Modelling of viscoacoustic wave propagation in transversely isotropic media using decoupled fractional Laplacians

Abstract

A new wave equation is derived for modelling viscoacoustic wave propagation in transversely isotropic media under acoustic transverse isotropy approximation. The formulas expressed by fractional Laplacian operators can well model the constant‐Q (i.e. frequency‐independent quality factor) attenuation, anisotropic attenuation, decoupled amplitude loss and velocity dispersion behaviours. The proposed viscoacoustic anisotropic equation can keep consistent velocity and attenuation anisotropy effects with that of qP‐wave in the constant‐Q viscoelastic anisotropic theory. For numerical simulations, the staggered‐grid pseudo‐spectral method is implemented to solve the velocity–stress formulation of wave equation in the time domain. The constant fractional‐order Laplacian approximation method is used to cope with spatial variable‐order fractional Laplacians for efficient modelling in heterogeneous velocity and Q media. Simulation results for a homogeneous model show the decoupling of velocity dispersion and amplitude loss effects of the constant‐Q equation, and illustrate the influence of anisotropic attenuation on seismic wavefields. The modelling example of a layered model illustrates the accuracy of the constant fractional‐order Laplacian approximation method. Finally, the Hess vertical transversely isotropic model is used to validate the applicability of the formulation and algorithm for heterogeneous media.

Seismic migration in viscoacoustic vertical transversely isotropic media using a pure qP wave equation

“…According to the acoustic approximation of transversely isotropy medium (Duveneck & Bakker, 2011), the relaxation matrix and quality factor matrix of viscoacoustic vertical transversely isotropic (VTI) medium can be expressed as (Hosten et al., 1987; Qiao et al., 2020) $$$$\begin{equation}\rm{\psi} = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{3}{c}@{}} {{\rm {\psi} }_{11}}&{{\rm{\psi} }_{11}}&{{\rm{\psi} }_{13}}\\ {{\rm{\psi} }_{11}}&{{\rm{\psi} }_{11}}&{{\rm{\psi} }_{13}}\\ {{\rm{\psi} }_{13}}&{{\rm{\psi} }_{13}}&{{\rm{\psi} }_{33}} \end{array} } \right],\end{equation}$$$$ $$$$\begin{equation}\bf{Q} = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{3}{c}@{}} {{{Q}}_{\textit{11}}}&{{{Q}}_{\textit{11}}}&{{{Q}}_{\textit{13}}}\\ {{{Q}}_{\textit{11}}}&{{{Q}}_{\textit{11}}}&{{{Q}}_{\textit{13}}}\\ {{{Q}}_{\textit{13}}}&{{{Q}}_{\textit{13}}}&{{{Q}}_{\textit{33}}} \end{array} } \right],\end{equation}$$$$where $${Q}_{\textit{11}} = \frac{{{Q}_{\textit{33}}}}{{1 + {\varepsilon }_Q}}$$, $${Q}_{\textit{13}} = \frac{{2{Q}_{\textit{33}}}}{{{\delta }_Q{C}_{33}/C_{13}^2 + 2}}$$, …”

“…Here, we implement a numerical example of wavefield modelling in the homogeneous medium above, where the reference frequency is 150 Hz, and the source is in centre of the model, which is a Ricker wavelet with dominant frequency 25 Hz. First, we compare the qP wave equation (Equation ) and pseudo qP wave equation (Qiao et al., 2020, equation 24) in viscoacoustic VTI media. As the snapshots shown in Figure 2, the qSV wave interference in part (b) does not appear in part (a).…”

“…(2019) developed two wave equations involving the fractional time derivatives or fractional Laplacian for simulating elastic wave propagation in viscoelastic anisotropic media, and by setting the on‐axis velocity of SV wave to zero only, Qiao et al. (2020) derived a pseudo qP wave equation expressed by fractional Laplacian operators for modelling viscoacoustic wave propagation in transversely isotropy media. On the basis of these studies, and considering the efficiency and accuracy of the imaging algorithm, we further introduce a pure qP wave equation in viscoacoustic VTI media as the propagator into the RTM and LSRTM in this paper.…”

Viscoelastic and viscoacoustic modelling using the Lie product formula

“…For the case in which q = δ q = 0, it corresponds to the special case (Q 11 = Q 33 = Q 13 ) that P-wave attenuation is independent of propagation directions (i.e. isotropic attenuation) even for arbitrary velocity anisotropy (Qiao et al, 2020) and thus equation ( 14) reduces…”

“…For the case in which $$\epsilon _q=\delta _{q}=0$$, it corresponds to the special case ($$Q_{11}=Q_{33}=Q_{13}$$) that P‐wave attenuation is independent of propagation directions (i.e. isotropic attenuation) even for arbitrary velocity anisotropy (Qiao et al ., 2020) and thus equation () reduces …”