Journal of Geodesy

, Volume 93, Issue 5, pp 723–747 | Cite as

On the computation of gravitational effects for tesseroids with constant and linearly varying density

  • Miao LinEmail author
  • Heiner Denker
Original Article


The accurate computation of gravitational effects from topographic and atmospheric masses is one of the core issues in gravity field modeling. Using gravity forward modeling based on Newton’s integral, mass distributions are generally decomposed into regular mass bodies, which can be represented by rectangular prisms or polyhedral bodies in a rectangular coordinate system, or tesseroids in a spherical coordinate system. In this study, we prefer the latter representation because it can directly take the Earth’s curvature into account, which is particularly beneficial for regional and global applications. Since the volume integral cannot be solved analytically in the case of tesseroids, approximation solutions are applied. However, one well-recognized issue of these solutions is that the accuracy decreases as the computation point approaches the tesseroid. To overcome this problem, we develop a method that can precisely compute the gravitational potential \(\left( V\right) \) and vector \(\left( V_x, V_y, V_z\right) \) on the tesseroid surface. In addition to considering a constant density for the tesseroid, we further derive formulas for a linearly varying density. In the near zone (up to a spherical distance of 15 times the horizontal tesseroid dimension from the computation point), the gravitational effects of the tesseroids are computed by Gauss–Legendre quadrature using a two-dimensional adaptive subdivision technique to ensure high accuracy. The tesseroids outside this region are evaluated by means of expanding the integral kernel in a Taylor series up to the second order. The method is validated by synthetic tests of spherical shells with constant and linearly varying density, and the resulting approximation error is less than \(10^{-4}\,\hbox {m}^2\,\hbox {s}^{-2}\) for V, \(10^{-5}\,\hbox {mGal}\) for \(V_x\), \(10^{-7}\,\hbox {mGal}\) for \(V_y\), and \(10^{-4}\,\hbox {mGal}\) for \(V_z\). Its practical applicability is then demonstrated through the computation of topographic reductions in the White Sands test area and of global atmospheric effects on the Earth’s surface using the US Standard Atmosphere 1976.


Tesseroid Gravity forward modeling Adaptive subdivision Topographic reduction Atmospheric effect Linearly varying density 



We thank the anonymous reviewers for their constructive comments that helped to significantly improve the manuscript. This work was financially supported by the German Research Foundation (DFG) within CRC 1128 “Relativistic Geodesy and Gravimetry with Quantum Sensors (geo-Q)”, project C04. Most of the figures were plotted by the Generic Mapping Tools (GMT; Wessel and Smith 1998).


  1. Amante C, Eakins BW (2009) ETOPO1 1 arc-minute global relief model: procedures, data sources and analysis. NOAA technical memorandum NESDIS NGDC-24. National Geophysical Data Center, NOAA.
  2. Anderson EG (1976) The effect of topography on solutions of Stokes’ problem. UNISURV S-14 report, School of Surveying, University of New South Wales, Kensington, AustraliaGoogle Scholar
  3. Asgharzadeh MF, Von Frese RRB, Kim HR, Leftwich TE, Kim JW (2007) Spherical prism gravity effects by Gauss-Legendre quadrature integration. Geophys J Int 169:1–11. CrossRefGoogle Scholar
  4. Benedek J, Papp G (2009) Geophysical inversion of on board satellite gradiometer data—a feasibility study in the ALPACA region, central Europe. Acta Geod Geophys Hung 44:179–190. CrossRefGoogle Scholar
  5. Benedek J, Papp G, Kalmár J (2018) Generalization techniques to reduce the number of volume elements for terrain effect calculations in fully analytical gravitational modelling. J Geod 92:361–381. CrossRefGoogle Scholar
  6. Chai Y, Hinze WJ (1988) Gravity inversion of an interface above which the density contrast varies exponentially with depth. Geophysics 53:837–845. CrossRefGoogle Scholar
  7. Conway JT (2015) Analytical solution from vector potentials for the gravitational field of a general polyhedron. Celest Mech Dyn Astron 121:17–38. CrossRefGoogle Scholar
  8. Deng X, Grombein T, Shen W, Heck B, Seitz K (2016) Corrections to “A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling” (Heck and Seitz, 2007) and “Optimized formulas for the gravitational field of a tesseroid” (Grombein et al., 2013). J Geod 90:585–587. CrossRefGoogle Scholar
  9. Denker H (2013) Regional gravity field modeling: theory and practical results. In: Xu G (ed) Sciences of geodesy—II. Springer, Berlin, Heidelberg, pp 185–291. CrossRefGoogle Scholar
  10. D’Urso MG (2013) On the evaluation of the gravity effects of polyhedral bodies and a consistent treatment of related singularities. J Geod 87:239–252. CrossRefGoogle Scholar
  11. D’Urso MG (2014a) Analytical computation of gravity effects for polyhedral bodies. J Geod 88:13–29. CrossRefGoogle Scholar
  12. D’Urso MG (2014b) Gravity effects of polyhedral bodies with linearly varying density. Celest Mech Dyn Astron 120:349–372. CrossRefGoogle Scholar
  13. D’Urso MG, Trotta S (2017) Gravity anomaly of polyhedral bodies having a polynomial density contrast. Surv Geophys 38:781–832. CrossRefGoogle Scholar
  14. Forsberg R (1984) A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modelling. OSU report 355, Department of Geodetic Science and Surveying, Ohio State University, Columbus, Ohio, USAGoogle Scholar
  15. Fukushima T (2018) Accurate computation of gravitational field of a tesseroid. J Geod. Google Scholar
  16. García-Abdeslem J (1992) Gravitational attraction of a rectangular prism with depth-dependent density. Geophysics 57:470–473. CrossRefGoogle Scholar
  17. García-Abdeslem J (2005) Gravitational attraction of a right rectangular prism with density varying with depth following a cubic polynomial. Geophysics 70:J39–J42. CrossRefGoogle Scholar
  18. Götze HJ, Lahmeyer B (1988) Application of three-dimensional interactive modeling in gravity and magnetics. Geophysics 53:1096–1108. CrossRefGoogle Scholar
  19. Gradshteyn IS, Ryzhik IM (2007) Table of integrals, series, and products, 7th edn. Academic Press, New YorkGoogle Scholar
  20. Grombein T, Seitz K, Heck B (2013) Optimized formulas for the gravitational field of a tesseroid. J Geod 87:645–660. CrossRefGoogle Scholar
  21. Grüninger W (1990) Zur topographisch-isostatischen Reduktion der Schwere. PhD thesis, Universität KarlsruheGoogle Scholar
  22. Hamayun, Prutkin I, Tenzer R (2009) The optimum expression for the gravitational potential of polyhedral bodies having a linearly varying density distribution. J Geod 83:1163–1170.
  23. Hansen RO (1999) An analytical expression for the gravity field of a polyhedral body with linearly varying density. Geophysics 64:75–77. CrossRefGoogle Scholar
  24. Heck B, Seitz K (2007) A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling. J Geod 81:121–136. CrossRefGoogle Scholar
  25. Holstein H (2003) Gravimagnetic anomaly formulas for polyhedra of spatially linear media. Geophysics 68:157–167. CrossRefGoogle Scholar
  26. Holstein H, Ketteridge B (1996) Gravimetric analysis of uniform polyhedra. Geophysics 61:357–364. CrossRefGoogle Scholar
  27. Holstein H, Schürholz P, Starr AJ, Chakraborty M (1999) Comparison of gravimetric formulas for uniform polyhedra. Geophysics 64:1438–1446. CrossRefGoogle Scholar
  28. Jiang L, Zhang J, Feng Z (2017) A versatile solution for the gravity anomaly of 3D prism-meshed bodies with depth-dependent density contrast. Geophysics 82:G77–G86. CrossRefGoogle Scholar
  29. Ku CC (1977) A direct computation of gravity and magnetic anomalies caused by 2- and 3-dimensional bodies of arbitrary shape and arbitrary magnetic polarization by equivalent-point method and a simplified cubic spline. Geophysics 42:610–622. CrossRefGoogle Scholar
  30. Kuhn M (2003) Geoid determination with density hypotheses from isostatic models and geological information. J Geod 77:50–65. CrossRefGoogle Scholar
  31. Li Z, Hao T, Xu Y, Xu Y (2011) An efficient and adaptive approach for modeling gravity effects in spherical coordinates. J Appl Geophys 73:221–231. CrossRefGoogle Scholar
  32. MacMillan WD (1930) Theoretical mechanics, vol 2: the theory of the potential. McGraw-Hill, New YorkGoogle Scholar
  33. Mader K (1951) Das Newtonsche Raumpotential prismatischer Körper und seine Ableitungen bis zur dritten Ordnung. Österreichische Zeitschrift für Vermessungswesen Sonderheft 11Google Scholar
  34. Makhloof AA, Ilk KH (2008) Effects of topographic–isostatic masses on gravitational functionals at the earth’s surface and at airborne and satellite altitudes. J Geod 82:93–111. CrossRefGoogle Scholar
  35. Marotta AM, Barzaghi R (2017) A new methodology to compute the gravitational contribution of a spherical tesseroid based on the analytical solution of a sector of a spherical zonal band. J Geod 91:1207–1224. CrossRefGoogle Scholar
  36. Martinec Z, Vaníc̆ek P (1994) Direct topographical effect of Helmert’s condensation for a spherical approximation of the geoid. Manuscr Geod 19:257–268Google Scholar
  37. Mikuška J, Marušiak I, Pašteka R, Karcol R, Beňo J (2008) The effect of topography in calculating the atmospheric correction in gravimetry. In: SEG Las Vegas annual meeting, expanded abstracts, pp 784–788.
  38. Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geod 74:552–560. CrossRefGoogle Scholar
  39. Nagy D, Papp G, Benedek J (2002) Corrections to “The gravitational potential and its derivatives for the prism”. J Geod 76:475. CrossRefGoogle Scholar
  40. Novák P, Grafarend EW (2005) Ellipsoidal representation of the topographical potential and its vertical gradient. J Geod 78:691–706. CrossRefGoogle Scholar
  41. Okabe M (1979) Analytical expressions for gravity anomalies due to homogeneous polyhedral bodies and translations into magnetic anomalies. Geophysics 44:730–741. CrossRefGoogle Scholar
  42. Pohánka V (1988) Optimum expression for computation of the gravity field of a homogeneous polyhedral body. Geophys Prospect 36:733–751. CrossRefGoogle Scholar
  43. Pohánka V (1998) Optimum expression for computation of the gravity field of a polyhedral body with linearly increasing density. Geophys Prospect 46:391–404. CrossRefGoogle Scholar
  44. Ren Z, Chen C, Pan K, Kalscheuer T, Maurer H, Tang J (2017) Gravity anomalies of arbitrary 3D polyhedral bodies with Horizontal and vertical mass contrasts. Surv Geophys 38:479–502. CrossRefGoogle Scholar
  45. Ren Z, Zhong Y, Chen C, Tang J, Pan K (2018) Gravity anomalies of arbitrary 3D polyhedral bodies with horizontal and vertical mass contrasts up to cubic order. Geophysics 83:G1–G13. CrossRefGoogle Scholar
  46. Roussel C, Verdun J, Cali J, Masson F (2015) Complete gravity field of an ellipsoidal prism by Gauss–Legendre quadrature. Geophys J Int 203:2220–2236. CrossRefGoogle Scholar
  47. Shen W, Deng X (2016) Evaluation of the fourth-order tesseroid formula and new combination approach to precisely determine gravitational potential. Stud Geophys Geod 60:583–607. CrossRefGoogle Scholar
  48. Stroud AH, Secrest D (1966) Gaussian quadrature formulas. Prentice-Hall, New JerseyGoogle Scholar
  49. Tenzer R, Vajda P, Hamayun (2009) Global atmospheric effects on the gravity field quantities. Contrib Geophys Geod 39:221–236. CrossRefGoogle Scholar
  50. Tsoulis D (1999) Analytical and numerical methods in gravity field modelling of ideal and real masses. Reihe C, Heft Nr. 510, Deutsche Geodätische Kommission, MünchenGoogle Scholar
  51. Tsoulis D (2012) Analytical computation of the full gravity tensor of a homogeneous arbitrarily shaped polyhedral source using line integrals. Geophysics 77:F1–F11. CrossRefGoogle Scholar
  52. Tsoulis D, Novák P, Kadlec M (2009) Evaluation of precise terrain effects using high-resolution digital elevation models. J Geophys Res 114:B02404. CrossRefGoogle Scholar
  53. Uieda L, Barbosa VCF, Braitenberg C (2016) Tesseroids: forward-modeling gravitational fields in spherical coordinates. Geophysics 81:F41–F48. CrossRefGoogle Scholar
  54. Wessel P, Smith WHF (1998) New, improved version of generic mapping tools released. Eos Trans AGU 79:579. CrossRefGoogle Scholar
  55. Wild-Pfeiffer F (2008) A comparison of different mass elements for use in gravity gradiometry. J Geod 82:637–653. CrossRefGoogle Scholar
  56. Wu L, Chen L (2016) Fourier forward modeling of vector and tensor gravity fields due to prismatic bodies with variable density contrast. Geophysics 81:G13–G26. CrossRefGoogle Scholar
  57. Zhou X (2009) 3D vector gravity potential and line integrals for the gravity anomaly of a rectangular prism with 3D variable density contrast. Geophysics 74:I43–I53. CrossRefGoogle Scholar
  58. Zhou X (2010) Analytic solution of the gravity anomaly of irregular 2D masses with density contrast varying as a 2D polynomial function. Geophysics 75:I11–I19. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Erdmessung (IfE)Leibniz Universität HannoverHannoverGermany

Personalised recommendations