Introduction

The isotropic acoustic assumption for seismic processing may not always be appropriate. This fact has been recognized in areas such as the North Sea and the Canadian Foothills (Etgen et al., 2009). Conventional isotropic methods may cause errors in anisotropic media, which result in low resolution and misplaced images of subsurface structures (Zhou et al., 2006b). Therefore, imaging of surveys in the presence of anisotropy requires a migration method that can handle general anisotropic media to obtain a significant improvement in image quality, clarity, and positioning.

Some ordering in the structure of rocks, such as fine-layering and parallel cracks can induce anisotropic effects in the wave propagation. VTI RTM is a good simplification to image such structures since they have similar properties with a VTI medium (Crampin, 1984). However, the VTI assumption may not be satisfied in imaging under steeply dipping anisotropic overburdens. For example, if the sedimentary layering is not horizontal, such as shale masses overlying dipping salt flanks, the symmetry axis is most likely to be tilted. Ignoring the titled symmetry not only causes image blurring and mispositioning of the salt flank, but also degrades and distorts the base of salt and subsalt images (Zhang and Zhang, 2009). In other words, a local symmetry assumption instead of a global one is more realistic.

Alkhalifah (1998,2000) started from the dispersion relation and proposed a pseudo-acoustic wave equation in TI media by setting the shear-wave velocity along the symmetry axis to be zero. Based on Alkhalifah's pseudo-acoustic approximation, a number of variations of pseudo-acoustic wave equation have been developed to account for VTI media (Zhou et al., 2006a; Du et al., 2008). Assuming the symmetry axis is non-vertical and locally variable, extensions from VTI to TTI have been developed (Zhang and Zhang, 2008; Fletcher et al., 2008; Zhou et al., 2006b). This allows the anisotropy to conform to spatially variable structure.

Tilting the symmetry axis relative to the coordinates does not add any new physics, just more algebraic complexity (Fowler et al., 2010). However, in practice, not only does TTI RTM require significantly more computer resources than isotropic and VTI cases, but also non-vertical symmetry axes can cause severe numerical dispersion and subsequent instabilities (Zhang and Zhang, 2008).

Fletcher et al. (2009) proposed to add non-zero shear wave velocity terms to overcome the instability problem. In their method, SV wavefront triplications are removed by using a small value for $\sigma = v_{pz}^2/v_{sz}^2(\varepsilon - \delta)$ (choosing large $v_{sz}$ ), which yields stable wave propagation even with a highly variable tilt-axis orientation. Here, $v_{pz}$ and $v_{sz}$ represent the P-wave and SV velocity along the symmetry axis, respectively (Tsvankin, 2001; Fletcher et al., 2009). By doing so, further numerical complexity to the TTI coupled equations and extra computational cost are introduced (Zhang and Zhang, 2009). Instead of incorporating large $v_{sz}$ , Yoon et al. (2010) proposed a selective anisotropic parameter matching technique with less pain to reduce the instability by making $\varepsilon = \delta$ in regions with rapid dip angle variation. This method does not complicate the TTI formulations, and no more computational cost is introduced.

In this report, I present a pseudo-Fourier implementation of the TTI coupled equations using a hybrid FD and PS method, where the space-derivative is calculated in the wavenumber domain and time stepping is done by an explicit FD method. In areas of higher velocity values, the PS scheme is replaced by the FD method to compute the Laplacian operator for a more efficient computational performance. The instability problem is eased by using a selective anisotropic parameter matching technique.

This chapter is organized as follows. First I present the TTI coupled equations. Then two TTI RTM examples are presented using synthetic data associated with the 2D Hess salt model and the 2D BP TTI model. Several practical issues regarding the implementation are addressed as well. At the end, I draw some conclusions.