Fractional laplacians viscoelastic wave equation low-rank temporal extrapolation

Abstract

The fractional Laplacians constant-Q (FLCQ) viscoelastic wave equation can describe seismic wave propagation accurately in attenuating media. A staggered-grid pseudo-spectral (SGPS) method is usually applied to solve this wave equation but it is of only second-order accuracy in time, due to a second-order finite-difference (FD) time differentiation. Visible time dispersion and numerical instability could appear in the case of a large timestepping size. To resolve this problem, we develop a more accurate low-rank temporal extrapolation scheme for the FLCQ viscoelastic wave equation. We realize this goal by deriving an analytical time-marching formula from the general solution of the FLCQ wave equation. Compressional (P) and shear (S) wave velocities dependent k-space operators are involved in the formula and they can compensate for the time dispersion errors caused by the FD time differentiation. To implement the k-space operators efficiently in heterogeneous media, we adopt a low-rank approximation of these operators, which reduces the computational cost at each time step to several fast Fourier transforms (FFTs). Another benefit of the low-rank extrapolation is explicit separation of P and S waves, which is helpful for further developing vector wavefield-based seismic migration methods. Several numerical examples are presented to verify the higher accuracy and the less restrictive stability condition of the low-rank temporal extrapolation than the traditional SGPS extrapolation.