A constant fractional-order viscoelastic wave equation and its numerical simulation scheme

Abstract

Efficient modeling schemes currently exist to handle the spatially variable-order fractional Laplacians in the fractional Laplacian viscoacoustic wave equation. The simplest approach is to change the spatially variable-order fractional Laplacians into a linear combination of several constant fractional-order Laplacians. We generalize the constant fractional-order scheme to a spatially variable fractional-order viscoelastic wave equation and develop an almost-equivalent constant fractional-order viscoelastic wave equation. Our constant fractional-order scheme avoids the simulation error introduced by directly averaging the spatially varying fractional order; thus, our scheme simulates seismic wave propagation in viscoelastic media with sharp [Formula: see text] contrasts well. The fast Fourier transform is used in the approximation of the fractional Laplacians, which improves the spectral accuracy in space. Several simulation examples are performed to verify that the numerical solution of a homogeneous [Formula: see text] model obtained by solving our constant fractional-order viscoelastic wave equation agrees well with that obtained by solving the original viscoelastic wave equation. The numerical simulations for spatially varying [Formula: see text] models obtained by the new wave equation are more straightforward than those currently in use and match the reference solutions obtained by accurate, but inefficient, methods. This match of simulation results verifies the accuracy of our viscoelastic wave equation.