Advertisement

Journal of Earth System Science

, Volume 110, Issue 3, pp 205–213 | Cite as

Scattering of a spherical pulse from a small inhomogeneity: Dilatation and rotation

  • M. D. Sharma
Article
  • 37 Downloads

Abstract

Perturbations in elastic constants and density distinguish a volume inhomogeneity from its homogeneous surroundings. The equation of motion for the first order scattering is studied in the perturbed medium. The scattered waves are generated by the interaction between the primary waves and the inhomogeneity. First order scattering theory is modified to include the source term generating the primary waves. The body force equivalent to the scattering source is presented in a convenient form involving the perturbations in wave velocities and gradient of density perturbation.

A procedure is presented to study the scattering of a spherical pulse from a small inhomogeneity, in time domain. The size of inhomogeneity is assumed small as compared to its distance from source and receiver. No restrictions are placed on the positions of source, receiver and inhomogeneity. The dilatation and rotations are calculated for a pulse scattered from an arbitrary point in a spherical volume. The aggregate of the scattered phases from all the points of the inhomogeneity, reaching at a fixed receiver, gives the amount of scattering from the inhomogeneity. The interaction of bothP andS waves with inhomogeneity are considered. Dilatation and rotations for scattering are obtained as integral expressions over the solid angle of inhomogeneity. These expressions are computed numerically, for hypothetical models. The effects of source (unit force) orientations, velocity and density perturbations, and size of inhomogeneity, on the scattered phases, are discussed.

Keywords

Scattering inhomogeneity spherical pulse perturbations dilatation rotation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aki K and Richards P G 1980Quantitative seismology, theory and methods, (New York: Freeman)Google Scholar
  2. Benites R, Aki K and Yomogida K 1992 Multiple scattering of SH waves in 2-D media with many cavities;PAGEOPH 138 353–390CrossRefGoogle Scholar
  3. Chapmann C H and Drummond R 1982 Body-wave seismograms in inhomogeneous media using Maslov asymptotic theory;Bull. Seism. Soc. Am. 72 277–317Google Scholar
  4. Chen G and Zhou H 1994 Boundary element modeling of nondispersive and dispersive waves;Geophysics 59 113–118CrossRefGoogle Scholar
  5. Fehler M and Aki K 1978 Numerical study of diffraction of plane elastic waves by a finite crack with application to location of magma lens;Bull. Seism. Soc. Am. 68 573–598Google Scholar
  6. Gajewski D 1993 Radiation from point sources in general anisotropic media;Geophys. J. Int. 113 299–317CrossRefGoogle Scholar
  7. Gritto R, Korneev V A and Johnson L R 1995 Low frequency elastic wave scattering by an inclusion: limits of applications;Geophys. J. Int. 120 677–692CrossRefGoogle Scholar
  8. Gubernatis J E, Domany E, Krumhansl J A and Huberman M 1977 The Born approximation of the theory of scattering of ultrasound by flaws in elastic materials;J. Appl. Phys. 48 2812–2819CrossRefGoogle Scholar
  9. Gurevich B, Sadovnichaja A P, Lopatnikov S L and Shapiro S A 1992 The Born approximation in the problem of elastic wave scattering by a spherical inhomogeneity in a fluid saturated porous media;Appl. Phys. Lett. 61 1275–1277CrossRefGoogle Scholar
  10. Gurevich B, Sadovnichaja A P, Lopatnikov S L and Shapiro S A 1998 Scattering of a compressional wave in a poroelastic medium by an ellipsoidal inclusion;Geophys. J. Int. 133 91–103CrossRefGoogle Scholar
  11. Hinders M, Bogan S D and Fang T 1992 Spherical wave scattering from an elastic sphere;Il Nuovo Cimento,107B #10, 1215Google Scholar
  12. Korneev V A and Johnson L R 1993a Scattering of elastic waves by a spherical inclusion-I: Theory and numerical results;Geophys. J. Int. 115 230–250.CrossRefGoogle Scholar
  13. Korneev V A and Johnson L R 1993b Scattering of elastic waves by a spherical inclusion-II: Limitations of asymptotic solutions;Geophys. J. Int. 115 251–263CrossRefGoogle Scholar
  14. Leary P C 1995 Quantifying crustal fracture heterogeneity by seismic scattering;Geophys. J. Int. 122 125–142CrossRefGoogle Scholar
  15. Lysmer J and Drake L A 1972 A finite element method for seismology, Methods of Computational Physics. Vol.11. (New York: Academic Press)Google Scholar
  16. Neuberg J and Pointer T 1995 Modelling seismic reflections fromD′' using the Kirchoff method;Phys. Earth Planet. Inter. 90 273–281CrossRefGoogle Scholar
  17. Piquette P C 1986 Spherical wave scattering by an elastic solid cylinder — A numerical comparison of an approximate theory with the exact theory;J. Acoust. Soc. Am. 82 699–702CrossRefGoogle Scholar
  18. Piquette P C 1987 Spherical wave scattering by an elastic solid cylinder of infinite length;J. Acoust. Soc. Am. 79 1248–59CrossRefGoogle Scholar
  19. Paulsson B N P, Meredith J A, Wang Z and Fairborn J W 1994 The Steepbank crosswell seismic project: Reservoir definition and evaluation of steam flood technology in Alberta tar sands;The Leading Edge 13 737–747CrossRefGoogle Scholar
  20. Rose J H 1989 Elastic wave inverse scattering in nondestructive evaluation;PAGEOPH 131 715–739CrossRefGoogle Scholar
  21. Sato H 1984 Attenuation and envelope formation of three component seismograms of small local earthquakes in randomly inhomogeneous lithosphere;J. Geophys. Res. 89 1221–1242Google Scholar
  22. Tadeu A J B, Kausel E and Vrettos C 1996 Scattering of waves by subterranean structures via the boundary element method;Soil Dynamics and Earthq. Engg. 15 387–397CrossRefGoogle Scholar
  23. Wu R S 1989 The perturbation method in elastic wave scattering;PAGEOPH 131 605–637CrossRefGoogle Scholar
  24. Wu R S 1994 Wide angle elastic wave one-way propagation in heterogeneous media and an elastic wave complexscreen method;J. Geophys. Res. 99 751–766CrossRefGoogle Scholar
  25. Wu R S and Aki K 1985a Scattering characteristics of elastic waves by an elastic heterogeneity;Geophysics 50 582–595CrossRefGoogle Scholar
  26. Wu R S and Aki K 1985b Elastic wave scattering by a random medium and the small scale inhomogeneities in the lithosphere;J. Geophys. Res. 90 10261–10273CrossRefGoogle Scholar
  27. Wu R S and Aki K 1988 Introduction: Seismic wave scattering in three-dimensionally heterogeneous earth;PAGEOPH 128 1–6CrossRefGoogle Scholar

Copyright information

© Printed in India 2001

Authors and Affiliations

  • M. D. Sharma
    • 1
  1. 1.Department of MathematicsKurukshetra UniversityKurukshetraIndia

Personalised recommendations