The calculation of first-arrival P-wave traveltime for attenuating TTI media

Abstract

First-arrival P-wave traveltimes play a crucial role in many geophysical applications such as migration and tomography. And the solution of Eikonal equation has been proven effective and accurate for approximating the traveltimes. Eikonal equation for attenuative case can provide not only the information of phase arrival, but also the decay in wave amplitudes resulted from energy absorption by media. By applying the rotation operator to the eikonal equation in attenuating VTI (vertical transversely isotropic) media, the equation in attenuating TTI (titled transversely isotropic) media can be derived directly. However, numerical methods such as fast marching method that has been widely used for eikonal solutions, updating the traveltimes of each grid points by choosing the minimum of traveltimes, cannot be applied to attenuating case due to no minimum value between two complex numbers. In order to address this issue, the perturbation method can be introduced by decomposing the complex-valued eikonal equation into real and imaginary part, called zeroth- and first-order governing equation, respectively. The real part of the complex-valued travel-time corresponds to the phase of waves, while the imaginary part describes the magnitude of attenuation. Thus, by solving these governing equations successively, the real and imaginary part of travel-time can be obtained. Numerical modeling for homogeneous or SEG/ Hess attenuating TTI media demonstrate the effectiveness and accuracy of our technique.