Fractional Laplacian Viscoacoustic Wave Equation Low-Rank Temporal Extrapolation

Hanming Chen,Ying Rao,Shuqi Jiang,Hui Zhou

doi:10.1109/access.2019.2927760

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.

Highlights

Acoustic wave modeling is an essential part of the technique of acoustic imaging that has been widely used in the areas of nondestructive examination [1], biomedicine [2], geophysics [3], seismology [4] and so on
The results indicate that low-rank numerical solutions computed with δt = 1.5 ms are more accurate than the PSTD4 numerical solutions computed with the same time step
The temporal extrapolation is based on a low-rank approximation of the mixed-domain k-space operators that underlie the fractional Laplacian viscoacoustic wave equation

Summary

INTRODUCTION

Acoustic wave modeling is an essential part of the technique of acoustic imaging that has been widely used in the areas of nondestructive examination [1], biomedicine [2], geophysics [3], seismology [4] and so on. The Lax-Wendroff approach that transforms the high-order temporal derivatives into highorder spatial derivatives via the wave equation is feasible to improve the temporal approximation accuracy, while preserving the compactness of the three-step time-marching formula [14]. We develop a low-rank extrapolation scheme for the constant fractional-order Laplacian CQ wave equation in [42] to improve wavefield temporal extrapolation accuracy in lossy media. We first review the development of FDTD and PSTD numerical solvers in the introduction, and introduce the traditional PSTD temporal extrapolation for the fractional Laplacian viscoacosutic wave equation. This is followed by a description of the low-rank extrapolation. We review the construction of the temporal fourth-order PSTD (PSTD4) using the Lax-Wendroff approach in Appendix A

FIRST-ORDER K-SPACE OPERATORS

GRADIENT MODEL

CONCLUSION