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 [ ] 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 [ ] 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 [ ] 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.

Two staggered-grid finite-difference (SGFD) schemes with fourth- and sixth-order accuracies in time have been developed recently based on new SGFD stencils. The SGFD coefficients of the two schemes are determined by a Taylor-series expansion (TE) approach, which is accurate only near a zero wavenumber. We have adopted the same SGFD stencils and determined the SGFD coefficients by minimizing the errors between the wavenumber responses of the SGFD operators and the first-order $k$ (wavenumber)-space operator in a least-squares (LS) sense. We solved the LS problems by performing a weighted pseudoinverse of nonsquare matrices to obtain the SGFD coefficients and to yield LS-based SGFD methods. We have developed an efficient LS implementation by using a small number of representative wavenumbers, which makes the computational time for estimating the coefficients negligible. Dispersion analysis and numerical examples demonstrate that our LS-based SGFD methods can preserve the original temporal accuracy and achieve better spatial accuracy than the existing TE-based SGFD methods. Extensive numerical tests proved that our LS-based methods do not damage the stability of the corresponding TE-based methods significantly.

In this paper, the lattice Boltzmann method (LBM) is employed to simulate wave propagation in viscous media. LBM is a kind of microscopic method for modelling waves through tracking the evolution states of a large number of discrete particles. By choosing different relaxation times in LBM experiments and using spectrum ratio method, we can reveal the relationship between the quality factor Q and the parameter τ in LBM. A two-dimensional (2D) homogeneous model and a two-layered model are tested in the numerical experiments, and the LBM results are compared against the reference solution of the viscoacoustic equations based on the Kelvin-Voigt model calculated by finite difference method (FDM). The wavefields and amplitude spectra obtained by LBM coincide with those by FDM, which demonstrates the capability of the LBM with one relaxation time. The new scheme is relatively simple and efficient to implement compared with the traditional lattice methods. In addition, through a mass of experiments, we find that the relaxation time of LBM has a quantitative relationship with Q. Such a novel scheme offers an alternative forward modelling kernel for seismic inversion and a new model to describe the underground media.