Journal of Geodesy

, Volume 92, Issue 2, pp 205–218 | Cite as

Forward calculation of gravity and its gradient using polyhedral representation of density interfaces: an application of spherical or ellipsoidal topographic gravity effect

  • Yi Zhang
  • Chao Chen
Original Article


A density interface modeling method using polyhedral representation is proposed to construct 3-D models of spherical or ellipsoidal interfaces such as the terrain surface of the Earth and applied to forward calculating gravity effect of topography and bathymetry for regional or global applications. The method utilizes triangular facets to fit undulation of the target interface. The model maintains almost equal accuracy and resolution at different locations of the globe. Meanwhile, the exterior gravitational field of the model, including its gravity and gravity gradients, is obtained simultaneously using analytic solutions. Additionally, considering the effect of distant relief, an adaptive computation process is introduced to reduce the computational burden. Then features and errors of the method are analyzed. Subsequently, the method is applied to an area for the ellipsoidal Bouguer shell correction as an example and the result is compared to existing methods, which shows our method provides high accuracy and great computational efficiency. Suggestions for further developments and conclusions are drawn at last.


Density interface Polyhedron Gravity Gravity gradient Icosahedron 



We thank Dr. Walter Mooney, Dr. Mikhail Kaban and three anonymous reviewers for their precious suggestions for preparing and revising the manuscript. This study is supported by the China Scholarship Council (No. 1609130003) and the Natural Science Foundation of China (No. 41574070). The source code of the method is available upon request.


  1. Amante C, Eakins BW (2009) Etopo1 1arc-minute global relief model: procedures, data sources and analysis. In: NOAA Technical Memorandum NESDIS NGDC-24 Boulder, COGoogle Scholar
  2. An YL, Guo LH, Zhang MH (2015) High precision computation and numerical value characteristics of gravity emendation values arising from mass of the earth’s crust at the distance over 169 km from the observation point. Geophys Geochem Explor 39(1):1–11 (in Chinese)Google Scholar
  3. Ardalan AA, Safari A (2004) Ellipsoidal terrain correction based on multi-cylindrical equal-area map projection of the reference ellipsoid. J Geod 78:114–123Google Scholar
  4. Asgharzadeh MF, Frese RRB, Kim HR, Leftwich TE, Kim JW (2007) Spherical prism gravity effects by Gauss–Legendre quadrature integration. Geophys J Int 169:1–1CrossRefGoogle Scholar
  5. Balmino G, Vales N, Bonvalot S, Briais A (2011) Spherical harmonic modelling to ultra-high degree of Bouguer and isostatic anomalies. J Geod 86:499–520CrossRefGoogle Scholar
  6. Bouman J, Ebbing J, Fuchs M (2013) Reference frame transformation of satellite gravity gradients and topographic mass reduction. J Geophys Res 118(2):759–774CrossRefGoogle Scholar
  7. Bullard EC (1936) Gravity measurements in East Africa. Philos Trans R Soc 235:445–534CrossRefGoogle Scholar
  8. Çavsak H (2012) The effects of the Earth’s curvature on gravity and geoid calculations. Pure Appl Geophys 169(4):733–740CrossRefGoogle Scholar
  9. Chapin DA (1996) The theory of the Bouguer gravity anomaly: a tutorial. Lead Edge 15:361–363CrossRefGoogle Scholar
  10. Claessens SJ, Hirt C (2013) Ellipsoidal topographic potential: new solutions for spectral forward gravity modeling of topography with respect to a reference ellipsoid. J Geophys Res 118(11):5991–6002CrossRefGoogle Scholar
  11. Du JS, Chen C, Liang Q, Wang LS, Zhang Y, Wang QG (2012) Gravity anomaly calculation based on volume integral in spherical cap and comparison with the Tesseroid–Taylor series expansion approach. Acta Geodaetica Cartogr Sin 41(3):339–346 (in Chinese)Google Scholar
  12. Dutton GH (1999) A hierarchical coordinate system for geoprocessing and cartography. In: Lecture Notes in Earth Sciences, vol. 79. Springer, BerlinGoogle Scholar
  13. Eshagh M (2013) Numerical aspects of EGM08-based geoid computations in Fennoscandia regarding the applied reference surface and error propagation. J Appl Geophys 96:28–32CrossRefGoogle Scholar
  14. Eshagh M, Sjöberg LE (2009) Atmospheric effect on satellite gravity gradiometry data. J Geodyn 47:9–19CrossRefGoogle Scholar
  15. Fairhead JD, Green CM, Blitzkow D (2003) The use of GPS in gravity surveys. Lead Edge 22(10):954–959CrossRefGoogle Scholar
  16. Feather WE, Dentith MC (1997) A geodetic approach to gravity data reduction for geophysics. Comput Geosci 23(10):1063–1070Google Scholar
  17. Fekete G, Treinish L (1990) Sphere quadtrees: A new data structure to support the visualization of spherically distributed data. In: Extracting meaning from complex data: processing, display, interaction, SPIE, vol 1259, pp 242–250Google Scholar
  18. Förste C, Bruinsma S, Abrikosov O, Flechtner F, Marty JC, Lemoine JM, Biancale R (2014) EIGEN-6C4—the latest combined global gravity field model including GOCE data up to degree and order 1949 of GFZ Potsdam and GRGS Toulouse. In: EGU general assembly conference abstracts, vol 16, May 2014Google Scholar
  19. Fuller RB (1975) Synergetics. MacMillan, New YorkGoogle Scholar
  20. Gregory M (1999) Comparing inter-cell distance and cell wall midpoint criteria for discrete global grid systems. PhD thesis, Department of Geosciences, Oregon State UniversityGoogle Scholar
  21. Grombein T, Seitz K, Heck B (2013) Optimized formulas for the gravitational field of a tesseroid. J Geod 87(7):645–660CrossRefGoogle Scholar
  22. Heck B, Seitz K (2007) A comparison of tesseroid, prism and point-mass approaches for mass reductions in gravity field modeling. J Geophys 81:121–136Google Scholar
  23. Hensel EG (1992) Discussion on: ‘An exact solution for the gravity curvature (Bullard B) correction’ by T. R. Lafehr. Geophysics 57:1093–1094CrossRefGoogle Scholar
  24. Hirt C, Kuhn M, Featherstone WE, Gottl F (2012) Topographic/isostic evaluation of new-generation GOCE gravity field models. J Geophys Res 117(B5):B05407. doi: 10.1029/2011JB008878 CrossRefGoogle Scholar
  25. Kenner H (1976) Geodesic math and how to use it. University of California Press, BerkeleyGoogle Scholar
  26. Lafehr TR (1991a) An exact solution for the gravity curvature (Bullard B) correction. Geophysics 56(8):1179–1184CrossRefGoogle Scholar
  27. Lafehr TR (1991b) Standardisation in gravity reduction. Geophysics 56:1170–1178CrossRefGoogle Scholar
  28. Lafehr TR (1992) Discussion on: ‘An exact solution for the gravity curvature (Bullard B) correction’ by T. R. Lafehr. Geophysics 57:1094CrossRefGoogle Scholar
  29. Lafehr TR (1998) On Talwani’s ‘Errors in the total Bouguer reduction’. Geophysics 63:1131–1136CrossRefGoogle Scholar
  30. Lane R (2009) Some issues and insights for gravity and magnetic modeling at the region to continent scale. ASEG Ext Abstr 2009(1):1–11Google Scholar
  31. Leaman DE (1998) The gravity terrain correction—practical considerations. Explor Geophys 29:467–471CrossRefGoogle Scholar
  32. Lemoine, FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olson TR (1998) The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96. NASA/TP-1998-206861, July 1998Google Scholar
  33. Li CY, Sideris MG (1994) Improved gravimetric terrain corrections. Geophys J Int 119:740–752CrossRefGoogle Scholar
  34. Li X, Götze H (2001) Ellipsoid, geoid, gravity, geodesy, and geophysics. Geophysics 66(6):1660–1668CrossRefGoogle Scholar
  35. Mikuška J, Pašteka R, Marušiak I (2006) Estimation of distant relief effect in gravimetry. Geophysics 71(6):J59–J69CrossRefGoogle Scholar
  36. Nowell DAG (1999) Gravity terrain corrections—an overview. J Appl Geophys 42:117–134CrossRefGoogle Scholar
  37. Parker RL (1973) The rapid calculation of potential anomalies. Geophys J R Astron Soc 31(4):447–455CrossRefGoogle Scholar
  38. Parker RL (1975) Improved fourier terrain correction, Part 1. Geophysics 60(4):1007–1017CrossRefGoogle Scholar
  39. Roussel C, Verdun J, Cali J, Masson F (2015) Complete gravity field of an ellipsoidal prism by Gauss–Legendre quadrature. Geophys J Int 203(3):2220–2236CrossRefGoogle Scholar
  40. Sadourny R, Arakawa A, Mintz Y (1968) Integration of the nondivergent barotropic vorticity equation with an icosahedral-hexagonal grid for the sphere. Mon Weather Rev 96(6):351–356CrossRefGoogle Scholar
  41. Sahr K, White D, Kimerling JA (2003) Geodesic discrete global grid systems. Cartogr Geogr Inf Sci 30(2):121–134CrossRefGoogle Scholar
  42. Sjöberg LE (2001) Topographic and atmospheric corrections of gravimetric geoid determination with special emphasis on the effects of harmonics of degrees zero and one. J Geod 75(5–6):283–290Google Scholar
  43. Talwani M (1998) Errors in the total Bouguer reduction. Geophysics 63:1125–1130CrossRefGoogle Scholar
  44. Tenzer R, Hamayun K, Vajda P (2008) Global secondary indirect effects of topography, bathymetry, ice and sediments. Contrib Geophys Geod 38(2):209–216Google Scholar
  45. Tenzer R, Hamayun K, Vajda P (2009) Global maps of the CRUST 2.0 crustal components stripped gravity disturbances. J Geophys Res 114:B05408CrossRefGoogle Scholar
  46. Thuburn J (1997) A PV-based shallow-water model on a hexagonal-icosahedral grid. Mon Weather Rev 125:2328–2347CrossRefGoogle Scholar
  47. Tsoulis D (1998) A combination method for computing terrain corrections. Phys Chem Earth 23(1):53–58CrossRefGoogle Scholar
  48. Tsoulis D (2012) Analytical computation of the full gravity tensor of a homogeneous arbitrarily shaped polyhedral source using line integrals. Geophysics 77(2):F1–F14CrossRefGoogle Scholar
  49. Tsoulis D, Novák P, Kadlec M (2009a) Evaluation of precise terrain effects using high-resolution digital elevation models. J Appl Geophys 114(B02404):1–14Google Scholar
  50. Tsoulis D, Jamet O, Verdum J, Gonindard N (2009b) Recursive algorithms for the computation of the potential harmonic coefficients of a constant density polyhedron. J Geod 83:925–942CrossRefGoogle Scholar
  51. Tsoulis D, Petrović S (2001) Short note: on the singularities of the gravity field of a homogeneous polyhedral body. Geophysics 66(2):535–539CrossRefGoogle Scholar
  52. Tsoulis D, Wziontek H, Petrović S (2003) A bilinear approximation of the surface relief in terrain correction computations. J Geod 77:338–344CrossRefGoogle Scholar
  53. Uieda L, Bomfim EP, Braiterberg C, Molina E (2011) Optimal forward calculation method of the Marussi tensor due to a geologic structure at GOCE height. In: Proceedings of the 4th international GOCE user workshop, ESA SP-696, July 2011, MunichGoogle Scholar
  54. Uieda L, Barbosa VCF, Braitenberg C (2015) Tesseroids: forward-modeling gravitational fields in spherical coordinates. Geophysics 81(5):F41–F48CrossRefGoogle Scholar
  55. Urso MG (2013) On the evaluation of the gravity effects of polyhedral bodies and a consistent treatment of related singularities. J Geod 87:239–252CrossRefGoogle Scholar
  56. Wang HS, Wu P, Wang ZY (2006) Approach for spherical harmonic analysis of non-smooth data. Comput Geosci 32:1654–1668CrossRefGoogle Scholar
  57. Werner AR, Scheeres JD (1996) Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of Asteroid 4769 Castalia. Celest Mech Dyn Astron 65(3):313–344Google Scholar
  58. White D, Kimerling JA, Overton SW (1992) Cartographic and geometric components of a global sampling design for environmental monitoring. Cartogr Geogr Inf Syst 19(1):5–22Google Scholar
  59. White D, Kimerling JA, Sahr K, Song L (1998) Comparing area and shape distortion on polyhedral-based recursive partitions of the sphere. Int J Geogr Inf Sci 12:805–827CrossRefGoogle Scholar
  60. Wieczorek MA, Phillips RJ (1998) Potential anomalies on a sphere: applications to the thickness of the lunar crust. J Geophys Res 103(E1):1715–1724CrossRefGoogle Scholar
  61. Wild-Pfeiffer F (2008) A comparison of different mass elements for use in gravity gradiometry. J Geod 82:637–653CrossRefGoogle Scholar
  62. Williamson LD (1968) Integration of the barotropic vorticity equation on a spherical geodesic grid. Tellus 20(4):642–653CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Subsurface Multi-Scale Imaging Lab, Institute of Geophysics and GeomaticsChina University of GeosciencesWuhanChina
  2. 2.Earthquake Science CenterUnited States Geological SurveyMenlo ParkUSA

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