Two efficient modeling schemes for fractional Laplacian viscoacoustic wave equation

Recently, a decoupled fractional Laplacian viscoacoustic wave equation has been developed based on the constant-Q model to describe wave propagation in heterogeneous media. We have developed two efficient modeling schemes to solve the decoupled fractional Laplacian viscoacoustic wave equation. Both schemes can cope with spatial variable-order fractional Laplacians conveniently, and thus are applicable for modeling viscoacoustic wave propagation in heterogeneous media. Both schemes are based on fast Fourier transform, and have a spectral accuracy in space. The first scheme solves a modified wave equation with constant-order fractional Laplacians instead of spatial variable-order fractional Laplacians. Due to separate discretization of space and time, the first scheme has only first-order accuracy in time. Differently, the second scheme is based on an analytical wave propagator, and has a higher accuracy in time. To increase computational efficiency of the second modeling scheme, we have adopted the low-rank decomposition in heterogeneous media. We also evaluated the feasibility of applying an empirical approximation to approximate the fractional Laplacian that controls amplitude loss during wave propagation. When the empirical approximation is applied, our two modeling schemes become more efficient. With the help of numerical examples, we have verified the accuracy of our two modeling schemes with and without applying the empirical approximation, for a wide range of seismic quality factor (Q). We also compared computational efficiency of our two modeling schemes using numerical tests.