Carsten Mayer

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.

Multiscale deformation analysis by Cauchy-Navier wavelets

A geoscientifically relevant wavelet approach is established for the classical (inner) displacement problem corresponding to a regular surface (such as sphere, ellipsoid, and actual earth surface). Basic tools are the limit and jump relations of (linear) elastostatics. Scaling functions and wavelets are formulated within the framework of the vectorial Cauchy-Navier equation. Based on appropriate numerical integration rules, a pyramid scheme is developed providing fast wavelet transform (FWT). Finally, multiscale deformation analysis is investigated numerically for the case of a spherical boundary.

Multiscale Downward Continuation of CHAMP FGM-Data for Crustal Field Modelling

Crustal field determination and downward continuation from CHAMPFGM-data will be presented within a multiscale framework using scalar as well as vectorial radial basis functions. The basic idea is to formulate the problem in terms of integral equations relating the radial or tangential projections of the geomagnetic field at satellite height with the magnetic field at the Earth’s surface. In both, the radial as well as the tangential case, crustal field downward continuation turns out to be an exponentially ill-posed problem. As an appropriate solution method multiscale regularization in terms of scalar and vectorial wavelets is presented based on the knowledge of the singular systems corresponding to the aforementioned integral operators. Finally, first wavelet results of crustal field contributions at Earth level, calculated from CHAMP-FGM-data, will be presented.

Tree Algorithms in Wavelet Approximations by Helmholtz Potential Operators

Abstract By means of the limit and jump relations of classical potential theory with respect to the Helmholtz equation a wavelet approach is established on a regular surface. The multiscale procedure is constructed in such a way that the emerging potential kernels act as scaling functions, wavelets are defined via a canonical refinement equation. A tree algorithm for fast computation of a function discretely given on a regular surface is developed based on numerical integration rules. By virtue of the tree algorithm, an efficient numerical method for the solution of Fredholm integral equations involving boundary-value problems of the Helmholtz equation corresponding to (general) regular (boundary) surfaces is discussed in more detail.

Wavelet modelling of the spherical inverse source problem with application to geomagnetism

This paper is concerned with the modelling of ionospheric current systems from induced magnetic fields measured by satellites in a multiscale framework. Scaling functions and wavelets are used to realize a multiscale analysis of the function spaces under consideration and to establish a multiscale regularization procedure for the inversion of the considered vectorial operator equation. Based on the knowledge of the singular system a regularization technique in terms of certain product kernels and corresponding convolutions can be formed. In order to reconstruct ionospheric current systems from satellite magnetic field data, an inversion of the Biot-Savart law in terms of multiscale regularization is derived. The corresponding operator is formulated and the singular values are calculated. The method is tested on real magnetic field data of the satellite CHAMP and simulated data of the proposed satellite mission SWARM.

MODELING TANGENTIAL VECTOR FIELDS ON REGULAR SURFACES BY MEANS OF MIE POTENTIALS

By means of the limit and jump relations of classical potential theory with respect to the vectorial Helmholtz equation, a wavelet approach is established on a regular surface. The multiscale procedure is constructed in such a way that the emerging scalar, vectorial and tensorial potential kernels act as scaling functions. Corresponding wavelets are defined via a canonical refinement equation. A tree algorithm for fast decomposition of a tangential complex-valued vector field given on a regular surface is developed based on numerical integration rules. Some numerical test examples conclude the paper.

Multiscale Solution of Oblique Boundary-Value Problems by Layer Potentials

With the aid of classical results of potential theory, the limit- and jump-relations, a multiscale framework on geodetically relevant regular surfaces is established corresponding to oblique derivative data. By the oblique distance to the regular surface a scale factor in the kernel functions of the limit- and jump-operators is introduced, which connects these intergral kernels with the theory of scaling functions and wavelets.

Multiscale Determination of Radial Current Distribution from CHAMP FGM-Data

A multiscale method using spherical vectorial wavelets for the computation of radial current distributions from CHAMP-FGM data is presented. The considerations are essentially based on the Mie representation of solenoidal vector fields in terms of toroidal and poloidal parts as well as the Helmholtz decomposition of spherical vector fields. It will be shown how a vectorial wavelet representation of the geomagnetic field at the satellite’s orbit can be used to determine a multiscale approximation of the radial current distribution at that height by considering the connection of the toroidal magnetic field scalar with the radial part of the current distribution via the Beltrami differential equation. The applicability and efficiency of the multiresolution technique will be illustrated and first results obtained from CHAMP-FGM data will be presented.

Multiresolution Data Analysis — Numerical Realization by Use of Domain Decomposition Methods and Fast Multipole Techniques

This survey paper deals with multiresolution analysis of geodetically relevant data and its numerical realization for functions harmonic outside a (Bjerhammar) sphere inside the Earth. Harmonic wavelets are introduced within a suitable framework of a Sobolev-like Hilbert space. Scaling functions and wavelets are defined by means of convolutions. A pyramid scheme provides efficient implementation and economical computation. Essential tools are the multiplicative Schwarz alternating algorithm (providing domain decomposition procedures) and fast multipole techniques (accelerating iterative solvers of linear systems).