1887
Volume 58, Issue 4
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

This paper presents a Lebedev finite difference scheme on staggered grids for the numerical simulation of wave propagation in an arbitrary 3D anisotropic elastic media. The main concept of the scheme is the definition of all the components of each tensor (vector) appearing in the elastic wave equation at the corresponding grid points, i.e., all of the stresses are stored in one set of nodes while all of the velocity components are stored in another. Meanwhile, the derivatives with respect to the spatial directions are approximated to the second order on two‐point stencils. The second‐order scheme is presented for the sake of simplicity and it is easy to expand to a higher order.

Another approach, widely‐known as the rotated staggered grid scheme, is based on the same concept; therefore, this paper contains a detailed comparative analysis of the two schemes. It is shown that the dispersion condition of the Lebedev scheme is less restrictive than that of the rotated staggered grid scheme, while the stability criteria lead to approximately equal time stepping for the two approaches. The main advantage of the proposed scheme is its reduced computational memory requirements. Due to a less restrictive dispersion condition and the way the media parameters are stored, the Lebedev scheme requires only one‐third to two‐thirds of the computer memory required by the rotated staggered grid scheme. At the same time, the number of floating point operations performed by the Lebedev scheme is higher than that for the rotated staggered grid scheme.

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2010-01-26
2024-04-18
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References

  1. BackusG.E.1962. Long‐wave elastic anisotropy produced by horizontal layering. Journal of Geophysical Research67, 4427–4440.
    [Google Scholar]
  2. BansalR. and SenM.K.2008. Finite‐difference modelling of S‐wave splitting in anisotropic media. Geophysical Prospecting56, 293–312.
    [Google Scholar]
  3. BecacheE., FauqueuxS. and JolyP.2003. Stability of perfectly matched layers, group velocities and anisotropic waves. Journal of Computational Physics188, 399–433.
    [Google Scholar]
  4. BerengerJ.‐P.1994. A perfectly matched layer for the absorption of electromagnetic waves. Journal of Computational Physics114, 185–200.
    [Google Scholar]
  5. CohenG.1994. Méthodes numériques d'ordre élevé pour les ondes en régime transitoire . INRIA.
    [Google Scholar]
  6. CohenG. and JolyP.1996. Construction analysis of fourth‐order finite difference schemes for the acoustic wave equation in nonhomogeneous media. SIAM Journal on Numerical Analysis33, 1266–1302.
    [Google Scholar]
  7. DengX. and MaekawaH.1997. Compact high‐order accurate nonlinear schemes. Journal of Computational Physics130, 77–91.
    [Google Scholar]
  8. DeubelbeissY. and KausB.J.P.2008. Comparison of Eulerian and Lagrangian numerical techniques for the Stokes equations in the presence of strongly varying viscosity. Physics of the Earth and Planetary Interiors171, 92–111.
    [Google Scholar]
  9. FengK.‐A., TengC.‐H. and ChenM.‐H.2007. A pseudospectral penalty scheme for 2D isotropic elastic wave computations. Journal of Scientific Computing33, 313–348.
    [Google Scholar]
  10. GravesR.W.1996. Simulating seismic wave propagation in 3D elastic media using staggered‐grid finite difference. Bulletin of the Seismological Society of America86, 1091–1106.
    [Google Scholar]
  11. GrechkaV. and KachanovM.2006. Effective elasticity of rocks with closely spaced and intersecting cracks. Geophysics71, D85–D91.
    [Google Scholar]
  12. GustafssonB. and WahlundP.2004. Time compact difference methods for wave propagation in discontinuous media. SIAM Journal on Scientific Computing26, 272–293.
    [Google Scholar]
  13. IgelH., MoraP. and RiolletB.1995. Anisotropic wave propagation through finite‐difference grids. Geophysics60, 1203–1216.
    [Google Scholar]
  14. KaserM. and DumbserM.2006. An arbitrary high‐order discontinuous Galerkin method for elastic waves on unstructured meshes – I. The two‐dimensional isotropic case with external source terms. Geophysical Journal International166, 855–877.
    [Google Scholar]
  15. KrügerO.S., SaengerE.H., OatesS.J. and ShapiroS.A.2007. A numerical study on reflection coefficients of fractured media. Geophysics72, D61–D67.
    [Google Scholar]
  16. LebedevV.I.1964. Difference analogues of orthogonal decompositions of basic differential operators and some boundary value problems. I. USSR Computational Mathematics and Mathematical Physics4, 449–465.
    [Google Scholar]
  17. LevanderA.R.1988. Fourth‐order finite‐difference P‐SV seismograms. Geophysics53, 1425–1436.
    [Google Scholar]
  18. LisitsaV.2007. Lebedev scheme for anisotropic elastic problems. Proceedings of 8th International Conference on Theoretical and Computational Acoustics , 331–341. ISBN 9789608975842.
  19. LisitsaV. and LysE.2009. Reflectionless truncation of target area for axially symmetric anisotropic elasticity. Journal of Computational and Applied Mathematics. doi:10.1016/j.cam.2009.08.031
    [Google Scholar]
  20. LiuE., HudsonJ.A. and PointerT.2000. Equivalent medium representation of fractured rock. Journal of Geophysical Research-Solid Earth105, 2981–3000.
    [Google Scholar]
  21. Meza‐FajardoK.C. and PapageorgiouA.S.2008. A nonconvolutional, split‐field, perfectly matched layer for wave propagation in isotropic and anisotropic elastic media: Stability analysis. Bulletin of the Seismological Society of America98, 1811–1836.
    [Google Scholar]
  22. MoczoP., KristekJ., VavrycukV., ArchuletaR.J. and HaladaL.2002. 3D heterogeneous staggered‐grid finite‐difference modeling of seismic motion with volume harmonic and arithmetic averaging of elastic moduli and densities. Bulletin of the Seismological Society of America92, 3042–3066.
    [Google Scholar]
  23. PissarenkoD., ReshetovaG.V. and TcheverdaV.A.2009. 3D finite‐difference synthetic acoustic logging in cylindrical coordinates. Geophysical Prospecting57, 367–377.
    [Google Scholar]
  24. SaengerE.H. and BohlenT.2004. Finite‐difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid. Geophysics69, 583–591.
    [Google Scholar]
  25. SaengerE.H., GoldN. and ShapiroS.A.2000. Modeling the propagation of the elastic waves using a modified finite‐difference grid. Wave motion31, 77–92.
    [Google Scholar]
  26. SamarskiiA.A.2001. The Theory of Difference Schemes . CRC Press. ISBN 0824704681.
    [Google Scholar]
  27. SayersC.M.1999. Stress‐dependent seismic anisotropy of shales. Geophysics64, 93–98.
    [Google Scholar]
  28. SchoenbergM. and MuirF.1989. A calculus for finely layered anisotropic media. Geophysics54, 581–589.
    [Google Scholar]
  29. SchoenbergM. and SayersC.M.1995. Seismic anisotropy of fractured rock. Geophysics60, 204–211.
    [Google Scholar]
  30. SymesW.W., TerentyevI.S. and VdovinaT.W.2008. Gridding requirements for accurate finite difference simulation. 78th SEG meeting, Las Vegas , Nevada , USA , Expanded Abstracts, 2077–2080.
  31. SymesW. and VdovinaT.2009. Interface error analysis for numerical wave propagation. Computational Geosciences13, 363–371.
    [Google Scholar]
  32. ThomsenL.1986. Weak elastic anisotropy. Geophysics51, 1954–1966.
    [Google Scholar]
  33. TrompJ., KomatitschD. and LiuQ.2008. Spectral‐element and adjoint methods in seismology. Communications in Computational Physics3, 1–32.
    [Google Scholar]
  34. VirieuxJ.1986. P‐SV wave propagation in heterogeneous media: Velocity – stress finite‐difference method. Geophysics51, 889–901.
    [Google Scholar]
  35. WintersteinD.F.1990. Velocity anisotropy terminology for geophysicists. Geophysics55, 1070–1088.
    [Google Scholar]
  36. ZahradnikJ., MoczoP. and HornF.1993. Testing four elastic finite‐difference schemes for behavior at discontinuities. Bulletin of the Seismological Society of America83, 107–129.
    [Google Scholar]
  37. ZahradnikJ. and PrioloE.1995. Heterogeneous formulations of elastodynamic equations and finite‐difference schemes. Geophysical Journal International120, 663–676.
    [Google Scholar]
  38. ZhangJ. and VerschuurD.J.2002. Elastic wave propagation in heterogeneous anisotropic media using the lumped finite‐element method. Geophysics67, 625–638.
    [Google Scholar]
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  • Article Type: Research Article
Keyword(s): 3D anisotropic elastic media; Finite difference schemes; Wave propagation

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