Elastic wave-vector decomposition in heterogeneous anisotropic media

Low-rank approximations for wave-vector decomposition operator

For the wave-vector decomposition method, we focus on equation 14 and transform it to the space domain using the inverse Fourier transform as follows (Cheng and Fomel, 2014):

$$\mathbf{U}^{\alpha}\mathbf{(x)} = \int e^{i\mathbf{k}\cdot\mathbf... ...mathbf{x},\mathbf{\bar{k}})\mathbf{\widetilde{U}(\bar{k})}\right]d\mathbf{k} ~,$$ (15)

where $\mathbf{\widetilde{U}(k)}$ denotes the unseparated wavefield in the Fourier domain and $\mathbf{U}^{\alpha}\mathbf{(x)}$ denotes the decomposed wavefield of $\alpha$ wave mode in the space domain. Equation 15 indicates that the unseparated wavefield is projected onto polarization vector $\mathbf{a}^{\alpha}(\mathbf{x},\mathbf{\bar{k}})$ of $\alpha$ wave mode in the Fourier domain and subsequently transformed back to the space domain for the final decomposed wavefield. The elements of matrix $\mathbf{A}^{\alpha}(\mathbf{x},\mathbf{\bar{k}})$ are given by

$$\mathbf{A}^{\alpha}(\mathbf{x},\mathbf{\bar{k}}) = \begin{bmatrix... ...a}_x & a^{\alpha}_z a^{\alpha}_y & a^{\alpha}_z a^{\alpha}_z \\ \end{bmatrix}~,$$ (16)

where $\mathbf{a}^{\alpha}(\mathbf{x},\mathbf{\bar{k}}) = \{a^{\alpha}_x,a^{\alpha}_y,a^{\alpha}_z\}$ and $x,$ $y,$ and $z$ denoting different components.

Applying the low-rank approximation approach (Fomel et al., 2013), each element of $\mathbf{A}^{\alpha}(\mathbf{x},\mathbf{\bar{k}})$ for a specified $\alpha$ wave mode in equation 15 and 16 can be approximated as follows (Cheng and Fomel, 2014):

$$a^{\alpha}_i a^{\alpha}_j = A^{\alpha}_{ij}(\mathbf{x},\mathbf{\b... ...},\mathbf{\bar{k}}_m)\mathbf{W}_{mn}\mathbf{C}(\mathbf{x}_n,\mathbf{\bar{k}})~,$$ (17)

where $\mathbf{B}(\mathbf{x},\mathbf{\bar{k}}_m)$ and $\mathbf{C}(\mathbf{x}_n,\mathbf{\bar{k}})$ are mixed-domain matrices with reduced wavenumber and spatial dimensions respectively; $\mathbf{W}_{mn}$ is a $M\times N$ matrix with $M$ and $N$ representing the rank of this low-rank decomposition. One can view $\mathbf{B}$ as a submatrix of $A^{\alpha}_{ij}$ consisting of columns associated with ${\mathbf{\bar{k}}_m}$ , and $\mathbf{C}$ as a submatrix of $A^{\alpha}_{ij}$ consisting of rows associated with ${\mathbf{x}_n}$ . Physically, this process means that we only consider a selected few representative spatial locations ($N \ll N_x$ ) and representative wavenumbers ($M \ll N_x$ ), where $N_x$ is the size of the model, to build an effective approximation. As a result, the low-rank approximation reduces computational cost by transforming the Fourier integral operator in equation 15 for each component $i$ and $j$ denoting $x, y,$ and $z$ components to

$$\int e^{i\mathbf{k}\cdot\mathbf{x}}A^{\alpha}_{ij}(\mathbf{x},\ma... ...bf{x}_n,\mathbf{\bar{k}})\widetilde{U}_j(\mathbf{k})d\mathbf{k}\right)\right)~.$$ (18)

The computational cost of applying equation 18 is equivalent to the cost of $N$ inverse fast Fourier Transforms (FFT) (Cheng and Fomel, 2014).

Elastic wave-vector decomposition in heterogeneous anisotropic media