Commun. Comput. Phys.,
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Volume 3.


Retrieval of Elastic Green's Tensor near a Cylindrical Inhomogeneity from Vector Correlations

Juan A. Perez-Ruiz 1*, Francisco Luzon 1, Francisco J. Sanchez-Sesma 2

1 Grupo de Geofisica Aplicada, Departamento de Fisica Aplicada, Universidad de Almeria, Canada de San Urbano, 04120-Almeria, Spain.
2 Departamento de Sismologia, Instituto de Ingenieria, Universidad Nacional Autonoma de Mexico, Cd. Universitaria, Circuito Escolar, Coyoacan 04510, Mexico D. F., Mexico.

Received 30 January 2007; Accepted (in revised version) 6 June 2007
Available online 14 September 2007

Abstract

Multiple scattering of elastic waves in realistic media makes that average field intensities or energy densities follow diffusive processes. In such regime the successive $P$ to $S$ energy conversions by distributed random inhomogeneities give rise to equipartition which means that in the phase space the available elastic energy is distributed in average with equal amounts among the possible states of $P$ and $S$ waves. In such diffusive regime the $P$ to $S$ energy ratio equilibrates in an universal way independent of the particular details of the scattering. It has been demonstrated that averaging the cross correlations at any two points of an elastic medium subjected to diffuse elastic wavefields leads to the emergence of the Green function, which is the wave field that would be observed at one position if an impulsive load is applied at the other. In this work we study the problem of the retrieval of the 2D tensor elastodynamic Green function in an infinite elastic space containing a circular cylinder inclusion. We illuminate isotropically the elastic space with plane waves. We assume the spectra for both $P$ and $S$ waves uniform but such that the energy ratio $E_S/E_P = (\alpha/\beta)^2$, which is the one predicted by equipartition theory in two-dimensions. We then show that the Fourier transform of azimuthal average of the cross-correlation of motion between two points within an elastic medium is proportional to the imaginary part of the exact Green tensor function between these points. The numerical results presented here point out the possibility of detection and imaging of diffractors and resonant diffractors by cross correlation even in presence of attenuation exists.

AMS subject classifications: 86A15, 74J20, 74B05

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Key words: Multiple scattering, elastic waves, elastic Green's tensor.

*Corresponding author.
Email: japerez@ual.es (J. A. Perez-Ruiz), fluzon@ual.es (F. Luzon), sesma@servidor.unam.mx (F. J. Sanchez-Sesma)
 

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