Some ordering in the structure of rocks, such as fine-layering and parallel cracks can induce anisotropic effects in the wave propagation. Vertical transversely isotropic (VTI) reverse-time migration (RTM) is a good simplification for imaging such structures because they have properties similar to that of a VTI media (i.e., the medium is stack of thin isotropic layers that induce apparent anisotropic effects in the wave propagation). However, the VTI assumption may not be satisfied for imaging under a steeply dipping anisotropic overburden. 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 the salt and subsalt images (Zhang and Zhang, 2009). In other words, a local symmetry assumption instead of a global one is more realistic.
Alkhalifah (2000) started from the dispersion relation and proposed a pseudo-acoustic wave equation for 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; Duveneck et al., 2008; Du et al., 2008). Assuming the symmetry axis is non-vertical and locally variable, extensions from VTI to TTI (tilted transversely isotropic) have been developed (Zhang and Zhang, 2008; Fletcher et al., 2008; Zhou et al., 2006b). This allows the anisotropy to conform to spatially variable structures.
The main problem with previously published methods is that they are not really free of shear waves.
Simply setting the shear wave velocity along the symmetry axis to be zero
does not result in the vanishing of
the shear wave phase velocity everywhere in an acoustic TI media (Grechka et al., 2004).
The generated SV component is usually considered as a numerical artifact and
may cause numerical instabilities in modeling wave propagation in a TTI media.
It is well known that a small smoothly tapered circular region with
setting around the source (Duveneck et al., 2008)
can avoid shear wave artifacts generated from the source.
However, contrasts existed in anisotropic parameters (such as reflectors)
elsewhere still
produce shear wave artifacts.
To avoid the undesired SV wave mode, different approaches have recently been proposed to
model the pure P-wave mode (Pestana et al., 2011; Liu et al., 2009; Etgen and Brandsberg-Dahl, 2009) for the VTI case.
In this dissertation, I overview the TTI coupled equations first, then implement and validate the algorithm with numerical examples. After that, I present a set of new pseudo-differential TTI decoupled equations. Rather than following Alkhalifah's (2000) work, I start with a new derivation from the exact dispersion relation that was originally derived by Tsvankin (1996). Using a square root approximation, I obtain an approximation for the P-wave dispersion relation. 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. Modeling examples using the new equations have been shown. RTM examples associated with the BP 2D TTI benchmark data set are presented as well to validate the new algorithm.
The main challenge with solving the TTI decoupled equations is that it demands much more computational cost than standard TTI RTM algorithm using coupled equations. As a remedy, I describe a hybrid pseudospectral and finite-difference scheme to solve the decoupled equations. In the hybrid solution, most of the cost-consuming wavenumber terms in the decoupled equations are replaced by inexpensive finite-difference operators, which in turn accelerates the computation and reduces the computational cost. Both 2D and 3D computation examples are provided to demonstrate the uplift in computational efficiency using the hybrid solution.