P- and S-wave simulation using a Cole–Cole model to incorporate thermoelastic attenuation and dispersion

In thermoelastic wave attenuation, such as that caused by heterogeneities much smaller than the wavelength, e.g., Savage theory of spherical pores, the shape of the relaxation peak differs from that of the Zener (or standard linear solid) mechanical model. In these effective homogeneous media, the anelastic behavior is better represented by a stress-strain relation based on fractional derivatives; particularly, P- and S-wave dispersion and attenuation is well described by a Cole–Cole equation. We propose a time-domain algorithm for wave propagation based on the Grünwald–Letnikov numerical derivative and the Fourier pseudospectral method to compute the spatial derivatives. As an example, we consider Savage theory and verify the algorithm by comparison with the analytical solution in homogeneous media based on the frequency-domain Green function. Moreover, we illustrate the modeling performance with wave propagation in a two half-space medium where one section is lossless and the other is a Cole–Cole medium. This apparently simple example, which does not have an analytical solution, shows the complexity of the wavefield that characterizes a single flat interface.