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Book Review

Willi Freeden, M. Zuhair Nashed (Editors): Handbook of Mathematical Geodesy. Functional Analytic and Potential Theoretic Methods

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Title:

Handbook of Mathematical Geodesy. Functional Analytic and Potential Theoretic Methods

Author:

Willi Freeden, M. Zuhair Nashed (Editors)

Publisher:

Birkhäuser

ISBN:

978-3-319-57181-2

Year:

2018

Price:

€ 139.99 (Hardcover), € 118.99 (eBook)

Details:

1st edition, XIV, 932 pages 155 illustrations, 76 illustrations in color.

The Handbook of Mathematical Geodesy belongs to the Series ’Geosystems Mathematics’ with the Series editors W. Freeden and M. Z. Nashed. It deals with understanding the system Earth and is driven by the public concern about the future of our planet, its climate and its environment. ’Geosystems Mathematics’ tries to advance the cooperation between mathematics and geodisciplines.

The Handbook of Mathematical Geodesy starts with an Introduction by W. Freeden and then presents fifteen different contributions, which will be put forward in the following.

W. Freeden begins his introduction with F. R. Helmerts famous definition that geodesy is the science that deals with the measurement and the modeling of the Earth, including its gravity field. W. Freeden states that the goal of the Handbook of Mathematical Geodesy is twofold: to make the mathematicians aware of the tools, which geodesists are using for analyzing their measurements at the surface of the Earth or in space, and make the geodesists conscious of the abstract methods the mathematicians apply for handling measurements.

The first contribution ’Gauss as Scientific Mediator Between Mathematics and Geodesy from the Past to the Present’ with 163 pages was written by Willi Freeden, Thomas Sonar, and Bertold Witte. The authors begin with taking a look at his life in the introduction, and it is amazing what Gauss has accomplished. Then, they go from the Gaussian circle problem to geosampling, from Gaussian integration to geocubature, from the Gaussian theorem to geoidal determination. Stokes determined the geoid, and Molodensky introduced the height anomaly leading to the telluroid. Gaussian least squares adjustment and the inverse multiscale regularization follow with the graphical illustration of the shape of the Earth, and finally Gaussian geometry and geodetic surveying.

The second contribution to the Handbook of Mathematical Geodesy ’An Overview on Tools from Functional Analysis’ with 35 pages was written by Matthias Agustin, Sarah Eberle, and Martin Grothaus. The authors start with the basic concepts, which are metric spaces, formed spaces, Banach spaces, and linear operators. They move on to the function spaces with properties of convolutions, Hausdorff Measure, and Fourier Transformation. Finally, differential equations and reproducing kernel functions are presented.

The third contribution ’Ill-Posed Problems: Operator Methodologies of Resolution and Regularization,’ with 114 pages was written by Will Freedmen and M. Zuhair Nashed. This is a classical topic in the research of mathematical geodesy. The editors and publishers therefore decided to include this contribution although its content has been extracted from W. Freeden, M. Z. Nashed, Operator Theory and Regularization Approaches to Ill-Posed Problem, GEM Int. J. Geomath., Springer, 2017. The authors present first the solvability of ill-posed operator equations with the pseudoinverse, Tikhonov regularization, least squares problems and generalized inverses, then operator methodologies of resolution, and reconstruction methods and regularizing filters.

The fourth contribution ’Geodetic Observable and Their Mathematical Treatment in Multiscale Framework’ with 144 pages were written by Will Freeden and Helga Nutz. The current state of gravity field determination by gravity measurements, vertical deflections, satellite-to-satellite tracking, and satellite gravity gradiometry is reviewed first. The geodetically relevant Sobolev spaces, the pseudodifferential operators and geodetic nomenclature, the reproducing kernel structures and observational functionals, the ill-posedness of satellite problems, the geodetically oriented wavelet approximation, the bandlimited Runge–Walsh multiscale approximation, Meissl schemata are then introduced together with several illustrations. Appendix defines geodetic notions and explains, how they are observed.

The fifth contribution ’The Analysis of the Geodetic Boundary Value Problem: State and Perspectives’ with 31 pages was written by Fernando Sansò. The author explains in the first three lines of the introduction that the Geodetic Boundary Value Problem (GBVP) determines the figure of the Earth from as many as possible measurements on the gravity field and the least possible knowledge of the geometry. This was acceptable before the space era but is not realistic anymore, as the Global Positioning System (GPS) or the Global Navigation Satellite System (GNSS) determines in any point its coordinates. The geometry could be assumed as known already in 1992, when Koch and Pope proved the uniqueness and existence of the GBVP using the known surface of the Earth. F. Sansò continues by moving from the vector to the scalar GBVP, then to linearizing the scalar GBVP in geometry space, to the analysis of the simple Molodensky problem, and to the analysis of the linearized scalar GBVP. In the conclusions, he admits that the theory of the fixed GBVP requires much weaker conditions to obtain a theorem of existence uniqueness.

The sixth contribution ’Oblique Stochastic Boundary Value Problem’ with 26 pages was written by Martin Grothaus and Thomas Raskop. The aim of the contribution is finding weak solutions to oblique boundary value problems. After citing results for the existence of the deterministic problem, the relevant domains and function spaces are introduced, Poincaré inequality as key issue of the inner problem, fundamental results for the outer problem, and future directions. The authors conclude that they reached the limit for weak solutions under as weak assumptions as possible.

The seventh contribution ’About the Importance of the Runge–Walsh Concept of Gravitational Field Determination’ with 44 pages was written by Matthias Augustin, Willi Freeden, and Helga Nutz. The goal of the contribution is providing the conceptual setup of the Runge–Walsh theorem so that the geodetic expectation and the mathematical justification become transparent. Special function systems are defined, the Runge–Walsh closure theorems, and the Runge–Walsh solution of the geodetic boundary value problem together with several illustrations. The authors conclude that the contribution gives constructive realizations of the Runge–Walsh theorem to solve the boundary value problem.

The eighth contribution ’Geomathematical Advances in Satellite Gravity Gradiometry (SGG)’ with 44 pages is written by Willi Freeden, Helga Nutz, and Michael Schreiner. The Satellite Gravity Gradiometry (SGG) is a tool, from which the fine structure of the Earth’s gravity field can be expected. The authors deal first with potential theoretic aspects, followed by the functional analytic background, SGG as exponentially ill-posed problem, spline inversion, multiscale inversion, and a tree algorithm based on harmonic spline exact approximation. Several illustrations are added.

The ninetieth contribution ’Parameter Choices for Fast Harmonic Spline Approximation’ with 35 pages is written by Martin Gutting. Spherical splines have been developed by W. Freeden and generalized to harmonic splines. After the preliminaries, the multipole methods for splines are presented, the fast multipole method for splines with illustrations, and the parameter choice methods for spline approximation. The author concludes that for highly irregular distribution of the data the spline approach reaches its limit due to ill-conditioning.

The tenth contribution ’Inverse Gravimetry as an Ill-Posed Problem in Mathematical Geodesy’ with 45 pages is written by Willi Freeden and M. Zuhair Nashed. Gravimetric measurements are executed by very sensitive instruments, which determine the variations of the gravity field at the surface of the Earth caused by the density variations inside. This topic is a research area of geodesy and geophysics. To work with inverse gravimetry, the Newton volume integral is presented first, then the ill-posedness of the gravimetry problem, the mollifier methods, and the reproducing kernel Hilbert space methods.

While the preceding contribution dealt with the Inverse Gravimetry as an Ill-Posed Problem, the eleventh presents ’Gravimetry and Exploration’ with 65 pages. It is written by C. Blick, W. Freeden, and H. Nutz. Starting from gravity, gravitation, and gravimetry, where the gravity effect of a salt dome is shown, the surface horizontal/vertical derivatives of the gravity potential are presented, and the interior gravitational potential and density distribution, combined with many illustrations. The authors conclude that the multiscale approach, which is presented, breaks up the signal into a waveband signature with different resolutions.

The twelfth contribution ’Spherical Harmonics Based Special Function Systems and Constructive Approximation Methods’ with 67 pages is written by Willi Freeden, Volker Michel, and Frederic J. Simons. Spherical harmonics are generally used to represent the gravity field of the Earth. However, there is an increasing need for modeling local areas. The authors therefore begin with special function systems on sphere and ball, the spherical uncertainty principle is then introduced, and constructive approximations on the sphere, accompanied by many illustrations. The authors conclude that Slepian functions provide a valuable tool for regionally approximating a signal.

The thirteenth contribution ’Spherical Potential Theory: Tools and Applications’ with 33 pages is written by Christian Gerhards. Classical potential theory in three-dimensional space has been described in the preceding contributions. Now, the sphere does not represent a boundary but an underlying domain on which a problem is formulated. After the fundamental tools, the boundary value problems for the Beltrami operator are formulated, then the spherical decompositions and first-order differential equations, complete function systems, and applications in geoscience, all chapters with illustrations.

The fourteenth contribution ’Joint Inversion of Multiple Observations’ with 28 pages is written by Christian Gerhards, Sergiy Pereverzyev Jr., and Pavlo Tkachenko. Observations containing information about the gravity field of the Earth are now available based on different physical principles like satellite-to-satellite tracking or satellite gravity gradiometry. For combining these observations, their physical properties and their locations have to be considered. The global combination of satellite models is being dealt with first. Global combination of satellite and ground models follow, and then local combination of satellite and ground models, all chapters with illustrations.

The fifteenth contribution ’On the Non-uniqueness of Gravitational and Magnetic Field Data Inversion (Survey Article) with 37 pages, is written by Sarah Leweke, Volker Michel, and Roger Telschov. The inversion of the gravity observations and the magnetic field data detect hidden structures at the surface of the Earth or its interior. However, the inversion suffers from a non-uniqueness. The authors therefore present a generalized approach. They start with preliminaries, then treat the generalization of gravitational and magnetic field inversion, the investigation of the homogeneous problem with several illustrations, and the constraints for the uniqueness of the solution.

The Handbook of Mathematical Geodesy presents for the mathematicians a wealth of applications and for the geodesists a solid embedding of the fundamental concepts of physical geodesy into approximation theory. The book bridges the gap between the abstract work of the mathematicians and the practically oriented measurements of the geodesists. With fifteen contributions the book is broadly planned, and it presents the present state of knowledge. One could have wished to hear about Monte Carlo methods. They are nowadays more and more applied, for instance, for determining the covariance matrix of nonlinearly transformed vectors, which saves computing the derivatives. An index of 12 pages complements the Handbook of Mathematical Geodesy, for which the editors W. Freeden and M. Z. Nashed have to be congratulated publishing it.

Karl-Rudolf Koch