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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