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Spherical Harmonics, Splines, and Wavelets

Definitoric Constituents, Strategic Perspectives, Specific Applicability and Applications

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Mathematische Geodäsie/Mathematical Geodesy

Part of the book series: Springer Reference Naturwissenschaften ((SRN))

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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.

Zusammenfassung

Dieser Beitrag stellt eine geodätisch relevante Sammlung von besonders wertvollem Material in den diversen Approximationsgebieten dar, die mit Kugelfunktionen, Splines und Wavelets involviert sind, und zwar in einem konsistenten und vereinheitlichtem Gefüge. Das Ziel der Arbeit besteht darin vorzugsweise Geodäten zu überzeugen, dass sphärisch orientierte Approximation ein reiches mathematisches Füllhorn bereitstellt, welches viel für eine breite Palette von Anwendungen zu bieten hat. Geomathematisch spiegelt es sowohl die approximative Erdfigur als auch die typische Satellitengeometrie eines tief fliegenden Erdorbiters wider. Unser wesentliches Interesse liegt in den Charakteristiken der Rekonstruktion und Dekomposition der verschiedenen Datentypen auf Sphären und der natürlich im geodätischen Kontext auftretenden mannigfaltigen Observablen, soweit effiziente und ökonomische nummerische Realisationen gefordert sind. Ein weiteres Anliegen ist, eine Zusatzbibliothek für Interessenten in lokal sowie global geprägter sphärischer Approximationstheorie verfügbar zu machen.

This chapter is part of the series Handbuch der Geodäsie, volume “Mathematische Geodäsie/Mathematical Geodesy”, edited by Willi Freeden, Kaiserslautern.

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Acknowledgements

This contribution represents a synopsis of ideas and concepts presented in the textbook “Spherical Sampling”, Geosystem Mathematics, Birkhäuser, Basel [90]. For more mathematical details and algorithmic aspects the interested reader is referred to the contents of this work.

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  1. Geomathematics Group, Mathematics Department, University of Kaiserslautern, Kaiserslautern, Germany

    Willi Freeden

  2. Institute for Computational Engineering, University of Applied Sciences of Technology NTB, Buchs, Switzerland

    Michael Schreiner

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Correspondence to Willi Freeden .

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    Willi Freeden

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Freeden, W., Schreiner, M. (2020). Spherical Harmonics, Splines, and Wavelets. In: Freeden, W. (eds) Mathematische Geodäsie/Mathematical Geodesy. Springer Reference Naturwissenschaften . Springer Spektrum, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55854-6_101

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