Numerical simulation of seismic wave propagation in viscoelastic-anisotropic media using frequency-independent Q wave equation

Abstract

Seismic anisotropy is the fundamental phenomenon of wave propagation in the earth's interior. Numerical modeling of wave behavior is critical for exploration and global seismology studies. The full elastic (anisotropy) wave equation is often used to model the complexity of velocity anisotropy, but it ignores attenuation anisotropy. I have presented a time-domain displacement-stress formulation of the anisotropic-viscoelastic wave equation, which holds for arbitrarily anisotropic velocity and attenuation 1∕Q. The frequency-independent Q model is considered in the seismic frequency band; thus, anisotropic attenuation is mathematically expressed by way of fractional time derivatives, which are solved using the truncated Grünwald-Letnikov approximation. I evaluate the accuracy of numerical solutions in a homogeneous transversely isotropic (TI) medium by comparing with theoretical Q P and Q S values calculated from the Christoffel equation. Numerical modeling results show that the anisotropic attenuation is angle dependent and significantly different from the isotropic attenuation. In synthetic examples, I have proved its generality and feasibility by modeling wave propagation in a 2D TI inhomogeneous medium and a 3D orthorhombic inhomogeneous medium.

Viscoacoustic anisotropic wave equations

“…The wave equations 44 can be solved numerically by analogy with Carcione et al (2002) and Zhu (2017 (Podlubny, 1998). Finally, we may obtain the scalar viscoacoustic wave equation with fractional derivatives.…”

“…Extending some of these models to the anisotropic case can be seen in Carcione (2015), Bai and Tsvankin (2016), Bai et al (2017) Zhu (2017), Zhu and Bai (2019).…”

Efficient modeling of wave propagation in a vertical transversely isotropic attenuative medium based on fractional Laplacian

“…Before proceeding to a discussion of fractional Laplacian constitutive equations, it will be beneficial to briefly revisit the fractional time formulation of the anisotropic-viscoelastic wave equation (Zhu, 2017). This section will serve as the basis for later derivation.…”

“…The stress-strain relation is simplified from equation 3 to (Zhu, 2017). Zhu (2017) adopts the computational schemethe Grünwald-Letnikov approximation (Carcione et al, 2002) to calculate the fractional time derivative ∂ 2γ IJ t in equation 4. In his implementation, γ IJ is a spatial-dependent variable without the averaging approximation.…”

“…Following this, Ruud and Hestholm (2005) implement a 2D time-domain finite-difference modeling of orthorhombic attenuation, and Bai and Tsvankin (2016) present a detailed numerical demonstration and analysis of vertical transversely isotropic (VTI) attenuation by 2D time-domain finite-difference modeling. Zhu (2017) derives a viscoelastic-anisotropic wave equation based on frequency-independent Q (Kjartansson, 1979). Although his equation is able to parameterize general Q anisotropy and can be used to simulate wave simulation with arbitrarily anisotropic Q, the efficiency of his numerical computation may hamper large-scale applications.…”