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Spline functions that approximate (geostrophic) wind field or ocean circulation data are developed in a weighted Sobolev space setting on the (unit) sphere. Two problems are discussed in more detail: the modelling of the (geostrophic) wind field from (i)discrete scalar air pressure data and (ii) discrete vectorial velocity data. Domain decomposition methods based on the Schwarz alternating algorithm for positive definite symmetric matrices are described for solving large linear systems occuring in vectorial spline interpolation or smoothing of geostrophic flow.

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

A geoscientifically relevant wavelet approach is established for the classical (inner) displacement problem corresponding to a regular surface (such as sphere, ellipsoid, actual earth's 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.

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.

The basic idea behind selective multiscale reconstruction of functions from error-affected data is outlined on the sphere. The selective reconstruction mechanism is based on the premise that multiscale approximation can be well-represented in terms of only a relatively small number of expansion coefficients at various resolution levels. An attempt is made within a tree algorithm (pyramid scheme) to remove the noise component from each scale coefficient using a priori statistical information (provided by an error covariance kernel of a Gaussian, stationary stochastic model).

Spherical Tikhonov Regularization Wavelets in Satellite Gravity Gradiometry with Random Noise
(2000)

This paper considers a special class of regularization methods for satellite gravity gradiometry based on Tikhonov spherical regularization wavelets with particular emphasis on the case of data blurred by random noise. A convergence rate is proved for the regularized solution, and a method is discussed for choosing the regularization level a posteriori from the gradiometer data.

Abstract: The basic concepts of selective multiscale reconstruction of functions on the sphere from error-affected data is outlined for scalar functions. The selective reconstruction mechanism is based on the premise that multiscale approximation can be well-represented in terms of only a relatively small number of expansion coefficients at various resolution levels. A new pyramid scheme is presented to efficiently remove the noise at different scales using a priori statistical information.

Die Grundgleichungen der Physikalischen Geodäsie (in der klassischen Formulierung) werden einer Multiskalenformulierung mittels (sphärisch harmonischer) Wavelets unterzogen. Die Energieverteilung des Störpotentials wird in Auflösung nach Skala und Ort durch Verwendung von Waveletvarianzen beschrieben. Schließlich werden zur Modellierung der zeitlichen Variationen des Schwerefeldes zeit- und ortsgebundene Energiespektren zur Detektion lokaler sowie periodischer/saisonaler Strukturen eingeführt.

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) Dirichlet's or Neumann's boundary-value problem corresponding to regular surfaces.