Introduction

In geophysics, seismic imaging using the isotropic assumption is not always appropriate due to the existence of seismic anisotropy in the subsurface. Conventional isotropic wave propagation methods can lead to incorrectly timed and positioned wavefields when used for reverse time migration (RTM) in anisotropic media. These inaccurate propagations may result in misplaced images and low resolution of the target. Therefore, imaging seismic surveys in the presence of anisotropy requires migration methods that can correctly propagate waves in anisotropic media.

Fine vertical layering and horizontally aligned parallel cracks can induce anisotropic effects in seismic wave propagation. This type of vertical transversely isotropic (VTI) media is a good assumption in many areas and my migration algorithms must now include the effects and parameters of VTI media (Crampin, 1984) to produce correct images. However, the VTI assumption is not always satisfied when imaging under steeply dipping anisotropic overburdens, such as shale masses overlying dipping salt flanks, where the symmetry axis is likely to be tilted. Mispositioning of subsurface structures and image blurring are easily introduced due to the neglect of this kind of anisotropy or the inclusion of incorrect assumptions about the nature of anisotropy. In many areas it is more realistic to make a local symmetry assumption rather than using a global one such as VTI. This leads to the need for a spatially variable tilted symmetry axis for anisotropy, or tilted transversely isotropic (TTI) media.

Originally, efforts to describe acoustic anisotropic wave propagation in seismic imaging started with the dispersion relation of the elastic wave equation and simply defined the SV wave velocity as zero along the symmetry axis (Alkhalifah, 2000). This is a pseudo-acoustic wave equation assumption which is unphysical but useful for seismic imaging purposes. Based on this pseudo-acoustic approximation, a series of variations of pseudo-acoustic wave equations have been proposed to deal with VTI media (Zhou et al., 2006a; Duveneck et al., 2008; Du et al., 2008). Assuming the symmetry axis is not vertical but locally variable extends these developments from VTI to TTI (Zhang and Zhang, 2008; Fletcher et al., 2008; Zhou et al., 2006b). This allows imaging algorithms to include anisotropy due to spatially variable structures.

Tilting the symmetry axis from the vertical not only adds more computational effort than the isotropic and VTI cases, but also introduces severe numerical dispersion and instability from the introduction of the non-vertical symmetry axis. Zhang and Zhang (2008) showed that by smoothing the anisotropic parameter models before modeling or migration, the instability problem is alleviated. Instead of model smoothing, Yoon et al. (2010) proposed a similar approach to reduce the instability by making $\varepsilon = \delta$ in regions with rapid dip angle variations. While these methods reduce strong variations present in the model parameters and thus stabilize the wave propagation, they alter the wave kinematics. Fletcher et al. (2009) found that including a finite SV wave velocity in the coupled equations can partly solve the instability problem. In their method, SV wavefront triplications (which present as an incorrect diamond shape) are removed by including a non-zero SV wave velocity, which seems to reduce the instability. This method does not alter the wave kinematics. However, there are SV wave components present in the P wave simulation and instabilities still affect wave propagation at later times.

Simply setting the SV wave velocity to be zero along the symmetry axis in acoustic TI media does not lead to a total elimination of the shear wave phase (Grechka et al., 2004). The generated shear waves are usually considered as numerical artifacts in conventional P wave modeling and migration, and are likely the cause of numerical instabilities in TTI media. It is well known that setting $\epsilon = \delta$ around the source (Duveneck et al., 2008) can avoid shear wave artifacts generated at the source. However, contrasts existing in the anisotropic parameter models elsewhere still produce shear wave artifacts. To avoid the undesired SV wave mode completely, different approaches have recently been proposed to model just a pure P wave mode (Pestana et al., 2011; Liu et al., 2009; Etgen and Brandsberg-Dahl, 2009) for the VTI case.

In this paper, new pseudo-differential TTI decoupled equations are derived which are purely 2nd-order in time and have kinematics similar to acoustic P wave and SV wave solutions in a TTI medium. Rather than following Alkhalifah's (2000) work, I start with a derivation based on the exact dispersion relation that was originally derived by Tsvankin (1996). I follow Pestana et al. (2011) (but note that Fowler (2003) developed the same equations using a similar approach) who used a square root approximation to obtain completely decoupled P and SV wave equations for VTI, I now use the same equations as the basis and obtain new approximations for TTI. I use the pseudospectral method for the spatial response since the equations are best developed and then evaluated in the wavenumber domain. The rapid expansion method (REM) proposed by Pestana and Stoffa (2010) is chosen to propagate the wavefield in time since it has no numerical dispersion. Both 2D and 3D impulse responses for modeling using the new TTI decoupled equations are shown. Then modeling of a 2D wedge tests the instability issue. Finally, RTM examples associated with the BP 2D TTI benchmark data are presented to validate the new imaging algorithm. For comparison, results from the TTI coupled equations (Fletcher et al., 2009) solved by REM are also presented.