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An SVD in Spherical Surface Wave Tomography

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New Trends in Parameter Identification for Mathematical Models

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Abstract

In spherical surface wave tomography, one measures the integrals of a function defined on the sphere along great circle arcs. This forms a generalization of the Funk–Radon transform, which assigns to a function its integrals along full great circles. We show a singular value decomposition (SVD) for the surface wave tomography provided we have full data.

Since the inversion problem is overdetermined, we consider some special cases in which we only know the integrals along certain arcs. For the case of great circle arcs with fixed opening angle, we also obtain an SVD that implies the injectivity, generalizing a previous result for half circles in Groemer (Monatsh Math 126(2):117–124, 1998). Furthermore, we derive a numerical algorithm based on the SVD and illustrate its merchantability by numerical tests.

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Acknowledgements

The authors thank Volker Michel for pointing out the problem of spherical surface wave tomography at the Mecklenburg Workshop on Approximation Methods and Data Analysis 2016 and for fruitful conversations later on. Furthermore, we thank the anonymous reviewer for providing helpful comments and suggestions to improve this article.

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Authors and Affiliations

  1. Faculty of Mathematics, Chemnitz University of Technology, Reichenhainer Straße 39, 09126, Chemnitz, Germany

    Ralf Hielscher, Daniel Potts & Michael Quellmalz

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  1. Ralf Hielscher
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  2. Daniel Potts
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  3. Michael Quellmalz
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Correspondence to Michael Quellmalz .

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Editors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Chemnitz, Chemnitz, Germany

    Bernd Hofmann

  2. Department of Mathematics, Federal University of Santa Catarin, Florianopolis, Brazil

    Antonio Leitão

  3. Pura e Aplicada, Instituto Nacional de Matematica, Rio de Janeiro, Rio de Janeiro, Brazil

    Jorge P. Zubelli

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Hielscher, R., Potts, D., Quellmalz, M. (2018). An SVD in Spherical Surface Wave Tomography. In: Hofmann, B., Leitão, A., Zubelli, J. (eds) New Trends in Parameter Identification for Mathematical Models. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-70824-9_7

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