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## Über dieses Buch

Written by leading experts, this book provides a clear and comprehensive survey of the “status quo” of the interrelating process and cross-fertilization of structures and methods in mathematical geodesy. Starting with a foundation of functional analysis, potential theory, constructive approximation, special function theory, and inverse problems, readers are subsequently introduced to today’s least squares approximation, spherical harmonics reflected spline and wavelet concepts, boundary value problems, Runge-Walsh framework, geodetic observables, geoidal modeling, ill-posed problems and regularizations, inverse gravimetry, and satellite gravity gradiometry. All chapters are self-contained and can be studied individually, making the book an ideal resource for both graduate students and active researchers who want to acquaint themselves with the mathematical aspects of modern geodesy.

## Inhaltsverzeichnis

### Gauss as Scientific Mediator Between Mathematics and Geodesy from the Past to the Present

Abstract.
The objective of the paper is to document the pioneer dimension of Gauss’s ideas, concepts, and methods in a twofold direction based on selected case examples, namely to demonstrate his mediation function between mathematics and geodesy to explain the historic development over the past centuries from the initial Gaussian ignition to modern characteristics and tendencies.
Willi Freeden, Thomas Sonar, Bertold Witte

### An Overview on Tools from Functional Analysis

Abstract.
Many modern mathematical methods treat geodetic problems in terms of functions from certain spaces, proving convergence properties of such functions and regard the evaluation of such functions or their derivatives at given points as operators. In doing so, knowingly or unknowingly, they use the language of functional analysis.
This contribution aims at summarizing some fundamental concepts from functional analysis which are used throughout this book. In this way, it tries to add a layer of self-sufficiency and to act as supplement to other contributions for those readers who are not familiar with functional analytic tools. For this purpose, we introduce, among others, the general ideas of vector spaces, norms, metrics, inner products, orthogonality, completeness, Banach spaces, Hilbert spaces, functionals, linear operators, different notions of convergence. Then we show how functions can be interpreted as vectors in different kind of function spaces, e.g., spaces of continuous functions, Lebesgue spaces, or Sobolev spaces and how the more general concepts come into play here. Moreover, we have a brief glimpse at differential equations and how functional analytic tools provide the necessary background to discuss them, and at the idea of reproducing kernels and the corresponding reproducing kernel Hilbert spaces.
Matthias Augustin, Sarah Eberle, Martin Grothaus

### Ill-Posed Problems: Operator Methodologies of Resolution and Regularization

Abstract.
A general framework of regularization and approximation methods for ill-posed problems is developed. Three levels in the resolution processes are distinguished and emphasized: philosophy of resolution, regularizationapproximation schema, and regularization algorithms. Dilemmas and methodologies of resolution of ill-posed problems and their numerical implementations are examined with particular reference to the problem of finding numerically minimum weighted-norm least squares solutions of first kind integral equations (and more generally of linear operator equations with non-closed range). An emphasis is placed on the role of constraints, function space methods, the role of generalized inverses, and reproducing kernels in the regularization and stable computational resolution of these problems. The thrust of the contribution is devoted to the interdisciplinary character of operator-theoretic and regularization methods for ill-posed problems, in particular in mathematical geoscience.
Willi Freeden, M. Zuhair Nashed

### Geodetic Observables and Their Mathematical Treatment in Multiscale Framework

Abstract.
For the determination of the Earth’s gravitational field various types of observations are available nowadays, e.g., from terrestrial gravimetry, airborne gravimetry, satellite-to-satellite tracking, satellite gravity gradiometry, etc. The mathematical relation between these observables on the one hand and the gravitational field and the shape of the Earth on the other hand is called the integrated concept of physical geodesy. In this paper, an integrated concept of physical geodesy in terms of harmonic wavelets is presented. Essential tools for approximation are Runge–Walsh type integration formulas relating an integral over an internal sphere to suitable linear combinations of observational functionals, i.e., linear functionals representing the geodetic observables in terms of gravitational quantities on and outside the Earth. A scale discrete version of multiresolution is described for approximating the gravitational potential on and outside the Earth’s surface. Furthermore, an exact fully discrete wavelet approximation is developed for the case of bandlimited wavelets. A method for combined global outer harmonic and local harmonic wavelet modeling is proposed corresponding to realistic Earth’s models.
Willi Freeden, Helga Nutz

### The Analysis of the Geodetic Boundary Value Problem: State and Perspectives

Abstract.
The geodetic boundary value problem is mathematically a freeboundary, oblique derivative boundary value problem for the Laplace operator. The solution of the problem is the determination of the shape of the Earth and of its gravity field. The analysis of such a problem, specially for its non-linear formulation, is hard, so it started only in 1976 with a paper by L. Hörmander [13].
Since then the research has continued for both the non-linear and the linearized version, till recent years. In this article the author tries to give an overview on the subject, including a new result for the so-called Simple Molodensky Problem.
Fernando Sansò

### Oblique Stochastic Boundary Value Problem

Abstract.
Aim of this note is to report the current state of the analysis for weak solutions to oblique boundary problems for the Poisson equation. In this paper, as well deterministic as stochastic inhomogeneities are treated and existence and uniqueness results for corresponding weak solutions are presented. We consider the problem for inner bounded and outer unbounded domains in $$\mathbb{R}^n$$. Main tools for the deterministic inner problem are a Poincaré inequality and some analysis for Sobolev spaces on submanifolds, in order to use the Lax–Milgram Lemma. The Kelvin transformation enables us to translate the outer problem to a corresponding inner problem. Thus we can define a solution operator by using the solution operator of the inner problem. The extension to stochastic inhomogeneities is done with help of tensor product spaces of a probability space with the Sobolev spaces from the deterministic problems. We can prove a regularization result which shows that the weak solution fulfills the classical formulation for smooth data. A Ritz–Galerkin approximation method for numerical computations is available. Finally, we show that the results are applicable to geomathematical problems.

### About the Importance of the Runge–Walsh Concept for Gravitational Field Determination

Abstract.
On the one hand, the Runge–Walsh theorem plays a particular role in physical geodesy, because it allows to guarantee a uniform approximation of the Earth’s gravitational potential within arbitrary accuracy by a harmonic function showing a larger analyticity domain. On the other hand, there are some less transparent manifestations of the Runge–Walsh context in the geodetic literature that must be clarified in more detail. Indeed, some authors make the attempt to apply the Runge–Walsh idea to the gravity potential of a rotating Earth instead of the gravitational potential in non-rotating status. Others doubt about the convergence of series expansions approximating the Earth’s gravitational potential inside the whole outer space of the actual Earth.
The goal of this contribution is to provide the conceptual setup of the Runge–Walsh theorem such that geodetic expectation as well as mathematical justification become transparent and coincident. Even more, the Runge–Walsh concept in form of generalized Fourier expansions corresponding to certain harmonic trial functions (e.g., mono- and/or multi-poles) will be extended to the topology of Sobolev-like reproducing kernel Hilbert spaces thereby avoiding any need of (numerical) integration in the occurring spline solution process.
Matthias Augustin, Willi Freeden, Helga Nutz

Abstract.
A promising technique of globally establishing the fine structure and the characteristics of the external Earth’s gravitational field is satellite gravity gradiometry (SGG). Satellites such as ESA’s gradiometer satellite GOCE are able to provide sufficiently large data material of homogeneous quality and accuracy. In geodesy, traditionally the external Earth’s gravitational potential and its Hesse matrix are described using orthogonal (Fourier) expansions in terms of (outer) spherical harmonics. Spherical and outer harmonics are introduced for the global modeling of (scalar / tensor) fields. We briefly recapitulate the results interconnecting spherically the potential coefficients with respect to tensor spherical harmonics at Low Earth Orbiter’s (LEO) altitude to the corresponding coefficients with respect to scalar spherical harmonics at the Earth’s surface. The relation between the known tensorial measurements g (i.e., gradiometer data) and the gravitational potential F on the Earth’s surface is expressed by a linear integral equation of the first kind. This operator equation is discussed in the framework of pseudodifferential operators as an invertible mapping between Sobolev spaces under the assumption that the data are not erroneous. In reality, however, the data g are noisy such that the Sobolev reference space for the (noisy) tensorial data g must be embedded in a larger Sobolev space. Under these conditions, we base our inversion process on the fact that the reference Sobolev subspace is dense in the larger Sobolev space and that, e.g., a smoothing spline process or a signal-to-noise procedure in multiscale framework open appropriate perspectives to approximate F (in suitable accuracy) from noisy data g.
Willi Freeden, Helga Nutz, Michael Schreiner

### Parameter Choices for Fast Harmonic Spline Approximation

Abstract.
The approximation by harmonic trial functions allows the construction of the solution of boundary value problems in geoscience where the boundary is often the known surface of the Earth itself. Using harmonic splines such a solution can be approximated from discrete data on the surface. Due to their localizing properties regional modeling or the improvement of a global model in a part of the Earth’s surface is possible with splines.
Fast multipole methods have been developed for some cases of the occurring kernels to obtain a fast matrix-vector multiplication. The main idea of the fast multipole algorithm consists of a hierarchical decomposition of the computational domain into cubes and a kernel approximation for the more distant points. This reduces the numerical effort of the matrix-vector multiplication from quadratic to linear in reference to the number of points for a prescribed accuracy of the kernel approximation. In combination with an iterative solver this provides a fast computation of the spline coefficients.
The application of the fast multipole method to spline approximation which also allows the treatment of noisy data requires the choice of a smoothing parameter. We summarize several methods to (ideally automatically) choose this parameter with and without prior knowledge of the noise level.
Martin Gutting

### Inverse Gravimetry as an Ill-Posed Problem in Mathematical Geodesy

Abstract.
The gravitational potential of (a part of) the Earth is assumed to be available, the inverse gravimetry problem is to determine the density contrast function inside (the specified part of) the Earth from known potential values. This paper deals with the characteristic ill-posed features of transferring input gravitational information in the form of Newtonian volume integral values to geological output characteristics of the density contrast function. Some properties of the Newton volume integral are recapitulated. Different methodologies of the resolution of the inverse gravimetry problem and their numerical implementations are examined dependent on the data source. Three cases of input information may be distinguished, namely internal (borehole), terrestrial (surface), and/or external (spaceborne) gravitational data sets. Singular integral theory based inversion of the Newtonian integral equation such as Haar-type solutions are proposed in a multiscale framework to decorrelate specific geological signal signatures with respect to inherently available features. Reproducing kernel Hilbert space regularization techniques are studied (together with their transition to mollified variants) to provide geological contrast density distributions by “downward continuation” from terrestrial and/or spaceborne data. Finally, reproducing kernel Hilbert space solutions are formulated for use of gravimeter data, independent of a specifically chosen input area, i.e., in whole Euclidean space $$\mathbb{R}^3$$
Willi Freeden, M. Zuhair Nashed

### Gravimetry and Exploration

Abstract.
In this work we are especially concerned with the “mathematization” of gravimetric exploration and prospecting. We investigate the extractable information of the Earth’s gravitational potential and its observables obtained by gravimetry for gravitational modeling as well as geological interpretation. More explicitly, local gravimetric data sets are exploited to visualize multiscale reconstruction and decorrelation features to be found in geophysically and geologically relevant signature bands.
C. Blick, W. Freeden, H. Nutz

### Spherical Harmonics Based Special Function Systems and Constructive Approximation Methods

Abstract.
Special function systems are reviewed that reflect particular properties of the Legendre polynomials, such as spherical harmonics, zonal kernels, and Slepian functions. The uncertainty principle is the key to their classification with respect to their localization in space and frequency/momentum. Methods of constructive approximation are outlined such as spherical harmonic and Slepian expansions, spherical spline and wavelet concepts. Regularized Functional Matching Pursuit is described as an approximation technique of combining heterogeneous systems of trial functions to a kind of a ‘best basis’.
Willi Freeden, Volker Michel, Frederik J. Simons

### Spherical Potential Theory: Tools and Applications

Abstract.
In the current chapter, we transfer classical potential theoretic concepts from the Euclidean space $$\mathbb{R}^3$$ to a setting intrinsic on the sphere. We present uniqueness results for the Poisson equation on the sphere, explicitly construct Green functions for spherical caps and complete function systems for harmonic approximation, and elaborate on decompositions of vector fields on the sphere. Among the intended applications are problems from oceanography, geodesy, and geomagnetism. Some examples are presented at the end.
Christian Gerhards

### Joint Inversion of Multiple Observations

Abstract.
Joint inversion becomes increasingly important with the availability of various types of measurements related to the same quantity. Questions arising in this context are how to combine the different data sets in the first place and, secondly, how to choose the multiple parameters that naturally occur in such a combination. This chapter discusses some recently proposed techniques addressing these issues. Additionally, we distinguish the two cases when all underlying problems are ill posed (e.g., satellite data only) and when some of them are not ill posed (e.g., satellite data is complemented by data at the Earth surface). Theoretical discussions of the topics above are presented as well as numerical experiments with different settings of simulated data.
Christian Gerhards, Sergiy Pereverzyev, Pavlo Tkachenko

### On the Non-uniqueness of Gravitational and Magnetic Field Data Inversion (Survey Article)

Abstract.
The gravitational and the magnetic field of the Earth represent some of the most important observables of the geosystem. The inversion of these fields reveals hidden structures and dynamics at the surface or in the interior of the Earth (or other celestial bodies). However, the inversions of both fields suffer from a severe non-uniqueness of the solutions. In this paper, we present a generalized approach which includes the inversion of gravitational and magnetic field data. Amongst others, uniqueness constraints are proposed and compared. This includes the surface density ansatz (also known as the thin layer assumption). We characterize the null space of the considered class of inverse problems via an appropriate orthonormal basis system. Further, we expand the reconstructable part of the solution by means of orthonormal bases and reproducing kernels. One result is that information on the radial dependence of the solution is lost in the observables. As an illustration of the non-uniqueness, we show examples of anomalies which cannot be disclosed from the inversion of gravitational data. This paper is intended to be a theoretical reference work on the inversion of gravitational but also magnetic field data of the Earth.
Sarah Leweke, Volker Michel, Roger Telschow

### Backmatter

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