Elastic wave-mode separation for TTI media

Introduction

Although acoustic migration is currently the most common seismic imaging procedure, elastic imaging, with the addition of converted-waves, has been recognized to have potential advantages in seeing through gas-charged sediments and in structural and near surface imaging (Stewart et al., 2003). Two options are available for elastic imaging: 1) one can separate wave-modes at the surface and image with the separated PP and PS data using acoustic migration tools (Sun et al., 2004), or 2) one can extrapolate the recorded multicomponent data and image with the reconstructed elastic wavefields by applying an imaging condition to wave-modes separated in the vicinity of the image points (Yan and Sava, 2008). The first approach benefits from the simplicity of imaging with scalar waves, but it is based on the assumption that P and S modes can be successfully separated on the recording surface, which is difficult for complicated datasets. The second approach reconstructs elastic wavefields in the subsurface, thus capturing all possible wave-mode transmissions and reflections, although it increases the computational cost in elastic wavefields modeling. In addition, the elastic migration technique requires wave-mode separation before the application of an imaging condition to avoid crosstalk between different wave-modes.

In isotropic media, the P- and S-modes (shear waves do not split in isotropic media) can easily be separated by taking the divergence and curl of the elastic wavefield (Aki and Richards, 2002), and the procedure is effective in homogeneous as well as heterogeneous media. This is because, in the far field, P- and S-waves are polarized parallel and perpendicular to the wave vectors, respectively. The polarization directions of the P- and S-waves only depend on the wave propagation direction and are not altered by the medium. Therefore, the wave-mode separators are invariant with space, and divergence and curl can always be used to separate compressional (scalar) and shear (vector) wave-modes.

However, divergence and curl do not fully separate wave-modes in anisotropic media, because P- and S-waves are not polarized parallel and perpendicular to the wave vectors. Dellinger and Etgen (1990) separate wave-modes in homogeneous VTI (vertically transversely isotropic) media by projecting the vector wavefields onto the polarization vectors of each mode. In VTI media, the polarization vectors of P- and SV-waves depend on the anisotropy parameters $\epsilon$ and $\delta$  (Thomsen, 1986) and are spatially-varying when the medium is inhomogeneous. Therefore, Yan and Sava (2009) separate wave-modes in heterogeneous VTI media by filtering the wavefields with spatially varying separators in the space domain and show that separation is effective even for complex geology with high heterogeneity.

However, VTI models are suitable only for limited geological settings with horizontal layering. Many case studies have shown that TTI (tilted transversely isotropic) models better represent complex geologies like thrusts and fold belts, e.g., the Canadian Foothills (Godfrey, 1991). Using the VTI assumption to image structures characterized by TTI anisotropy introduces both kinematic and dynamical errors in migrated images. For example, Vestrum et al. (1999) and Isaac and Lawyer (1999) show that seismic structures can be mispositioned if isotropy, or even VTI anisotropy, is assumed when the medium above the imaging targets is TTI. To carry out elastic wave-equation migration for TTI models and apply the imaging condition that crosscorrelates the separated wave-modes, the wave-mode separation algorithm needs to be adapted to TTI media. For sedimentary layers bent under geological forces, TTI migration models usually incorporate locally varying tilts, and the local symmetry axes are assumed to be orthogonal to the reflectors throughout the model (C. et al., 2008; Alkhalifah and Sava, 2010). Therefore, in complex TI models, both the local anisotropy parameters $\epsilon$ and $\delta$ , and the local symmetry axes with tilt $\nu$ and azimuth $\alpha$ can be space-dependent.

This technique of separation by projecting the vector wavefields onto polarization vectors has been applied only to 2D VTI models (Yan and Sava, 2009; Dellinger, 1991) and for P-mode separation for 3D VTI models (Dellinger, 1991). For 3D models, the main challenge resides in the fact that fast and slow shear modes have non-linear polarizations along symmetry-axis propagation directions. It is possible to apply the 2D separation method to 3D TTI models using the following procedure. First, project the elastic wavefields onto symmetry planes (which contains P- and SV-modes) and their orthogonal directions (which contain the SH-mode only); then separate P- and SV-modes in the symmetry planes using divergence and curl operators for isotropic media or polarization vector projection for TI media. However, this approach is difficult as wavefields are usually constructed in Cartesian coordinates and symmetry planes of the models do not align with the Cartesian coordinates. Furthermore, for heterogeneous models, the symmetry planes change spatially, which makes projection of wavefields onto symmetry planes impossible. To avoid these problems, I propose a simpler and more straightforward solution to separate wave-modes with 3D operators, which eliminates the need for projecting the wavefields onto symmetry planes. The new approach constructs shear-wave filters by exploiting the mutual orthogonality of shear modes with the P mode, whose polarization vectors are computed by solving 3D Christoffel equations.

In this chapter, I briefly review wave-mode separation for 2D VTI media and then I extend the algorithm to symmetry planes of TTI media. Then, I generalize the wave-mode separation to 3D TI media. Finally, I demonstrate wave-mode separation in 2D with homogeneous and heterogeneous examples and separation in 3D with a homogeneous TTI example.

Elastic wave-mode separation for TTI media