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Reconstruction and Decomposition of Scalar and Vectorial Potential Fields on the Sphere

A Brief Overview
  • Christian GerhardsEmail author
  • Roger Telschow
Living reference work entry
Part of the Springer Reference Naturwissenschaften book series (SRN)

Zusammenfassung

Wir geben einen kurzen Überblick über Approximationsmethoden auf der Sphäre, welche Anwendung in verschiedenen geophysikalischen Fragestellungen finden. Im Speziellen geht es um Methoden mit Bezug zu Potentialfeldpro- blemen und Lokalisierung auf der Sphäre (z. B. Splines, Multiskalenmethoden und Slepian Funktionen). Des Weiteren führen wir zwei bekannte Vektorfeldzerlegungen (Helmholtz und Hardy-Hodge) ein und stellen die Verbindung zu einigen neueren Resultaten her. Abschliessend illustrieren wir unsere Ansätze an zwei geophysikalischen Beispielen: der Bestimmung des Störpotentials aus Lotabweichungen und der Approximation des Magnetfelds, welches durch Ozeangezeiten erzeugt wird.

Keywords

Approximation on the sphere Spatial localization on the sphere Spherical multiscale expansions Spherical function systems Spherical vector field decompositions Potential theory on the sphere 

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.

Notes

Acknowledgements

This work was partly supported by DFG grant GE 2781/1-1.

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Copyright information

© Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Computational Science CenterUniversity of ViennaViennaAustria

Section editors and affiliations

  • Willi Freeden
    • 1
  1. 1.Geomathematics GroupUniversity of KaiserslauternKaiserslauternGermany

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