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Tue, 07 Jun 2011 03:44:25 +0200Tue, 07 Jun 2011 03:44:25 +0200Mathematische Methoden in der Geothermie
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2669
Insbesondere bei der industriellen Nutzung tiefer geothermischer Systeme gibt es Risiken, die im Hinblick auf eine zukunftsträchtige Rolle der Ressource "Geothermie" innerhalb der Energiebranche eingeschätzt und minimiert werden müssen. Zur Förderung und Unterstützung dieses Prozesses kann die Mathematik einen entscheidenden Beitrag leisten. Um dies voranzutreiben haben wir zur Charakterisierung tiefer geothermischer Systeme ein Säulenmodell entwickelt, das die Bereiche Exploration, Bau und Produktion näher beleuchtet. Im Speziellen beinhalten die Säulen: Seismische Erkundung, Gravimetrie/Geomagnetik, Transportprozesse, Spannungsfeld.Matthias Augustin; Willi Freeden; Christian Gerhards; Sandra Möhringer; Isabel Ostermannpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2669Wed, 06 Jul 2011 03:44:25 +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 +0200On the Completeness and Closure of Vector and Tensor Spherical Harmonics
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1943
An intrinsically on the 2-sphere formulated proof of the closure and completeness of spherical harmonics is given in vectorial and tensorial framework. The considerations are essentially based on vector and tensor approximation in terms of zonal vector and tensor Bernstein kernels, respectively.Willi Freeden; Martin Guttingpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1943Mon, 07 Apr 2008 13:48:15 +0200Biorthogonal 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 Wavelet Approach to Time-Harmonic Maxwells Equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1699
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 complex-valued vector field given on a regular surface is developed based on numerical integration rules. By virtue of this tree algorithm, an effcient numerical method for the solution of vectorial Fredholm integral equations on regular surfaces is discussed in more detail. The resulting multiscale formulation is used to solve boundary-value problems for the time harmonic Maxwell's equations corresponding to regular surfaces.Willi Freeden; Carsten Mayerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1699Tue, 17 Jan 2006 15:54:18 +0100Wavelet Modelling of Regional and Temporal Variations of the Earth´s Gravitational Potential
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1641
This work is dedicated to the wavelet modelling of regional and temporal variations of the Earth's gravitational potential observed by GRACE. In the first part, all required mathematical tools and methods involving spherical wavelets are introduced. Then we apply our method to monthly GRACE gravity fields. A strong seasonal signal can be identified, which is restricted to areas, where large-scale redistributions of continental water mass are expected. This assumption is analyzed and verified by comparing the time series of regionally obtained wavelet coefficients of the gravitational signal originated from hydrology models and the gravitational potential observed by GRACE. The results are in good agreement to previous studies and illustrate that wavelets are an appropriate tool to investigate regional time-variable effects in the gravitational field.Martin J. Fengler; Willi Freeden; Annika Kohlhaas; Volker Michel; Thomas Peterspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1641Thu, 02 Jun 2005 14:48:00 +0200The Spherical Bernstein Wavelet
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1637
In this work we introduce a new bandlimited spherical wavelet: The Bernstein wavelet. It possesses a couple of interesting properties. To be specific, we are able to construct bandlimited wavelets free of oscillations. The scaling function of this wavelet is investigated with regard to the spherical uncertainty principle, i.e., its localization in the space domain as well as in the momentum domain is calculated and compared to the well-known Shannon scaling function. Surprisingly, they possess the same localization in space although one is highly oscillating whereas the other one shows no oscillatory behavior. Moreover, the Bernstein scaling function turns out to be the first bandlimited scaling function known to the literature whose uncertainty product tends to the minimal value 1.Martin J. Fengler; Willi Freeden; Martin Guttingpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1637Fri, 20 May 2005 12:07:21 +0200Local Multiscale Approximations of Geostrophic Flow: Theoretical Background and Aspects of Scientific Computing
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1634
In modern geoscience, understanding the climate depends on the information about the oceans. Covering two thirds of the Earth, oceans play an important role. Oceanic phenomena are, for example, oceanic circulation, water exchanges between atmosphere, land and ocean or temporal changes of the total water volume. All these features require new methods in constructive approximation, since they are regionally bounded and not globally observable. This article deals with methods of handling data with locally supported basis functions, modeling them in a multiscale scheme involving a wavelet approximation and presenting the main results for the dynamic topography and the geostrophic flow, e.g., in the Northern Atlantic. Further, it is demonstrated that compressional rates of the occurring wavelet transforms can be achieved by use of locally supported wavelets.Willi Freeden; Dominik Michel; Volker Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1634Mon, 09 May 2005 16:04:38 +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 +0100Multiscale Solution for the Molodensky Problem on Regular Telluroidal Surfaces
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1589
Based on the well-known results of classical potential theory, viz. the limit and jump relations for layer integrals, a numerically viable and e±cient multiscale method of approximating the disturbing potential from gravity anomalies is established on regular surfaces, i.e., on telluroids of ellipsoidal or even more structured geometric shape. The essential idea is to use scale dependent regularizations of the layer potentials occurring in the integral formulation of the linearized Molodensky problem to introduce scaling functions and wavelets on the telluroid. As an application of our multiscale approach some numerical examples are presented on an ellipsoidal telluroid.Willi Freeden; Carsten Mayerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1589Thu, 02 Dec 2004 16:53:33 +0100A 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 +0100SST-Regularisierung durch Multiresolutionstechniken auf der Basis von CHAMP-Daten
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1377
Die Bestimmung des Erdgravitationspotentials aus den Meßdaten des Forschungssatelliten CHAMP lässt sich als Operatorgleichung formulieren (SST-Problem). Dieser Ansatz geht davon aus, dass ein geometrischer Orbit des Satelliten CHAMP vorliegt. Mittels numerischer Differentiation unter Einsatz eines geeigneten Denoising Verfahrens kann dann aus dem geometrischen Orbit der Gradient des Potentials längs der Bahn bestimmt werden. Damit sind insbesondere die Radialableitung (und der Flächengradient) auf einem Punktgitter auf der Bahn bekannt. In einem erdfesten System stellt sich dies als eine nahezu vollständige Überdeckung der Erde (bis auf Polar Gaps) mit einem ziemlich dichten Datengitter auf Flughöhe des Satelliten dar. Die Lösung der SST-Operatorgleichung (Bestimmung des Potentials auf der Erdoberfläche aus Kenntnis der Radialableitung auf einem Datengitter auf Flughöhe) ist ein schlecht gestelltes inverses Problem, das mit einer geeigneten Regularisierungstechnik gelöst werden muß. Im vorliegenden Fall wurde eine solche Regularisierung mit Hilfe von nicht-bandlimitierten Regularisierungsskalierungsfunktionen und Regularisierungswavelets umgesetzt. Diese sind stark ortslokalisierend und führen daher auf ein Potentialmodell, welches eine Linearkombination stark ortslokalisierender Funktionen ist. Ein solches Modell kann als Lokalmodell auch aus nur lokalen Daten berechnet werden und bietet daher gegenüber Kugelfunktionsmodellen wie EGM96 erhebliche Vorteile für die moderne Geopotentialbestimmung. Die Diskretisierung und numerische Umsetzung der Berechnung eines solchen Modells erfolgt mit Splines, die hier ebenfalls Linearkombinationen stark ortslokalisierender Funktionen sind. Die großen linearen Gleichungssysteme, die zur Berechnung der glättenden oder interpolierenden Splines gelöst werden müssen, können auf schnelle und effiziente Weise mit dem Schwarzschen alternierenden Algorithmus in Verbindung mit schnellen Summationsverfahren (Fast Multipole Methods) gelöst werden. Eine Kombination des Schwarzschen alternierenden Algorithmus mit solchen schnellen Summationsverfahren ermöglicht eine weitere erhebliche Beschleunigung beim Lösen dieser Gleichungssysteme. Zur Bestimmung von Glättungsparametern (Spline-Smoothing) und Regularisierungsparametern kann die L-Curve Method zum Einsatz kommen.Willi Freeden; Thorsten Maierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1377Mon, 24 Feb 2003 14:15:34 +0100Satellite-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 +0100On the Multiscale Solution of Satellite Problems by Use of Locally Supported Kernel Functions Corresponding to Equidistributed Data on Spherical Orbits
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1117
Being interested in (rotation-)invariant pseudodi erential equations of satellite problems corresponding to spherical orbits, we are reasonably led to generating kernels that depend only on the spherical distance, i. e. in the language of modern constructive approximation form spherical radial basis functions. In this paper approximate identities generated by such (rotation-invariant) kernels which are additionally locally supported are investigated in detail from theoretical as well as numerical point of view. So-called spherical di erence wavelets are introduced. The wavelet transforms are evaluated by the use of a numerical integration rule, that is based on Weyl's law of equidistribution. This approximate formula is constructed such that it can cope with millions of (satellite) data. The approximation error is estimated on the orbital sphere. Finally, we apply the developed theory to the problems of satellite-to-satellite tracking (SST) and satellite gravity gradiometry (SGG).Willi Freeden; Kerstin Hessepreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1117Mon, 21 Aug 2000 00:00:00 +0200Basic 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 +0200Geophysical Field Modelling by Multiresolution Analysis
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/628
Wavelet transform originated in 1980's for the analysis of seismic signals has seen an explosion of applications in geophysics. However, almost all of the material is based on wavelets over Euclidean spaces. This paper deals with the generalization of the theory and algorithmic aspects of wavelets to a spherical earth's model and geophysically relevant vector fields such as the gravitational, magnetic, elastic field of the earth.A scale discrete wavelet approach is considered on the sphere thereby avoiding any type of tensor-valued 'basis (kernel) function'. The generators of the vector wavelets used for the fast evaluation are assumed to have compact supports. Thus the scale and detail spaces are finite-dimensional. As an important consequence, detail information of the vector field under consideration can be obtained only by a finite number of wavelet coefficients for each scale. Using integration formulas that are exact up to a prescribed polynomial degree, wavelet decomposition and reconstruction are investigated for bandlimited vector fields. A pyramid scheme for the recursive computation of the wavelet coefficients from level to level is described in detail. Finally, data compression is discussed for the EGM96 model of the earth's gravitational field.Michael Bayer; Stefan Beth; Willi Freedenpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/628Mon, 03 Apr 2000 00:00:00 +0200A: New Wavelet Methods for Approximating Harmonic Functions; B: Satellite Gradiometry - from Mathematical and Numerical Point of View
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/537
Some new approximation methods are described for harmonic functions corresponding to boundary values on the (unit) sphere. Starting from the usual Fourier (orthogonal) series approach, we propose here nonorthogonal expansions, i.e. series expansions in terms of overcomplete systems consisting of localizing functions. In detail, we are concerned with the so-called Gabor, Toeplitz, and wavelet expansions. Essential tools are modulations, rotations, and dilations of a mother wavelet. The Abel-Poisson kernel turns out to be the appropriate mother wavelet in approximation of harmonic functions from potential values on a spherical boundary.Willi Freeden; Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/537Mon, 03 Apr 2000 00:00:00 +0200Spherical Wavelet Transform and its Discretization
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/555
A continuous version of spherical multiresolution is described, starting from continuous wavelet transform on the sphere. Scale discretization enables us to construct spherical counterparts to Daubechies wavelets and wavelet packets (known from Euclidean theory). Essential tool is the theory of singular integrals on the sphere. It is shown that singular integral operators forming a semigroup of contraction operators of class (Co) (like Abel-Poisson or Gauß-Weierstraß operators) lead in canonical way to (pyramidal) algorithms.Willi Freeden; U. Windheuserpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/555Mon, 03 Apr 2000 00:00:00 +0200An Adaptive Hierarchical Approximation Method on the Sphere Using Axisymmetric Locally Supported Basis Functions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/561
The paper discusses the approximation of scattered data on the sphere which is one of the major tasks in geomathematics. Starting from the discretization of singular integrals on the sphere the authors devise a simple approximation method that employs locally supported spherical polynomials and does not require equidistributed grids. It is the basis for a hierarchical approximation algorithm using differently scaled basis functions, adaptivity and error control. The method is applied to two examples one of which is a digital terrain model of Australia.Willi Freeden; J. Fröhhlich; R. Brandpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/561Mon, 03 Apr 2000 00:00:00 +0200Deformation Analysis Using Navier Spline Interpolation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/567
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.Willi Freeden; E. Groten; Michael Schreiner; W. Söhhne; M. Tücckspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/567Mon, 03 Apr 2000 00:00:00 +0200Equidistribution on the Sphere
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/574
A concept of generalized discrepancy, which involves pseudodifferential operators to give a criterion of equidistributed pointsets, is developed on the sphere. A simply structured formula in terms of elementary functions is established for the computation of the generalized discrepancy. With the help of this formula five kinds of point systems on the sphere, namely lattices in polar coordinates, transformed 2-dimensional sequences, rotations on the sphere, triangulation, and sum of three squares sequence, are investigated. Quantitative tests are done, and the results are compared with each other. Our calculations exhibit different orders of convergence of the generalized discrepancy for different types of point systems.Willi Freeden; J. Cuipreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/574Mon, 03 Apr 2000 00:00:00 +0200Combined Spherical Harmonic and Wavelet Expansion - a Future Concepts in Earth" s Gravitational Determination
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/576
The basic theory of spherical singular integrals is recapitulated. Criteria are given for measuring the space-frequency localization of functions on the sphere. The trade off between space localization on the sphere and frequency localization in terms of spherical harmonics is described in form of an uncertainty principle. A continuous version of spherical multiresolution is introduced, starting from continuous wavelet transform corresponding to spherical wavelets with vanishing moments up to a certain order. The wavelet transform is characterized by least-squares properties. Scale discretization enables us to construct spherical counterparts of wavelet packets and scale discrete Daubechies" wavelets. It is shown that singular integral operators forming a semigroup of contraction operators of class (Co) (like Abel-Poisson or Gauß-Weierstraß operators) lead in canonical way to pyramyd algorithms. Fully discretized wavelet transforms are obtained via approximate integration rules on the sphere. Finally applications to (geo-)physical reality are discussed in more detail. A combined method is proposed for approximating the low frequency parts of a physical quantity by spherical harmonics and the high frequency parts by spherical wavelets. The particular significance of this combined concept is motivated for the situation of today" s physical geodesy, viz. the determination of the high frequency parts of the earth" s gravitational potential under explicit knowledge of the lower order part in terms of a spherical harmonic expansion.Willi Freeden; U. Windheuserpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/576Mon, 03 Apr 2000 00:00:00 +0200A Survey on Spherical Spline Approximation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/590
Spline functions that approximate data given on the sphere are developed in a weighted Sobolev space setting. The flexibility of the weights makes possible the choice of the approximating function in a way which emphasizes attributes desirable for the particular application area. Examples show that certain choices of the weight sequences yield known methods. A convergence theorem containing explicit constants yields a usable error bound. Our survey ends with the discussion of spherical splines in geodetically relevant pseudodifferential equations.Willi Freeden; Michael Schreiner; R. Frankepreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/590Mon, 03 Apr 2000 00:00:00 +0200Gradiometry - an Inverse Problem in Modern Satellite Geodesy
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/595
Satellite gradiometry and its instrumentation is an ultra-sensitive detection technique of the space gravitational gradient (i.e. the Hesse tensor of the gravitational potential). Gradeometry will be of great significance in inertial navigation, gravity survey, geodynamics and earthquake prediction research. In this paper, satellite gradiometry formulated as an inverse problem of satellite geodesy is discussed from two mathematical aspects: Firstly, satellite gradiometry is considered as a continuous problem of harmonic downward continuation. The space-borne gravity gradients are assumed to be known continuously over the satellite (orbit) surface. Our purpose is to specify sufficient conditions under which uniqueness and existence can be guaranteed. It is shown that, in a spherical context, uniqueness results are obtainable by decomposition of the Hesse matrix in terms of tensor spherical harmonics. In particular, the gravitational potential is proved to be uniquely determined if second order radial derivatives are prescribed at satellite height. This information leads us to a reformulation of satellite gradiometry as a (Fredholm) pseudodifferential equation of first kind. Secondly, for a numerical realization, we assume the gravitational gradients to be known for a finite number of discrete points. The discrete problem is dealt with classical regularization methods, based on filtering techniques by means of spherical wavelets. A spherical singular integral-like approach to regularization methods is established, regularization wavelets are developed which allow the regularization in form of a multiresolution analysis. Moreover, a combined spherical harmonic and spherical regularization wavelet solution is derived as an appropriate tool in future (global and local) high-presision resolution of the earth" s gravitational potential.Willi Freeden; F. Schneider; Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/595Mon, 03 Apr 2000 00:00:00 +0200