Role of the inhomogeneity angle in anisotropic attenuation analysis

The inhomogeneity angle (the angle between the real and imaginary parts of the wave vector) is seldom taken into account in estimating attenuation coefficients from seismic data. Wave propagation through the subsurface, however, can result in relatively large inhomogeneity angles ξ, especially for models with significant attenuation contrasts across layer boundaries. Here we study the influence of the angle ξ on phase and group attenuation in arbitrarily anisotropic media using the first-order perturbation theory verified by exact numerical modeling. Application of the spectral-ratio method to transmitted or reflected waves yields the normalized group attenuation coefficient Ag, which is responsible for amplitude decay along seismic rays. Our analytic solutions show that for a wide range of inhomogeneity angles, the coefficient Ag is close to the normalized phase attenuation coefficient A computed for ξ=0°(∣A∣ξ=0°). The coefficient∣A∣ξ=0° can be inverted directly for the attenuation-anisotropy parameters, so no knowledge of the inhomogeneity angle is required for attenuation analysis of seismic data. This conclusion remains valid even for uncommonly high attenuation with the quality factor Q less than 10 and strong velocity and attenuation anisotropy. However, the relationship between group and phase attenuation coefficients becomes more complicated for relatively large inhomogeneity angles approaching so-called “forbidden directions.” We also demonstrate that the velocity function remains practically independent of attenuation for a wide range of small and moderate angles ξ. In principle, estimation of the attenuation-anisotropy parameters from the coefficient ∣A∣ξ=0° requires computation of the phase angle, which depends on the anisotropic velocity field. For moderately anisotropic models, however, the difference between the phase and group directions should not significantly distort the results of attenuation analysis.