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Inverse Gravimetry: Density Signatures from Gravitational Potential Data

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Handbuch der Geodäsie

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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. (eds) Handbook of Geomathematics, 2nd edn, vol 1, 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. (eds) Handbook of Mathematical Geodesy. Geosystems Mathematics, 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. (2018) GEM Int J Geomath. https://doi.org/10.1007/s13137-018-0103-5 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.

Zusammenfassung

Diese Arbeit stellt eine erweiterte Version der Publikationen Freeden W. (2015) Geomathematics: its role, its aim, and its potential. In: Freeden W., Nashed M.Z., Sonar T. (eds) Handbook of Geomathematics, 2nd edn, vol 1, 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. (eds) Handbook of Mathematical Geodesy. Geosystems Mathematics, 641–685. Birkhäuser/Springer, Basel/New-York/Heidelberg ,,Handbook of Mathematical Geodesy“, Birkhäuser, International Springer Publishing, 2018, und, insbesondere, (Freeden W., Nashed M.Z. (2018) GEM Int J Geomath. https://doi.org/10.1007/s13137-018-0103-5) dar, deren theoretisches Gerüst vollständig übernommen wurde. Ziel der Arbeit ist die Beschäftigung mit dem schlecht-gestellten Problem, gravitative Potential-Input-Information in Form von Newtonschen Volumenintegralwerten in geologische Output-Charakteristika der Dichtekontrastfunktion umzuwandeln.

Einige wesentliche Eigenschaften des Newtonschen Volumenintegrals werden rekapituliert. Verschiedene Methoden zur Lösung des inversen Gravimetrieproblems sowie numerische Implementierungen werden abhängig von der Datenquelle untersucht. Drei Typen von Inputdaten-Information lassen sich unterscheiden, nämlich innere (Bohrloch), terrestrische (Erdoberfläche), und/oder äußere (Raum) Gravitationsdatensätze. Inversion der Newtonschen Integralgleichnung basierend auf singuärer Integrationstheorie, wie etwa Haar-Typ Inversion, wird in einem Multiskalengefüge zur Dekorrelation spezifischer geologischer Signalsignaturen mit inhärent gegebenen Merkmalen behandelt. Reproduzierende Kern-Hilbertraum-Regulasierungstechniken werden (zusammen mit ihrem Übergang zu bestimmten Mollifier-Varianten) studiert, um geologische Kontrastdichteverteilungen durch ,,Fortsetzen nach unten“ aus terrestrischen und/oder räumlichen Daten zu erhalten. Für die Anwendung auf Gravimeterdatensysteme werden reproduzierende Hilbertraum-Kern-Spline-Lösungen numerisch in Form Gaußscher approximierender Summen formuliert.

This contribution is based on the article: Freeden, W. & Nashed, M.Z. Int J Geomath (2018). https://doi.org/10.1007/s13137-018-0103-5

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

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Acknowledgements

The first author thanks the “Federal Ministry for Economic Affairs and Energy, Berlin” and the “Project Management Jülich” (corporate manager Dr. S. Schreiber) for funding the projects “GEOFÜND” (funding reference number: 0325512A, PI Prof. Dr. W. Freeden, University of Kaiserslautern, Germany) and “SPE” (funding reference number: 0324061, PI Prof. Dr. W. Freeden, CBM – Gesellschaft für Consulting, Business und Management mbH, Bexbach, Germany, corporate manager Prof. Dr. mult. M. Bauer).

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

    Willi Freeden

  2. Department of Mathematics, University of Central Florida, Orlando, FL, USA

    M. Zuhair Nashed

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  1. Willi Freeden
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  2. M. Zuhair Nashed
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Correspondence to Willi Freeden .

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  1. Lehr- und Forschungsgebiet "Geomathemati, TU Kaiserslautern, Kaiserslautern, Germany

    Willi Freeden

  2. Physikalische Geodäsie, TU München Inst. Astronomische und, München, Germany

    Reiner Rummel

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

    Willi Freeden

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Freeden, W., Nashed, M.Z. (2018). Inverse Gravimetry: Density Signatures from Gravitational Potential Data. In: Freeden, W., Rummel, R. (eds) Handbuch der Geodäsie. Springer Reference Naturwissenschaften . Springer Spektrum, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46900-2_96-1

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