Fractional-order velocity-dilatation-rotation viscoelastic wave equation and numerical solution based on constant-Q model

The viscoelastic wave equations based on the constant- Q (CQ) model can accurately describe the amplitude dissipation and phase distortion of waves in anelastic medium. However, only three velocity or displacement components can be obtained directly by solving such equations. Starting from the time-domain second-order displacement viscoelastic wave equation, we derived the decoupled P- and S-wave displacement vector viscoelastic wave equation by using the polarization difference of P- and S- waves propagation in isotropic media. The equation can be transformed into the velocity-dilatation-rotation viscoelastic wave equation containing the first-order temporal derivative and fractional Laplacian operators which can be solved directly by using the staggered-grid finite-difference and pseudo-spectral methods. We use the low-rank decomposition method to approximate the derived mixed space-wavenumber domain fractional Laplacian operators for modeling wave propagation in heterogeneous attenuating medium. We also demonstrated the precision of the proposed equation by comparing the numerical solutions with the analytical solutions. Furthermore, compared with the conventional velocity-stress viscoelastic wave equation, experimental results demonstrate that the proposed equation can separate the pure P- and S-waves from the mixed wavefield during wavefield continuation, but also be decoupled to the equation containing predominantly amplitude attenuation or phase distortion term.