Skip to main content
SpringerLink
Log in

Seismic wave fields in continuously inhomogeneous media with variable wave velocity profiles

  • Special
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

In this work, elastic wave motion in a continuously inhomogeneous geological medium under anti-plane strain conditions is numerically investigated using the boundary integral equation method (BIEM). More specifically, the geological medium possesses a variable velocity profile, in addition to the presence of either parallel or non-parallel graded layers, of surface relief, and of buried cavities and tunnels. This complex continuum is swept either by time-harmonic, free-traveling horizontally polarized shear waves or by incoming waves radiating from an embedded seismic source. The BIEM employs a novel type of analytically derived fundamental solution to the equation of motion defined in the frequency domain, by assuming a position-dependent shear modulus and a density of arbitrary variation in terms of the depth coordinate. This fundamental solution, and its spatial derivatives and asymptotic forms, are all derived in a closed-form by using an appropriate algebraic transformation for the displacement vector. The accuracy of the present BIEM numerical implementation is gauged by comparison with available results drawn from examples that appear in the literature. Following that, a series of parametric studies are conducted and numerical results are generated in the form of synthetic seismic signals for a number of geological deposits. This allows for an investigation of the seismic wave field sensitivity to the material gradient and the wave velocity variation in the medium, to the presence of layers, canyons and cavities, and to the frequency content of the incoming signal.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Acharya, H.K.: Field due to point source in an inhomogeneous elastic medium. J. Acoust. Soc. Am. 50, 172–175 (1971)

  2. Achenbach, J.D.: Wave Propagation in Elastic Solids. North-Holland, Amsterdam (1973)

    MATH  Google Scholar 

  3. Albuquerque, E.L., Sollero, P., Fedelinski, P.: Dual reciprocity boundary element method in Laplace domain applied to anisotropic dynamic crack problems. Comput. Struct. 81, 1703–1713 (2003)

    Article  Google Scholar 

  4. Ang, W.T., Clements, D.L., Vahdati, N.: A dual-reciprocity boundary element method for a class of elliptic boundary value problems for non-homogeneous anisotropic media. Eng. Anal. Bound. Elem. 27(1), 49–55 (2003)

    Article  MATH  Google Scholar 

  5. Alvarez-Rubio, S., Benito, J.J., Sanchez-Sesma, F.J., Alarcon, E.: The use of direct boundary element method for gaining insight into complex seismic site response. Comput. Struct. 83, 821–835 (2005)

    Article  Google Scholar 

  6. Alvarez-Rubio, S., Sanchez-Sesma, F.J., Benito, J.J., Alarcon, E.: The direct boundary element method: 2D site effects assessment on laterally varying layered media. Soil Dyn. Earthq. Eng. 24, 167–180 (2004)

    Article  Google Scholar 

  7. Beskos, D.E.: Boundary element methods in dynamic analysis. Appl. Mech. Rev. 40(1), 1–23 (1987)

    Article  Google Scholar 

  8. Beskos, D.E.: Boundary element methods in dynamic analysis, Part II, 1986–1996. Appl. Mech. Rev. 50(3), 149–197 (1997)

    Article  Google Scholar 

  9. Bouchon, M., Coutant, O.: Calculation of synthetic seismograms in a laterally varying medium by the boundary element-discrete wave number method. Bull. Seismol. Soc. Am. 84(6), 1869–1881 (1994)

    Google Scholar 

  10. Brekhovskikh, L.M., Godin, O.A.: Acoustics of Layered Media I. Springer, Berlin (1990)

    Book  Google Scholar 

  11. Chen, J.T., Lee, J.W., Wu, C.F., Chen, I.L.: SH-wave diffraction by a semi-circular hill revisited: a null-field boundary integral equation method using degenerate kernels. Soil Dyn. Earthq. Eng. 31, 729–736 (2011)

    Article  MATH  Google Scholar 

  12. Daros, C.H.: A fundamental solution for SH-waves in a class of inhomogeneous anisotropic media. Int. J. Eng. Sci. 46, 809–817 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Daros, C.H.: Green’s function for SH-waves in inhomogeneous anisotropic elastic solid with power-function velocity variation. Wave Motion 50, 101–110 (2013)

    Article  MathSciNet  Google Scholar 

  14. Dineva, P., Manolis, G.D., Rangelov, T.V., Wuttke, F.: SH-wave scattering in the orthotropic half-plane weakened by cavities using BIEM. Acta Acust. United Acust. 100(2), 266–276 (2014)

    Article  Google Scholar 

  15. Dravinski, M., Yu, M.C.: Scattering of plane harmonic SH-waves by multiple inclusions. Geophys. J. Int. 186(3), 1331–1346 (2011)

    Article  Google Scholar 

  16. Dineva, P., Rangelov, T.V., Gross, D.: BIEM for 2D steady-state problems in cracked anisotropic materials. Eng. Anal. Bound. Elem. 29(7), 689–698 (2005)

    Article  MATH  Google Scholar 

  17. Dominguez, J.: Boundary Elements in Dynamics. Computational Mechanics Publications, Southampton (1993)

    MATH  Google Scholar 

  18. Dravinski, M., Yu, M.C.: The effect of impedance contrast upon surface motion due to scattering of plane harmonic P, SV, and Rayleigh waves by a randomly corrugated elastic inclusion. J. Seismol. 17, 281–295 (2013)

    Article  Google Scholar 

  19. Ewing, W.M., Jardetsky, W.S., Press, F.: Elastic Waves in Layered Media. McGraw-Hill, New York (1957)

    MATH  Google Scholar 

  20. Fontara, I.-K., Parvanova, S., Dineva, P., Rangelov, T.V., Wuttke, F.: Seismic signals synthesis for laterally inhomogeneous geological media via BIEM. In: Brebbia, C.A. (ed.) Proceedings of 37th International Conference on Boundary Elements and other Mesh Reduction Methods, 8–10 September 2014, The New Forest, UK, pp. 265–278. WIT Press, Southampton (2014)

    Google Scholar 

  21. Gatmiri, B., Arson, C.: Seismic site effects by an optimized 2D BE/FE method. II. Quantification of site effects in two-dimensional sedimentary valleys. Soil Dyn. Earthq. Eng. 28(8), 646–661 (2008)

    Article  Google Scholar 

  22. Gatmiri, B., Arson, C., Nguyen, K.V.: Seismic site effects by an optimized 2D BE/FE method I. Theory, numerical optimization and application to topographical irregularities. Soil Dyn. Earthq. Eng. 28, 632–645 (2008)

    Article  Google Scholar 

  23. Gatmiri, B., Maghoul, P., Arson, C.: Site-specific spectral response of seismic movement due to geometrical and geotechnical characteristics of sites. Soil Dyn. Earthq. Eng. 29(1), 51–70 (2009)

    Article  Google Scholar 

  24. Ge, Z.: Simulation of the seismic response of sedimentary basins with constant-gradient velocity along arbitrary direction using boundary element method: SH case. Earthq. Sci. 23, 149–155 (2010)

    Article  Google Scholar 

  25. Hanyga, A.: Seismic Wave Propagation in the Earth. Elsevier, Amsterdam (1985)

    Google Scholar 

  26. Hirose, S., Zhang, Ch., Wang, C.-Y.: A comparative study on two time domain BEM/BIEM for transient dynamic crack analysis of anisotropic solids. In: Yao, Z., Aliabadi, M.H. (eds.) Proceedings 3rd BETEQ International Conference, pp. 106–112. Tsinghua University Press, Beijing (2002)

    Google Scholar 

  27. Kakar, R., Kakar, S.: Propagation of Love waves in a non-homogeneous elastic media. J. Acad. Indus. Res. 1(6), 323–328 (2012)

    Google Scholar 

  28. Karakostas, C.Z., Manolis, G.D.: Transient signal simulation due to explosion in heterogeneous soil media. Int. J. BEM Commun. 8, 160–167 (1997)

    Google Scholar 

  29. Kausel, E., Manolis, G.D. (eds.): Wave Motion Problems in Earthquake Engineering. WIT Press, Southampton (2000)

  30. Kuvashinov, B., Mulder, W.A.: The exact solution of the time-harmonic wave equation for linear velocity profile. Geophys. J. Int. 167, 659–662 (2006)

    Article  Google Scholar 

  31. Kobayashi, S.: Elastodynamics, Chapter 4 in Boundary Element Methods in Mechanics, edited by DE Beskos. North-Holland, Amsterdam (1987)

  32. Liu, E., Zhang, Zh, Yue, J., Dobson, A.: Boundary integral modeling of elastic wave propagation in multi-layered 2D media with irregular interfaces. Commun. Comput. Phys. 3(1), 52–62 (2008)

    MATH  Google Scholar 

  33. Luco, J.E., Barros, C.P.: Dynamic displacements and stresses in the vicinity of a cylindrical cavity embedded in a half-space. Earthq. Eng. Struct. Dyn. 23, 321–340 (1994)

    Article  Google Scholar 

  34. Luzon, F., Ramirez, L., Sanchez-Sesma, F.J., Posadas, A.: Propagation of SH elastic waves in deep sedimentary basins with an oblique velocity gradient. Wave Motion 38, 11–23 (2003)

    Article  MATH  Google Scholar 

  35. Luzon, F., Ramirez, L., Sanchez-Sesma, F.J., Posadas, A.: Simulation of the seismic response of sedimentary basins with vertical constant-gradient of velocity. Pure Appl. Geophys. 12, 1533–1547 (2004)

    Google Scholar 

  36. Luzon, F., Sanchez-Sesma, F.J., Perez-Ruiz, A., Ramirez, L., Pech, A.: In-plane seismic response of inhomogeneous alluvial valleys with vertical gradients of velocities and constant Poisson ration. Soil Dyn. Earthq. Eng. 29, 994–1004 (2009)

    Article  Google Scholar 

  37. Manolis, G.D., Beskos, D.E.: Boundary Element Methods in Elastodynamics. Allen and Unwin, London (1988)

    Google Scholar 

  38. Manolis, G.D., Shaw, R.: Harmonic wave propagation through viscoelastic heterogeneous media exhibiting mild stochasticity-I. Fundamental solutions. Soil Dyn. Earthq. Eng. 15, 119–127 (1996)

    Article  Google Scholar 

  39. Manolis, G.D., Shaw, R.P.: Harmonic wave propagation through viscoelastic heterogeneous media exhibiting mild stochasticity-II. Applications. Soil Dyn. Earthq. Eng. 15, 129–139 (1996)

    Article  Google Scholar 

  40. Manolis, G.D., Shaw, R.P.: Green’s function for a vector wave equation in a mildly heterogeneous continuum. Wave Motion 24, 59–83 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  41. Manolis, G.D., Shaw, R.P., Pavlou, S.: Elastic waves in non-homogeneous media under 2D conditions: I. Fundamental solutions. Soil Dyn. Earthq. Eng. 18(1), 19–30 (1999)

    Article  Google Scholar 

  42. Manolis, G.D., Rangelov, T.V., Shaw, R.P.: Conformal mapping methods for variable parameter elastodynamics. Wave Motion 36, 185–202 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  43. Manolis, G.D., Dineva, P.S., Rangelov, T.V.: Wave scattering by cracks in inhomogeneous continua using BIEM. Int. J. Solids Str. 41, 3905–3927 (2004)

    Article  MATH  Google Scholar 

  44. Manolis, G.D., Dineva, P.S., Rangelov, T.V.: Dynamic fracture analysis of a smoothly inhomogeneous plane containing defects by BEM. Eng. Anal. Bound. Elem. 36, 727–737 (2012)

    Article  MathSciNet  Google Scholar 

  45. MATLAB: The Language of Technical Computing, Version 7.7. The Math-Works, Inc., Natick (2008)

  46. Nayfeh, A.H.: Wave Propagation in Layered Anisotropic Media. North-Holland, Amsterdam (1995)

    MATH  Google Scholar 

  47. Pao, Y.H., Mow, C.C.: Diffraction of Elastic Waves and Dynamic Stress Concentration. Crane Russak, New York (1971)

    Google Scholar 

  48. Parvanova, S., Dineva, P., Manolis, G.D., Wuttke, F.: Seismic response of lined tunnels in the half-plane with surface topography. Bull. Earthq. Eng. 12, 981–1005 (2014a)

    Article  Google Scholar 

  49. Parvanova, S., Dineva, P., Manolis, G.D.: Elastic wave fields in a half-plane with free surface relief, tunnels and multiple buried inclusions. Acta Mech. 225, 1843–1865 (2014b)

    Article  MathSciNet  MATH  Google Scholar 

  50. Rodriguez-Castellanos, A., Sanchez-Sesma, F.J., Luzon, F., Martin, R.: Multiple scattering of elastic waves by subsurface fractures and cavities. Bull. Seismol. Soc. Am. 96(4A), 1359–1374 (2006)

    Article  Google Scholar 

  51. Sanchez-Sesma, F.J., Campillo, M.: Topographic effects for incident P, SV and Rayleigh waves. Tectonophysics 218(1–3), 113–125 (1993)

    Article  Google Scholar 

  52. Sanchez-Seisma, F.J., Madariaga, R., Irikura, K.: An approximate elastic 2-D Green’s function for a constant-gradient medium. Geophys. J. Int. 146, 237–248 (2001)

    Article  Google Scholar 

  53. Schanz, M.: Wave Propagation in Viscoelastic and Poroelastic Continua: A Boundary Element Approach, Lecture Notes in Applied Mechanics, Vol. 2, Series Editor F Pfeiffer. Springer, Berlin (2001)

  54. Schanz, M.: Poroelastodynamics: linear models, analytical solutions and numerical methods. Appl. Mech. Rev. 62(3), 030803-1–030803-15 (2009)

    Article  Google Scholar 

  55. Tadeu, A.J.B., Kausel, E., Vrettos, C.: Scattering of waves by subterranean structure via the boundary element method. Soil Dyn. Earthq. Eng. 15(6), 387–397 (1996)

    Article  Google Scholar 

  56. Trifunac, M.D.: Scattering of plane SH waves by semi-cylindrical canyon. Earthq. Eng. Struct. Dyn. 1, 267–281 (1973)

    Article  Google Scholar 

  57. Tsvankin, : Seismic Signatures and Analysis of Reflection Data in Anisotropic Media. Elsevier Applied Science, Amsterdam (2005)

    Google Scholar 

  58. Udias, A.: Principles of Seismology: Waves in Layered Media. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  59. Vrettos, C.: Dispersive SH-surface waves in soil deposits of variable shear modulus. Soil Dyn. Earthq. Eng. 9, 255–264 (1990)

    Article  Google Scholar 

  60. Vrettos, C.: Forced anti-plane vibrations at the surface of an inhomogeneous half-space. Soil Dyn. Earthq. Eng. 10, 230–235 (1991)

    Article  Google Scholar 

  61. Wang, C.D., Lin, Y.T., Jeng, Y.S., Ruan, Z.W.: Wave propagation in an inhomogeneous cross-anisotropic medium. Int. J. Numer. Anal. Meth. Geomech. 34, 711–732 (2010)

    Article  MATH  Google Scholar 

  62. Watanabe, K.: Transient response of an inhomogeneous elastic solid to an impulsive SH-source (variable SH-wave velocity) Bull. JSME 25, 315–320 (1982)

    Article  Google Scholar 

  63. Watanabe, K., Payton, R.G.: Green’s function and its non-wave nature for SH-wave in inhomogeneous elastic solid. Int. J. Eng. Sci. 42, 2087–2106 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  64. Wuttke, F., Fontara, I.-K., Dineva, P., Rangelov, T.V.: SH-wave propagation in a continuously inhomogeneous half-plane with free-surface relief by BIEM. ZAMM 1–16 (2014). doi:10.1002/zamm.201300198

  65. Wuttke, F.: Beitrag zur Standortidentifzierung mit Oberfachenwellen. PhD thesis, Bauhaus Universität, Weimar, Germany (2005)

  66. Yu, M.C., Dravinski, M.: Scattering of plane harmonic \(P\), SV or Rayleigh waves by a completely embedded corrugated cavity. Geophys. J. Int. 178(1), 479–487 (2009)

    Article  Google Scholar 

  67. Zhang, Ch., Gross, D.: On Wave Propagation in Elastic Solids with Cracks. Computational Mechanics Publication, Southampton (1998)

    MATH  Google Scholar 

  68. Zhang, Ch.: A 2D hypersingular time-domain traction BEM for transient elastodynamic crack analysis. Wave Motion 35, 17–40 (2002)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors wish to acknowledge support provided by DFG Grant No. DFG-Wu 496/5-1. Also, author PSD wishes to acknowledge support from the bilateral Bulgarian-Greek, Personnel Exchange Program, Cooperation Program ‘SCIG: Synthesis of Seismic Signals in Continuously Inhomogeneous Geological Media’, BAS-AUTH, 2014.

Author information

Authors and Affiliations

  1. Institute of Applied Geoscience, Christian-Albrechts-Universität (CAU), Kiel, Germany

    Ioanna-Kleoniki M. Fontara & Frank Wuttke

  2. Institute of Mechanics, Bulgarian Academy of Sciences (BAS), Sofia, Bulgaria

    Petia S. Dineva

  3. Department of Civil Engineering, Aristotle University (AUTH), Thessaloniki, Greece

    George D. Manolis

  4. Department of Civil Engineering, University of Architecture, Civil Engineering and Geodesy (UACEG), Sofia, Bulgaria

    Sonia L. Parvanova

Authors
  1. Ioanna-Kleoniki M. Fontara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petia S. Dineva
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. George D. Manolis
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sonia L. Parvanova
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Frank Wuttke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to George D. Manolis.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fontara, IK.M., Dineva, P.S., Manolis, G.D. et al. Seismic wave fields in continuously inhomogeneous media with variable wave velocity profiles. Arch Appl Mech 86, 65–88 (2016). https://doi.org/10.1007/s00419-015-1094-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-015-1094-4

Keywords

Search

Navigation