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Fri, 04 Jul 2008 17:08:11 +0200Fri, 04 Jul 2008 17:08:11 +0200On the Local Multiscale Determination of the Earth`s Disturbing Potential From Discrete Deflections of the Vertical
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1947
As a first approximation the Earth is a sphere; as a second approximation it may be considered an ellipsoid of revolution. The deviations of the actual Earth's gravity field from the ellipsoidal 'normal' field are so small that they can be understood to be linear. The splitting of the Earth's gravity field into a 'normal' and a remaining small 'disturbing' field considerably simplifies the problem of its determination. Under the assumption of an ellipsoidal Earth model high observational accuracy is achievable only if the deviation (deflection of the vertical) of the physical plumb line, to which measurements refer, from the ellipsoidal normal is not ignored. Hence, the determination of the disturbing potential from known deflections of the vertical is a central problem of physical geodesy. In this paper we propose a new, well-promising method for modelling the disturbing potential locally from the deflections of the vertical. Essential tools are integral formulae on the sphere based on Green's function of the Beltrami operator. The determination of the disturbing potential from deflections of the vertical is formulated as a multiscale procedure involving scale-dependent regularized versions of the surface gradient of the Green function. The modelling process is based on a multiscale framework by use of locally supported surface curl-free vector wavelets.Willi Freeden; Thomas Fehlinger; Carsten Mayer; Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1947Mon, 07 Apr 2008 17:08:11 +0200Local Modelling of Sea Surface Topography from (Geostrophic) Ocean Flow
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1836
This paper deals with the problem of determining the sea surface topography from geostrophic flow of ocean currents on local domains of the spherical Earth. In mathematical context the problem amounts to the solution of a spherical differential equation relating the surface curl gradient of a scalar field (sea surface topography) to a surface divergence-free vector field(geostrophic ocean flow). At first, a continuous solution theory is presented in the framework of an integral formula involving Green’s function of the spherical Beltrami operator. Different criteria derived from spherical vector analysis are given to investigate uniqueness. Second, for practical applications Green’s function is replaced by a regularized counterpart. The solution is obtained by a convolution of the flow field with a scaled version of the regularized Green function. Calculating locally without boundary correction would lead to errors near the boundary. To avoid these Gibbs phenomenona we additionally consider the boundary integral of the corresponding region on the sphere which occurs in the integral formula of the solution. For reasons of simplicity we discuss a spherical cap first, that means we consider a continuously differentiable (regular) boundary curve. In a second step we concentrate on a more complicated domain with a non continuously differentiable boundary curve, namely a rectangular region. It will turn out that the boundary integral provides a major part for stabilizing and reconstructing the approximation of the solution in our multiscale procedure.Thomas Fehlinger; Willi Freeden; Simone Gramsch; Carsten Mayer; Dominik Michel; Michael Schreinerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1836Sun, 11 Feb 2007 15:03:15 +0100Biorthogonal Locally Supported Wavelets on the Sphere Based on Zonal Kernel Functions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1787
This paper presents a method for approximating spherical functions from discrete data of a block-wise grid structure. The essential ingredients of the approach are scaling and wavelet functions within a biorthogonalisation process generated by locally supported zonal kernel functions. In consequence, geophysically and geodetically relevant problems involving rotation-invariant pseudodifferential operators become attackable. A multiresolution analysis is formulated enabling a fast wavelet transform similar to the algorithms known from one-dimensional Euclidean theory.Willi Freeden; Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1787Tue, 24 Oct 2006 22:02:39 +0200A Tree Algorithm for Isotropic Finite Elements on the Sphere
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1447
The Earth's surface is an almost perfect sphere. Deviations from its spherical shape are less than 0,4% of its radius and essentially arise from its rotation. All equipotential surfaces are nearly spherical, too. In consequence, multiscale modelling of geoscientifically relevant data on the sphere involving rotational symmetry of the trial functions used for the approximation plays an important role. In this paper we deal with isotropic kernel functions showing local support and (one-dimensional) polynomial structure (briefly called isotropic finite elements) for reconstructing square--integrable functions on the sphere. Essential tool is the concept of multiresolution analysis by virtue of the spherical up function. The main result is a tree algorithm in terms of (low--order) isotropic finite elements.Frank Bauer; Willi Freeden; Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1447Mon, 10 Nov 2003 10:47:32 +0100Multiresolution Analysis by Spherical Up Functions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1446
A new class of locally supported radial basis functions on the (unit) sphere is introduced by forming an infinite number of convolutions of ''isotropic finite elements''. The resulting up functions show useful properties: They are locally supported and are infinitely often differentiable. The main properties of these kernels are studied in detail. In particular, the development of a multiresolution analysis within the reference space of square--integrable functions over the sphere is given. Altogether, the paper presents a mathematically significant and numerically efficient introduction to multiscale approximation by locally supported radial basis functions on the sphere.Willi Freeden; Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1446Mon, 10 Nov 2003 10:42:17 +0100Weighted Particles in the Finite Pointset method
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/698
Using particle methods to solve the Boltzmann equation for rarefied gases numerically, in realistic streaming problems, huge differences in the total number of particles per cell arise. In order to overcome the resulting numerical difficulties the application of a weighted particle concept is well-suited. The underlying idea is to use different particle masses in different cells depending on the macroscopic density of the gas. Discrepance estimates and numerical results are given.Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/698Sun, 09 Jul 2000 00:00:00 +0200A Pyramid Scheme for Spherical Wavelets
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/605
We consider a scale discrete wavelet approach on the sphere based on spherical radial basis functions. If the generators of the wavelets have a compact support, the scale and detail spaces are finite-dimensional, so that the detail information of a function is determined by only finitely many wavelet coefficients for each scale. We describe a pyramid scheme for the recursive determination of the wavelet coefficients from level to level, starting from an initial approximation of a given function. Basic tools are integration formulas which are exact for functions up to a given polynomial degree and spherical convolutions.Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/605Sat, 24 Jun 2000 00:00:00 +0200The C Programmes for "Numerical Methods (Programmes and Implementation)"
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/594
Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/594Fri, 23 Jun 2000 00:00:00 +0200Locally Supported Kernels for Spherical Spline Interpolation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/572
By the use of locally supported basis functions for spherical spline interpolation the applicability of this approximation method is spread out since the resulting interpolation matrix is sparse and thus efficient solvers can be used. In this paper we study locally supported kernels in detail. Investigations on the Legendre coefficients allow a characterization of the underlying Hilbert space structure. We show now spherical spline interpolation with polynomial precision can be managed with locally supported kernels, thus giving the possibility to combine approximation techniques based on spherical harmonic expansions with those based on locally supported kernels.Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/572Wed, 07 Jun 2000 00:00:00 +0200On a New Condition for Strictly Positive Definite Functions on Spheres
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/573
Recently, Xu and Cheney (1992) have proved that if all the Legendre coefficients of a zonal function defined on a sphere are positive then the function is strictly positive definite. It will be shown in this paper, that even if finitely many of the Legendre coefficients are zero, the strict positive definiteness can be assured. The results are based on approximation properties of singular integrals, and provide also a completely different proof of the results ofXu and Cheney.Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/573Wed, 07 Jun 2000 00:00:00 +0200