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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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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 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.