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Computational Geosciences

, Volume 20, Issue 1, pp 245–264 | Cite as

Finite-difference strategy for elastic wave modelling on curved staggered grids

  • C. A. Pérez Solano
  • D. Donno
  • H. Chauris
ORIGINAL PAPER

Abstract

Waveform modelling is essential for seismic imaging and inversion. Because including more physical characteristics can potentially yield more accurate Earth models, we analyse strategies for elastic seismic wave propagation modelling including topography. We focus on using finite differences on modified staggered grids. Computational grids can be curved to fit the topography using distribution functions. With the chain rule, the elasto-dynamic formulation is adapted to be solved directly on curved staggered grids. The chain-rule approach is computationally less expensive than the tensorial approach for finite differences below the 6th order, but more expensive than the classical approach for flat topography (i.e. rectangular staggered grids). Free-surface conditions are evaluated and implemented according to the stress image method. Non-reflective boundary conditions are simulated via a Convolutional Perfect Matching Layer. This implementation does not generate spurious diffractions when the free-surface topography is not horizontal, as long as the topography is smoothly curved. Optimal results are obtained when the angle between grid lines at the free surface is orthogonal. The chain-rule implementation shows high accuracy when compared to the analytical solution in the case of the Lamb’s problem, Garvin’s problem and elastic interface.

Keywords

Elastic wave propagation Seismic modelling Topography Curved grids 

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References

  1. 1.
    Aki, K., Richards, P.G.: Quantitative seismology. W.H. Freeman & Co (1980)Google Scholar
  2. 2.
    Appelö, D., Petersson, N.A.: A stable finite difference method for the elastic wave equation on complex geometries with free surfaces. Commun. Comput. Phys. 5(1), 86–107 (2009)Google Scholar
  3. 3.
    Bérenger, J.P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200 (1994)CrossRefGoogle Scholar
  4. 4.
    Berg, P., If, P., Nilsen, P., Skovgaard, O.: Analytical reference solutions: advanced seismic modeling. In: Helbig, K. (ed.) Modeling the Earth for Oil Exploration, pp. 421–427. Pergamon Press (1994)Google Scholar
  5. 5.
    Bohlen, T., Saenger, E.H.: Accuracy of heterogeneous staggered-grid finite-difference modeling of Rayleigh waves. Geophysics 71(4), 109–115 (2006)CrossRefGoogle Scholar
  6. 6.
    Brossier, R., Virieux, J., Operto, S.: Parsimonious finite-volume frequency-domain method for 2-D p-SV-wave modelling. Geophys. J. Int. 175, 541–559 (2008)CrossRefGoogle Scholar
  7. 7.
    Chapman, C.H.: Fundamental of seismic waves propagation. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  8. 8.
    Cohen, G.: Méthodes numériques d’ordre élevé pour les ondes en régime transitoire. INRIA (1994)Google Scholar
  9. 9.
    Collino, F., Tsogka, C.: Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media. Geophysics 66, 294–307 (2001)CrossRefGoogle Scholar
  10. 10.
    de la Puente, J., Ferrer, M., Hanzich, M., Castillo, J.E., Cela, J.M.: Mimetic seismic wave modelling including topography on deformed staggered grids. Geophysics 79(3), T125–T141 (2014)CrossRefGoogle Scholar
  11. 11.
    Dovgilovich, L., Sofronov, I.: High-accuracy finite-difference schemes for solving elastodynamic problems in curvilinear coordinates within multiblock approach. Appl. Numer. Math. 93, 176–194 (2015)CrossRefGoogle Scholar
  12. 12.
    Dunkin, J.W.: Computation of modal solutions in layered elastic media at high frequencies. Bull. Seismol. Soc. Am. 55, 335–358 (1965)Google Scholar
  13. 13.
    Garvin, W.: Exact transient solution of buried line source problem. Proc. R. Soc. Lond. 234, 528–541 (1956)CrossRefGoogle Scholar
  14. 14.
    Hestholm, S., Ruud, B.: 2D finite-difference elastic wave modeling including surface topography. Geophys. J. Int. 118(2), 643–670 (1994)CrossRefGoogle Scholar
  15. 15.
    Hestholm, S., Ruud, B.: 3-D finite-difference elastic wave modeling including surface topography. Geophysics 63(2), 613–622 (1998)CrossRefGoogle Scholar
  16. 16.
    Hestholm, S., Ruud, B.: 2D surface topography boundary conditions in seismic wave modelling. Geophys. Prospect. 49, 445–460 (2001)CrossRefGoogle Scholar
  17. 17.
    de Hoop, A.T.: A modification of Cagniard’s method for solving seismic pulse problems. Appl. Sci. Res. B 8, 349–356 (1960)CrossRefGoogle Scholar
  18. 18.
    Kaser, M., Dumbser, M.: An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - I The two-dimensional isotropic case with external source terms. Geophys. J. Int. 166, 855–877 (2006)CrossRefGoogle Scholar
  19. 19.
    Kaser, M., Igel, H.: Numerical simulation of 2D wave propagation on unstructured grids using explicit differential operators. Geophys. Prospect. 49, 607–619 (2001)CrossRefGoogle Scholar
  20. 20.
    Komatitsch, D., Martin, R.: An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation. Geophysics 72(5), 155–167 (2007)CrossRefGoogle Scholar
  21. 21.
    Komatitsch, D., Vilotte, J.P.: The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures. Bull. Seismol. Soc. Am. 88(2), 368–392 (1998)Google Scholar
  22. 22.
    Komatitsch, D., Coutel, F., Mora, P.: Tensorial formulation of the wave equation for interfaces. Geophys. J. Int. 127, 156–168 (1996)CrossRefGoogle Scholar
  23. 23.
    Kozdon, J.E., Dunham, E.M., Nordström, J.: Simulation of dynamic earthquake ruptures in complex geometries using high-order finite difference methods. J. Sci. Comput. 55(1), 92–124 (2013)CrossRefGoogle Scholar
  24. 24.
    Kristek, J., Moczo, P., Archuleta, R.J.: Efficient methods to simulate planar free surface in the 2D 4th-order staggered-grid finite-difference schemes. Stud. Geophys. Geod. 46, 355–381 (2002)CrossRefGoogle Scholar
  25. 25.
    Kristekova, M., Kristek, J., Moczco, P., Day, S.: Misfit criteria for quantitative comparison of seismograms. Bull. Seismol. Soc. Am. 96(5), 1836–1850 (2006)CrossRefGoogle Scholar
  26. 26.
    Lamb, H.: Exact transient solution of buried line source problem. Philos. Trans. R. Soc. Lond. 204, 1–42 (1904)CrossRefGoogle Scholar
  27. 27.
    Levander, A.R.: Fourth-order finite-difference p-SV seismograms. Geophysics 53, 1425–1436 (1988)CrossRefGoogle Scholar
  28. 28.
    Lisitsa, V., Vishnevsky, D.: Lebedev scheme for the numerical simulation of wave propagation in 3D anisotropic elasticity. Geophys. Prospect. 58, 619–635 (2010)CrossRefGoogle Scholar
  29. 29.
    Lisitsa, V., Vishnevsky, D., Tcheverda, V.: Numerical simulation of seismic waves in models with anisotropic formations: coupling Virieux and Lebedev finite difference schemes. Comput. Geosci. 16, 1135–1152 (2012)CrossRefGoogle Scholar
  30. 30.
    Lombard, B., Piraux, J.: Numerical treatment of two-dimensional interfaces for acoustic and elastic waves. J. Comput. Phys. 195, 90–116 (2004)CrossRefGoogle Scholar
  31. 31.
    Lombard, B., Piraux, J., Gélis, C., Virieux, J.: Free and smooth boundaries in 2-D finite-difference schemes for transient elastic waves. Geophys. J. Int. 172, 252–261 (2007)CrossRefGoogle Scholar
  32. 32.
    Malvern, L.E.: Introduction to the mechanics of a continuous medium. Prentice-Hall, Series in Engineering of the Physical Sciences, New Jersey (1969)Google Scholar
  33. 33.
    McConnell, A.J.: Applications of tensor analysis. Dover Publications, USA (1957)Google Scholar
  34. 34.
    Mittet, R.: Free-surface boundary conditions for elastic staggered-grid modeling schemes. Geophysics 67, 1616–1623 (2002)CrossRefGoogle Scholar
  35. 35.
    Moczo, P., Robertsson, J.O.A., Eisner, L.: The finite-difference time-domain method for modeling of seismic wave propagation Advances in Wave Propagation in Heterogeneous Earth, vol. 48. Academic Press, UK (2007)Google Scholar
  36. 36.
    Oppenheim, A.V., Schafer, R.W., Buck, J.R.: Discrete-time signal processing. Academic Press, New Jersey (1998)Google Scholar
  37. 37.
    Pérez Solano, C.A., Donno, D., Chauris, H.: Alternative waveform inversion for surface wave analysis in 2-D media. Geophys. J. Int. 198, 1359–1372 (2014)CrossRefGoogle Scholar
  38. 38.
    Pujol, J.: Elastic wave propagation and generation in seismology (2003)Google Scholar
  39. 39.
    Robertsson, J.: A numerical free-surface condition for elastic/viscoelastic finite-difference modeling in the presence of topography. Geophysics 61, 1921–1934 (1996)CrossRefGoogle Scholar
  40. 40.
    Rojas, O., Otero, B., Castillo, J.E., Day, S.M.: Low dispersive modelling of Rayleigh waves on partly-staggered grids. Comput. Geosci. 18, 29–43 (2014)CrossRefGoogle Scholar
  41. 41.
    Saenger, E.H., Gold, N., Shapiro, S.A.: Modeling the propagation of elastic waves using a modified finite-difference grid. Wave Motion 31, 77–92 (2000)CrossRefGoogle Scholar
  42. 42.
    Tarrass, I., Giraud, L., Thore, P.: New curvilinear scheme for elastic wave propagation in presence of curved topography. Geophys. Prospect. 59, 889–906 (2011)CrossRefGoogle Scholar
  43. 43.
    Vinokur, M.: Conservation equations of gasdynamics in curvilinear coordinates systems. J. Comput. Phys. 14, 105–125 (1974)CrossRefGoogle Scholar
  44. 44.
    Virieux, J.: P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method. Geophysics 51, 889–901 (1986)CrossRefGoogle Scholar
  45. 45.
    Virieux, J., Calandra, H., Plessix, R É: A review of the spectral, pseudo-spectral, finite-difference and finite-element modelling techniques for geophysical imaging. Geophys. Prospect. 59, 794–813 (2011)CrossRefGoogle Scholar
  46. 46.
    Xu Y, Xia J, Miller RD: Numerical investigation of implementation of air-earth boundary by acoustic-elastic boundary approach. Geophysics 75(5), SM147–SM153 (2007)Google Scholar
  47. 47.
    Zhang, J.: Quadrangle-grid velocity-stress finite-difference method for elastic-wave-propagation simulation. Geophys. J. Int. 131, 127–134 (1997)CrossRefGoogle Scholar
  48. 48.
    Zhang, W., Chen, X.: Traction image method for irregular free surface boundaries in finite difference seismic wave simulation. Geophys. J. Int. 167, 337–353 (2006)CrossRefGoogle Scholar
  49. 49.
    Zhang, W., Zhang, Z., Chen, X.: Three-dimensional elastic wave numerical modelling in the presence of surface topography by a collocated-grid finite-difference method on curvilinear grids. Geophys. J. Int. 190, 358–378 (2012)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Centre de GéosciencesMINES ParisTech PSL Research UniversityFontainebleauFrance
  2. 2.Shell Global Solutions International B.V.RijswijkThe Netherlands

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