Modeling of pseudo-acoustic P-waves in orthorhombic media with a lowrank approximation

Introduction

Nowadays, a growing number of seismic modeling and imaging techniques are being developed to handle wave propagation in transversely isotropic media (TI). Such anisotropic phenomena are typical in sedimentary rocks, in which the process of lithification usually produces identifiable layering. In anisotropic media, the velocity is no longer described by a single parameter. Equations for anisotropic wave propagation are more complicated, even for simple cases. Although exact expressions for phase velocities in VTI media involve four independent parameters, It has been observed that only three parameters influence wave propagation and are of interest to surface seismic processing (Alkhalifah and Tsvankin, 1995). Different approximations have been developed to simplify anisotropic equations, such as the weak-anisotropy approximation (Thomsen, 1986), elliptical approximations (Helbig, 1983; Dellinger and Muir, 1988), acoustic approximations (Alkhalifah, 1998,2000) and anelliptic approximations (Fomel, 2004; Muir, 1985; Dellinger et al., 1993). Tectonic movement of the crust may rotate the rocks and tilt the natural vertical orientation of the symmetry axis (VTI), causing a tilted TI (TTI) anisotropy. In addition, tectonic stresses may also fracture rocks, inducing another TI with a symmetry axis parallel to the stress direction and usually normal to the sedimentation-based TI. The combination of these effects can be represented by an orthorhombic model with three mutually orthogonal planes of mirror symmetry; the P-waves in each symmetry plane can be described kinematically as an independent TI model. Realization of the importance of orthorhombic models mainly comes from observation of azimuthal velocity variations in flat-layered rocks, which may indicate valuable fracture properties of reservoirs (Tsvankin and Grechka, 2011).

Wavefields in anisotropic media are well described by the anisotropic elastic-wave equation. However, in practice, we often have little information about shear waves and prefer to deal with scalar wavefields, especially for conventional imaging of subsurface structure. Alkhalifah (2000) derived an acoustic scalar wave equation for VTI media by careful reparametrization followed by setting the shear velocity along the symmetry axis to zero, which provided accurate kinematics for the conventional elastic wavefield. Later on, Alkhalifah (2003) followed the same approach and introduced an acoustic wave equation of the sixth order in axis-aligned orthorhombic media. Fowler and King (2011) presented coupled systems of partial differential equations for pseudo-acoustic wave propagation in orthorhombic media by extending their previous work in TI media (Fowler et al., 2010). Zhang and Zhang (2011) extended self-adjoint differential operators in TTI media (Duveneck and Bakker, 2011; Zhang et al., 2011) to orthorhombic media.

Pseudo-acoustic P-wave modeling with coupled equations may have shear-wave numerical artifacts in the simulated wavefield (Zhang et al., 2009; Grechka et al., 2004; Duveneck et al., 2008). Those artifacts as well as sharp changes in symmetry axis tilting may introduce severe numerical dispersion and instability in modeling. Yoon et al. (2010) proposed to reduce the instability by making $\epsilon \,=\, \delta$ in regions with rapid tilt changes. Fletcher et al. (2009) suggested that including a finite shear-wave velocity enhances the stability when solving the coupled equations. These methods can alleviate the instability problem; however, they may alter the wave propagation kinematics or still leave shear-wave components in the P-wave simulation. Shear-wave artifacts can be removed from the P-wavefield in the phase-shift extrapolation method because the P- and S-wave solutions lie in a different part of the wavenumber spectrum (Bale, 2007). A number of spectral methods are proposed to provide solutions which can completely avoid the shear-wave artifacts (Fowler and Lapilli, 2012; Song and Fomel, 2011; Etgen and Brandsberg-Dahl, 2009; Chu and Stoffa, 2011; Fomel et al., 2012; Song et al., 2013; Zhan et al., 2012).

In this paper, we adopt a dispersion relation for orthorhombic anisotropic media (Alkhalifah, 2003) and introduce a mixed-domain acoustic wave extrapolator for time marching in orthorhombic media. We use the lowrank approximation (Fomel et al., 2010,2012) to handle this mixed-domain operator. We demonstrate by numerical examples that our method is kinematically accurate. Furthermore, there is no coupling of quasi-P and quasi-SV waves in the wavefield and no constraints on Thomsen's parameters required for stability.

Modeling of pseudo-acoustic P-waves in orthorhombic media with a lowrank approximation