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Tue, 18 Nov 2008 13:50:49 +0100Tue, 18 Nov 2008 13:50:49 +0100Klassische Erdschwerefeldbestimmung aus der Sicht moderner Geomathematik
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2040
Gegenstand dieser Arbeit ist die kanonische Verbindung klassischer globaler Schwerefeldmodellierung in der Konzeption von Stokes (1849) und Neumann (1887) und moderner lokaler Multiskalenberechnung mittels lokalkompakter adaptiver Wavelets. Besonderes Anliegen ist die "Zoom-in"-Ermittlung von Geoidhöhen aus lokal gegebenen Schwereanomalien bzw. Schwerestörungen.Willi Freeden; Kerstin Wolfreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2040Tue, 18 Nov 2008 13:50:49 +0100On 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 +0200Wavelet Deformation Analysis for Spherical Bodies
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1612
In this paper we introduce a multiscale technique for the analysis of deformation phenomena of the Earth. Classically, the basis functions under use are globally defined and show polynomial character. In consequence, only a global analysis of deformations is possible such that, for example, the water load of an artificial reservoir is hardly to model in that way. Up till now, the alternative to realize a local analysis can only be established by assuming the investigated region to be flat. In what follows we propose a local analysis based on tools (Navier scaling functions and wavelets) taking the (spherical) surface of the Earth into account. Our approach, in particular, enables us to perform a zooming-in procedure. In fact, the concept of Navier wavelets is formulated in such a way that subregions with larger or smaller data density can accordingly be modelled with a higher or lower resolution of the model, respectively.Willi Freeden; Volker Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1612Thu, 17 Feb 2005 18:43:34 +0100Regularized Multiresolution Recovery of the Mass Density Distribution From Satellite Data of the Earth´s Gravitational Field
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1586
The inverse problem of recovering the Earth's density distribution from data of the first or second derivative of the gravitational potential at satellite orbit height is discussed for a ball-shaped Earth. This problem is exponentially ill-posed. In this paper a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part the second radial derivative of the gravitational potential at 200 km orbitheight is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG (satellite gravity gradiometry) satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earth's crust. Moreover, the noise sensitivity of the regularization technique is analyzed numerically.Volker Michelreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1586Tue, 16 Nov 2004 14:25:33 +0100Regularized Multiresolution Recovery of the Mass Density Distribution from Satellite Data of the Earth's Gravitational Field
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1413
The inverse problem of recovering the Earth's density distribution from satellite data of the first or second derivative of the gravitational potential at orbit height is discussed. This problem is exponentially ill-posed. In this paper a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part the second radial derivative of the gravitational potential at 200 km orbit height is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earth's crust.Volker Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1413Tue, 15 Jul 2003 13:04:31 +0200Satellite-to-Satellite Tracking and Satellite Gravity Gradiometry
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1134
The purpose of satellite-to-satellite tracking (SST) and/or satellite gravity gradiometry (SGG) is to determine the gravitational field on and outside the Earth's 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 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.Willi Freeden; Volker Michel; Helga Nutzpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1134Thu, 08 Mar 2001 00:00:00 +0100Basic Aspects of Geopotential Field Approximation From Satellite-to-Satellite Tracking Data
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1116
The satellite-to-satellite tracking (SST) problems are characterized from mathematical point of view. Uniqueness results are formulated. Moreover, the basic relations are developed between (scalar) approximation of the earth's gravitational potential by "scalar basis systems" and (vectorial) approximation of the gravitational eld by "vectorial basis systems". Finally, the mathematical justication is given for approximating the external geopotential field by finite linear combinations of certain gradient fields (for example, gradient fields of multi-poles) consistent to a given set of SST data.Willi Freeden; Volker Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1116Mon, 21 Aug 2000 00:00:00 +0200