pure and applied geophysics

, Volume 138, Issue 3, pp 353–390 | Cite as

Multiple scattering ofSH waves in 2-D media with many cavities

  • Rafael Benites
  • Keiiti Aki
  • Kiyoshi Yomogida


The full waveform synthetic seismogram of multiple scatteredSH waves by many cylindrical cavities in two-dimensional homogeneous elastic media is computed. We used the so-called “single-layer potential” integral representation of the scattered field and a discretization scheme with line source distribution for each cavity. The total field is the sum of the incident wave plus the field radiated from all sources, each multiplied by an unknown complex constant representing its strength. These constants are determined by imposing the appropriate boundary conditions in the least-squares sense. Here we solve scattering problems involving one, two, four, twelve and fifty cavities regularly distributed in a half-space. The seismograms computed along the free-surface show regions where the incident wave is strongly attenuated, as well as the arrivals of all multiple scattered phases. The accuray of the method is estimated from the degree of agreement of our solution for one cavity with the corresponding analytical solution, and also from the magnitude of the residual tractions along the boundaries of two cavities separated at various distances. Finally we apply the method to compute the case of fifty cylindrical cavities, each of radiusa, randomly distributed in a region 80a wide by 30a deep in a half-space. The value of scattering loss is obtained from the amplitude decay of the primary wave with distance for wavelengths in the range from 1.7a to 13.3a, using the synthetic seismogram calculated for the same distribution of 50 cavities as above, but in full-space.


Primary Wave Incident Wave Elastic Medium Line Source Discretization Scheme 


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  1. Benites, R., andAki, K. (1989),Boundary Integral-Gaussian Beam Method for Seismic Wave Scattering: SH in Two-dimensional Media, J. Acoust. Soc. Am.86 (1), 375–386.Google Scholar
  2. Brebbia, C. A.,The Boundary Element Method for Engineers (Pentech Press, London 1978).Google Scholar
  3. Burton, A. J., andMiller, G. F. (1971),The Application of Integral Equation Methods to the Numerical Solutions of Some Exterior Boundary-value Problems, Proc. Roy. Soc. Ser. A323, 201–210.Google Scholar
  4. Chaterjee, A. K., Mal, A. K., andKnopoff, L. (1978),Elastic Moduli of Two-component Systems, J. Geophys. Res.83, 1785–1792.Google Scholar
  5. Chernov, L. A.,Wave Propagation in a Random Medium (McGraw-Hill, New York, 1960).Google Scholar
  6. Copley, L. G. (1967),Integral Equation Method for Radiation from Vibrating Surfaces, J. Acoust. Soc. Am.41, 807–816.Google Scholar
  7. Coutant, O. (1986),Numerical Study of the Diffraction of Elastic Waves by Fluid-filled Cracks, J. Geophys. Res.94, 17,805–17,818.Google Scholar
  8. Dainty, A. M. (1984),High-frequency Acoustic Backscattering and Seismic Attenuation, J. Geophys. Res.89, 3172–3196.Google Scholar
  9. Dravinski, M. (1982),Scattering of SH Waves by Subsurface Topography, J. Eng. Mech. Div., ASCE108, 1–17.Google Scholar
  10. Dravinski, M. (1983),Ground Motion Amplification Due to Elastic Inclusions in a Half-space, Earthquake Eng. and Struct. Dyn.11, 313–335.Google Scholar
  11. Foldy, L. L. (1945),The Multiple Scattering of Waves, I. General Theory for Isotropic Scattering by Randomly Distributed Scatterers, Phys. Rev.67, 107–119.Google Scholar
  12. Frankel, A., andClayton, R. 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, J. Geophys. Res.86, 6465–6489.Google Scholar
  13. Herraiz, M., andEspinosa, A. F. (1987),Coda Waves: A Review, Pure and Appl. Geophys.125, 499–577.Google Scholar
  14. Kawase, H. (1988),Time-domain Response of a Semicircular Canyon for Incident SV, P, and Rayleigh Waves Calculated by the Discrete Wave Number Boundary Element Method, Bull. Seismol. Soc. Am.78, 1415–1437.Google Scholar
  15. Keller, J. B. (1962),Geometrical Theory of Diffraction, J. Opt. Soc. Am.52, 116–130.Google Scholar
  16. Kellogg, O. D.,Foundations of Potential Theory (New York, Dover 1953).Google Scholar
  17. Kikuchi, M. (1981a).Dispersion and Attenuation of Elastic Waves due to Multiple Scattering from Inclusions, Phys. Earth Planet. Inter.25, 159–162.Google Scholar
  18. Kikuchi, M. (1981b),Dispersion and Attenuation of Elastic Waves due to Multiple Scattering from Cracks, Phys. Earth Planet. Inter.27, 100–105.Google Scholar
  19. Kristensson, G., andStröm, S.,The T-matrix approach to scattering from buried inhomogeneities. InAcoustic, Electromagnetic and Elastic Waves—Focus on the T-Matrix Approach (eds. Varadan, V. K., and Varadan, V. V.) (Pergamon, New York 1978) pp. 135–167.Google Scholar
  20. Kupradze, V. D.,Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, North Holland series inApplied Mathematics and Mechanics, 25 (ed. Kupradze, V. D.) (North-Holland 1979), 300 pp.Google Scholar
  21. Lax, M. (1951),Multiple Scattering of Waves, Revs. Modern Phys.23, 287–310.Google Scholar
  22. Lee, V. W. (1977),On Deformations near Circular Underground Cavity Subjected to Incident Plane SH Waves, Proceedings of the Application of Computer Methods in Engineering Conference, Vol. II, University of Southern California, Los Angeles, Calif., 951–962.Google Scholar
  23. Manolis, G. D., andBeskos, D. E.,Boundary Element Methods in Elastodynamics (Academic Division of Unwin Hyman Ltd, London, UK 1988).Google Scholar
  24. Matsunami, K. (1990),Laboratory Measurements of Spatial Fluctuations and Attenuation of Elastic Waves by Scattering due to Random Heterogeneities, Pure and Appl. Geophys.132, 197–220.Google Scholar
  25. Menke, W., Witte, D., andChen, R. (1985),Laborotory Test of Apparent Attenuation Formulas, Bull. Seismol. Soc. Am.75, 1383–1393.Google Scholar
  26. Mow, C. C., andPao, Y. H.,The Diffraction of Elastic Waves and Dynamic Stress Concentrations (The Rand Corporation, USA 1971).Google Scholar
  27. Ricker, N. H.,Transient Waves in Visco-elastic Media (Elsevier Scientific Publishing Co., Amsterdam. Holland 1977).Google Scholar
  28. Sánchez-Sesma, F. J., andRosenblueth, E. (1979),Ground Motions at Canyons of Arbitrary Shapes under Incident SH Waves, Earthquake Eng. Struct. Dyn.7, 441–450.Google Scholar
  29. Sánchez-Sesma, F. J., andEsquivel, J. A. (1979),Ground Motion on Alluvial Valleys under Incident Plane SH Waves, Bull. Seismol. Soc. Am.69, 1107–1120.Google Scholar
  30. Sato, H. (1982),Amplitude Attenuation of Impulsive Waves in Random Media Based on Travel Time Corrected Mean Wave Formalism, J. Acoust. Soc. Am.71, 559–564.Google Scholar
  31. Smirnov, V. I.,A Course in Higher Mathematics, vol. 4 (Pergamon, London 1964).Google Scholar
  32. Ursell, F. (1973),On the Exterior Problems of Acoustics, Proc. Camb. Phil. Soc.74, 117–125.Google Scholar
  33. Varadan, V. K., Varadan, V. V., andPao, Yih-Hsing (1978),Multiple Scattering of Elastic Waves by Cylinders of Arbitrary Cross Section. I. SH Waves, J. Acoust. Soc. Am.63 (5), 1310–1319.Google Scholar
  34. Varadan, V. K., Ma, Y., andVaradan, V. V. (1989),Scattering and Attenuation of Elastic Waves in Random Media, Pure and Appl. Geophys.131, 577–603.Google Scholar
  35. Waterman, P. C. (1969),New Formulation of Acoustic Scattering, J. Acoust. Soc. Am.45, 1417–1428.Google Scholar
  36. Wu, R.-S. (1982),Attenuation of Short Period Seismic Waves due to Scattering, Geophys. Res. Lett.9, 9–12.Google Scholar
  37. Wu, R.-S.,Seismic wave scattering. InThe Encyclopedia of Solid Earth Geophysics, Encyclopedia of Earth Sciences Series (ed. James, E. D.) (Van Nostrand, Reinhold 1989) pp. 1166–1186.Google Scholar

Copyright information

© Birkhäuser Verlag 1992

Authors and Affiliations

  • Rafael Benites
    • 1
  • Keiiti Aki
    • 1
  • Kiyoshi Yomogida
    • 2
  1. 1.Department of Geological SciencesUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Department of Earth and Planetary Systems Science, Faculty of ScienceUniversity of HiroshimaHigashi-HiroshimaJapan

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