Scattering of Seismic Waves by Cracks with the Boundary Integral Method

  • Kiyoshi Yomogida
  • Rafael Benites

Abstract

Wedevelop a new scheme to compute 2-D SH seismograms for media with many flat cracks, based on the boundary integral method. A dry or traction-free boundary condition is applied to crack surfaces although other kinds of cracks such as wet or fluid-saturated cracks can be treated simply by assigning different boundary conditions. While body forces are distributed for cavities or inclusions to express scattered wave, dislocations (or displacement discontinuities between the top and the bottom surfaces of each crack) are used as fictitious sources along crack surfaces. With these dislocations as unknown coefficients, the scattered wave is expressed by the normal derivative of Green’s function along the crack surface, which is called “double-layer potentials” in the boundary integral method, while we used “single-layer potentials” for cavities or inclusions. These unknowns are determined so that boundary conditions or crack surfaces are satisfied in the least-squared sense, for example, traction-free for dry cracks. Seismograms with plane-wave incidence are synthesized for homogeneous media with many cracks. First, we check the accuracy of our scheme for a medium with one long crack. All the predicted phases such as reflected wave, diffraction from a crack tip and shadow behind the crack are simulated quite accurately, under the same criterion as in the case for cavities or inclusions. Next, we compute seismograms for 50 randomly distributed cracks and compare them with those for circular cavities. When cracks are randomly oriented, waveforms and the strength of scattering attenuation are similar to the cavity case in a frequency range higher thankd≃2 where the size of scatterersd(i.e., crack length or cavity diameter) is comparable with the wavelength considered(kis the wavenumber). On the other hand, the scattering attenuation for cracks becomes much smaller in a lower frequency range(kd <2) because only the volume but not detail geometry of scatterers becomes important with wavelength much longer than each scatterer. When all the cracks are oriented in a fixed direction, the scattering attenuation depends strongly on the incident angle to the crack surface as frequency increases(kd >2): scattering becomes weak for cracks oriented parallel to the direction of the incident wave, while it gets close to the cavity case for cracks aligned perpendicular to the incident wave.

Key words

Scattering crack media attenuation boundary method 

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References

  1. ACHENBACH, J. D.Wave Propagation in Elastic Solids(North-Holland, Amsterdam 1973).Google Scholar
  2. AKI, K. (1973)Scattering of P Waves under the Montana LasaJ. Geophys. Res., 78, 1334–1346.CrossRefGoogle Scholar
  3. AKI, K., and RICHARDS, P. G.Quantitative Seismology, Theory and Methodsvol. 1 (W. H. Freeman and Company, San Francisco 1980).Google Scholar
  4. BENITES, R., AKI, K., and YOMOGIDA, K. (1992)Multiple Scattering of SH Waves in 2-D Media with Many CavitiesPure appl. geophys. 138, 353–390.CrossRefGoogle Scholar
  5. BENITES, R., YOMOGIDA, K., ROBERTS, P. M., and FEHLER, M. (1997)Scattering of Elastic Waves in 2-D Composite Media I: Theory and Test, Phys. Earth Planet. Inter. 104, 161–173.CrossRefGoogle Scholar
  6. BOUCHON, M. (1987)Diffraction to Elastic Waves by Cracks or Cavities Using the Discrete Wave Number MethodJ. Acoust. Soc. Am. 81, 1671–1676.CrossRefGoogle Scholar
  7. CHERNOV, J. D.Wave Propagation in a Random Medium(McGraw-Hill, New York 1960).Google Scholar
  8. COUTANT, O. (1989)Numerical Study of the Diffraction of Elastic Waves by Fluid-filled CracksJ. Geophys. Res. 94, 17,805–17,818.Google Scholar
  9. CRAMPIN, S. and LOVELL, H. (1991)A Decade of Shear-wave Splitting in the Earth’s Crust: What Does it Mean? What Use Can we Make of it? and What Should we Do NextGeophys. J. Int. 107, 387–407.Google Scholar
  10. HUDSON, J. A. (1981)Wave Speeds and Attenuation of Elastic Waves in Material Containing CracksGeophys. J. Roy. Astr. Soc. 64, 133–150.CrossRefGoogle Scholar
  11. KAWAHARA, J. (1998)Seismic Scattering by Cracks Containing LiquidsProc. 4th SEGJ Inter. Symp., 161–167.Google Scholar
  12. KAWAHARA, J., and YAMASHITA, T. (1992)Scattering of Elastic Waves by a Fracture Zone Containing Randomly Distributed CracksPure app]. geophys. 139, 121–144.CrossRefGoogle Scholar
  13. Li, Y.-G., LEARY, P. C., ADAMS, D., and HASEMI, A. (1994)Seismic Guided Waves Trapped in the Fault Zone of the Landers, California, Earthquake of 1992J. Geophys. Res. 99, 11,705–11,722.Google Scholar
  14. MAL, A. K. (1970), Interaction of Elastic Waves with a Griffith Crack, Int. J. Eng. Sci. 8, 763–776.CrossRefGoogle Scholar
  15. MATSUNAMI, K. (1990)Laboratory Measurements of Spatial Fluctuations and Attenuation of Elastic Waves by Scattering due to Random HeterogeneitiesPure app]. geophys. 132, 197–220.CrossRefGoogle Scholar
  16. MURAI, Y., KAWAHARA, J., and YAMASHITA, T. (1995)Multiple Scattering of SH Waves in 2-D Elastic Media with Distributed Cracks, Geophys. J. Int. 122, 925–937.Google Scholar
  17. NIsHIZAwA, O., SATOH, T., LEI, X., and KUWAHARA, Y. (1997)Laboratory Studies of Seismic Wave Propagation in Inhomogeneous Media Using a Laser Doppler VibrometerBull. Seismol. Soc. Am. 87, 809–823.Google Scholar
  18. SATO, H., and FEHLER, M. C.Seismic Wave Propagation and Scattering in the Heterogeneous Earth(Springer-Verlag, New York 1998).CrossRefGoogle Scholar
  19. URSELL, F. (1973)On the Exterior Problems of AcousticsProc. Camb. Phil. Soc. 74, 117–125.CrossRefGoogle Scholar
  20. VARADAN, V. K., VARADAN, V. V., and PAO, Y.-H. (1978)Multiple Scattering of Elastic Waves by Cylinders of Arbitrary Cross Section. I. SH Waves, J. Acoust. Soc. Am. 63, 1310–1319.CrossRefGoogle Scholar
  21. Wu, R. S. (1982)Attenuation of Short-period Seismic Waves due to Scattering, Geophys. Res. Lett. 9, 9–12.Google Scholar
  22. YAMASHITA, T. (1990)Attenuation and Dispersion of SH Waves due to Scattering by Randomly Distributed CracksPure app]. geophys. 132, 545–568.CrossRefGoogle Scholar
  23. YOMOGIDA, K., and BENITES, R. (1995)Relation between Direct Wave Q and Coda Q: A Numerical ApproachGeophys. J. Int. 123, 471–483.Google Scholar
  24. YOMOGIDA, K., BENITES, R., ROBERTS, P. M., and FEHLER, M. (1997)Scattering of Elastic Waves in 2-D Composite Media II: Waveforms and SpectraPhys. Earth Planet. Inter. 104, 175–192.Google Scholar

Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Kiyoshi Yomogida
    • 1
  • Rafael Benites
    • 2
  1. 1.Division of Earth and Planetary Sciences, Graduate School of ScienceHokkaido UniversitySapporoJapan
  2. 2.Institute of Geological and Nuclear SciencesLower HuttNew Zealand

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