Fast algorithms for elastic-wave-mode separation and vector decomposition using low-rank approximation for anisotropic media

Elastic wave mode separation

Using the Helmholtz decomposition theory (Aki and Richards, 1980; Morse and Feshbach, 1953), a vector wavefield $\mathbf{U}=\{U_{x},U_{y},U_{z}\}$ can be decomposed into a curl-free P-wavefield and a divergence-free S-wavefield: $\mathbf{U} = \mathbf{U}^{P} + \mathbf{U}^{S}$ . The P- and S-waves satisfy, respectively,

$$\nabla\times\mathbf{U}^{P} = 0, \qquad \nabla\cdot\mathbf{U} = \nabla\cdot\mathbf{U}^{P},$$ (1)

and

$$\nabla\cdot\mathbf{U}^{S} = 0, \qquad \nabla\times\mathbf{U} = \nabla\times\mathbf{U}^{S}.$$ (2)

These equations imply that the divergence and curl operations pass P- and S-wave modes respectively. In the Fourier-domain, equivalent operations are expressed as follows:

$$\tilde{P}(\mathbf{k}) = i\mathbf{k}\cdot\tilde{\mathbf{U}}(\mathbf{k}), \qquad \widetilde{S}(\mathbf{k}) = i\mathbf{k}\times\tilde{\mathbf{U}}(\mathbf{k}),$$ (3)

where $\mathbf{k}=\{k_{x}, k_{y}, k_{z}\}$ represents the wave vector and $\tilde{U}(k_{x}, k_{y}, k_{z})$ is the 3D wavefield in the wavenumber domain. These operations essentially project the elastic wavefield onto the wave vector or its orthogonal directions, thus separate P- and S-waves successfully. In anisotropic media, however, qP- and qS-waves are not generally polarized parallel and perpendicular to the wave vector. Dellinger and Etgen (1990) extended wave mode separation to anisotropic media with the following divergence-like and curl-like operators in the wavenumber-domain,

$$\widetilde{qP}(\mathbf{k}) = i\mathbf{a}_{p}(\mathbf{k})\cdot\tilde{\mathbf{U}}(\mathbf{k}), \qquad \widetilde{qS}(\mathbf{k}) = i\mathbf{a}_{p}(\mathbf{k})\times\tilde{\mathbf{U}}(\mathbf{k}),$$ (4)

where $\mathbf{a}_{p}(\mathbf{k})$ stands for the normalized polarization vector of qP wave in the wavenumber domain, calculated from Christoffel equation. Note that the second equation of equation 4 separates only the shear part of the elastic wavefields, which contain the fast and low S-waves, i.e., $qS_{1}$ - and $qS_{2}$ modes. Unlike the well-behaved qP mode, the two qS modes do not consistently polarize as a function of the propagation direction (or wavenumber) and thus cannot be designated as SV and SH waves, except in isotropic and TI media (Crampin, 1991; Zhang and McMechan, 2010; Dellinger, 1991; Winterstein, 1990). In this paper, the approaches to separate and decompose qS-waves are restricted to TI anisotropy.

For TI media, one can separate scalar qSV and SH waves by projecting the elastic wavefield onto their polarization directions using

$$\widetilde{qSV}(\mathbf{k}) = i\mathbf{a}_{sv}(\mathbf{k})\cdot\tilde{\mathbf{U}}(\mathbf{k}), \qquad \widetilde{SH}(\mathbf{k}) = i\mathbf{a}_{sh}(\mathbf{k})\cdot\tilde{\mathbf{U}}(\mathbf{k}),$$ (5)

where $\mathbf{a}_{sv}(\mathbf{k})$ and $\mathbf{a}_{sh}(\mathbf{k})$ represent normalized polarization vectors of the qSV and SH waves, respectively. For heterogeneous TI media, these operations can be expressed as nonstationary filtering in the space domain (Yan and Sava, 2009b). In fact, the cost may become prohibitive in 3D because it is proportional to the number of grids in the model and the size of each filter (Yan and Sava, 2011).

In general, we can determine polarization vectors by solving the Christoffel equation:

$$(\widetilde{\mathbf{G}}-\rho{V_{n}}^2\mathbf{I})\mathbf{a}_{n}=0,$$ (6)

where $\widetilde{\mathbf{G}}$ represents the Christoffel tensor in the Voigt notation with $\widetilde{G}_{ij}=c_{ijkl}n_{j}n_{l}, c_{ijkl}$ as the stiffness tensor, and $n_{j}$ and $n_{l}$ are the normalized wave vector components in $j$ and $l$ directions, with $i,j,k,l=1,2,3$ . The parameter $V_{n} (n=qP, qS_{1}, qS_{2})$ represents phase velocities of $qP$ -, $qS_{1}$ - and $qS_{2}$ -wave modes. The Christoffel equation poses a standard $3\times3$ eigenvalue problem, the three eigenvalues of which correspond to phase velocities of the three wave modes and the corresponding eigenvector $\mathbf{a}_{n}$ represents polarization direction of the given mode. When shear singularities appear, the coincidence of the longitudinal and transverse polarizations prevents us from constructing 3D global operators to separate qSV and SH waves on the base of the Christoffel solution, and the polarization discontinuity will cause the two modes to leak energy into each other (Yan and Sava, 2009a; Zhang and McMechan, 2010; Dellinger, 1991; Yan and Sava, 2011). Following Yan and Sava (2009a,2011), we mitigate the kiss singularity at $k_{z}=\pm1$ in 3D TI media by using relative qP-qSV-SH mode polarization orthogonality and scaling the polarizations of the qSV- and SH-waves by $\sin{\phi}$ , with $\phi$ being the polar angle.

Fast algorithms for elastic-wave-mode separation and vector decomposition using low-rank approximation for anisotropic media