1887
Volume 49, Issue 4
  • E-ISSN: 1365-2478

Abstract

New formulations of boundary conditions at an arbitrary two‐dimensional (2D) free‐surface topography are derived. The top of a curved grid represents the free‐surface topography while the grid's interior represents the physical medium. The velocity–stress version of the viscoelastic wave equations is assumed to be valid in this grid. However, the rectangular grid version attained by grid transformation is used to model wave propagation in this work in order to achieve the numerical discretization. We show the detailed solution of the particle velocities at the free surface resulting from discretizing the boundary conditions by second‐order finite‐differences (FDs). The resulting system of equations is spatially unconditionally stable. The FD order is gradually increased with depth up to eighth order inside the medium. Staggered grids are used in both space and time, and the second‐order leap‐frog and Crank–Nicholson methods are used for time‐stepping. We simulate point sources at the surface of a homogeneous medium with a plane free surface containing a hill and a trench. Applying parameters representing exploration surveys, we present examples with a randomly realized surface topography generated by a 1D von Kármán function of order 1. Viscoelastic simulations are presented using this surface with a homogeneous medium and with a layered, randomized medium realization, all generating significant scattering.

Loading

Article metrics loading...

/content/journals/10.1046/j.1365-2478.2001.00268.x
2001-12-21
2024-03-28
Loading full text...

Full text loading...

References

  1. BlanchJ.O., RobertssonJ.O.A., SymesW.W.1995. Modelling of a constant Q: methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique. Geophysics60, 176–184.
    [Google Scholar]
  2. BouchonM., SchultzC.A., ToksözM.N.1996. Effect of three‐dimensional topography on seismic motion. Journal of Geophysical Research101, 5835–5846.
    [Google Scholar]
  3. CarcioneJ.M., KosloffD., KosloffR.1988. Wave propagation simulation in a linear viscoelastic medium. Geophysical Journal of the Royal Astronomical Society95, 597–611.
    [Google Scholar]
  4. CerjanC., KosloffD., KosloffR., ReshefM.1985. A nonreflecting boundary condition for discrete acoustic and elastic wave equations. Geophysics50, 705–708.
    [Google Scholar]
  5. CharretteE.E.I.1991. Elastic wave scattering in laterally inhomogeneous media. PhD thesis, Massachusetts Institute of Technology.
  6. FornbergB.1988a. Generation of finite difference formulas on arbitrary spaced grids. Mathematics of Computation51, 699–706.
    [Google Scholar]
  7. FornbergB.1988b. The pseudospectral method: accurate representation of interfaces in elastic wave calculations. Geophysics53, 625–637.
    [Google Scholar]
  8. FrankelA. & ClaytonR.W.1986. Finite difference simulations of seismic scattering: implications for the propagation of short‐period seismic waves in the crust and models of crustal heterogeneity. Journal of Geophysical Research91, 6465–6489.
    [Google Scholar]
  9. HestholmS.O., HusebyeE.S., RuudB.O.1994. Seismic wave propagation in complex crust–upper mantle media using 2D finite‐difference synthetics. Geophysical Journal International118, 643–670.
    [Google Scholar]
  10. HestholmS.O. & RuudB.O.1994. 2D finite‐difference elastic wave modelling including surface topography. Geophysical Prospecting42, 371–390.
    [Google Scholar]
  11. HestholmS.O. & RuudB.O.1998. 3D finite‐difference elastic wave modelling including surface topography. Geophysics63, 613–622.
    [Google Scholar]
  12. HestholmS.O. & RuudB.O.2000. 2D finite‐difference viscoelastic wave modelling including surface topography. Geophysical Prospecting48, 341–373.DOI: 10.1046/j.1365-2478.2000.00185.x
    [Google Scholar]
  13. HigdonR.L.1990. Radiation boundary conditions for elastic wave propagation. SIAM Journal of Numerical Analysis27, 831–870.
    [Google Scholar]
  14. HolligerK. & LevanderA.R.1992. A stochastic view of lower crustal fabric based on evidence from the ivrea zone. Geophysical Research Letters19, 1153–1156.
    [Google Scholar]
  15. ImhofM.G.1996. Scattering of elastic waves using non‐orthogonal expansions. PhD thesis, Massachusetts Institute of Technology.
  16. JihR.S., McLaughlinK.L., Der ZA.1988. Free‐boundary conditions of arbitrary topography in a two‐dimensional explicit finite difference scheme. Geophysics53, 1045–1055.
    [Google Scholar]
  17. KindelanM., KamelA., SguazzeroP.1990. On the construction and efficiency of staggered numerical differentiators for the wave equation. Geophysics55, 107–110.
    [Google Scholar]
  18. KomatitschM., CoutelF., MoraP.1996. Tensorial formulation of the wave equation for modelling curved interfaces. Geophysical Journal International127, 156–168.
    [Google Scholar]
  19. LevanderA.R. & HolligerK.1992. Small‐scale heterogeneity and large‐scale velocity structure of the continental crust. Journal of Geophysical Research97, 8797–8804.
    [Google Scholar]
  20. OhminatoT. & ChouetB.A.1997. A free‐surface boundary condition for including 3D topography in the finite difference method. Bulletin of the Seismological Society of America87, 494–515.
    [Google Scholar]
  21. PengC. & ToksözM.N.1994. An optimal absorbing boundary condition for finite difference modelling of acoustic and elastic wave propagation. Journal of the Acoustical Society of America95, 733–745.
    [Google Scholar]
  22. RenautR.A. & PetersenJ.1989. Stability of wide‐angle absorbing boundary conditions for the wave equation. Geophysics54, 1153–1163.
    [Google Scholar]
  23. RobertssonJ.O.A.1996. A numerical free‐surface condition for elastic/viscoelastic finite‐difference modelling in the presence of topography. Geophysics61, 1921–1934.
    [Google Scholar]
  24. RobertssonJ.O.A., BlanchJ.O., SymesW.W.1994. Viscoelastic finite‐difference modelling. Geophysics59, 1444–1456.
    [Google Scholar]
  25. RuudB.O., HusebyeE.S., HestholmS.O.1993. Rg observations from four continents: inverse‐ and forward‐modelling experiments. Geophysical Journal International114, 465–472.
    [Google Scholar]
  26. Sanchez‐SesmaF.J. & CampilloM.1991. Diffraction of P, SV and Rayleigh waves by topographic features: a boundary integral formulation. Bulletin of the Seismological Society of America81, 2234–2253.
    [Google Scholar]
  27. SimoneA. & HestholmS.1998. Instabilities in applying absorbing boundary conditions to high order seismic modelling algorithms. Geophysics63, 1017–1023.
    [Google Scholar]
  28. TessmerE. & KosloffD.1994. 3D elastic modelling with surface topography by a Chebychev spectral method. Geophysics59, 464–473.
    [Google Scholar]
  29. TessmerE., KosloffD., BehleA.1992. Elastic wave propagation simulation in the presence of surface topography. Geophysical Journal International108, 621–632.
    [Google Scholar]
  30. XuH., DayS.M., MinsterJ.B.H.1999. Two‐dimensional linear and nonlinear wave propagation in a half‐space. Bulletin of the Seismological Society of America89, 903–917.
    [Google Scholar]
  31. XuT. & McMechanG.A.1995. Composite memory variables for viscoelastic synthetic seismograms. Geophysical Journal International121, 634–639.
    [Google Scholar]
  32. XuT. & McMechanG.A.1998. Efficient 3D viscoelastic modelling with application to near‐surface land seismic data. Geophysics63, 601–612.
    [Google Scholar]
http://instance.metastore.ingenta.com/content/journals/10.1046/j.1365-2478.2001.00268.x
Loading
/content/journals/10.1046/j.1365-2478.2001.00268.x
Loading

Data & Media loading...

  • Article Type: Research Article

Most Cited This Month Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error