Elastic wave-vector decomposition in heterogeneous anisotropic media

Locating singularities

Crampin (1991,1984) showed that kiss and point singularities are the only possible kinds of singularity in orthorhombic or any other lower-symmetry media. In order to get rid of the planar artifacts caused by polarization discontinuity at such locations, we need first to locate the singularities. Locations of singularities correspond to the locations where the Christoffel matrix $\mathbf{G}$ degenerates and the phase velocities of both S waves become equal. In low-symmetry anisotropic media, S waves have the following well-known explicit expressions for phase velocity (Tsvankin, 2012; Schoenberg and Helbig, 1997; Tsvankin, 1997):

$$v^2_{S1} (c_{ij},\mathbf{n}) = 2\sqrt{\frac{-d}{3}}\cos(\frac{\nu}{3}+\frac{2\pi}{3})-\frac{a}{3} ~,$$ (21)

$$v^2_{S2} (c_{ij},\mathbf{n}) = 2\sqrt{\frac{-d}{3}}\cos(\frac{\nu}{3}+\frac{4\pi}{3})-\frac{a}{3} ~,$$ (22)

where

$$\nu~(c_{ij},\mathbf{n}) = \arccos \left(\frac{-q}{2\sqrt{(-d/3)^3}}\right)~,$$

$$q~(c_{ij},\mathbf{n}) = 2\left(\frac{a}{3}\right)^3 - \frac{ab}{3}+c~,~~~d~~=~~-\frac{a^2}{3} + b~,$$

$$a~(c_{ij},\mathbf{n}) = -(G_{11}+G_{22}+G_{33})~,$$

$$b~(c_{ij},\mathbf{n}) = G_{11}G_{22}+G_{11}G_{33}+G_{22}G_{33}-G^2_{12}-G^2_{13}-G^2_{23}~,$$

$$c~(c_{ij},\mathbf{n}) = G_{11}G^2_{23}+G_{22}G^2_{13}+G_{33}G^2_{12}-G_{11}G_{22}G_{33}-2G_{12}G_{13}G_{23}~.$$

$G_{ij}$ are given in equation A-2 for the most general case of triclinic media and are functions of local stiffness $c_{ij}$ and phase direction $\mathbf{n} = \mathbf{\bar{k}} = \mathbf{k}/\vert\mathbf{k}\vert$ . Equations 21 and 22 can be derived from the generic solution of a cubic equation for $v^2$ , which corresponds to the Christoffel equation. They are valid for any locally homogeneous anisotropic medium with associated $G_{ij}$ (Schoenberg and Helbig, 1997). With these analytical expressions, we can derive the sufficient condition for the occurence of a S-wave singularity. From $v^2_{S1} = v^2_{S2}$ and equations 21 and 22, it follows that in the singular direction $\mathbf{\hat{n}}$ ,

$$\cos(\frac{\nu~(c_{ij},\mathbf{\hat{n}})}{3}+\frac{2\pi}{3}) = \cos(\frac{\nu~ (c_{ij},\mathbf{\hat{n}})}{3}+\frac{4\pi}{3})~,$$ (23)

or doing trigonometric expansions,

$$\cos(\frac{\nu~(c_{ij},\mathbf{\hat{n}})}{3})\cos(\frac{2\pi}{3})... ...4\pi}{3}) - \sin(\frac{\nu~(c_{ij},\mathbf{\hat{n}})}{3})\sin(\frac{4\pi}{3})~,$$ (24)

which leads to the condition of

$$\sin\left(\frac{\nu~(c_{ij},\mathbf{\hat{n}})}{3}\right) = 0~.$$ (25)

We propose to apply equation 25 to numerically detect the proximity of the point singularity for a given phase direction. This condition also allows us to create a filter with an adjustable effective area to smooth the polarization vectors around the singularity in order to attenuate the planar artifacts in wave-mode separation and wave vector decomposition. The variation of values of the left-hand side in equation 25 with different phase directions ( $\mathbf {n}$ ) in the case of the example orthorhombic model is shown in Figure 3b. Similar plots for the triclinic model are shown in Figures 4 and 5.

ortoctant,orthos
Figure 3. a) Polarization vectors in the orthorhombic model of S1 (blue) and S2 (red) plotted on phase slowness surfaces to show how they rapidly rotate in the vicinity of point singularities. b) $\sin\big(\nu~(c_{ij},\mathbf{n})/3\big)$ for different phase directions ( $\mathbf {n}$ ) plotted on S1 phase slowness surface. Notice that the values turn zero at the locations corresponding to point singularities in circles and that there are two point singularities in the ${[x,z]}$ plane, one in ${[y,z]}$ planes, and one in the plane cut along $x$ = $y$ in this octant.

t090p090s,t090p90180s,t090p180270s,t090p270360s
Figure 4. $\sin\big(\nu~(c_{ij},\mathbf{n})/3\big)$ for different phase directions ( $\mathbf {n}$ ) plotted on S1 phase slowness surface the zenith angle $\theta =0^\circ -90^\circ$ measured from vertical and from the azimuthal angle a) $\phi =0^\circ -90^\circ$ , b) $\phi =90^\circ -180^\circ$ , c) $\phi =180^\circ -270^\circ$ , and d) $\phi =270^\circ -360^\circ$ measured with respect to $x$ -axis. Notice that the values turn zero at the locations corresponding to singularities in circles.

t90180p090s,t90180p90180s,t90180p180270s,t90180p270360s
Figure 5. $\sin\big(\nu~(c_{ij},\mathbf{n})/3\big)$ for different phase directions ( $\mathbf {n}$ ) plotted on S1 phase slowness surface the zenith angle $\theta =90^\circ -180^\circ$ measured from vertical and from the azimuthal angle a) $\phi =0^\circ -90^\circ$ , b) $\phi =90^\circ -180^\circ$ , c) $\phi =180^\circ -270^\circ$ , and d) $\phi =270^\circ -360^\circ$ measured with respect to $x$ -axis. Notice that the values turn zero at the locations corresponding to singularities in circles.

Elastic wave-vector decomposition in heterogeneous anisotropic media