Year of publication
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- Orthogonal and non-orthogonal multiresolution analysis, scale discrete and exact fully discrete wavelet transform on the sphere (1996)
- Based on a new definition of delation a scale discrete version of spherical multiresolution is described, starting from a scale discrete wavelet transform on the sphere. Depending on the type of application, different families of wavelets are chosen. In particular, spherical Shannon wavelets are constructed that form an orthogonal multiresolution analysis. Finally fully discrete wavelet approximation is discussed in case of band-limited wavelets.
- Wavelet Approximations on Closed Surfaces and their Application to Boundary-Value Problems of Potential Theory (1998)
- 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.
- An integrated wavelet concept of physical geodesy (1998)
- For the determination of the earth" s gravity field many types of observations are available nowadays, e.g., terrestrial gravimetry, airborne gravimetry, satellite-to-satellite tracking, satellite gradiometry etc. The mathematical connection between these observables on the one hand and gravity field and shape of the earth on the other hand, is called the integrated concept of physical geodesy. In this paper harmonic wavelets are introduced by which the gravitational part of the gravity field can be approximated progressively better and better, reflecting an increasing flow of observations. An integrated concept of physical geodesy in terms of harmonic wavelets is presented. Essential tools for approximation are integration formulas relating an integral over an internal sphere to suitable linear combinations of observation functionals, i.e., linear functionals representing the geodetic observables. A scale discrete version of multiresolution is described for approximating the gravitational potential outside and on the earth" s surface. Furthermore, an exact fully discrete wavelet approximation is developed for the case of band-limited wavelets. A method for combined global outer harmonic and local harmonic wavelet modelling is proposed corresponding to realistic earth" s models. As examples, the role of wavelets is discussed for the classical Stokes problem, the oblique derivative problem, satellite-to-satellite tracking, satellite gravity gradiometry, and combined satellite-to-satellite tracking and gradiometry.
- Regularization Wavelets and Multiresolution (1997)
- 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.
- Runge-Walsh Wavelet Approximation for (1997)
- Metaharmonic wavelets are introduced for constructing the solution of theHelmholtz equation (reduced wave equation) corresponding to Dirichlet's orNeumann's boundary values on a closed surface approach leading to exactreconstruction formulas is considered in more detail. A scale discrete version ofmultiresolution is described for potential functions metaharmonic outside theclosed surface and satisfying the radiation condition at infinity. Moreover, wediscuss fully discrete wavelet representations of band-limited metaharmonicpotentials. Finally, a decomposition and reconstruction (pyramid) scheme foreconomical numerical implementation is presented for Runge-Walsh waveletapproximation.
- Spherical panel clustering and its numerical aspects (1997)
- 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.
- Multiscale Gravitational Field Recovery from GPS-Satellite-to-Satellite Tracking (1999)
- The purpose of GPS-satellite-to-satellite tracking (GPS-SST) is to determine the gravitational potential at the earth's surface from measured ranges (geometrical distances) between a low-flying satellite and the high-flying satellites of the Global Posittioning System (GPS). In this paper GPS-satellite-to-satellite tracking is reformulated as the problem of determining the gravitational potential of the earth from given gradients at satellite altitude. Uniqueness and stability of the solution are investigated. The essential tool is to split the gradient field into a normal part (i.e. the first order radial derivative) and a tangential part (i.e. the surface gradient). Uniqueness is proved for polar, circular orbits corresponding to both types of data (first radial derivative and/or surface gradient). In both cases gravity recovery based on satellite-to-satellite tracking turns out to be an exponentially ill-posed problem. As an appropriate solution method regularization in terms of spherical wavelets is proposed based on the knowledge of the singular system. Finally, the extension of this method is generalized to a non-spherical earth and a non-spherical orbital surface based on combined terrestrial and satellite data material.
- A General Hilbert Space Approach to Wavelets and Its Application in Geopotential Determination (1999)
- A general approach to wavelets is presented within a framework of a separable functional Hilbert space H. Basic tool is the construction of H-product kernels by use of Fourier analysis with respect to an orthonormal basis in H. Scaling function and wavelet are defined in terms of H-product kernels. Wavelets are shown to be 'building blocks' that decorrelate the data. A pyramid scheme provides fast computation. Finally, the determination of the earth's gravitational potential from single and multipole expressions is organized as an example of wavelet approximation in Hilbert space structure.
- Constructive Approximation and Numerical Methods in Geodetic; Research Today - An Attempt of a Categorization Based on anUncertainty Principle (1999)
- This review article reports current activities and recent progress on constructive approximation and numerical analysis in physical geodesy. The paper focuses on two major topics of interest, 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.
- Tensor Spherical Harmonics and Tensor Spherical Splines (1993)
- 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.