Simulating propagation of separated wave modes in general anisotropic media, Part II: qS-wave propagators

Pure-mode SH-wave equation

Pure SH-waves horizontally polarize in the planes perpendicular to the symmetry axis of VTI media with $u_z\equiv0$ , so we introduce a similarity transformation to the Christoffel matrix ignoring the vertical component, i.e.,

$$\widetilde{\overline{\mathbf{\Gamma}}}_\mathbf{m} = \mathbf{M}\widetilde{\mathbf{\Gamma}}_2\mathbf{M}^{-1},$$ (11)

with a generally invertible $2\times2$ matrix $\mathbf{M}$ related to the reference polarization direction $\mathbf{e}_2$ :

$$\mathbf{M}= \begin{pmatrix}-{k_y} & 0 \cr 0 & {k_x} \end{pmatrix},$$ (12)

and

$$\widetilde{\Gamma}_2=\begin{pmatrix}C_{11}{k_x}^2+C_{66}{k_y}^2+C... ...12}+C_{66}){k_x}{k_y} & C_{66}{k_x}^2+C_{22}{k_y}^2+C_{44}{k_z}^2\end{pmatrix}.$$ (13)

Accordingly, we derive a transformed Christoffel equation,

$$\widetilde{\overline{\mathbf{\Gamma}}}_\mathbf{m}\widetilde{\overline{\mathbf{u}}}= \rho{\omega}^2\widetilde{\overline{\mathbf{u}}},$$ (14)

for the SH-wave mode:

$$\widetilde{\overline{\mathbf{u}}}=\mathbf{M}\widetilde{\mathbf{u}}_2,$$ (15)

in which $\widetilde{\mathbf{u}}_2=(\widetilde{u}_x,\widetilde{u}_y)^{\top}$ represents the horizontal components of the original elastic wavefields, and $\widetilde{\overline{\mathbf{u}}}=(\widetilde{\overline{u}}_x,\widetilde{\overline{u}}_y)^{\top}$ represents the horizontal components of the transformed wavefields. Note that the matrix $\mathbf{M}$ will be not invertible when $k_x=0$ or/and $k_y=0$ . These special directions don't affect the derivation of the pseudo-pure-mode wave equation for the following reasons: First, we don't directly project the elastic wavefield into the wavenumber-domain, but instead apply the similarity transformation to the Christoffel equation and eventually inverse the transformed Christoffell equation back into the time-space-domain. Second, the original Christofell matrix $\widetilde{\Gamma}_2$ automatically becomes a diagonal matrix in these directions, so the similarity transformation is not actually needed for the corresponding wavenumber components.

Note the similarity transformation does not change the eigenvalue of the Christoffel matrix corresponding to the SH-wave and, thus, introduces no kinematic error for this wave mode. We also can obtain a kinematically equivalent Christoffel equation if $\mathbf{M}$ is constructed using the normalized form of $\mathbf{e}_2$ to ensure all spatial frequencies are uniformly scaled. For a locally smooth medium, applying an inverse Fourier transform to equation 14, we obtain a linear second-order system in the time-space domain:

$$\rho\partial_{tt}\overline{\mathbf{u}} = \overline{\mathbf{\Gamma}}_\mathbf{m}\overline{\mathbf{u}},$$ (16)

or in its extended form:

$$\begin{split}\rho\partial_{tt}\overline{u}_x &= C_{11}\partia... ..._y} - (C_{11}-C_{66})\partial_{xx}{\overline{u}_x}, \end{split}$$ (17)

where $\overline{\mathbf{u}}=(\overline{u}_x, \overline{u}_y)^{\top}$ represents the horizontal components of SH-wave in time-space domain, and $\overline{\mathbf{\Gamma}}_\mathbf{m}$ represents the Christoffel differential-operator matrix after the similarity transformation.

Due to the cylindrical symmetry of a TI material, the two equations in equation 17 may be summed to produce a scalar wave equation in terms of $\overline{u}$ :

$$\rho\partial_{tt}\overline{u} = C_{66}(\partial_{xx}+\partial_{yy}){\overline{u}}+C_{44}\partial_{zz}{\overline{u}},$$ (18)

with $\overline{u}=\overline{u}_{x}+\overline{u}_{y}$ representing the total horizontal components of the transformed SH-wave fields. This is consistent with the fact that only $C_{44}$ and $C_{66}$ affect the kinematic signatures of the SH-wave in VTI media (Tsvankin, 2001). In addition, the derived equation naturally reduces to the acoustic wave equation if we apply the isotropic assumption by setting $C_{44}=C_{66}=\rho{V_s}^2$ with $V_s$ representing the velocity of the isotropic shear wave.

Simulating propagation of separated wave modes in general anisotropic media, Part II: qS-wave propagators