Surveys in Geophysics

, Volume 39, Issue 3, pp 337–363 | Cite as

Analytic Expressions for the Gravity Gradient Tensor of 3D Prisms with Depth-Dependent Density

  • Li Jiang
  • Jie Liu
  • Jianzhong Zhang
  • Zhibing Feng


Variable-density sources have been paid more attention in gravity modeling. We conduct the computation of gravity gradient tensor of given mass sources with variable density in this paper. 3D rectangular prisms, as simple building blocks, can be used to approximate well 3D irregular-shaped sources. A polynomial function of depth can represent flexibly the complicated density variations in each prism. Hence, we derive the analytic expressions in closed form for computing all components of the gravity gradient tensor due to a 3D right rectangular prism with an arbitrary-order polynomial density function of depth. The singularity of the expressions is analyzed. The singular points distribute at the corners of the prism or on some of the lines through the edges of the prism in the lower semi-space containing the prism. The expressions are validated, and their numerical stability is also evaluated through numerical tests. The numerical examples with variable-density prism and basin models show that the expressions within their range of numerical stability are superior in computational accuracy and efficiency to the common solution that sums up the effects of a collection of uniform subprisms, and provide an effective method for computing gravity gradient tensor of 3D irregular-shaped sources with complicated density variation. In addition, the tensor computed with variable density is different in magnitude from that with constant density. It demonstrates the importance of the gravity gradient tensor modeling with variable density.


Gravity gradient tensor 3D right rectangular prism Polynomial density function 



This work was supported by the National Natural Science Foundation of China (Grant Nos. 41074077 and 41230318) and the Key Research and Development Program of Shandong Province (Grant No. 2017GSF16103). We thank Editor in Chief Michael J. Rycroft and an anonymous reviewer for their constructive comments that greatly improved this manuscript.


  1. Athy LF (1930) Density, porosity, and compaction of sedimentary rocks. AAPG Bull 14(1):1–24Google Scholar
  2. Barnes G, Barraud J (2012) Imaging geologic surfaces by inverting gravity gradient data with depth horizons. Geophysics 77(1):G1–G11CrossRefGoogle Scholar
  3. Barnes G, Lumley J (2011) Processing gravity gradient data. Geophysics 76(2):I33–I47CrossRefGoogle Scholar
  4. Barraud J (2013) Improving identification of valid depth estimates from gravity gradient data using curvature and geometry analysis. First Break 31(4):87–92Google Scholar
  5. Beiki M (2011) Analytic signals of gravity gradient tensor and their application to estimate source location. Geophysics 75(6):I59–I74CrossRefGoogle Scholar
  6. Beiki M, Pedersen LB (2010) Eigenvector analysis of gravity gradient tensor to locate geologic bodies. Geophysics 75(6):I37–I49CrossRefGoogle Scholar
  7. Beiki M, Pedersen LB (2011) Window constrained inversion of gravity gradient tensor data using dike and contact models. Geophysics 76(6):I59–I72CrossRefGoogle Scholar
  8. Beiki M, Keating P, Clark DA (2014) Interpretation of magnetic and gravity gradient tensor data using normalized source strength—a case study from McFaulds Lake, Northern Ontario, Canada. Geophys Prospect 62(5):1180–1192CrossRefGoogle Scholar
  9. Blakely RJ (1995) Potential theory in gravity and magnetic applications. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  10. Bouman J, Ebbing J, Meekes S, Fattah RA, Fuchs M, Gradmann S, Haagmans R, Lieb A, Schmidt M, Dettmering D, Bosch W (2015) GOCE gravity gradient data for lithospheric modeling. Int J Appl Earth Obs Geoinf 35:16–30CrossRefGoogle Scholar
  11. Bowin C, Scheer E, Smith W (1986) Depth estimates from ratios of gravity, geoid, and gravity gradient anomalies. Geophysics 51(1):123–136CrossRefGoogle Scholar
  12. Capriotti J, Li Y (2014) Gravity and gravity gradient data: understanding their information content through joint inversions. SEG Denver 2014 annual meeting, pp. 1329–1333Google Scholar
  13. Cevallos C, Kovac P, Lowe SJ (2013) Application of curvatures to airborne gravity gradient data in oil exploration. Geophysics 78(4):G81–G88CrossRefGoogle Scholar
  14. Chai Y, Hinze WJ (1988) Gravity inversion of an interface above which the density contrast varies exponentially with depth. Geophysics 53(6):837–845CrossRefGoogle Scholar
  15. Chakravarthi V, Raghuram HM, Singh SB (2002) 3-D forward gravity modeling of basement interfaces above which the density contrast varies continuously with depth. Comput Geosci 28(1):53–57CrossRefGoogle Scholar
  16. Chappell A, Kusznir N (2008) An algorithm to calculate the gravity anomaly of sedimentary basins with exponential density-depth relationships. Geophys Prospect 56(2):249–258CrossRefGoogle Scholar
  17. Conway J (2015) Analytical solution from vector potentials for the gravitational field of a general polyhedron. Celest Mech Dyn Astron 121(1):17–38CrossRefGoogle Scholar
  18. Čuma M, Wilson GA, Zhdanov MS (2012) Large-scale 3d inversion of potential field data. Geophys Prospect 60(6):1186–1199CrossRefGoogle Scholar
  19. D’Urso MG (2012) New expressions of the gravitational potential and its derivatives for the prism. In: Sneeuw N, Novak P, Crespi M, Sansò F (eds) 7th Hotine–Marussi International Symposium on Mathematical Geodesy. International Association of Geodesy Symposia, SpringerGoogle Scholar
  20. D’Urso MG (2014a) Analytical computation of gravity effects for polyhedral bodies. J Geodesy 88(1):13–29CrossRefGoogle Scholar
  21. D’Urso MG (2014b) Gravity effects of polyhedral bodies with linearly varying density. Celest Mech Dyn Astron 120(4):349–372CrossRefGoogle Scholar
  22. D’Urso MG (2015) The gravity anomaly of a 2D polygonal body having density contrast given by polynomial functions. Surv Geophys 36(3):391–425CrossRefGoogle Scholar
  23. D’Urso MG (2016) A remark on the computation of the gravitational potential of masses with linearly varying density. In: Sneeuw N, Novak P, Crespi M, Sansò F (eds) 8th Hotine–Marussi international symposium on mathematical geodesy. International Association of Geodesy Symposia, SpringerGoogle Scholar
  24. D’Urso MG, Trotta S (2017) Gravity anomaly of polyhedral bodies having a polynomial density contrast. Surv Geophys 38(4):781–832CrossRefGoogle Scholar
  25. Droujinine A, Vasilevsky A, Evans R (2007) Feasibility of using full tensor gradient (FTG) data for detection of local lateral density contrasts during reservoir monitoring. Geophys J Int 169(3):795–820CrossRefGoogle Scholar
  26. Dubey CP, Tiwari VM (2016) Computation of the gravity field and its gradient: some applications. Comput Geosci 88:83–96CrossRefGoogle Scholar
  27. Forsberg R (1984) A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modelling. Ohio State Univ Columbus Dept Of Geodetic Science and Surveying, No. OSU/DGSS-355Google Scholar
  28. Gallardo-Delgado LA, Pérez-Flores MA, Gómez-Treviño E (2003) A versatile algorithm for joint 3D inversion of gravity and magnetic data. Geophysics 68(3):949–959CrossRefGoogle Scholar
  29. García-Abdeslem J (1992) Gravitational attraction of a rectangular prism with depth-dependent density. Geophysics 57(3):470–473CrossRefGoogle Scholar
  30. García-Abdeslem J (2005) The gravitational attraction of a right rectangular prism with density varying with depth following a cubic polynomial. Geophysics 70(6):J39–J42CrossRefGoogle Scholar
  31. Geng MX, Huang DN, Yang QJ, Liu YP (2014) 3D inversion of airborne gravity-gradiometry data using cokriging. Geophysics 79(4):G37–G47CrossRefGoogle Scholar
  32. Guo ZH, Guan ZN, Xiong SQ (2004) Cuboid ∆T and its gradient forward theoretical expressions without analytic odd points. Chin J Geophys 47(6):1277–1285CrossRefGoogle Scholar
  33. Haáz IB (1953) Relations between the potential of the attraction of the mass contained in a finite rectangular prism and its first and second derivatives. Geophys Trans II 7:57–66Google Scholar
  34. Hayes TJ, Tiampo KF, Fernández J, Rundle JB (2008) A gravity gradient method for characterizing the post-seismic deformation field for a finite fault. Geophys J Int 173(3):802–805CrossRefGoogle Scholar
  35. Holstein H (2003) Gravimagnetic anomaly formulas for polyhedra of spatially linear media. Geophysics 68(1):157–167CrossRefGoogle Scholar
  36. Holstein H, Ketteridge B (1996) Gravimetric analysis of uniform polyhedra. Geophysics 61(2):357–364CrossRefGoogle Scholar
  37. Hou ZL, Wei XH, Huang DN (2016) Fast density inversion solution for full tensor gravity gradiometry data. Pure Appl Geophys 173(2):509–523CrossRefGoogle Scholar
  38. Hudec MR, Jackson MPA, Schultz-Ela DD (2006) The paradox of minibasin subsidence into salt: clues to the evolution of crustal basins. Geol Soc Am Bull 121(1–2):201–221Google Scholar
  39. Jekeli C, Zhu L (2006) Comparison of methods to model the gravitational gradients from topographic data bases. Geophys J Int 166(3):999–1014CrossRefGoogle Scholar
  40. Jiang L, Zhang J, Feng ZB (2017) A versatile solution for the gravity anomaly of 3D prism-meshed bodies with depth-dependent density contrast. Geophysics 82(4):G77–G86CrossRefGoogle Scholar
  41. Jorgensen GJ, Kisabeth JL, Huffman AR, Sinton JB, Bell DW (2002) Method for gravity and magnetic data inversion using vector and tensor data with seismic imaging and geopressure prediction for oil, gas and mineral exploration and production. US, US6502037Google Scholar
  42. Kim S, Wessel P (2016) New analytic solutions for modeling vertical gravity gradient anomalies. Geochem Geophys Geosyst 17(5):1915–1924CrossRefGoogle Scholar
  43. Kwok YK (1991) Singularities in gravity computation for vertical cylinders and prisms. Geophys J Int 104(1):1–10CrossRefGoogle Scholar
  44. LaFehr TR, Nabighian MN (2012) Fundamentals of gravity exploration, society of exploration geophysicists.
  45. Leonardo U, Barbosa VCF (2012) Robust 3D gravity gradient inversion by planting anomalous densities. Geophysics 77(4):G55–G66CrossRefGoogle Scholar
  46. Li X, Chouteau M (1998) Three-dimensional gravity modeling in all space. Surv Geophys 19(4):339–368CrossRefGoogle Scholar
  47. Lu W, Qian J (2015) A local level-set method for 3D inversion of gravity-gradient data. Geophysics 80(1):G35–G51CrossRefGoogle Scholar
  48. Luo Y, Yao CL (2007) Forward modeling of gravity, gravity gradients, and magnetic anomalies due to complex bodies. J Chin Univ Geosci 18(3):280–286CrossRefGoogle Scholar
  49. Martinez C, Li Y (2011) Inversion of regional gravity gradient data over the Vredefort Impact Structure, South Africa. In: SEG technical program expanded abstracts 2011. Society of Exploration Geophysicists, pp. 841–845Google Scholar
  50. Martinez C, Li Y (2016) Denoising of gravity gradient data using an equivalent source technique. Geophysics 81(4):G67–G79CrossRefGoogle Scholar
  51. Martinez C, Li Y, Krahenbuhl R, Braga MA (2013) 3D inversion of airborne gravity gradiometry data in mineral exploration: a case study in the Quadrilatero Ferrifero, Brazil. Geophysics 78(1):B1–B11CrossRefGoogle Scholar
  52. Mataragio J, Kikley J (2009) Application of full tensor gradient invariants in detection of intrusion-hosted sulphide mineralization: implications for deposition mechanisms. First Break 27(7):95–98Google Scholar
  53. Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geodesy 74(7):552–560CrossRefGoogle Scholar
  54. Okabe M (1979) Analytical expressions for gravity anomalies due to homogeneous polyhedral bodies and translations into magnetic anomalies. Geophysics 44(4):730–741CrossRefGoogle Scholar
  55. Oliveira VC, Barbosa VCF (2013) 3-D radial gravity gradient inversion. Geophys J Int 195(2):883–902CrossRefGoogle Scholar
  56. Oruç B (2010) Depth estimation of simple causative sources from gravity gradient tensor invariants and vertical component. Pure appl Geophys 167(10):1259–1272CrossRefGoogle Scholar
  57. Oruç B (2011) Edge detection and depth estimation using a tilt angle map from gravity gradient data of the Kozaklı-Central Anatolia Region, Turkey. Pure Appl Geophys 168(10):1769–1780CrossRefGoogle Scholar
  58. Oruç B, Sertçelik İ, Kafadar Ö, Selim HH (2013) Structural interpretation of the Erzurum basin, eastern Turkey, using curvature gravity gradient tensor and gravity inversion of basement relief. J Appl Geophys 88(1):105–113CrossRefGoogle Scholar
  59. Pan Q, Liu DJ, Geng M, Cheng X, Wang X (2016) Euler deconvolution of the analytic signals of gravity gradient tensor for underground horizontal pipeline. In: 78th EAGE Conference and Exhibition 2016, EAGE, Extended abstract.
  60. Pedersen LB (1990) The gradient tensor of potential field anomalies: some implications on data collection and data processing of maps. Geophysics 55(12):1558–1566CrossRefGoogle Scholar
  61. Petrović S (1996) Determination of the potential of homogeneous polyhedral bodies using line integrals. J Geodesy 71(1):44–52CrossRefGoogle Scholar
  62. Pilkington M (2014) Evaluating the utility of gravity gradient tensor components. Geophysics 79(1):G1–G14CrossRefGoogle Scholar
  63. Pohanka V (1988) Optimum expression for computation of the gravity field of a homogeneous polyhedral body. Geophys Prospect 36(7):733–751CrossRefGoogle Scholar
  64. Prutkin I, Tenzer R (2009) The optimum expression for the gravitational potential of polyhedral bodies having a linearly varying density distribution. J Geodesy 83(12):1163–1170CrossRefGoogle Scholar
  65. Qin P, Huang D, Yuan Y, Geng M, Liu J (2016) Integrated gravity and gravity gradient 3D inversion using the non-linear conjugate gradient. J Appl Geophys 126:52–73CrossRefGoogle Scholar
  66. Rao DB (1990) Analysis of gravity anomalies of sedimentary basins by an asymmetrical trapezoidal model with quadratic density function. Geophysics 55(2):226–231CrossRefGoogle Scholar
  67. Rao CV, Raju ML, Chakravarthi V (1995) Gravity modelling of an interface above which the density contrast decreases hyperbolically with depth. J Appl Geophys 34(1):63–67CrossRefGoogle Scholar
  68. Ren ZY, Chen CJ, Pan KJ, Kalscheuer T, Maurer H, Tang J (2017) Gravity anomalies of arbitrary 3D polyhedral bodies with horizontal and vertical mass contrasts. Surv Geophys 38(2):479–502CrossRefGoogle Scholar
  69. Rim H, Li Y (2012) Single-hole imaging using borehole gravity gradiometry. Geophysics 77(5):G67–G76CrossRefGoogle Scholar
  70. Rim H, Li Y (2016) Gravity gradient tensor due to a cylinder. Geophysics 81(4):G59–G66CrossRefGoogle Scholar
  71. Saad AH (2006) Understanding gravity gradients—a tutorial. Lead Edge 25(8):942–949CrossRefGoogle Scholar
  72. Sastry RG, Gokula A (2016) Full gravity gradient tensor of a vertical pyramid model of flat top & bottom with depth-wish linear density variation II[C]//Symposium on the application of geophysics to engineering and environmental problems 2015. Soc Explor Geophys Environ Eng Geophys Soc 2016:294–301. Google Scholar
  73. Shi L, Li YH, Zhang EH (2015) A new approach for density contrast interface inversion based on the parabolic density function in the frequency domain. J Appl Geophys 116:1–9CrossRefGoogle Scholar
  74. Sun Y, Yang W, Zeng X, Zhang Z (2016) Edge enhancement of potential field data using spectral moments. Geophysics 81(1):G1–G11CrossRefGoogle Scholar
  75. Sykes TJS (1996) A correction for sediment load upon the ocean floor: uniform versus varying sediment density estimations—implications for isostatic correction. Mar Geol 133(1–2):35–49CrossRefGoogle Scholar
  76. Tsoulis D (2000) A note on the gravitational field of the right rectangular prism. Bollettino di Geodesia e Scienze Affini 59(1):21–35Google Scholar
  77. Tsoulis D (2012) Analytical computation of the full gravity tensor of a homogeneous arbitrarily shaped polyhedral source using line integrals. Geophysics 77(2):F1–F11CrossRefGoogle Scholar
  78. Tsoulis D, Petrovic S (2001) On the singularities of the gravity field of a homogeneous polyhedral body. Geophysics 66(2):535–539CrossRefGoogle Scholar
  79. Uieda L, Barbosa VCF (2012) 3D gravity gradient inversion by planting density anomalies. In: Eage conference and exhibition incorporating Spe Europec, pp 1–5Google Scholar
  80. Vasco DW (1989) Resolution and variance operators of gravity and gravity gradiometry. Geophysics 54(7):889–899CrossRefGoogle Scholar
  81. Verweij JM, Boxem TAP, Nelskamp S (2016) 3D spatial variation in vertical stress in on- and offshore Netherlands; integration of density log measurements and basin modeling results. Mar Pet Geol 78:870–882CrossRefGoogle Scholar
  82. Werner RA (2017) The solid angle hidden in polyhedron gravitation formulations. J Geod 91:307–328CrossRefGoogle Scholar
  83. While J, Biegert E, Jackson A (2009) Generalized sampling interpolation of noisy gravity/gravity gradient data. Geophys J Int 178(2):638–650CrossRefGoogle Scholar
  84. Wu LY, Chen LW (2016) Fourier forward modeling of vector and tensor gravity fields due to prismatic bodies with variable density contrast. Geophysics 81(1):G13–G26CrossRefGoogle Scholar
  85. Yuan Y, Yu QL (2015) Edge detection in potential-field gradient tensor data by use of improved horizontal analytical signal methods. Pure appl Geophys 172(2):461–472CrossRefGoogle Scholar
  86. 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(2):1176–1190CrossRefGoogle Scholar
  87. Zhang C, Mushayandebvu MF, Reid AB, Fairhead JD, Odegard ME (2000) Euler deconvolution of gravity tensor gradient data. Geophysics 65(2):512–520CrossRefGoogle Scholar
  88. Zhang J, Zhong BS, Zhou XX, Dai Y (2001) Gravity anomalies of 2-D bodies with variable density contrast. Geophysics 66(3):809–813CrossRefGoogle Scholar
  89. Zhdanov MS, Ellis R, Mukherjee S (2004) Three-dimensional regularized focusing inversion of gravity gradient tensor component data. Geophysics 69(4):1–4CrossRefGoogle Scholar
  90. Zhou XB (2009) 3D vector gravity potential and line integrals for the gravity anomaly of a rectangular prism with 3D variable density contrast. Geophysics 74(6):I43–I53CrossRefGoogle Scholar
  91. Zhou XB (2010) Analytic solution of the gravity anomaly of irregular 2D masses with density contrast varying as a 2D polynomial function. Geophysics 75(2):I11–I19CrossRefGoogle Scholar
  92. Zhou W (2015) Normalized full gradient of full tensor gravity gradient based on adaptive iterative Tikhonov regularization downward continuation. J Appl Geophys 118:75–83CrossRefGoogle Scholar
  93. Zwillinger D (2011) CRC standard mathematical tables and formulae, 32nd edn. CRC Press, Boca RatonCrossRefGoogle Scholar

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Authors and Affiliations

  • Li Jiang
    • 1
  • Jie Liu
    • 1
  • Jianzhong Zhang
    • 1
    • 2
  • Zhibing Feng
    • 1
  1. 1.Key Laboratory of Submarine Geosciences and Prospecting Techniques of the Ministry of Education, College of Marine GeosciencesOcean University of ChinaQingdaoChina
  2. 2.Evaluation and Detection Technology Laboratory of Marine Mineral Resources, Qingdao National Laboratory for Marine Science and TechnologyQingdaoChina

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