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Local Multiscale Modelling of Geoid Undulations from Deflections of the Vertical

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Abstract

This paper deals with the problem of determining a scalar spherical field from its surface gradient, i.e., the modelling of geoid undulations from deflections of the vertical. Essential tools are integral formulae on the sphere based on Green’s function of the Beltrami operator. The determination of geoid undulations from deflections of the vertical is formulated as multiscale procedure involving scale-dependent regularized versions of the surface gradient of Green’s function. An advantage of the presented approach is that the multiscale method is based on locally supported wavelets. In consequence, local modelling of geoid undulations are calculable from locally available deflections of the vertical

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References

  • Bayer M, Beth S, Freeden W (1998) Geophysical field modelling by multiresolution analysis. Acta Geod et Geophys Hungarica 33:289–319

    Google Scholar 

  • Beth S (2000) Multiscale approximation by vector radial basis functions on the sphere. PhD thesis, Geomathematics Group, University of Kaiserslautern, Shaker, Aachen

    Google Scholar 

  • Bruns EH (1878) Die Figur der Erde. Publikation Königl. Preussisch. Geodätisches Institut. P. Stankiewicz uchdruckerei, Berlin

  • Featherstone WE, Rüeger JM (2000) The importance of using deviations of the vertical for the reduction of survey data to a geocentric datum. Aust Surveyor 45(2):46–61 (Erratum: Austr Surveyor 47(1):7)

    Google Scholar 

  • Freeden W (1979) Über eine Klasse von Integralformeln der Mathematischen Geodäsie. Habilitation Thesis, Veröffentlichung des Geodätischen Instituts der RWTH Aachen, No. 27

  • Freeden W, Michel V (2004) Multiscale potential theory (with applications to geoscience). Birkhäuser, Basel

    Google Scholar 

  • Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere (with applications to geomathematics). Oxford Science Publications, Clarendon Press, Oxford

    Google Scholar 

  • Freeden W, Maier T (2002) On multiscale denoising of spherical functions: basic theory and numerical aspects. Electr Trans Numer Analysis (ETNA) 14:40–62

    Google Scholar 

  • Freeden W, Maier T (2003) Spectral and multiscale signal-to-noise thresholding of spherical vector fields. Comput Geosci 7(3):215–250

    Article  Google Scholar 

  • Freeden W, Schreiner M (2005a) Multiresolution analysis by spherical up functions. Constr Approx (in press)

  • Freeden W, Schreiner M (2005b) Spaceborne gravitational field determination by means of locally supported wavelets. J Geod 79:431–446

    Article  Google Scholar 

  • Fubini G (1958) Sugli integrali multipli. Opere scelte 2:243–249

    Google Scholar 

  • Grafarend EW (2001) The spherical horizontal and spherical vertical boundary value problem – vertical deflections and geoidal undulations – the completed Meissl diagram. J Geod 75:363–390

    Article  Google Scholar 

  • Groten E (1979) Geodesy and the Earth’s gravity field I, II. Dümmler, Bonn

    Google Scholar 

  • Groten E (1981) Local and global gravity field representation. Rev Geophys Space Phys 19:407–414

    Article  Google Scholar 

  • Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San Francisco

    Google Scholar 

  • Jekeli C (1999) An analysis of vertical deflections derived from high-degree spherical harmonic models. J Geod 73(1):10–22

    Article  Google Scholar 

  • Meissl P (1971a) A study of covariance functions related to the Earth’s disturbing potential. Report 151, Department of Geodetic Science, The Ohio State University Columbus

  • Meissl, P (1971b) On the linearization of the geodetic boundary value problem. Report 152, Department of Geodetic Science, The Ohio State University Columbus

  • Nutz H (2001) A unified setup of gravitational field observables. PhD Thesis, Geomathematics Group, University of Kaiserslautern, Shaker, Aachen

    Google Scholar 

  • Pizzetti P (1910) Sopra il calcoba tesrico delle deviazioni del geoide dall’ ellissoide. Att R Acad Sci Torino 46:331–350

    Google Scholar 

  • Rummel R (1992) Fysische Geodesie I, II. Collegediktaat, Faculty of Geodesy, Delft University of Technology, Delft

    Google Scholar 

  • Rummel R, van Gelderen M (1992) Spectral analysis of the full gravity tensor. Geophys J Int 111:159–169

    Article  Google Scholar 

  • Torge W (1991) Geodesy. de Gruyter, Berlin

    Google Scholar 

  • Stokes GG (1849) On the variation of gravity at the surface of the Earth. Trans Cambridge Phil Soc 8:672–712. In: Mathematical and physical papers by George Gabriel Stokes, vol II. Johnson Reprint Corporation, New York, pp 131–171

    Google Scholar 

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Authors and Affiliations

  1. Geomathematics Group, University of Kaiserslautern, 67653, Kaiserslautern, P.O. Box 3049, Germany

    W. Freeden

  2. Laboratory for Industrial Mathematics, University of Buchs, Werdenbergstrasse 4, Buchs, 9471, Switzerland

    M. Schreiner

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  1. W. Freeden
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  2. M. Schreiner
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Correspondence to W. Freeden.

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Freeden, W., Schreiner, M. Local Multiscale Modelling of Geoid Undulations from Deflections of the Vertical. J Geodesy 79, 641–651 (2006). https://doi.org/10.1007/s00190-005-0017-5

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  • DOI: https://doi.org/10.1007/s00190-005-0017-5

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