Journal of Geodesy

, Volume 93, Issue 10, pp 2123–2144 | Cite as

A spectral-domain approach for gravity forward modelling of 2D bodies

  • Cheng Chen
  • Shaofeng BianEmail author
  • Houpu Li
Original Article


We provide the spectral-domain solutions for the gravity forward modelling of a 2D body with constant, polynomial and exponential density distributions, including the gravitational attraction and its arbitrary-order derivatives. The gravity effects may be directly obtained from the derivatives of a constructed scalar quantity and the surface integral of the density function over the 2D body. Then, the surface integrals of the spectral expansion coefficients of the scalar and the density function are converted to the line integrals along the boundary of the mass body using the Gauss divergence theorem and thus can be evaluated by numerical integration. The surface integrals for the exponential density model are expressed as the infinite expansions of line integrals and converge fast. Approximating the mass body to a 2D polygon, the surface integrals can be evaluated by simple analytic formulas for the constant density model and by linear recursive relations for the polynomial density model. The numerical implementation shows that the spectral-domain algorithm of this paper can produce high accurate forward results, e.g. 13–16 digits achievable precision for the gravity vector. Although the spectral-domain method is only suitable for forward computation of the external gravity effects of the 2D body, it is numerically stable for arbitrary observation point outside the smallest enclosed circle. The closed-form solutions for the 2D body with constant or polynomial density distribution are high precision to evaluate the gravity anomaly at the point near or inside the body, but may be lower precision when the computation point moves away from the body. The spectral-domain algorithm can handle the polynomial and exponential density model in both horizontal and vertical directions.


Gravity effect 2D mass representation Spectral-domain method Polynomial density Exponential density Polygon 



We are grateful to the editors and two anonymous reviewers for their valuable comments and suggestions that improved the manuscript. This work is supported by National Natural Science Foundation of China under Grant Nos. 41631072, 41774021 and 41771487.


  1. Artemjev M, Kaban M, Kucherinenko V, Demyanov G, Taranov V (1994) Subcrustal density inhomogeneities of Northern Eurasia as derived from the gravity data and isostatic models of the lithosphere. Tectonophysics 240:249–280Google Scholar
  2. Balmino G (1994) Gravitational potential harmonics from the shape of an homogeneous body. Celest Mech Dyn Astron 60:331–364Google Scholar
  3. Bott M (1960) The use of rapid digital computing methods for direct gravity interpretation of sedimentary basins. Geophys J Int 3:63–67Google Scholar
  4. Chai Y, Hinze WJ (1988) Gravity inversion of an interface above which the density contrast varies exponentially with depth. Geophysics 53:837–845Google Scholar
  5. Chakravarthi V, Sundararajan N (2004) Ridge-regression algorithm for gravity inversion of fault structures with variable density. Geophysics 69:1394–1404Google Scholar
  6. Chao BF, Rubincam DP (1989) The gravitational field of phobos. Geophys Res Lett 16:859–862Google Scholar
  7. Chappell A, Kusznir N (2008) An algorithm to calculate the gravity anomaly of sedimentary basins with exponential density-depth relationships. Geophys Prospect 56:249–258Google Scholar
  8. Chen C, Chen Y, Bian S (2019a) Evaluation of the spherical harmonic coefficients for the external potential of a polyhedral body with linearly varying density. Celest Mech Dyn Astron 131:1–28Google Scholar
  9. Chen C, Ouyang Y, Bian S (2019b) Spherical harmonic expansions for the gravitational field of a polyhedral body with polynomial density contrast. Surv Geophys 40:197–246Google Scholar
  10. Cordell L (1973) Gravity analysis using an exponential density-depth function; San Jacinto Graben, California. Geophysics 38:684–690Google Scholar
  11. Cunningham LE (1970) On the computation of the spherical harmonic terms needed during the numerical integration of the orbital motion of an artificial satellite. Celest Mech 2:207–216Google Scholar
  12. D’Urso MG (2012) New expressions of the gravitational potential and its derivatives for the prism. In: VII Hotine-Marussi symposium on mathematical geodesy. Springer, Berlin, pp 251–256Google Scholar
  13. D’Urso MG (2013) On the evaluation of the gravity effects of polyhedral bodies and a consistent treatment of related singularities. J Geod 87:239–252Google Scholar
  14. D’Urso MG (2014a) Analytical computation of gravity effects for polyhedral bodies. J Geod 88:13–29Google Scholar
  15. D’Urso MG (2014b) Gravity effects of polyhedral bodies with linearly varying density. Celest Mech Dyn Astron 120:349–372Google Scholar
  16. D’Urso MG (2015) The gravity anomaly of a 2D polygonal body having density contrast given by polynomial functions. Surv Geophys 36:391–425Google Scholar
  17. D’Urso MG, Trotta S (2015) Comparative assessment of linear and bilinear prism-based strategies for terrain correction computations. J Geod 89:199–215Google Scholar
  18. D’Urso MG, Trotta S (2017) Gravity anomaly of polyhedral bodies having a polynomial density contrast. Surv Geophys 38:781–832Google Scholar
  19. Fukushima T (2016) Numerical integration of gravitational field for general three-dimensional objects and its application to gravitational study of grand design spiral arm structure. Mon Not R Astron Soc 463:1500–1517Google Scholar
  20. Fukushima T (2017) Precise and fast computation of the gravitational field of a general finite body and its application to the gravitational study of asteroid eros. Astron J 154:145Google Scholar
  21. Fukushima T (2018a) Accurate computation of gravitational field of a tesseroid. J Geod 92:1371–1386Google Scholar
  22. Fukushima T (2018b) Recursive computation of gravitational field of a right rectangular parallelepiped with density varying vertically by following an arbitrary degree polynomial. Geophys J Int 215:864–879Google Scholar
  23. García-Abdeslem J (1992) Gravitational attraction of a rectangular prism with depth-dependent density. Geophysics 57:470–473Google Scholar
  24. García-Abdeslem J (2005) The gravitational attraction of a right rectangular prism with density varying with depth following a cubic polynomial. Geophysics 70:J39–J42Google Scholar
  25. Gradshteyn IS, Ryzhik IM (2007) Table of integrals, series, and products, 7th edn. Academic Press, LondonGoogle Scholar
  26. Grombein T, Seitz K, Heck B (2013) Optimized formulas for the gravitational field of a tesseroid. J Geod 87:645–660Google Scholar
  27. Hamayun Prutkin I, Tenzer R (2009) The optimum expression for the gravitational potential of polyhedral bodies having a linearly varying density distribution. J Geod 83:1163–1170Google Scholar
  28. Hansen R (1999) An analytical expression for the gravity field of a polyhedral body with linearly varying density. Geophysics 64:75–77Google Scholar
  29. Hansen R, Wang X (1988) Simplified frequency-domain expressions for potential fields of arbitrary three-dimensional bodies. Geophysics 53:365–374Google Scholar
  30. Heck B, Seitz K (2007) A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling. J Geod 81:121–136Google Scholar
  31. Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San FranciscoGoogle Scholar
  32. Hofmann-Wellenhof B, Moritz H (2006) Physical geodesy, 2nd edn. Springer, BerlinGoogle Scholar
  33. Holstein H (2003) Gravimagnetic anomaly formulas for polyhedra of spatially linear media. Geophysics 68(1):157–167Google Scholar
  34. Holstein H, Ketteridge B (1996) Gravimetric analysis of uniform polyhedra. Geophysics 61(2):357–364Google Scholar
  35. Holstein H, Sherratt E, Anastasiades C (2007) Gravimagnetic anomaly formulae for triangulated homogeneous polyhedra. In: 69th EAGE conference and exhibition incorporating SPE EUROPEC 2007Google Scholar
  36. Hubbert MK (1948) A line-integral method of computing the gravimetric effects of two-dimensional masses. Geophysics 13:215–225Google Scholar
  37. Jamet O, Thomas E (2004) A linear algorithm for computing the spherical harmonic coefficients of the gravitational potential from a constant density polyhedron. In: Proceedings of the 2nd international GOCE user workshop, GOCE. The Geoid and Oceanography, ESA-ESRIN, Frascati, Italy, Citeseer, pp 8–10Google Scholar
  38. 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–G86Google Scholar
  39. Jiang L, Liu J, Zhang J, Feng Z (2018) Analytic expressions for the gravity gradient tensor of 3D prisms with depth-dependent density. Surv Geophys 39:337–363Google Scholar
  40. Kellogg OD (1929) Foundations of potential theory. Springer, BerlinGoogle Scholar
  41. Laurie DP (1997) Calculation of Gauss–Kronrod quadrature rules. Math Comput 66:1133–1145Google Scholar
  42. Litinsky VA (1989) Concept of effective density: key to gravity depth determinations for sedimentary basins. Geophysics 54:1474–1482Google Scholar
  43. Martín-Atíenza B, Garcia-Abdeslem J (1999) 2-D gravity modeling with analytically defined geometry and quadratic polynomial density functions. Geophysics 64:1730–1734Google Scholar
  44. Martinec Z, Pěč K, Burša M (1989) The phobos gravitational field modeled on the basis of its topography. Earth Moon Planets 45(3):219–235Google Scholar
  45. Murthy IR, Rao DB (1979) Gravity anomalies of two-dimensional bodies of irregular cross-section with density contrast varying with depth. Geophysics 44:1525–1530Google Scholar
  46. Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geod 74:552–560Google Scholar
  47. Netteton LL (1940) Geophysical prospecting for oil. McGraw-Hill Book Company Inc., New YorkGoogle Scholar
  48. Pedersen LB (1978) Wavenumber domain expressions for potential fields from arbitrary 2-, 21/2-, and 3-dimensional bodies. Geophysics 43:626–630Google Scholar
  49. Petrovskaya M, Vershkov A (2010) Construction of spherical harmonic series for the potential derivatives of arbitrary orders in the geocentric earth-fixed reference frame. J Geod 84:165–178Google Scholar
  50. Pohánka V (1998) Optimum expression for computation of the gravity field of a polyhedral body with linearly increasing density. Geophys Prospect 46:391–404Google Scholar
  51. Rao DB (1985) Analysis of gravity anomalies over an inclined fault with quadratic density function. Pure Appl Geophys 123:250–260Google Scholar
  52. Rao DB (1986a) Gravity anomalies of a trapezoidal model with quadratic density function. Proc Indian Acad Sci Earth Planet Sci 95:275–284Google Scholar
  53. Rao DB (1986b) Modelling of sedimentary basins from gravity anomalies with variable density contrast. Geophys J R Astron Soc 84:207–212Google Scholar
  54. Rao DB (1990) Analysis of gravity anomalies of sedimentary basins by an asymmetrical trapezoidal model with quadratic density function. Geophysics 55:226–231Google Scholar
  55. Rao DB, Prakash M, Babu NR (1993) Gravity interpretation using Fourier transforms and simple geometrical models with exponential density contrast. Geophysics 58:1074–1083Google Scholar
  56. Rao CV, Chakravarthi V, Raju M (1994) Forward modeling: gravity anomalies of two-dimensional bodies of arbitrary shape with hyperbolic and parabolic density functions. Comput Geosci 20:873–880Google Scholar
  57. Rao CV, Raju M, Chakravarthi V (1995) Gravity modelling of an interface above which the density contrast decreases hyperbolically with depth. J Appl Geophys 34:63–67Google Scholar
  58. Ren Z, Chen C, Pan K, Kalscheuer T, Maurer H, Tang J (2017a) Gravity anomalies of arbitrary 3D polyhedral bodies with horizontal and vertical mass contrasts. Surv Geophys 38:479–502Google Scholar
  59. Ren Z, Zhong Y, Chen C, Tang J, Pan K (2017b) Gravity anomalies of arbitrary 3D polyhedral bodies with horizontal and vertical mass contrasts up to cubic order. Geophysics 83:G1–G13Google Scholar
  60. Ren Z, Zhong Y, Chen C, Tang J, Kalscheuer T, Maurer H, Li Y (2018) Gravity gradient tensor of arbitrary 3D polyhedral bodies with up to third-order polynomial horizontal and vertical mass contrasts. Surv Geophys 39:901–935Google Scholar
  61. Riley KF, Hobson MP, Bence SJ (2006) Mathematical methods for physics and engineering: a comprehensive guide. Cambridge University Press, CambridgeGoogle Scholar
  62. Ruotoistenmäki T (1992) The gravity anomaly of two-dimensional sources with continuous density distribution and bounded by continuous surfaces. Geophysics 57:623–628Google Scholar
  63. Silva JB, Costa DC, Barbosa VC (2006) Gravity inversion of basement relief and estimation of density contrast variation with depth. Geophysics 71:J51–J58Google Scholar
  64. Talwani M, Worzel JL, Landisman M (1959) Rapid gravity computations for two-dimensional bodies with application to the Mendocino submarine fracture zone. J Geophys Res 64:49–59Google Scholar
  65. Tsoulis D (2000) A note on the gravitational field of the right rectangular prism. Boll Geod Sci Affin 59:21–35Google Scholar
  66. Tsoulis D, Petrović S (2001) On the singularities of the gravity field of a homogeneous polyhedral body. Geophysics 66:535–539Google Scholar
  67. Tsoulis D, Jamet O, Verdun J, Gonindard N (2009) Recursive algorithms for the computation of the potential harmonic coefficients of a constant density polyhedron. J Geod 83:925–942Google Scholar
  68. Uieda L, Barbosa VC, Braitenberg C (2016) Tesseroids: forward-modeling gravitational fields in spherical coordinate. Geophysics 81:F41–F48Google Scholar
  69. van Gelderen M (1992) The geodetic boundary value problem in two dimensions and its iterative solution. PhD thesis, Faculty of Civil Engineering and Geosciences, Technische Universiteit Delft, DelftGoogle Scholar
  70. Vermeille H (2011) An analytical method to transform geocentric into geodetic coordinates. J Geod 85:105–117Google Scholar
  71. Werner RA (1997) Spherical harmonic coefficients for the potential of a constant-density polyhedron. Comput Geosci 23:1071–1077Google Scholar
  72. Werner RA, Scheeres DJ (1997) Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 castalia. Celest Mech Dyn Astron 65:313–344Google Scholar
  73. Wu L (2018a) Comparison of 3-D Fourier forward algorithms for gravity modelling of prismatic bodies with polynomial density distribution. Geophys J Int 215:1865–1886Google Scholar
  74. Wu L (2018b) Efficient modeling of gravity fields caused by sources with arbitrary geometry and arbitrary density distribution. Surv Geophys 39:401–434Google Scholar
  75. Wu L (2019) Fourier-domain modeling of gravity effects caused by polyhedral bodies. J Geod 93:635–653Google Scholar
  76. Wu L, Chen L (2016) Fourier forward modeling of vector and tensor gravity fields due to prismatic bodies with variable density contrast variable density contrast. Geophysics 81:G13–G26Google Scholar
  77. Zhang Y, Chen C (2018) Forward calculation of gravity and its gradient using polyhedral representation of density interfaces: an application of spherical or ellipsoidal topographic gravity effect. J Geod 92:205–218Google Scholar
  78. Zhang J, Jiang L (2017) Analytical expressions for the gravitational vector field of a 3-D rectangular prism with density varying as an arbitrary-order polynomial function. Geophys J Int 210:1176–1190Google Scholar
  79. Zhang J, Zhong B, Zhou X, Dai Y (2001) Gravity anomalies of 2-D bodies with variable density contrast. Geophysics 66:809–813Google Scholar
  80. Zhou X (2008) 2D vector gravity potential and line integrals for the gravity anomaly caused by a 2D mass of depth-dependent density contrast. Geophysics 73:I43–I50Google Scholar
  81. Zhou X (2009a) 3D vector gravity potential and line integrals for the gravity anomaly of a rectangular prism with 3D variable density contrast. Geophysics 74:I43–I53Google Scholar
  82. Zhou X (2009b) General line integrals for gravity anomalies of irregular 2D masses with horizontally and vertically dependent density contrast. Geophysics 74:I1–I7Google Scholar
  83. 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–I19Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Ministry of Education Key Laboratory of Geological Survey and EvaluationChina University of GeosciencesWuhanChina
  2. 2.Department of NavigationNaval University of EngineeringWuhanChina

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