Elastic wave-vector decomposition in heterogeneous anisotropic media

qS-wave polarization vectors in homogeneous low-symmetry anisotropic media

The wave-mode separation and wave-vector decomposition methods discussed in the previous sections rely on the knowledge of polarization vectors of wave modes in anisotropic media. Polarization vectors can be found from the Christoffel equation as described in Appendix A. The directions in which the two S waves have equal phase velocity are referred to as S-wave singularities. In the case of TI media, two kinds of S-wave singularities are possible: kiss and intersection singularities (Crampin, 1991; Crampin and Yedlin, 1981; Crampin, 1984). The former is located along the symmetry axis, whereas the latter can be located at a non-zero polar angle depending on the values of the stiffness tensor coefficients. The problems of wave-mode separation and wave vector decomposition in this kind of media have been studied previously (Yan and Sava, 2009,2012; Zhang and McMechan, 2010). The polarization vectors of both S modes rotate rapidly in the vicinity of the kiss singularity, forcing the modes to be coupled. On the other hand, there is no ambiguity in distinguishing between the two S modes at the intersection singularity, because each mode (SV and SH) must have continuous polarization vectors with respect to the change in phase directions and thus, follow different paths on the slowness surface. This behavior at the intersection singularity is similar to that observed when considering a 2D plane cut through a point singularity (Vavrycuk, 2001).

In general, P-wave polarization vectors are close to the phase direction and correspond to the largest phase velocity (Crampin, 1984). Therefore, P wave is generally well-separated from the S waves. Polarization vectors of the two S waves, however, cannot be fully decoupled from each other, which makes them cumbersome to compute. Orthorhombic or anisotropic media of lower symmetry, such as monoclinic or triclinic, can have both the kiss singularity and the third kind of singularity, point (conical) singularity, where the two S-wave slowness surfaces become continuous through the vertices of cone-shaped projections from the surface (Crampin and Yedlin, 1981). In the 2D plane where point singularities occur, the behavior of the two S modes appear similar to that in the case of intersection singularity in TI media (Vavrycuk, 2001). However, in 3D, the slowness surfaces of both S modes do not form a consistent line intersection as in the case of intersection singularity in TI media (Figures 1 and 2) but rather touch at several locations that define S-wave singularities with a rapid rotation of S-wave polarizations around such locations (Figure 3a). Therefore, the generalization of the definitions of SV and SH wave modes from the 2D plane are not valid in orthorhombic or lower-symmetry media (Tsvankin, 2012).

To define the wave modes in such media, we recognize that the three modes must be unique single-valued continuous functions of phase velocity versus phase direction. The most intuitive way to define S modes is to sort them based on the magnitude of the phase velocity (Dellinger, 1991). Note that, in some cases, where only 2D cuts of the wavefield are considered, it may still make sense to use notations of P, SV, and SH even though they might be misleading. For example, in the symmetry planes of orthorhombic media where the point singularities can potentially happen, these notations are acceptable because the polarization vectors behave as if they follow the analogous intersection singularity in TI media.

Dellinger (1991) showed that sorting the modes based on the magnitude of the phase velocity involves the problem of particle-motion discontinuity at the S-wave singularities, and therefore, does not lead to a clear separation. In the Fourier domain, this local discontinuity behaves similarly to a delta function and generates a strong planar artifact in the space domain. A similar artifact can be seen when S-wave modes with intersection singularity are being separated in 2D TI media based on their phase velocities. In the case of TI media, this planar artifact can be eliminated because of the use of SV and SH definitions that ensure polarization continuity in all phase directions. However, as described above, the same approach is not applicable in the case of low-symmetry anisotropic media. In order to successfully separate the wavefields of S-wave modes in such media, it is necessary to address the discontinuity problem.

In this paper, for the analysis on the S-wave polarization vectors and singularities (Figures 1-6), we adopt an example of orthorhombic medium from Schoenberg and Helbig (1997), which has the following density-normalized stiffness coefficients (in $km^2/s^2$ ):

$$c_{ij} = \begin{bmatrix}9 & 3.6 & 2.25 & 0 & 0 & 0 \\ 3.6 & 9.84 ... ...& 0 & 0 \\ 0 & 0 & 0 & 0 & 1.6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2.182 \end{bmatrix}~.$$ (19)

For the triclinic example, we use the stiffnesses given by Mah and Schmitt (2003):

$$c_{ij} = \begin{bmatrix}14.9 & 6.3 & 5.2 & 0.7 & 0.9 & -0.5 \\ 6.... ...& 0.8 & -0.1 & 3.0 & 0.2 \\ -0.5 & -0.4 & 0.1 & 0.1 & 0.2 & 3.7 \end{bmatrix}~.$$ (20)

ortcircles1,ortcircles2
Figure 1. Polarization vectors of a) S1 (blue) and b) S2 (red) modes in the orthorhombic model (equation 19) plotted on a unit sphere. Notice the sudden polarization vectors flip in the ${[x,z]}$ and ${[y,z]}$ planes and in the plane along $x$ = $y$ . These effects are observed as these planes cut through point singularities and they resemble jumping between modes at the intersection singularity.

tricircles1,tricircles2
Figure 2. Polarization vectors of a) S1 (blue) and b) S2 (red) modes in the triclinic model (equation 20) plotted on a unit sphere. Notice the patternless variation of polarizations with respect to phase directions due to the presence of only point symmetry.

Elastic wave-vector decomposition in heterogeneous anisotropic media