Applied Geophysics

, Volume 15, Issue 3–4, pp 513–523 | Cite as

Three-dimensional numerical modeling of gravity anomalies based on Poisson equation in space-wavenumber mixed domain

  • Shi-Kun Dai
  • Dong-Dong ZhaoEmail author
  • Qian-Jiang Zhang
  • Kun Li
  • Qing-Rui Chen
  • Xu-Long Wang


In gravity-anomaly-based prospecting, the computational and memory requirements for practical numerical modeling are potentially enormous. Achieving an efficient and precise inversion for gravity anomaly imaging over large-scale and complex terrain requires additional methods. To this end, we have proposed a new topography-capable 3D numerical modeling method for gravity anomalies in space-wavenumber mixed domain. By performing a two-dimensional Fourier transform in the horizontal directions, threedimensional partial differential equations in the spatial domain were transformed into a group of independent, one-dimensional differential equations engaged with different wave numbers. These independent differential equations are highly parallel across different wave numbers. This method preserves the vertical component in the space domain, which is beneficial when modeling complex topography. The finite element method was used to solve the transformed differential equations with different wave numbers, and the efficiency of solving fixedbandwidth linear equations was further improved by a chasing method. In a synthetic test, a prism model was used to verify the accuracy and reliability of the proposed algorithm by comparing the numerical solution with the analytical solution. We studied the computational precision and efficiency with and without topography using different Fourier transform methods. The results showed that the Guass-FFT method has higher numerical precision, while the standard FFT method is superior, in terms of computation time, for inversion and quantitative interpretation under complicated terrain.


Topography gravity anomaly space-wavenumber mixing domain threedimensional numerical modeling 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors are very grateful to the three reviewers and editor-in-chief Fan Weicui for their critiques, helpful comments, and valuable suggestions which improved this manuscript significantly. In addition, we would like to thank Prof. Xu Yungui and Associate Prof. Chen Longwei for their guidance and help during the development of this paper.


  1. Blakely, R. J., 1996, Potential theory in gravity and magnetic applications: Cambridge University Press, London, 128–153.Google Scholar
  2. Cai, Y., and Wang, C. Y., 2005, Fast finite–element calculation of gravity anomaly in complex geological regions: Geophysical Journal of the Royal Astronomical Society, 162(3), 696–708.Google Scholar
  3. Cg Farquharson, C. M., 2009, Three–dimensional modelling of gravity data using finite differences: Journal of Applied Geophysics, 68(3), 417–422.Google Scholar
  4. Chakravarthi, V., Raghuram, H. M., and Singh, S. B., 2002, 3–d forward gravity modeling of basement interfaces above which the density contrast varies continuously with depth: Computers & Geosciences, 28(1), 53–57.Google Scholar
  5. Chai, Y., and Hinze, W. J., 1988, Gravity inversion of interface above which the density contrast waries exponentially with depth: Geophysics, 53(6), 837–845.Google Scholar
  6. Chai, Y. P., 1997, Shift sampling theory and its applications: Petroleum Industry Press, Beijing, 15–74.Google Scholar
  7. Feng, R., 1986, The potential field calculation of the three–dimensional physical distribution: Chinese J. Geophys. (in Chinese), 29(4), 399–406.Google Scholar
  8. Forsberg, R., 1985, Gravity field terrain effect computations by fft: Bulletin Géodésique, 59(4), 342–360.Google Scholar
  9. 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–J42.Google Scholar
  10. Granser, H., 1987, Nonlinear inversion of gravity data using the schmidt–lichtenstein approach: Geophysics, 52(1), 88–93.Google Scholar
  11. Holstein, H., 2002, Gravimagnetic similarity in anomaly formulas for uniform polyhedra: Geophysics, 67(4), 1125–1133.Google Scholar
  12. Jahandari, H., and Farquharson, C. G., 2013, Forward modeling of gravity data using finite–volume and finite–element methods on unstructured grids: Geophysics, 78(3), G69–G80.Google Scholar
  13. Jian, Z., Wang, C. Y., Shi, Y., Cai, Y., Chi, W. C., and Douglas, D., 2004, Three–dimensional crustal structure in central taiwan from gravity inversion with a parallel genetic algorithm: Geophysics, 69(4), 917–924.Google Scholar
  14. Nagy, D., 1966, The gravitational attraction of a right rectangular prism: Geophysics, 31(2), 362–371.Google Scholar
  15. Nagy, D., Papp, G., and Benedek, J., 2000, The gravitational potential and its derivatives for the prism: Journal of Geodesy, 74(7–8), 552–560.Google Scholar
  16. Okabe, M., 1979, Analytical expressions for gravity anomalies due to homogeneous polyhedral bodies and translations into magnetic anomalies: Geophysics, 44(4), 730–741.Google Scholar
  17. Paul, M. K., 1974, The gravity effect of a homogeneous polyhedron for three–dimensional interpretation: Pure & Applied Geophysics, 112(3), 553–561.Google Scholar
  18. Pedersen, L. B., 1978a, A statistical analysis of potential fields using a vertical circular cylinder and a dike: Geophysics, 43(5), 943–953.Google Scholar
  19. Pedersen, L. B., 1978b, Wavenumber domain expressions for potential fields from arbitrary 2–, 21/2 and 3–dimensional bodies: Geophysics, 43(3), 626–630.Google Scholar
  20. Pedersen, L. B., 1985, The gravity and magnetic feilds from ellipsoidal bodies in the wavenumber domain: Geophysical Prospecting, 33(2), 263–281.Google Scholar
  21. Plouff, D., 1976, Gravity and magnetic fields of polygonal prisms and application to magnetic terrain corrections: Geophysics, 41(4), 727–741.Google Scholar
  22. Rao, D. B., Prakash, M. J., and Babu, N. R., 1993, Gravity interpretation using fourier–transforms and simple geometrical models with exponential density: Geophysics, 58(8), 1074–1083.Google Scholar
  23. Singh, B., and Guptasarma, D., 2001, New method for fast computation of gravity and magnetic anomalies from arbitrary polyhedra: Geophysics, 66(66), 521–526.Google Scholar
  24. Talwani, M., 1960, Rapid computation of gravitational attraction of three–dimensional bodies of arbitrary shape: Geophysics, 25(1), 203–225.Google Scholar
  25. Tontini F. C., Cocchi, L., and Carmisciano, C., 2009, Rapid 3–D forward model of fields with application to the palinuro seamount gravity anomaly(sourthern tyrrhenian sea, Italy): Journal of Geophysical Research: Solid Earth (1978–2012), 114(B2), 1205–1222.Google Scholar
  26. Wu, X. Z., 1983, The computation of spectrum of potential field due to 3–D arbitrary bodies with physical parameters varying with depth: Chinese J. Geophys. (in Chinese), 26(02), 177–187.Google Scholar
  27. Wu, L. Y., and Tian, G., 2014, High–precision Fourier forward modeling of potential field: Geophysics, 79(5), G59–G68.Google Scholar
  28. Wu, L. Y, 2016, Efficient modelling of gravity effects due to topographic masses using the Gauss–FFT method: Geophysical Journal International, 205(1), 160–178.Google Scholar
  29. Xiong, G. C., 1984, Some problems about 3–D fourier transforms of the gravity and magnetic fields: Chinese J. Geophys (in Chinese), 1(01), 103–109.Google Scholar
  30. Xu, S. Z., 1994, The Finite Element Method in Geophysics: Science Press, Beijing, 6–98.Google Scholar
  31. Zhao, S. K., and Yedlin, M. J., 1991, Chebyshev expansions for the solution of the forward gravity problem: Geophysical Prospecting, 39(6), 783–802.Google Scholar
  32. Zeng, H. L., 2005, Gravity field and gravity exploration: Geological Publishing House, Beijing, 1–35.Google Scholar

Copyright information

© Editorial Office of Applied Geophysics and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Shi-Kun Dai
    • 1
    • 2
  • Dong-Dong Zhao
    • 1
    • 2
    Email author
  • Qian-Jiang Zhang
    • 3
  • Kun Li
    • 1
    • 2
  • Qing-Rui Chen
    • 1
    • 2
  • Xu-Long Wang
    • 1
    • 2
  1. 1.School of Geosciences and Info-physicsCentral South UniversityChangshaChina
  2. 2.Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment MonitoringMinistry of Education (Central South University)ChangshaChina
  3. 3.School of College of Earth Sciences of Guilin university of technologyGuilinChina

Personalised recommendations