Modeling viscoacoustic wave propagation using a new spatial variable-order fractional Laplacian wave equation

Abstract

We propose a new time-domain viscoacoustic wave equation for simulating wave propagation in anelastic media. The new wave equation is derived by inserting the complex-valued phase velocity described by the Kjartansson attenuation model into the frequency-wavenumber domain acoustic wave equation. Our wave equation includes one second-order temporal derivative and two spatial variable-order fractional Laplacian operators. The two fractional Laplacian operators describe the phase dispersion and amplitude attenuation effects, respectively. To facilitate the numerical solution for the proposed wave equation, we use the arbitrary-order Taylor series expansion (TSE) to approximate the mixed domain fractional Laplacians and achieve the decoupling of the wavenumber and the fractional order. Then the proposed viscoacoustic wave equation can be directly solved using the pseudospectral method (PSM). We adopt a hybrid pseudospectral/finite-difference method (HPSFDM) to stably simulate wave propagation in arbitrarily complex media. We validate the high accuracy of the proposed approximate dispersion term and approximate dissipation term in comparison with the accurate dispersion term and accurate dissipation term. The accuracy of numerical solutions is evaluated by comparison with the analytical solutions in homogeneous media. Theory analysis and simulation results show that our viscoacoustic wave equation has higher precision than the traditional fractional viscoacoustic wave equation in describing constant- Q attenuation. For a model with Q < 10, the calculation cost for solving the new wave equation with TSE HPSFDM is lower than that for solving the traditional fractional-order wave equation with TSE HPSFDM under the high numerical simulation precision. Furthermore, we demonstrate the accuracy of HPSFDM in heterogeneous media by several numerical examples.

Multidimensional Q‐compensated reverse time migration using a high‐efficient decoupled viscoacoustic wave equation

“…Hence, the crucial kernel of Q ‐RTM is the high‐efficient time‐domain viscoacoustic wave equation with decoupled amplitude loss and phase dispersion terms. However, the majority of current decoupled viscoacoustic wave equations suitable for Q ‐RTM are based on Kjartansson's constant‐ Q model (Chen et al., 2016; Hao & Greenhalgh, 2021; Hao et al., 2022; Kjartansson, 1979; Li et al., 2016; Mao et al., 2023; Mu et al., 2021; Wang et al., 2018, 2020; Yang & Zhu, 2018b; Zhang et al., 2010; Zhu & Harris, 2014). These wave equations need to be solved using the pseudo‐spectral method (PSM) (Carcione, 2010; Chen & Holm, 2004).…”

Fractional laplacians viscoelastic wave equation low-rank temporal extrapolation

“…The space average strategy by only offers reasonable accuracy in the case of a smooth Q model but introduces large simulation errors in the case of a sharp Q variation. Several different approximations have been developed to transform the variable-order fractional Laplacians into constantorder fractional Laplacians Yang and Zhu, 2018;Xing and Zhu, 2019;Mu et al, 2021), which facilitates the FFT simulations, significantly. Although the FFT differentiation achieves a spectral accuracy in space, the tradtional pseudo-spectral (PS) method uses a second-order FD operator to approximate the time derivative, which limits the temporal extrapolation accuracy to second-order.…”

Multiparameter least‐squares reverse time migration using the viscoacoustic‐wave equation

“…They showed that the proposed 𝑘-space approach is better than the traditional finite difference method in the second-order precision scheme. Mu et al (2021) developed a viscoacoustic wave equation in the time domain to simulate wave propagation in anelastic media. They proposed a new viscoacoustic wave equation inserting the complex-valued phase velocity derived from the Kjartansson attenuation model in the frequency-wavenumber domain in the acoustic wave equation.…”