Willi Freeden

Spherical wavelet transform and its discretization

A continuous version of spherical multiresolution is described, starting from continuous wavelet transform on the sphere. Scale discretization enables us to construct spherical counterparts to wavelet packets and scale discrete wavelets. The essential tool is the theory of singular integrals on the sphere. It is shown that singular integral operators forming a semigroup of contraction operators of class (C0) (like Abel-Poisson or Gauß-Weierstraß operators) lead in a canonical way to (pyramidal) algorithms.

Spherical spline interpolation—basic theory and computational aspects

Abstract The purpose of the paper is to adapt to the spherical case the basic theory and the computational method known from surface spline interpolation in Euclidean spaces. Spline functions are defined on the sphere. The solution process is made simple and efficient for numerical computation. In addition, the convergence of the solution obtained by spherical spline interpolation is developed using estimates for Legendre polynomials.

Regularization wavelets and multiresolution

Many problems arising in (geo)physics and technology can be formulated as compact operator equations of the first kind \(A F = G\). Due to the ill-posedness of the equation a variety of regularization methods are in discussion for an approximate solution, where particular emphasize must be put on balancing the data and the approximation error. In doing so one is interested in optimal parameter choice strategies. In this paper our interest lies in an efficient algorithmic realization of a special class of regularization methods. More precisely, we implement regularization methods based on filtered singular value decomposition as a wavelet analysis. This enables us to perform, e.g., Tikhonov-Philips regularization as multiresolution. In other words, we are able to pass over from one regularized solution to another one by adding or subtracting so-called detail information in terms of wavelets. It is shown that regularization wavelets as proposed here are efficiently applicable to a future problem in satellite geodesy, viz. satellite gravity gradiometry.

Constructive approximation and numerical methods in geodetic research today – an attempt at a categorization based on an uncertainty principle

Abstract. Current activities and recent progress on constructive approximation and numerical analysis in physical geodesy are reported upon. Two major topics of interest are focused upon, namely trial systems for purposes of global and local approximation and methods for adequate geodetic application. A fundamental tool is an uncertainty principle, which gives appropriate bounds for the quantification of space and momentum localization of trial functions. The essential outcome is a better understanding of constructive approximation in terms of radial basis functions such as splines and wavelets.

Satellite-to-satellite tracking and satellite gravity gradiometry (Advanced techniques for high-resolution geopotential field determination)

The purpose of satellite-to-satellite tracking (SST) and/or satellite gravity gradiometry (SGG) is to determine the gravitational field on and outside the Earths surface from given gradients of the gravitational potential and/or the gravitational field at satellite altitude. In this paper both satellite techniques are analysed and characterized from a mathematical point of view. Uniqueness results are formulated. The justification is given for approximating the external gravitational field by finite linear combination of certain gradient fields (for example, gradient fields of single-poles or multi-poles) consistent to a given set of SGG and/or SST data. A strategy of modelling the gravitational field from satellite data within a multiscale concept is described; illustrations based on the EGM96 model are given.

Geomathematically oriented potential theory

PRELIMINARIES Three-Dimensional Euclidean Space R3 Basic Notation Integral Theorems Two-Dimensional Sphere OMEGA Basic Notation Integral Theorems (Scalar) Spherical Harmonics (Scalar) Circular Harmonics Vector Spherical Harmonics Tensor Spherical Harmonics POTENTIAL THEORY IN THE EUCLIDEAN SPACE R3 Basic Concepts Background Material Volume Potentials Surface Potentials Boundary-Value Problems Locally and Globally Uniform Approximation Gravitation Oblique Derivative Problem Satellite Problems Gravimetry Problem Geomagnetism Geomagnetic Background Mie and Helmholtz Decomposition Gauss Representation and Uniqueness Separation of Sources Ionospheric Current Systems POTENTIAL THEORY ON THE UNIT SPHERE OMEGA Basic Concepts Background Material Surface Potentials Curve Potentials Boundary-Value Problems Differential Equations for Surface Gradient and Surface Curl Gradient Locally and Globally Uniform Approximation Gravitation Disturbing Potential Linear Regularization Method Multiscale Solution Geomagnetics Mie and Helmholtz Decomposition Higher-Order Regularization Methods Separation of Sources Ionospheric Current Systems Bibliography Index Exercises appear at the end of each chapter.

On the approximation of external gravitational potential with closed systems of (trial) functions

SummaryLet S be the (regular) boundary-surface of an exterior regionEe in Euclidean space ℜ3 (for instance: sphere, ellipsoid, geoid, earths surface). Denote by {φn} a countable, linearly independent system of trial functions (e.g., solid spherical harmonics or certain singularity functions) which are harmonic in some domain containingEe ∪ S. It is the purpose of this paper to show that the restrictions {ϕn} of the functions {φn} onS form a closed system in the spaceC (S), i.e. any functionf, defined and continuous onS, can be approximated uniformly by a linear combination of the functions ϕn.Consequences of this result are versions of Runge and Keldysh-Lavrentiev theorems adapted to the chosen system {φn} and the mathematical justification of the use of trial functions in numerical (especially: collocational) procedures.

On Integral Formulas of the (Unit) Sphere and Their Application to Numerical Computation of Integrals

The aim of this paper is the study of integral formulas and methods for optimal numerical approximation of integrals over the (unit) sphere and coefficients being required in connection with series expansions into spherical harmonics. An explicit estimate of the (absolute) error of exact and approximating integral value is given by use of the theory of Greens (surface) functions on the sphere.ZusammenfassungGegenstand dieser Arbeit ist die Behandlung von Integralformeln und Verfahren zur optimalen numerischen Approximation von Integralen über die Einheitssphäre und von Koeffizienten in Reihenentwicklungen nach Kugelfunktionen. Durch Benutzung der Theorie Greenscher (Flächen-) Funktionen wird eine explizite Abschätzung des (absoluten) Fehlers von exaktem und approximierendem Integralwert angegeben.

Wavelets generated by layer potentials

By means of the limit and jump relations of classical potential theory the framework of a wavelet approach on a regular surface is established. The properties of a multiresolution analysis are verified, and a tree algorithm for fast computation is developed based on numerical integration. As applications of the wavelet approach some numerical examples are presented, including the zoom-in property as well as the detection of high frequency perturbations. At the end we discuss a fast multiscale representation of the solution of (exterior) Dirichlets or Neumanns boundary-value problem corresponding to regular surfaces.

A vector wavelet approach to iono- and magnetospheric geomagnetic satellite data

Abstract A multiscale method is introduced using spherical (vector) wavelets for the computation of the earths magnetic field within source regions of ionospheric and magnetospheric currents. The considerations are essentially based on two geomathematical keystones, namely (i) the Mie representation of solenoidal vector fields in terms of toroidal and poloidal parts and (ii) the Helmholtz decomposition of spherical (tangential) vector fields. Vector wavelets are shown to provide adequate tools for multiscale geomagnetic modelling in form of a multiresolution analysis, thereby completely circumventing the numerical obstacles caused by vector spherical harmonics. The applicability and efficiency of the multiresolution technique is tested with real satellite data.