Modeling complex-valued traveltimes is helpful for developing attenuation-associated techniques for seismic data processing. Transverse isotropy can explain the directional variation of velocity and attenuation anisotropy of long-wavelength seismic waves in many sedimentary rocks. The acoustic attenuating transversely isotropic eikonal equation can be used to accurately calculate P-wave complex-valued traveltimes under such a geologic condition. However, no ray-tracing system for this eikonal equation could be found in the literature until now. We have developed a ray perturbation method to solve this eikonal equation. Unlike all existing ray perturbation methods, our newly proposed method does not perturb the exact ray-tracing system but splits the acoustic attenuating transversely isotropic eikonal equation into a nonlinear partial differential equation (PDE) and a first-order PDE. These two PDEs can be solved by the method of characteristics. This gives rise to two sets of ray-tracing equations for the real and imaginary parts of the complex-valued traveltimes, respectively. Both sets of ray-tracing equations share the same raypath, which allows us to merge them into a complete ray-tracing system for complex-valued traveltimes. Numerical examples are used to demonstrate the high accuracy of the newly proposed ray-tracing system, analyze the complex-valued traveltimes of diving P waves, and compare the modeled complex-valued traveltimes with those extracted from constant-Q viscoelastic waveforms.

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