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In viscoelastic anisotropic media, the elastic moduli, slowness vector, phase and ray velocity are all complex-valued quantities in frequency domain. Solving the complex eikonal equation becomes computationally complex and time consuming. We present two approximate methods to effectively calculate the ray velocity vector, attenuation, and quality factor in viscoelastic transversely isotopic media with a vertical symmetry axis (VTI) and in the orthorhombic anisotropy (ORT). The first method is based on the perturbation theory (PER) under the assumption of a homogeneous complex ray vector, which is obtained by applying the elastic background and viscoelastic perturbations to the real and imaginary components of modulus tensor, respectively. The perturbations of the slowness vectors of the three wave modes (qP, qSV, qSH) are determined through the vanishing Hamiltonian function. The second method is derived by applying a real slowness direction (RSD) to the inhomogeneous complex slowness vector and then approximately calculate the complex ray velocity vector with the condition of homogeneous complex vector. The numerical results verify that the two approaches are able to produce accurate ray velocity vector, attenuation, and quality factors of the qP-wave in viscoelastic VTI and ORT media. The RSD method can yield high accuracies of ray velocity for the qSV- and qSH-wave in viscoelastic VTI models even at triplication of the qSV wavefronts, as well as qS1 and qS2 in a weak ORT medium ( Q > 20), except for near the cusp of the qS1 wavefronts (errors about 6%) where the PER has more than 10% error.

In a viscoelastic anisotropic medium, velocity anisotropy and wave energy attenuation occur and are often observed in seismic data applications. Numerical investigation of seismic wave propagation in complex viscoelastic anisotropic media is greatly helpful in understanding seismic data and reconstructing subsurface structures. Seismic ray tracing is one of the effective means to study the propagation characteristics of high-frequency seismic waves. Unfortunately, most seismic ray-tracing methods and traveltime tomographic inversion algorithms only deal with elastic media and ignore the effect of viscoelasticity on seismic raypath. We present a method to find the complex ray velocity that gives the seismic ray speed and attenuation in an arbitrary viscoelastic anisotropic medium, and then incorporate them with the modified shortest-path method to determine the raypath and calculate the real and imaginary traveltime (wave energy attenuation) simultaneously. We demonstrate that the complex ray-tracing method is applicable to arbitrary 2-D/3-D viscoelastic anisotropic media in a complex geological model, and the computational errors of the real and imaginary traveltime are less than 0.36% and 0.59% respectively. The numerical examples verify that the new method is an effective and powerful tool for accomplishing seismic complex ray tracing in heterogeneous viscoelastic anisotropic media.

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