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Far-zone gravity field contributions corrected for the effect of topography by means of molodensky’s truncation coefficients

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Abstract

A spectral representation of the topographic corrections to gravity field quantities is formulated in terms of spherical height functions. When computing the far-zone contributions to the topographic corrections, various types of the truncation coefficients are applied to a spectral representation of Newton’s integral. In this study we utilise Molodensky’s truncation coefficients in deriving the expressions for the far-zone contributions to topographic corrections. The expressions for computing the far-zone gravity field contributions corrected for the effect of topography are then obtained by combining the expressions for the far-zone contributions to the gravity field quantities and to the respective topographic corrections, both expressed in terms of Molodensky’s truncation coefficients. The numerical examples of the far-zone contributions to the topographic corrections and to the topography-corrected gravity field quantities are given over the study area situated in the Canadian Rocky Mountains with adjacent planes. Coefficients of the global elevation and geopotential models are used as the input data.

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References

  • Abramowitz M. and Stegun I.A., 1972. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York.

    Google Scholar 

  • Eshagh M. and Sjöberg L.E., 2008. Impact of topographic and atmospheric masses over Iran on validation and inversion of GOCE gradiometric data. J. Earth Space. Phys., 34, 15–30.

    Google Scholar 

  • Eshagh M. and Sjöberg L.E., 2009. Topographic and atmospheric effects on GOCE gradiometric data in a local north-oriented frame: A case study in Fennoscandia and Iran. Stud. Geophys. Geod., 53, 61–80.

    Article  Google Scholar 

  • Evans J.D. and Featherstone W.E., 2001. Improved convergence rates for the truncation error in gravimetric geoid determination. J. Geodesy, 74, 239–248.

    Article  Google Scholar 

  • Hagiwara Y., 1975. A new formula for evaluating the truncation error coefficient. J. Geodesy, 50, 131–135; DOI: 10.1007/BF02522312.

    Google Scholar 

  • Heck B., 2003. On Helmert’s methods of condensation. J. Geodesy, 7, 155–170.

    Article  Google Scholar 

  • Heiskanen W.H. and Moritz H., 1967. Physical Geodesy. W.H. Freeman and Co., San Francisco.

    Google Scholar 

  • Hinze W.J., 2003. Bouguer reduction density, why 2.67? Geophysics, 68, 1559–1560.

    Article  Google Scholar 

  • Hirvonen R.A. and Moritz H., 1963. Practical Computation of Gravity at High Altitudes. Report 27, Inst. Geod. Photogram. and Cartogr., Ohio State University, Columbus.

    Google Scholar 

  • Hobson E.W., 1931. The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, Cambridge, U.K.

    Google Scholar 

  • Huang J., 2002. Computational Methods for the Discrete Downward Continuation of the Earth Gravity and Effects of Lateral Topographical Mass Density Variation on Gravity and the Geoid. Technical Report 216, Dept. of Geodesy and Geomatics Engineering, University of New Brunswick, Fredericton, Canada.

    Google Scholar 

  • Lemoine F.G., Kenyon S.C., Factor J.K., Trimmer R.G., Pavlis N.K., Chinn D.S., Cox C.M., Klosko S.M., Luthcke S.B., Torrence M.H., Wang Y.M., Williamson R.G., Pavlis E.C., Rapp R.H. and Olson T.R., 1998. The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96. NASA Technical Report TP-1998-206861, National Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt, Maryland, USA.

    Google Scholar 

  • Makhloof A.A., 2007. The Use of Topographic-Isostatic Mass Information in Geodetic Application. Dissertation D98, Institute of Geodesy and Geoinformation, Bonn, Germany.

    Google Scholar 

  • Martinec Z., Matyska C., Grafarend E.W. and Vaníček P., 1993. On Helmert’s 2nd condensation method. Manuscr. Geod., 18, 417–421.

    Google Scholar 

  • Martinec Z. and Vaníček P., 1994. Direct topographical effect of Helmert’s condensation for a spherical geoid. Manuscr. Geod., 19, 257–268.

    Google Scholar 

  • Martinec Z., 1996. Stability investigations of a discrete downward continuation problem for geoid determination in the Canadian Rocky Mountains. J. Geodesy, 70, 805–828.

    Google Scholar 

  • Molodensky M.S., Yeremeev V.F. and Yurkina M.I, 1960. Methods for Study of the External Gravitational Field and Figure of the Earth. TRUDY Ts NIIGAiK, 131, Geodezizdat, Moscow (English translat.: Israel Program for Scientific Translation, Jerusalem 1962).

  • Moritz H., 1980. Advanced Physical Geodesy. Herbert Wichmann Verlag, Karlsruhe, Germany.

    Google Scholar 

  • Novák P., 2000. Evaluation of Gravity Data for the Stokes-Helmert Solution to the Geodetic Boundary-Value Problem. Technical Report 207, Dept. of Geodesy and Geomatics Engineering, University of New Brunswick, Fredericton, Canada.

    Google Scholar 

  • Novák P., Vaníček P., Martinec Z. and Veronneau M., 2001. Effects of the spherical terrain on gravity and the geoid. J. Geodesy, 75, 491–504.

    Article  Google Scholar 

  • Novák P. and Grafarend E.W., 2005. The ellipsoidal representation of the topographical potential and its vertical gradient. J. Geodesy, 78, 691–706.

    Article  Google Scholar 

  • Novák P. and Grafarend E.W., 2006. The effect of topographical and atmospheric masses on spaceborne gravimetric and gradiometric data. Stud. Geophys. Geod., 50, 549–582; DOI: 10.1007/s11200-006-0035-7.

    Article  Google Scholar 

  • Novák P., 2009. High resolution constituents of the Earth gravitational field. Surv. Geophys., 31, 1–21.

    Article  Google Scholar 

  • Novotn Ó., 1982. On the additional theorem for Legendre polynomials. Travaux Geophysics, 30, 33–45.

    Google Scholar 

  • Paul M., 1973. A method of evaluating the truncation error coefficients for geoidal heights. Geodesique, 110, 413–425.

    Article  Google Scholar 

  • Pavlis N.K., Holmes S.A., Kenyon S.C. and Factor J.K., 2008. An Earth Gravitational Model to Degree 2160: EGM2008. Geophysical Research Abstracts, 10, 2-2-2008. Full version released by National Geospatial-Intelligence Agency, Bethesda, MD, (http://www.dgfi.badw.de/typo3_mt/fileadmin/2kolloquium_muc/2008-10-08/Bosch/EGM2008.pdf).

  • Sjöberg L.E., 1998. The ellipsoidal corrections to the topographic geoid effects. J. Geodesy, 77, 804–808.

    Article  Google Scholar 

  • Sjöberg L.E. and Nahavandchi H., 1999. On the indirect effect in the Stokes-Helmert method of geoid determination. J. Geodesy, 73, 87–93.

    Article  Google Scholar 

  • Sjöberg L.E., 2000. Topographic effects by the Stokes-Helmert method of geoid and quasi-geoid determinations. J. Geodesy, 74, 255–268.

    Article  Google Scholar 

  • Sjöberg L.E., 2001. Topographic and atmospheric corrections of gravimetric geoid determination with special emphasis on the effects of harmonics of degrees zero and one. J. Geodesy, 75, 283–290.

    Article  Google Scholar 

  • Sjöberg L.E. and Hunegnaw A., 2001. Some modifications of Stokes’s formula that account for truncation and potential coefficient errors. J. Geodesy, 74, 232–238.

    Google Scholar 

  • Sjöberg L.E., 2007. Topographic bias by analytical continuation in physical geodesy. J. Geodesy, 81, 345–350.

    Article  Google Scholar 

  • Sünkel H., 1968. Global topographic-isostatic models. In: Sünkel H. (Ed.), Mathematical and Numerical Techniques in Physical Geodesy. Lecture Notes in Earth Sciences, 7, Springer-Verlag, Heidelberg, Germany, 417–462.

    Chapter  Google Scholar 

  • Tenzer R., 2005. Spectral domain of Newton’s integral. Bollettino di Geodesia e Scienze Affini, 2, 61–73.

    Google Scholar 

  • Tenzer R., Novák P., Prutkin I., Ellmann A. and Vajda P., 2009. Far-zone contributions to the gravity field quantities by means of Molodensky’s truncation coefficients. Stud. Geophys. Geod., 53, 157–167.

    Article  Google Scholar 

  • Tsoulis D., 1999. Spherical Harmonic Computations with Topographic/Isostatic Coefficients. Report No. 3 (ISBN 3-934205-02-X) in the IAPG/FESG series (ISSN 1437-8280). Institute of Astronomical and Physical Geodesy, Technical University of Munich, Munich, Germany.

    Google Scholar 

  • Tsoulis D., 2001. A Comparison between the Airy-Heiskanen and the Pratt-Hayford isostatic models for the computation of potential harmonic coefficients. J. Geodesy, 74, 637–643; DOI:10.1007/s001900000124.

    Article  Google Scholar 

  • Vaníček P. and Sjöberg L.E., 1991. Reformulation of Stokes’s theory for higher than second-degree reference field and a modification of integration kernels. J. Geophys. Res., 96(B4), 6529–6539.

    Article  Google Scholar 

  • Vaníček P., Najafi M., Martinec Z., Harrie L. and Sjöberg L.E., 1995. Higher-degree reference field in the generalised Stokes-Helmert scheme for geoid computation. J. Geodesy, 70, 176–182.

    Article  Google Scholar 

  • Wild F. and Heck B., 2004. Effects of topographic and isostatic masses in satellite gravity gradiometry. In: Lacoste H. (Ed.), Proceedings of the Second International GOCE User Workshop, The Geoid and Oceanography. ESA SP-569, ESA Publications Division, Noordwijk, The Netherlands, ISBN: 92-9092-880-8.

    Google Scholar 

  • Witte L., 1967. Truncation errors in the vertical extension of gravity anomalies by Poisson’s integral theorem. Bulletin Geodesique, 83, 41–52.

    Article  Google Scholar 

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Authors and Affiliations

  1. National School of Surveying, Faculty of Sciences, University of Otago, Dunedin, New Zealand

    Robert Tenzer & Ahmed Abdalla

  2. Department of Mathematics, University of West Bohemia, Plzeň, Czech Republic

    Pavel Novák

  3. Geophysical Institute, Slovak Academy of Sciences, Bratislava, Slovak Republic

    Peter Vajda

  4. Department of Civil Engineering, Tallinn University of Technology, Tallinn, Estonia

    Artu Ellmann

Authors
  1. Robert Tenzer
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  2. Pavel Novák
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  3. Peter Vajda
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  4. Artu Ellmann
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  5. Ahmed Abdalla
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Correspondence to Robert Tenzer.

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Tenzer, R., Novák, P., Vajda, P. et al. Far-zone gravity field contributions corrected for the effect of topography by means of molodensky’s truncation coefficients. Stud Geophys Geod 55, 55–71 (2011). https://doi.org/10.1007/s11200-011-0004-7

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  • DOI: https://doi.org/10.1007/s11200-011-0004-7

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