A ray perturbation method for the acoustic attenuating transversely isotropic eikonal equation

Modeling complex-valued traveltimes is helpful for devel-oping attenuation-associated techniques for seismic data processing. Transverse isotropy can explain the directional variation of velocity and attenuation anisotropy of long -wave-length seismic waves in many sedimentary rocks. The acous-tic attenuating transversely isotropic eikonal equation can be used to accurately calculate P-wave complex-valued travel -times under such a geologic condition. However, no ray -trac-ing 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 non-linear partial differential equation (PDE) and a first-order PDE. These two PDEs can be solved by the method of char-acteristics. This gives rise to two sets of ray-tracing equations for the real and imaginary parts of the complex-valued trav-eltimes, respectively. Both sets of ray-tracing equations share the same raypath, which allows us to merge them into a com-plete ray-tracing system for complex-valued traveltimes. Numerical examples are used to demonstrate the high accu-racy 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.