Inhomogeneous plane waves in dissipative, isotropic and anisotropic media. Reflection/transmission coefficients

Johana Brokesova & Vlastislav Cerveny

Summary

To understand well the propagation of high-frequency seismic body waves in dissipative, 3-D laterally varying isotropic and anisotropic structures, containing curved structural interfaces, it is useful to study first the behavior of plane waves in relevant homogeneous media with planar interfaces. The ray methods represent, in fact, a local application of plane waves in more complex structures. For example, the reflection/transmission coefficients of plane waves at plane interfaces between two homogeneous halfspaces may be locally applied even to high-frequency seismic body waves incident at a curved interface between two inhomogeneous media. This is the reason why this contribution is devoted to plane waves in dissipative homogeneous media with planar interfaces.

Inhomogeneous, time-harmonic, plane waves, propagating in dissipative, isotropic and anisotropic media are investigated. The elastodynamic equation for the dissipative homogeneous media is solved in terms of eigenvalues and eigenvectors of the 3x3 Christoffel matrix Gamma, with complex-valued elements Gammaik = aijklpjpl, where aijkl are complex-valued density normalized elastic moduli, and pj the complex-valued components of the slowness vector p. The eigenvalues of the Christoffel matrix can be used to determine the components of the slowness vector, and the eigenvectors to determine the (elliptical) polarization of individual inhomogeneous plane waves. Contrary to the traditionally used method based on Lame's elastic potentials, the used method can be applied quite generally, including anisotropic media. It can be also easily applied to the solution of the reflection/transmission problem at a plane interface between two dissipative halfspaces, or at a thin transition layer, simulated by a system of thin homogeneous parallel sublayers.

As pi are complex-valued, pi = Pi + iAi, the inhomogeneous plane waves are characterized by two planes: the plane of constant phases, Pixi=const, and plane of constant amplitudes, Aixi=const. The real-valued vector P is called the propagation vector, and the real-valued vector A the attenuation vector. The angle between them, called the attenuation angle and denoted by gamma, may be an arbitrary acute angle different from 90o in dissipative media. The plane wave with gamma= 0o is called homogeneous. The reflection/transmission problem is solved here for an inhomogeneous incident plane wave, with an arbitrary attenuation angle gamma, both for isotropic and anisotropic media. The homogeneous incident plane wave is included as a special case (gamma=0o).

A computer routine COEF52 is described, designed to compute arbitrary frequency-dependent R/T coefficients on a plane interface between two dissipative isotropic halfspaces, with a constant-Q dispersion relations, assuming an incident plane wave with an arbitrary attenuation angle gamma. Routine COEF52 is a generalization of routine COEF51, described by Brokesova and Cerveny (1997), where a special choice of the attenuation angle gamma of the incident wave was used. The routine COEF52 may be also used to compute R/T coefficients from transition layers.

Results of extensive computations of "reference R/T coefficients" on a plane interface between two dissipative isotropic halfspaces are presented and discussed. In all cases, they are also compared with those for non-dissipative media. Under reference R/T coefficients, we understand the R/T coefficients for dissipative media, computed for the fixed frequency, equal to the reference frequency fr, for which the complex-valued elastic moduli in the model are specified. The dependence of R/T coefficients on the attenuation angle gamma of the incident wave is studied in detail. For the computations of R/T coefficients, see Section 6, and for the conclusions Section 7. In general, the differences between the moduli of the R/T coefficients for dissipative and non-dissipative models are usually small for realistic Q's and gamma's. Exceptions are discussed in Section 6. The differences between the moduli are more distinct mainly in critical and post-critical regions of angles of incidence for the reflection coefficients S1S1 and P1P1, particularly for small Q's, large contrast of Q's across the interface, and for great |gamma|'s (say, |gamma|> 50o). The differences between the phases of R/T coefficients for dissipative and non-dissipative media may be greater. In certain situations, for some gamma's, the signs of the phases of R/T coefficients for dissipative and non-dissipative media may be roughly opposite, even for very high Q's.We speak of anomalous phases, see Section 6.2 for a detailed description.

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In: Seismic Waves in Complex 3-D Structures, Report 7, pp. 57-146, Dep. Geophys., Charles Univ., Prague, 1998.
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