The attenuation vector plays an important role in investigating waves propagating in dissipative, isotropic or anisotropic media. It is defined as the imaginary part of the complex-valued travel-time gradient. The real part of the complex-valued travel-time gradient is referred to as the propagation vector. In this paper, simple expressions for the attenuation vector are derived for seismic body waves propagating in heterogeneous, weakly dissipative, anisotropic media. The ray-theory perturbation method is used, in which the dissipative medium is considered to be a small perturbation of a perfectly elastic medium. The general perturbation procedure proposed by Klimes (2002) is very suitable for this purpose, as it does not require ray tracing in dissipative media; the computation of rays in the reference perfectly elastic medium, followed by quadratures along them, is quite sufficient. It is shown that the waves propagating in a heterogeneous, weakly dissipative, anisotropic or isotropic medium are, in general, inhomogeneous. This means that their attenuation vector is not parallel to the propagation vector. This holds even for waves generated by a point-source in a homogeneous anisotropic weakly dissipative medium. An exception is a wave generated by a point-source in a homogeneous isotropic dissipative medium, where the generated wave is homogeneous. Thus, the commonly used concept of homogeneous waves can be applied neither to heterogeneous nor to anisotropic dissipative media. The situation is different for plane waves propagating in homogeneous dissipative isotropic or anisotropic media. In this case, the homogeneity or inhomogeneity of the plane wave may be chosen freely. Besides the attenuation vector, we also study the complex-valued ray-velocity vector. In heterogeneous media, the complex-valued ray-velocity vector is generally inhomogeneous, that is, its real and imaginary parts are not parallel. It is also shown that twice the scalar product of the attenuation vector with the ray-velocity vector in the reference medium yields 1/Q, where Q is the position- and direction-dependent quality factor. Quality factor Q does not depend on the inhomogeneity of the wave under consideration and offers a convenient measure of intrinsic material dissipation.
Seismic anisotropy, seismic attenuation, theoretical seismology, wave propagation.
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