Asymptotic wave quantities such as ray velocity and ray attenuation are calculated in anisotropic viscoelastic media by using a stationary slowness vector. This vector generally is complex valued and inhomogeneous, and it predicts the complex energy velocity parallel to a ray. To compute the stationary slowness vector, one must find two independent, real-valued unit vectors that specify the directions of its real and imaginary parts. The slowness-vector inhomogeneity affects asymptotic wave quantities and complicates their computation. The critical quantities are attenuation and quality factor (Q-factor); these can vary significantly with the slowness-vector inhomogeneity. If the inhomogeneity is neglected, the attenuation and the Q-factor can be distorted distinctly by errors commensurate to the strength of the velocity anisotropy -- as much as tens of percent for sedimentary rocks. The distortion applies to strongly as well as to weakly attenuative media. On the contrary, the ray velocity, which defines the wavefronts and physically corresponds to the energy velocity of a high-frequency signal propagating along a ray, is almost insensitive to the slowness-vector inhomogeneity. Hence, wavefronts can be calculated in a simplified way except for media with extremely strong anisotropy and attenuation.
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