Applied Geophysics

, Volume 15, Issue 3–4, pp 420–431 | Cite as

Irregular surface seismic forward modeling by a body-fitted rotated–staggered-grid finite-difference method

  • Jing-Wang ChengEmail author
  • Na Fan
  • You-Yuan Zhang
  • Xiao-Chun Lü


Finite-difference (FD) methods are widely used in seismic forward modeling owing to their computational efficiency but are not readily applicable to irregular topographies. Thus, several FD methods based on the transformation to curvilinear coordinates using body-fitted grids have been proposed, e.g., stand staggered grid (SSG) with interpolation, nonstaggered grid, rotated staggered grid (RSG), and fully staggered. The FD based on the RSG is somewhat superior to others because it satisfies the spatial distribution of the wave equation without additional memory and computational requirements; furthermore, it is simpler to implement. We use the RSG FD method to transform the firstorder stress–velocity equation in the curvilinear coordinates system and introduce the highprecision adaptive, unilateral mimetic finite-difference (UMFD) method to process the freeboundary conditions of an irregular surface. The numerical results suggest that the precision of the solution is higher than that of the vacuum formalism. When the minimum wavelength is low, UMFD avoids the surface wave dispersion. We compare FD methods based on RSG, SEM, and nonstaggered grid and infer that all simulation results are consistent but the computational efficiency of the RSG FD method is higher than the rest.


Finite difference forward modeling grid staggered rotated body-fitted surface free boundary 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



We thank Komatitsch et al. for providing the open source code of the 2D spectral element method (SPECFEM2D) (


  1. Appelo, D., and Petersson, N. A., 2009, A Stable Finite Difference Method for the Elastic Wave Equation on Complex Geometries with Free Surfaces: Communications in Computational Physics, 5(1), 84–107.Google Scholar
  2. Bogey, C., and Bailly, C., 2004, A family of low dispersive and low dissipative explicit schemes for flow and noise computations: Journal of Computational Physics, 194(1), 194–214.Google Scholar
  3. Bohlen, T., and Saenger, E. H., 2006, Accuracy of heterogeneous staggered–grid finite–difference modeling of Rayleigh waves: Geophysics, 71(4), T109–T115.Google Scholar
  4. Castillo, J. E., Hyman, J. M., Shashkov, M., and Steinberg, S., 2001, Fourth–and sixth–order conservative finite difference approximations of the divergence and gradient: Applied Numerical Mathematics, 37(1–2), 171–187.Google Scholar
  5. Chaljub, E., Komatitsch, D., and Vilotte, J. P., 1998, The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures: Bulletin of the Seismological Society of America, 88(2), 368–392.Google Scholar
  6. Chen, H., Wang, X., and Zhao, H., 2006, A rotated staggered grid finite–difference with the absorbing boundary condition of a perfectly matched layer: Chinese Science Bulletin, 51(19), 2304–2314.Google Scholar
  7. Fan, N., Zhao, L. F., Xie, X. B., Ge, Z., and Yao, Z. X., 2016, Two–dimensional time–domain finite–difference modeling for viscoelastic seismic wave propagation: Geophysical Journal International, 206(3), 1539–1551.Google Scholar
  8. Hestholm, S., 2003, Elastic wave modeling with free surfaces: Stability of long simulation: Geophysics, 68(1), 314–321.Google Scholar
  9. Hestholm, S., and Ruud, B., 1998, 3–D finite–difference elastic wave modeling including surface topography: Geophysics, 63(2), 613–622.Google Scholar
  10. Huang, J. P., Qu, Y. M., Li, Q. Y., Li, Z. C., Li, D. L., and Bu, C. C., 2015, Variable–coordinate forward modeling of irregular surface based on dual–variable grid: Applied Geophysics, 12(1), 101–110.Google Scholar
  11. Komatitsch, D., and Tromp, J., 1999, Introduction to the spectral element method for three–dimensional seismic wave propagation: Geophysical Journal of the Royal Astronomical Society, 139(3), 806–822.Google Scholar
  12. Lan, H., and Zhang, Z., 2012a, Three–Dimensional Wave–Field Simulation in Heterogeneous Transversely Isotropic Medium with Irregular Free Surface: Bulletin of the Seismological Society of America, 101(3), 1354–1370.Google Scholar
  13. Lan, H. Q., and Zhang, Z. J., 2012b, Seismic wavefield modeling in media with fluid–filled fractures and surface topography: Applied Geophysics, 9(3), 301–312.Google Scholar
  14. Levander, A. R., 1988, Fourth–order finite–difference P–SV seismograms: Geophysics, 53(11), 1425–1436.Google Scholar
  15. Li, Q. Y., Huang, J. P., Li, Z. C., Li, N., Wang, C. Q., and Zhang, Y. Y., 2015, Undulating surface body–fitted grid seismic modeling based on fully staggered–grid mimetic finite difference: Oil Geophysical Prospecting, 50(4), 633–642.Google Scholar
  16. Lisitsa, V., and Vishnevsky, D., 2010, Lebedev scheme for the numerical simulation of wave propagation in 3D anisotropic elasticity double dagger: Geophysical Prospecting, 58(4), 619–635.Google Scholar
  17. Pitarka, A., and Irikura, K., 1996, Modeling 3D surface topography by finite–difference method: Kobe–JMA Station Site, Japan, Case Study: Geophysical Research Letters, 23(20), 2729–2732.Google Scholar
  18. Qiu, L., Tian, G., Shi, Z. J., and Shen, H. L., 2012, Finitedifference method for seismic wave numerical simulation in presence of topography—In generally orthogonal curvilinear coordinate system: Journal of Zhejiang University, 46(10), 1923–1930.Google Scholar
  19. Robertsson, J. O. A., 1996, A numerical free–surface condition for elastic/viscoelastic finite–difference modeling in the presence of topography: Geophysics, 61(6), 1921–1934.Google Scholar
  20. Rojas, O., Day, S., Castillo, J., and Dalguer, L. A., 2008, Modelling of rupture propagation using high–order mimetic finite differences: Geophysical Journal of the Royal Astronomical Society, 172(2), 631–650.Google Scholar
  21. Rojas, O., Otero, B., Castillo, J. E., and Day, S. M., 2014, Low dispersive modeling of Rayleigh waves on partly staggered grids: Computational Geosciences, 18(1), 29–43.Google Scholar
  22. Saenger, E. H., and Bohlen, T., 2004, Finite–difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid: Geophysics, 69(2), 583–591.Google Scholar
  23. Saenger, E. H., Gold, N., & Shapiro, S. A., 2000, Modeling the propagation of elastic waves using a modified finitedifference grid: Wave Motion, 31(1), 77–92.Google Scholar
  24. Tarrass, I., Giraud, L., and Thore, P., 2011, New curvilinear scheme for elastic wave propagation in presence of curved topography: Geophysical Prospecting, 59(5), 889–906.Google Scholar
  25. Thompson, J. F., Warsi, Z. U. A., and Mastin, C. W., 1985, Numerical grid generation: foundations and applications: Elsevier Science Publishing Co Inc, North–Holland, 1–300.Google Scholar
  26. Yan, H. Y., and Liu, Y., 2012, High–Order Finite–Difference Numerical Modeling of Wave Propagation in Viscoelastic TTI Media Using Rotated Staggered Grid: Chinese Journal of Geophysics, 55(2), 252–265.Google Scholar
  27. Yang, Y., Huang, J. P., Lei, J. S., Li, Z. C., Tian, K., and Li, Q. Y., 2016, Numerical simulation of Lebedev grid for viscoelastic media with irregular free–surface: Oil Geophysical Prospecting, 51(4), 698–706.Google Scholar
  28. Zhang, W., and Chen, X., 2006, Traction image method for irregular free surface boundaries in finite difference seismic wave simulation: Geophysical Journal International, 167(1), 337–353.Google Scholar
  29. Zhang, W., Zhang, Z., and Chen, X., 2012, Threedimensional elastic wave numerical modelling in the presence of surface topography by a collocatedgrid finite–difference method on curvilinear grids: Geophysical Journal International, 190(1), 358–378.Google Scholar
  30. Zhang, Y., Ping, P., and Zhang, S. X., 2017, Finitedifference modeling of surface waves in poroelastic media and stress mirror conditions: Applied Geophysics, 14(1), 105–114.Google Scholar

Copyright information

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

Authors and Affiliations

  • Jing-Wang Cheng
    • 1
    • 2
    Email author
  • Na Fan
    • 1
  • You-Yuan Zhang
    • 1
  • Xiao-Chun Lü
    • 3
  1. 1.Geophysics and Oil Resource InstituteYangtze UniversityWuhanChina
  2. 2.Key Laboratory of Exploration Technologies for oil and Gas Resources, Ministry of EducationYangtze UniversityWuhanChina
  3. 3.College of Resources and EnvironmentNorth China University of Water Resources and Electric PowerZhengzhouChina

Personalised recommendations