for realistic earth models is determinied by spline interpolation using Navier splines. Major emphasis is on the derteminination of the elastic field caused by water loads from surface tractions on the (real) earth" s surface. Finally the elastic deflection of an artificial lake assuming a homogeneous isotropic crust is compared for both evaluation methods.

Wavelets on closed surfaces in Euclidean space R3 are introduced starting from a scale discrete wavelet transform for potentials harmonic down to a spherical boundary. Essential tools for approximation are integration formulas relating an integral over the sphere to suitable linear combinations of functional values (resp. normal derivatives) on the closed surface under consideration. A scale discrete version of multiresolution is described for potential functions harmonic outside the closed surface and regular at infinity. Furthermore, an exact fully discrete wavelet approximation is developed in case of band-limited wavelets. Finally, the role of wavelets is discussed in three problems, namely (i) the representation of a function on a closed surface from discretely given data, (ii) the (discrete) solution of the exterior Dirichlet problem, and (iii) the (discrete) solution of the exterior Neumann problem.