Ellipsoidal area mean gravity anomalies — precise computation of gravity anomaly reference fields for remove-compute-restore geoid determination

  • Christian HirtEmail author
  • Sten J. Claessens


Gravity anomaly reference fields, required e.g. in remove-compute-restore (RCR) geoid computation, are obtained from global geopotential models (GGM) through harmonic synthesis. Usually, the gravity anomalies are computed as point values or area mean values in spherical approximation, or point values in ellipsoidal approximation. The present study proposes a method for computation of area mean gravity anomalies in ellipsoidal approximation (‘ellipsoidal area means’) by applying a simple ellipsoidal correction to area means in spherical approximation. Ellipsoidal area means offer better consistency with GGM quasigeoid heights. The method is numerically validated with ellipsoidal area mean gravity derived from very fine grids of gravity point values in ellipsoidal approximation. Signal strengths of (i) the ellipsoidal effect (i.e., difference ellipsoidal vs. spherical approximation), (ii) the area mean effect (i.e., difference area mean vs. point gravity) and (iii) the ellipsoidal area mean effect (i.e., differences between ellipsoidal area means and point gravity in spherical approximation) are investigated in test areas in New Zealand and the Himalaya mountains. The impact of both the area mean and the ellipsoidal effect on quasigeoid heights is in the order of several centimetres. The proposed new gravity data type not only allows more accurate RCR-based geoid computation, but may also be of some value for the GGM validation using terrestrial gravity anomalies that are available as area mean values.


global geopotential model (GGM) remove-compute-restore (RCR) geoid computation point gravity area mean gravity ellipsoidal area mean 


  1. Claessens S.J., 2006. Solutions to Ellipsoidal Boundary Value Problems for Gravity Field Modelling. PhD Thesis, Curtin University of Technology, Department of Spatial Sciences, Perth, Australia.Google Scholar
  2. Claessens S.J., Hirt C., Amos M.J., Featherstone W.E. and Kirby J.F., 2011. The NZGEOID09 model of New Zealand. Surv. Rev., 43, 2–15, DOI: 10.1179/003962610X12747001420780.CrossRefGoogle Scholar
  3. Cruz J.Y., 1986. Ellipsoidal Corrections to Potential Coefficients Obtained from Gravity Anomaly Data on the Ellipsoid. Report No. 371, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, Ohio.Google Scholar
  4. Featherstone W.E., Evans J.D. and Olliver J.G., 1998. A Meissl-modified Vaníček and Kleusberg kernel to reduce the truncation error in gravimetric geoid computations. J. Geodesy, 72, 154–160, DOI: 10.1007/s001900050157.CrossRefGoogle Scholar
  5. Featherstone W.E., Kirby J.F., Kearsley A.H.W., Gilliland J.R., Johnston G.M., Steed J., Forsberg R. and Sideris M.G., 2001. The AUSGeoid98 geoid model of Australia: data treatment, computations and comparisons with GPS-levelling data. J. Geodesy, 75, 313–330.CrossRefGoogle Scholar
  6. Featherstone W.E., Holmes S.A., Kirby J.F. and Kuhn M., 2004. Comparison of remove-computerestore and University of New Brunswick techniques to geoid determination over Australia, and inclusion of Wiener-type filters in reference field contribution. J. Surv. Eng., 130, 40–47.CrossRefGoogle Scholar
  7. Featherstone W.E., Kirby J.F., Hirt C., Filmer M.S., Claessens S.J., Brown N.J., Hu G. and Johnston G.M., 2011. The AUSGeoid09 model of the Australian Height Datum. J. Geodesy, 85, 133–150, DOI: 10.1007/s00190-010-0422-2.CrossRefGoogle Scholar
  8. Gleason D.M., 1988. Comparing ellipsoidal corrections to the transformation between the geopotential’s spherical and ellipsoidal spectrums. Manuscripta Geodaetica, 13, 114–129.Google Scholar
  9. Grafarend E.W., Ardalan A. and Sideris M.G., 1999. The spheroidal fixed-free two-boundary-value problem for geoid determination (the spheroidal Bruns’ transform). J. Geodesy, 73, 513–533.CrossRefGoogle Scholar
  10. Haagmans R., de Min E. and van Gelderen M., 1993. Fast evaluation of convolution integrals on the sphere using 1D FFT, and a comparison with existing methods for Stokes’ integral. Manuscripta Geodaetica, 18, 227–241.Google Scholar
  11. Heck B., 1991. On the Linearized Boundary Value Problems of Physical Geodesy, Report No. 407, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, Ohio.Google Scholar
  12. Heiskanen W.A. and Moritz H., 1967. Physical Geodesy. W.H. Freeman and Co., San Francisco.Google Scholar
  13. Hipkin R., 2004. Ellipsoidal geoid computation. J. Geodesy, 78, 167–179.CrossRefGoogle Scholar
  14. Hirt C., Featherstone W.E. and Claessens S.J., 2011. On the accurate numerical evaluation of geodetic convolution integrals. J. Geodesy, 85, 519–538, DOI: 10.1007/s00190-011-0451-5.CrossRefGoogle Scholar
  15. Holmes S.A., 2002. High-Degree Spherical Harmonic Synthesis: New Algorithms and Applications. PhD Thesis, Curtin University of Technology, Department of Spatial Sciences, Perth, Australia.Google Scholar
  16. Holmes S.A. and Pavlis N.K., 2008. Spherical Harmonic Synthesis Software harmonic_synth. (
  17. Jarvis A., Reuter H.I., Nelson A. and Guevara E., 2008. Hole-Filled SRTM for the Globe Version 4. Available from the CGIAR-SXI SRTM 90m database (
  18. Jekeli C., 1981. The Downward Continuation to the Earth’s Surface of Truncated Spherical and Ellipsoidal Harmonic Series of the Gravity and Height Anomalies. Report No. 323, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, Ohio.Google Scholar
  19. Jekeli C., 2006. Geometric Reference Systems in Geodesy. Division of Geodesy and Geospatial Science, School of Earth Sciences, Ohio State University, Columbus, Ohio (
  20. 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
  21. Paul M.K., 1978. Recurrence relations for integrals of associated Legendre functions. Bull. Geod., 52, 177–190.CrossRefGoogle Scholar
  22. Pavlis N.K., Holmes S.A., Kenyon S.C. and Factor J.K., 2008. An Earth Gravitational Model to Degree 2160: EGM2008. (
  23. Rapp R.H., 1997. Use of potential coefficient models for geoid undulation determinations using a spherical harmonic representation of the height anomaly/geoid undulation difference. J. Geodesy, 71, 282–289.CrossRefGoogle Scholar
  24. Sjöberg L., 2005. A discussion on the approximations made in the practical implementation of the remove-compute-restore technique in regional geoid modelling. J. Geodesy, 78, 645–653.CrossRefGoogle Scholar
  25. Smith D.A., 1998. There is no such thing as “The” EGM96 geoid: Subtle points on the use of a global geopotential model. IGeS Bull., 8, 17–28.Google Scholar
  26. Torge W., 2001. Geodesy. 3rd Edition. de Gruyter, Berlin, New York.CrossRefGoogle Scholar
  27. Vaníček P., Huang J., Novák P., Pagiatakis S., Véronneau M., Martinec Z. and Featherstone W.E., 1999. Determination of the boundary values for the Stokes-Helmert problem. J. Geodesy, 73, 180–192.CrossRefGoogle Scholar
  28. Vaníček P. and Featherstone W.E., 1998. Performance of three types of Stokes’s kernel in the combined solution for the geoid. J. Geodesy, 72, 684–697.CrossRefGoogle Scholar
  29. Wenzel H.-G., 1985. Hochauflösende Kugelfunktionsmodelle für das Gravitationspotential der Erde. Wissenschaftliche Arbeiten der Fachrichtung Vermesssungswesen der Universität Hannover No. 137, Hannover, Germany (in German).Google Scholar
  30. Wessel P. and Smith W.H.F., 1998. New, improved version of the Generic Mapping Tools released. EOS Trans. AGU, 79, 579.CrossRefGoogle Scholar
  31. Wolf K.-I., 2007. Kombination globaler Potentialmodelle mit terrestrischen Schweredaten für die Berechnung der zweiten Ableitungen des Gravitationspotentials in Satellitenbahnhöhe. Wissenschaftliche Arbeiten der Fachrichtung Geodäsie und Geoinformatik an der Universität Hannover No. 264, Hannover, Germany (in German).Google Scholar

Copyright information

© Institute of Geophysics of the ASCR, v.v.i 2011

Authors and Affiliations

  1. 1.Western Australian Centre for Geodesy & The Institute for Geoscience ResearchCurtin University of TechnologyPerthAustralia
  2. 2.The Institute for Geoscience Research & The Institute for Geoscience ResearchCurtin University of TechnologyPerthAustralia

Personalised recommendations