The time-accurate solution of the fractional Laplacian-based viscoacoustic wave equation based on the hybrid rapid expansion method

Abstract In recent years, the attention paid to the viscoacoustic wave equation with decoupled fractional Laplacian (DFL) operators has increased remarkably. This is due to the equation's unique decoupling property and its ability to accurately characterize the quality factor. The DFL viscoacoustic equation is typically computed numerically employing the finite difference (FD) and Fourier pseudo-spectral (PS) methods with respect to the derivatives of time and space, respectively. Although the FD-PS method provides spectral accuracy in the spatial domain, its time accuracy is limited to second order. In scenarios with a larger time sampling interval, the FD-PS method might encounter significant time dispersion. Although reducing the time sampling interval can alleviate this dispersion, it significantly boosts computational cost. To correct the time error introduced by time dispersion, we propose a hybrid rapid expansion method (HREM) to rectify the time error in DFL viscoacoustic equation. This method mitigates the time error by compensating for the dispersion-dominated term that exerts a considerable influence on the time error. HREM has been validated to effectively correct the time error through theoretical analysis and numerical experiments. Compared to the traditional FD-PS method, the stability condition of HREM is looser, thereby conferring enhanced flexibility in the selection of sampling parameters. Furthermore, HREM allows for the accurate estimation of the wavefield with larger time steps. The application of a three-dimensional overthrust model illustrates its effectiveness for large-scale seismic modelling.