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## Über dieses Buch

Selbstkonsistente Darstellung von Schlüssel- und Transfermethodologien vom Realitätsraum geodätischer Messungen und Beobachtungen in den Modellraum mathematischer Strukturen und Lösungen und zurück, neue Perspektiven und Forschungstrends im Bereich Mathematischer Geodäsie.

## Inhaltsverzeichnis

### 1. Mathematical Geodesy

Its Role, Its Aim, and Its Potential
Abstract
During the last decades, geodesy was influenced by two essential scenarios: First, the technological progress has completely changed the observational and measurement techniques. Modern high-speed computers and satellite-based techniques were more and more entering. Second, there was a growing public concern about the future of our planet, i.e., the change of its climate, the obligations of its environment, and about an expected shortage of its natural resources.
Simultaneously, all these aspects implied and imply the strong need of adequate mathematical structures, tools, and methods, i.e., geomathematics.
This contribution deals with today’s methodological components of the circuit mathematical geodesy characterizing the interrelations between geodesy and geomathematics with respect to origin and foundation, constituting ingredients, scientific role as well as perspective potential.
This introductory contribution represents a collection of known ideas and concepts from different sources in geodetic and geomathematical literature, however, in a new consistent setup and innovatively structured form.
Willi Freeden, Michael Schreiner

### 2. Inverse Probleme der Geodäsie

Ein Abriss mathematischer Lösungsstrategien
Zusammenfassung
Der vorliegende Beitrag beschftigt sich mit mathematischen Lösungspotentialen und Strategien für inverse Probleme in der Geodsie. Die Dilemmata hinsichtlich Existenz, Eindeutigkeit und Stabilität eines Lösungsprozesses werden beschrieben. Die Notwendigkeit zur Regularisierung wird herausgestellt, spezifische Eigenschaften der Regularisierungsverfahren werden kurz skizziert.
Willi Freeden, Helga Nutz

### 3. Up and Down Through the Gravity Field

Abstract
The knowledge of the gravity field has widespread applications in geosciences, in particular in Geodesy and Geophysics. The point of view of the paper is to describe the properties of the propagation of the potential, or of its relevant functionals, while moving upward or downward. The upward propagation is always a properly posed problem, in fact a smoothing and somehow related to the Newton integral and to the solution of boundary value problems (BVP). The downward propagation is always improperly posed, not only due to its intrinsic numerical instability but also because of the non-uniqueness that is created as soon as we penetrate layers of unknown mass density.
So the paper focuses on recent results on the Geodetic Boundary Value Problems on the one side and on the inverse gravimetric problem on the other, trying to highlight the significance of mathematical theory to numerical applications. Hence, on the one hand we examine the application of the BVP theory to the construction of global gravity models, on the other hand the inverse gravimetric problem is studied for layers together with proper regularization techniques.
The level of the mathematics employed in the paper is willingly kept at medium level, often recursing to spherical examples in support to the theory. Most of the material is already present in literature but for a few parts concerning global models and the inverse gravimetric problem for layers.
F. Sansó, M. Capponi, D. Sampietro

### 4. Spherical Harmonics, Splines, and Wavelets

Definitoric Constituents, Strategic Perspectives, Specific Applicability and Applications
Abstract
This contribution substantially represents a geodetically relevant collection of particularly valuable material in the diverse approximation areas involving spherical harmonics, splines, and wavelets, thereby establishing a consistent and unified setup. The goal of the work is to preferably convince members from geodesy that spherically oriented approximation provides a rich mathematical cornucopia that has much to offer to a large palette of applications. Geomathematically it reflects both the approximate shape of the Earth’s surface and the typical satellite geometry of a low Earth orbiter (LEO). Our essential interest is in reconstruction and decomposition characteristics corresponding to different types of data on spheres and various observables naturally occurring in geodetic context, when efficient and economic numerical realizations are required. Another objective is to provide an addition to the library of any individual interested in geodetically reflected local as well as global spherical approximation theory.
Willi Freeden, Michael Schreiner

### 5. A Mathematical View on Spin-Weighted Spherical Harmonics and Their Applications in Geodesy

Abstract
The spin-weighted spherical harmonics (by Newman and Penrose) form an orthonormal basis of L2(Ω) on the unit sphere Ω and have a huge field of applications. Mainly, they are used in quantum mechanics and geophysics for the theory of gravitation and in early universe and classical cosmology. Furthermore, they have also applications in geodesy. The quantity of formulations conditioned this huge spectrum of versatility. Formulations we use are for example given by the Wigner D-function, by a spin raising and spin lowering operator or as a function of spin weight.
We present a unified mathematical theory which implies the collection of already known properties of the spin-weighted spherical harmonics. We recapitulate this in a mathematical way and connect it to the notation of the theory of spherical harmonics. Here, the fact that the spherical harmonics are the spin-weighted spherical harmonics with spin weight zero is useful.
Furthermore, our novel mathematical approach enables us to prove some previously unknown properties. For example, we can formulate new recursion relations and a Christoffel-Darboux formula. Moreover, it is known that the spin-weighted spherical harmonics are the eigenfunctions of a differential operator. In this context, we found Green’s second surface identity for this differential operator and the fact that the spin-weighted spherical harmonics are the only eigenfunctions of this differential operator.
Volker Michel, Katrin Seibert

### 6. Reconstruction and Decomposition of Scalar and Vectorial Potential Fields on the Sphere

A Brief Overview
Abstract
We give a brief overview on approximation methods on the sphere that can be used in a variety of geophysical setups. A particular focus is on methods related to potential field problems and spatial localization, such as spherical splines, multiscale methods, and Slepian functions. Furthermore, we introduce the common Helmholtz and Hardy-Hodge decompositions of spherical vector fields together with some related recent results. The methods are illustrate for two different examples: determination of the disturbing potential from deflections of the vertical and approximation of magnetic fields induced by oceanic tides.
Christian Gerhards, Roger Telschow

### 7. Ellipsoidal-Spheroidal Representation of the Gravity Field

Abstract
We begin with a chapter on motivation, namely why the Earth cannot be a ball due to Earth rotation which we daily experience. In contrast, the Earth’s gravity field is axially symmetric as a first order approximation, not spherically symmetric. The same axially symmetric gravity field applies to all planets and mini-planets, of course the Moon, the Sun and other space objects which intrinsically rotate. The second chapter is therefore devoted to the definition of ellipsoidal-spheroidal coordinate which allow separation of variables. The mixed elliptic-trigonometric elliptic coordinates are generated by the intersection by a family of confocal, oblate spheroids, a family of confocal half hyperboloids and a family of half planes: in this coordinate system {λ, ϕ, u} we inject to forward transformation of spheroidal coordinates into Cartesian coordinates {x, y, z} and the uniquely inverted ones into the backward transformation {x, y, z}→{λ, ϕ, u}. In such a coordinate system we represent the eigenspace of the potential field in terms of the gravitational field being harmonic as well as the centrifugal potential field being anharmonic. Such an eigenspace is being described by normalized associated Legendre functions of first and second kind. The normalization is based on the global area element of the spheroid $$\mathbb {E}_{a,b}^2$$. The third chapter is a short introduction into the Somigliana-Pizetti level ellipsoid in terms of its semi-major axis and its semi-minor axis as well as best estimation of the fundamental Geodetic Parameters {W0, GM, J2, Ω} approximating the Physical Surface of the Planet Earth, namely the Gauss-Listing Geoid. These parameters determine the World Geodetic Datum for a fixed reference epoch. These parameters are called (i) the potential value of the equilibrium figure close to Mean Sea Level, (ii) the gravitational mass, (iii) the second kind, zero order (2, 0) of the gravitational field and finally (iv) the Mean Rotation Speed. These numerical values of the Planet Earth are numerically given. The best estimations of the form parameters derived from two constraints are presented for the Somigliana-Pizzetti Level Ellipsoid. In case of real observations we have to decide whether or not to reduce the constant tide effect. For this reason we have computed the “zero-frequency tidal reference system” and the “tide free reference system” which differ about 40 cm. The radii are {a = 6,378,136.572 m, b = 6,356,751.920 m} for the tide-free Geoid of Reference, but {a = 6,378,136.602 m, b = 6,356,751.860 m} for the zero-frequency tide Geoid of Reference. These results presented in the Datum 2000 differ significantly from the data of the Standard Geodetic Reference System 1980. The geostationary orbit balances the gravitational force and the centrifugal force to zero, the so-called Null Space. Its value of 42,164 km distance from the Earth Center has been calculated in the quasi-spherical referenced coordinate system introduced by T. Krarup. This Null Space evaluates the degree/order term (0, 0) of the gravitational field and the degree/order terms (0, 0) and (2, 0) of the centrifugal field. A careful treatment of the axial symmetric gravity field representing this gravitational and centrifugal field of this degree/order amounts to solve a polynomial equation of order ten. The intersection point of these two forces has been calculated with a lot of efforts! Referring to the Somigliana-Pizzetti Reference Gravity Field we compute in all detail Molodensky heights. In using the World Geodetic Datum 2000 we have presented the Telluroid, telluroid heights and the highlight “Molodensky Heights”. The highlight is our Quasi-geoid Map of East Germany, based on the minimum distance of the Physical Surface of the Earth to the Somigliana-Pizzetti telluroid. We build up the theory of the time-varying gravity field of excitation functions of various types: (i) tidal potential, (ii) loading potential, (iii) centrifugal potential and (iv) transverse stress. The mass density variation in time, namely caused by (i) initial mass density and (ii) the divergence of the time displacement vectors, is represented in terms (i) radial, (ii) spheroidal and (iii) toroidal displacement coefficients in terms of the spherical Love-Shida hypothesis. For the various excitation functions we compute those coefficients.
Erik. W. Grafarend

### 8. Monte Carlo Methods

Abstract
Monte Carlo methods deal with generating random variates from probability density functions in order to estimate unknown parameters or general functions of unknown parameters and to compute their expected values, variances and covariances. One generally works with the multivariate normal distribution due to the central limit theorem. However, if random variables with the normal distribution and random variables with a different distribution are combined, the normal distribution is not valid anymore. The Monte Carlo method is then needed to get the expected values, variances and covariances for the random variables with distributions different from the normal distribution.
The error propagation by the Monte Carlo method is discussed and methods for generating random variates from the multivariate normal distribution and from the multivariate uniform distribution. The Monte Carlo integration is presented leading to the sampling-importance-resampling (SIR) algorithm. Markov Chain Monte Carlo methods provide by the Metropolis algorithm and the Gibbs sampler additional ways of generating random variates. A special topic is the Gibbs sampler for computing and propagating large covariance matrices. This task arises when the geopotential is determined from satellite observations. The example of the minimal detectable outlier shows, how the Monte Carlo method is used to determine the power of a hypothesis test.
Karl-Rudolf Koch

### 9. Parameter Estimation, Variance Components and Statistical Analysis in Errors-in-Variables Models

Abstract
This chapter discusses statistical and numerical aspects of constrained and unconstrained errors-in-variables (EIV) models. The parameters in an EIV model can often be estimated by using three categories of methods: the conventional weighted least squares (LS) method, normed orthogonal regression, and the weighted total least squares (TLS) method. The conventional weighted LS method is of significantly computational advantage but not rigorous statistically. We systematically investigate the effects of random errors in the design matrix on the weighted LS estimates of parameters and variance components, construct the N-calibrated almost unbiased weighted LS estimator of parameters and derive almost unbiased estimates for the variance of unit weight. Although orthogonal regression can be used to estimate the parameters in an EIV model, it is not statistically optimal either. The weighted TLS method is most rigorous and optimal to statistically estimate the parameters in an EIV model at the cost of substantially increasing computation. We reformulate an EIV model as a nonlinear adjustment model without constraints and investigate the statistical effects of nonlinearity on the nonlinear TLS estimate, including the first order approximation of accuracy, nonlinear confidence region and bias of the nonlinear TLS estimate. Closed form solutions to coordinate transformation have been presented as well. Finally, we prove that variance components in an EIV model with the simplest stochastic structure are not estimable.
Peiliang Xu

### 10. Fast Harmonic/Spherical Splines and Parameter Choice Methods

Abstract
Solutions to boundary value problems in geoscience where the boundary is the Earth’s surface can be constructed in terms of harmonic splines. These are localizing trial functions that make use of a reproducing kernel. Splines allow regional modeling or the improvement of a global model in a part of the Earth’s surface.
For certain cases of the reproducing kernels a fast matrix-vector multiplication using the fast multipole method (FMM) is available. The main idea of the fast multipole algorithm consists of two parts: First, the hierarchical decomposition of the three-dimensional computational domain into cubes. Second, an approximation instead of the actual kernel is used for the more distant points which allows to consider many distant points at once. The numerical effort of the matrix-vector multiplication becomes linear in reference to the number of points for a prescribed accuracy of the approximation of the reproducing kernel.
This fast multiplication is used in spline approximation for the solution of the occurring linear systems which also allows the treatment of noisy data requires the choice of a smoothing parameter. Several methods are presented which ideally automatically choose this parameter with and without prior knowledge of the noise level. Using a fast solution algorithm we no longer have access to the whole matrix or its singular values whose computation requires a much larger numerical effort. This situation must be reflected by the parameter choice methods.
Martin Gutting

### 11. Numerical Methods for Solving the Oblique Derivative Boundary Value Problems in Geodesy

Abstract
We present various numerical approaches for solving the oblique derivative boundary value problem. At first, we describe a numerical solution by the boundary element method where the oblique derivative is treated by its decomposition into the normal and tangential components. The derived boundary integral equation is discretized using the collocation technique with linear basis functions. Then we present solution by the finite volume method on and above the Earth’s surface. In this case, the oblique derivative in the boundary condition is treated in three different ways, namely (i) by an approach where the oblique derivative is decomposed into normal and two tangential components which are then approximated by means of numerical solution values (ii) by an approach based on the first order upwind scheme; and finally (iii) by a method for constructing non-uniform hexahedron 3D grids above the Earth’s surface and the higher order upwind scheme. Every of proposed approaches is tested by the so-called experimental order of convergence. Numerical experiments on synthetic data aim to demonstrate their efficiency.
Róbert Čunderlík, Marek Macák, Matej Medl’a, Karol Mikula, Zuzana Minarechová

### 12. Geodetic Methods for Monitoring Crustal Motion and Deformation

Abstract
The use of geodetic data for crustal deformation studies is studied for the two possible cases: (a) The comparison of shape at two epochs for which station coordinates are available in order to compute, at any desired point, invariant planar deformation parameters (strain parameters), such as principal strains, principal elongations, dilatation and shear. (b) The utilization of coordinates and velocities at a particular epoch for the computation of the time derivatives of the deformation parameters (strain rate parameters). The classical approximate “infinitesimal” theory is presented as well as the widely used finite element method with triangular elements for the interpolation of station displacements and velocities. In addition, a new completely rigorous planar deformation theory, based on the singular value decomposition of the deformation gradient matrix, is presented for both strain and strain rate parameter computation. The invariance characteristics of all the above deformation parameters, under changes of the involved reference systems, are studied, from a purely geodetic point of view different from that in classical mechanics. Emphasis is given to the separation of rigid motion of independent tectonic regions from their internal deformation, utilizing the concept of a discrete Tisserand reference system that best fits the geodetic subnetwork covering the relevant region. Interpolation of displacements or velocities using stochastic minimum mean square error prediction (known as collocation or kriging) is also examined with emphasis on how it can become statistically relevant and rigorous based on sample covariance and cross-covariance functions. It is also shown how the planar deformation can be adapted to the study of surface deformation, with applications to the study of shell-like constructions in geodetic engineering and the deformation of the physical surface of the earth. The most important application presented, is the study of horizontal deformation on the surface of the reference ellipsoid, utilizing either differences of geodetic coordinates between two epochs, or the horizontal components of station velocities. Finally, it is also shown how the rigorous theory of planar deformation can be extended to the three-dimensional case.
Athanasios Dermanis

### 13. Theory and Realization of Reference Systems

Abstract
After a short introduction on the basics of reference system theory and its application for the description of earth rotation, the problem of establishing a reference system for the discrete stations of a geodetic network is studied, from both a theoretical and a practical – implementation point of view.
First the case of rigid networks is examined, which covers also the case of deformable networks with data collected within a time span, small enough for the network shape to remain practically unaltered. The problem of how to analyze observations, which are invariant under particular changes of the reference system, is examined within the framework of least squares estimation theory, with a rank deficiency in the design matrix. The complete theory is presented, including all necessary proofs. Not only of the usual statistical results for the rank deficient linear Gauss-Markov model, but also those of the rich geodetic theory are presented, based on the fact that the physical cause of the rank deficiency is known to be the lack of definition of the reference system. The additional geodetic results are based on the fact that one can easily construct a matrix with columns that are a basis of the null space of the design matrix. Insights are presented into the geometric characteristics of the problem and its relation to the theory of generalized inverses. Passing into deformable networks, a deterministic mathematical model is presented, based of the concept of geodesic lines which are the shortest between linear shape manifolds, associated with the network shape at each instant. Reference system optimality for a discrete network is related to the relevant ideas of Tisserand, developed for the continuum of the earth masses.
The practical problem of choosing a reference system for a coordinate time series is examined, for the case where a linear-in-time model is adopted for the temporal variation of coordinates. The choice of reference system is related to the choice of minimal constraints for obtaining one out of the infinitely many least squares solutions, corresponding to descriptions in different reference systems of the same sequence of network shapes. The a-posteriori change of the reference system is examined, where one moves from one least squares solution to another one, satisfying particular minimal constraints. Kinematic minimal constraints are also introduced, leading to coordinates that demonstrate the minimum coordinate variation and are thus connected to the ideas of Tisserand for reference system optimality. It is also shown how to convert a reference system of a geodetic network to one for the whole continuous earth, or at least the lithosphere, utilizing additional geophysical information.
The last item is the combination of data from four space techniques (VLBI, SLR, GPS, DORIS) in order to establish a global reference system realized though a number of parameters that constitute the International Terrestrial Reference Frame. After a theoretical exposition of the basics of data combination, the various methods of spatial data combination are presented, for both coordinate and Earth Orientation Parameter time series, while alternatives are presented for the choice of the origin (geocenter) and the network scale from the scale of VLBI and SLR. Finally, existing and new methodologies are presented for building post linear models, describing the temporal variation of station coordinates.
Athanasios Dermanis

### 14. From Gaussian Least Squares Approximation to Today’s Operator-Theoretic Regularization of Ill-Posed Problems

Abstract
The aim of this contribution is to document the pioneer dimension of Gauss’s method of least squares approximation and to demonstrate its mediation role to today’s regularization processes of pseudoinverses in ill-posed inverse problems.
Willi Freeden, Bertold Witte

### 15. The Numerical Treatment of Covariance Stationary Processes in Least Squares Collocation

Abstract
Digital sensors provide long series of equispaced and often strongly correlated measurements. A rigorous treatment of this huge set of correlated measurements in a collocation approach is a big challenge. Standard procedures – applied in a thoughtless brute force approach – fail because these techniques are not suitable to handle such huge systems.
In this article two different strategies, denoted as covariance approach and filter approach, to handle such huge systems are contrasted. In the covariance approach various decorrelation strategies based on different Cholesky approaches to factorize the variance/covariance matrices are reviewed. The focus is on arbitrary distributed data sets with a finite number of data. But also extensions to sparse systems resulting from finite covariance functions and on exploiting the Toeplitz structure which results in the case of equispaced systems are elaborated.
Apart from that filter approaches are discussed to perform a prewhitening strategy for the data and rearrange the whole model to work with this filtered data in a rigorous way. Here, the special focus is on autoregressive processes to model the correlations. Finite, causal, non-recursive filters are constructed as prewhitening filters for the data as well as the model. This approach is extreme efficient, but can only deal with infinite equispaced data sets.
In real data scenarios, finite sequences and data gaps must be handled as well. For the covariance approach this is straightforward but it is a serious problem for the filter approach. Therefore a combination of these approaches is constructed to select the best properties from each. Covariance matrices of equispaced data sets designed by recursively defined covariance sequences are represented by AR processes as well as by Cholesky factorized matrices. It is shown, that it is possible to switch between both strategies to get data gaps and the warm up phase for the filter approach under control.
Wolf-Dieter Schuh, Jan Martin Brockmann

### 16. Inverse Gravimetry: Density Signatures from Gravitational Potential Data

Abstract
This paper represents an extended version of the publications Freeden W. (2015) Geomathematics: its role, its aim, and its potential. In: Freeden W., Nashed M.Z., Sonar T. (Hrsg.) Handbook of Geomathematics, Bd. 1, 2. Aufl., S. 3–78. Springer, New York/Heidelberg, “Handbook of Geomathematics”, Springer, 2015, Freeden W., Nashed M.Z. (2018) Inverse gravimetry as an ill-posed problem in mathematical geodesy. In: Freeden W., Nashed M.Z. (Hrsg.) Handbook of Mathematical Geodesy. Geosystems Mathematics, S. 641–685. Birkhäuser/Springer, Basel/New York/Heidelberg “Handbook of Mathematical Geodesy”, Birkhäuser, International Springer Publishing, 2018, and, in particular, Freeden W., Nashed M.Z. GEM Int. J. Geomath. 9, 199–264, 2018, from which all the theoretical framework is taken over to this work. The aim of the paper is to deal with the ill-posed problem of transferring input gravitational potential information in the form of Newtonian volume integral values to geological output characteristics of the density contrast function.
Some essential properties of the Newton volume integral are recapitulated. Different methodologies of the resolution of the inverse gravimetry problem and their numerical implementations are examined including their dependence on the data source. Three types 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 a Haar-type solution is handled in a multiscale framework to decorrelate specific geological signal signatures with respect to inherently given features. Reproducing kernel Hilbert space regularization techniques are studied (together with their transition to certain mollified variants) to provide geological contrast density distributions by “downward continuation” from terrestrial and/or spaceborne data. Numerically, reproducing kernel Hilbert space spline solutions are formulated in terms of Gaussian approximating sums for use of gravimeter data systems.
Willi Freeden, M. Zuhair Nashed

### 17. From Newton’s Law of Gravitation to Multiscale Geoidal Determination and Ocean Circulation Modeling

Abstract
The objective of this contribution is the documentation of the pioneer dimension of Newton’s work to demonstrate his mediating role between classical gravitational theory and today’s multiscale concepts of geoidal determination and ocean circulation modeling.
Willi Freeden, Helga Nutz

### 18. Gravimetric Measurements, Gravity Anomalies, Geoid, Quasigeoid: Theoretical Background and Multiscale Modeling

Abstract
The methodical aspects of gravimetry are investigated from observational as well as mathematical/physical point of view. Local gravimetric data sets are exploited to visualize multiscale features in geophysically relevant signature bands of gravity anomalies and quasigeoidal heights. Wavelet decorrelation is illustrated for a certain area of Rhineland-Palatinate.
Gerhard Berg, Christian Blick, Matthias Cieslack, Willi Freeden, Zita Hauler, Helga Nutz

### Methods for Satellite and Space Techniques

#### Frontmatter

Abstract
Satellite Gravitational Gradiometry (SGG) is an observational technique of globally establishing the fine structure and the characteristics of the external Earth’s gravitational field. The “Gravity field and steady-state Ocean Circulation Explorer” GOCE (2009–2013) was the first satellite of ESA’s satellite program intended to realize the principle of SGG and to deliver useful SGG-data sets. In fact, GOCE was capable to provide suitable data material of homogeneous quality and high data density.
Mathematically, SGG demands the determination of the gravitational potential in the exterior of the Earth including its surface from given data of the gravitational Hesse tensor along the satellite orbit. For purposes of modeling we are led to invert the “upward continuation”-operator resulting from the Abel–Poisson integral formula of potential theory. This approach requires the solution of a tensorial Fredholm integral equation of the first kind relating the desired Earth’s gravitational potential to the measured orbital gravitational gradient acceleration. The integral equation constitutes an exponentially ill-posed problem of the theory of inverse problems, which inevitably needs two regularization processes, namely “downward continuation” and (weak or strong) “error regularization” in the case of noisy data.
This contribution deals with two different SGG-multiscale regularization methods, one in space domain and the other in frequency domain. Both procedures provide the gravitational potential as derived from tensorial SGG-data along the satellite orbit on the real Earth’s surface as required from the view point of geodesy.
Willi Freeden, Helga Nutz, Reiner Rummel, Michael Schreiner

### 20. Very Long Baseline Interferometry

Abstract
This chapter describes the theory and the individual operational steps and components needed to carry out geodetic and astrometric Very Long Baseline Interferometry (VLBI) measurements. Pairs of radio telescopes are employed to observe far distant compact radio galaxies for the determination of the differences of the arrival times at the telescopes. From multiple observations of time delays of different radio sources, geodetic parameters of interest such as telescope coordinates, Earth orientation parameters, and radio source positions are inferred. The VLBI operation’s scheme generally consists of scheduling, observing session, correlation, and data analysis.
Axel Nothnagel

### 21. Elementary Mathematical Models for GNSS Positioning

Abstract
In 1984, the author got the opportunity to visit the US National Geodetic Survey near Washington, D.C., and, guided by Dr. Benjamin W. Remondi, could contribute to the development of civilian software for processing data of the primarily military Global Positioning System (GPS), the US version of a Global Navigation Satellite System (GNSS). In parallel, the former Soviet Union developed its own military Global Navigation Satellite System (GLONASS). In these early days of development, the two systems were far from completion which occurred for GPS in late 1995 and for GLONASS in early 1996.
After his return, the author was not only impressed by the incredibly innovative potential of satellite-based navigation, but also decided to “simply write a book on GPS”. The reader might ask why GLONASS was not an issue; the answer is that in those days it was difficult to get official information on the Russian system. Thus, the intention was realized and “GPS – theory and practice”, coauthored by Herbert Lichtenegger and Jim Collins, was published by Springer, Wien New York in 1992. Nine years later, by 2001, the fifth edition was released.
In the early years of the new millenium, additional systems were either concepted or realized, e.g., the Chinese BeiDou system as a two-step approach with the first step as a regional and the second step as a global system and the European Galileo, a remarkable development of a system under civilian control. The Galileo definition phase was completed in 2003. These new developments made necessary a complete revision of the “GPS-book”; consequently, the author together with the coauthors Herbert Lichtenegger and Elmar Wasle published “GNSS – GPS, GLONASS, Galileo & more” again by Springer Wien New York in 2008.
Now the story approaches its end; when the author got the invitation to contribute to the Springer Handbook of Mathematical Geodesy the mathematical models for GNSS positioning, he argued that on the one hand that he had published this issue several years ago and on the other hand that essentially the same topics and much more is contained in the just recently released Springer Handbook of Global Navigation Satellite Systems. Nevertheless, the editors of this handbook claimed that “this is just what we want and need”. Therefore, the author felt honored and contacted the Publishing Company Springer and asked if it is possible to republish parts of his book on GNSS in this Springer Handbook of Mathematical Geodesy volume mainly unchanged but updated where requested and supplemented by measurement examples – and he got the permission. Thus, essentially, this contribution is extracted from “GNSS - GPS, GLONASS, Galileo & more”. Due to the fact that these chapters were originally written by the author only and the coauthors did not contribute to them, the names of the coauthors have been omitted for the current publication.
Since almost a decade has passed by with respect to the release of the GNSS book, much more experience has been gained by applying these mathematical models. This enables the author to spoil the readers with numerical examples for some of the models. These examples should justify the re-publication of the well-established models. The measurements, the processing of the data and the production of the respective figures were carried out by Mathias Duregger, a young and very talented student assistant at our institute at the Graz University of Technology. His very industrious support is gratefully acknowledged.
Bernhard Hofmann-Wellenhof

### Backmatter

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