Accurate and efficient modeling of global seismic wave propagation for an attenuative Earth model including the center

We propose a method for modeling global seismic wave propagation through an attenuative Earth model including the center. This method enables accurate and efficient computations since it is based on the 2.5-D approach, which solves wave equations only on a 2-D cross section of the whole Earth and can correctly model 3-D geometrical spreading. We extend a numerical scheme for the elastic waves in spherical coordinates using the finite-difference method (FDM), to solve the viscoelastodynamic equation. For computation of realistic seismic wave propagation, incorporation of anelastic attenuation is crucial. Since the nature of Earth material is both elastic solid and viscous fluid, we should solve stress-strain relations of viscoelastic material, including attenuative structures. These relations represent the stress as a convolution integral in time, which has had difficulty treating viscoelasticity in time-domain computation such as the FDM. However, we now have a method using so-called memory variables, invented in the 1980s, followed by improvements in Cartesian coordinates. Arbitrary values of the quality factor (Q) can be incorporated into the wave equation via an array of Zener bodies. We also introduce the multi-domain, an FD grid of several layers with different grid spacings, into our FDM scheme. This allows wider lateral grid spacings with depth, so as not to perturb the FD stability criterion around the Earth center. In addition, we propose a technique to avoid the singularity problem of the wave equation in spherical coordinates at the Earth center. We develop a scheme to calculate wavefield variables on this point, based on linear interpolation for the velocity-stress, staggered-grid FDM. This scheme is validated through a comparison of synthetic seismograms with those obtained by the Direct Solution Method for a spherically symmetric Earth model, showing excellent accuracy for our FDM scheme. As a numerical example, we apply the method to simulate seismic waves affected by hemispherical variations of P-wavespeed and attenuation in the top 300. km of the inner core.