Calculating complex‐valued P‐wave first‐arrival traveltimes in attenuative vertical transversely isotropic media with an irregular surface

ABSTRACT The complex‐valued first‐arrival traveltime can be used to describe the properties of both velocity and attenuation as seismic waves propagate in attenuative elastic media. The real part of the complex‐valued traveltime corresponds to phase arrival and the imaginary part is associated with the amplitude decay due to energy absorption. The eikonal equation for attenuative vertical transversely isotropic media discretized with rectangular grids has been proven effective and precise to calculate the complex‐valued traveltime, but less accurate and efficient for irregular models. By using the perturbation method, the complex‐valued eikonal equation can be decomposed into two real‐valued equations, namely the zeroth‐ and first‐order traveltime governing equations. Here, we first present the topography‐dependent zeroth‐ and first‐order governing equations for attenuative VTI media, which are obtained by using the coordinate transformation from the Cartesian coordinates to the curvilinear coordinates. Then, we apply the Lax–Friedrichs sweeping method for solving the topography‐dependent traveltime governing equations in order to approximate the viscosity solutions, namely the real and imaginary parts of the complex‐valued traveltime. Several numerical tests demonstrate that the proposed scheme is efficient and accurate in calculating the complex‐valued P‐wave first‐arrival traveltime in attenuative VTI media with an irregular surface.