Fractional Laplacian Viscoacoustic Wave Equation Low-Rank Temporal Extrapolation

Abstract

Description of seismic wave attenuation is a hot topic in the geophysical area, and it is the basis of the attenuation-compensated seismic imaging technique, which aims to retrieve a high-resolution subsurface image for geological structure analysis and hydrocarbon reservoir prediction. The numerical simulation of viscoacoustic wave equation is an effective way to observe the seismic attenuation in lossy media. The existing study confirms that the seismic-quality-factor (Q) that is used to represent the strength of the viscous behavior of the earth is nearly independent of frequency, which is referred to as the constant-Q (CQ) model. The mathematical concept of fractional Laplacian is recently introduced to the geophysical area to form a compact CQ wave equation to describe seismic wave propagation. However, numerically solving the fractional Laplacian CQ wave equation by the traditional pseudospectral time-domain (PSTD) method suffers from a strict stability condition and great numerical dispersion due to a low-order temporal finite-difference (FD) approximation. To improve the temporal extrapolation accuracy, we derive the analytical wavenumber (k)-space domain propagators underlying the fractional Laplacian wave equation. We regard the k-space operators as mixed-domain matrices in the case of heterogeneous media and adopt a low-rank decomposition to approximate the matrices. With the low-rank approximation, an efficient time-marching formula is obtained for wavefield temporal extrapolation. We formulate the time-marching formula into a first-order equation system in terms of pressure and particle-velocity to welcome the perfectly matched layer (PML) absorbing boundary condition to eliminate the wraparound effects caused by the Fourier transform. A spatial-variable density is also incorporated to simulate more realistic amplitude variation. The numerical examples are carried out to verify the accuracy and stability of the viscoacoustic low-rank extrapolation.

Keywords

• Fractional Laplacian
• pseudospectral time-domain
• wave propagation
• low-rank
• lossy media