# Cross-Attention U-Net for Elastic Wavefield Decomposition in Anisotropic Media
## 2 Theoretical Framework
### Low-Rank Approximation for Wave Vector Decomposition  
Wave vector decomposition, as demonstrated in Section 2.1, projects the vector wavefield onto the polarization vectors obtained from the Christoffel equation [5]. However, the direct implementation of these projections requires solving the Christoffel equation at every spatial location and propagation direction, leading to high computational costs. This complexity arises because polarization vectors are spatially dependent and must be computed individually for each grid point in a heterogeneous anisotropic medium.  
Applying the inverse Fourier transform to Equation (4), the wavefield decomposition in the space domain is expressed as follows:\[\begin{split}&\mathbf{U}_{qP,\alpha}(\mathbf{x})=\int e^{i\mathbf{k}\cdot \mathbf{x}}\sum_{\beta}\left[p_{qP,\alpha}(\mathbf{x},\mathbf{k})p_{qP,\beta}( \mathbf{x},\mathbf{k})\right]\widetilde{U}_{\beta}(\mathbf{k})d\mathbf{k},\\ &\mathbf{U}_{qSV,\alpha}(\mathbf{x})=\int e^{i\mathbf{k}\cdot \mathbf{x}}\sum_{\beta}\left[p_{qSV,\alpha}(\mathbf{x},\mathbf{k})p_{qSV, \beta}(\mathbf{x},\mathbf{k})\right]\widetilde{U}_{\beta}(\mathbf{k})d\mathbf{ k},\\ &\mathbf{U}_{qSH,\alpha}(\mathbf{x})=\int e^{i\mathbf{k}\cdot \mathbf{x}}\sum_{\beta}\left[p_{qSV,\alpha}(\mathbf{x},\mathbf{k})p_{qSH, \beta}(\mathbf{x},\mathbf{k})\right]\widetilde{U}_{\beta}(\mathbf{k})d\mathbf{ k},\end{split} \tag{5}\]  
where \(\mathbf{p}_{qP}(\mathbf{x},\mathbf{k})\), \(\mathbf{p}_{qSV}(\mathbf{x},\mathbf{k})\), and \(\mathbf{p}_{qSH}(\mathbf{x},\mathbf{k})\) represent the polarization vectors for the qP, qSV, and qSH waves at position \(\mathbf{x}\) and wavenumber \(\mathbf{k}\), respectively. Each separated wavefield \(\mathbf{U}_{qP,\alpha}\) is computed by summing up the spatial indices \(\beta\in\{x,y,z\}\), where \(p_{qP,\alpha}(\mathbf{x},\mathbf{k})\) denotes the \(\alpha\)-th component of the polarization vector. This formulation allows the wave mode separation to be expressed as nonstationary filtering, but the computational cost remains high due to the need for grid-by-grid polarization vector computation.  
To further improve the computational efficiency, the wave mode decomposition operator can be reformulated using matrix notation as follows [2]:  
\[W_{\alpha}(\mathbf{x},\mathbf{k})\approx\sum_{m=1}^{M}\sum_{n=1}^{N}B(\mathbf{ x},\mathbf{k}_{m})A_{mm}C(\mathbf{x}_{n},\mathbf{k}), \tag{6}\]  
where \(B(\mathbf{x},\mathbf{k}_{m})\) and \(C(\mathbf{x}_{n},\mathbf{k})\) are mixed-domain matrices that capture variations in wavefield properties, \(A_{mm}\) is a low-rank matrix that approximates the decomposition operator, \(N_{x}\) is the total number of spatial grid points in the computational domain, and \(N\) is the numerical rank used in the low-rank decomposition.  
Since \(M\), \(N\ll Nx\), this decomposition significantly reduces the computational cost. The direct computation of wave mode decomposition has a complexity of O(\(N_{x}^{2}\)) due to the need for solving the Christoffel equation at every spatial location. However, low-rank decomposition reduces the complexity to O(\(NN_{x}logN_{x}\)). This decomposition is implemented using pivoted QR (orthogonal-triangular) decomposition, enabling the precomputation of mode separation operators in a compact form. While low-rank approximation improves efficiency, the rank N controls the balance between accuracy and cost. A higher rank provides greater accuracy but increases computational demands, whereas a lower rank accelerates calculations at the expense of some accuracy loss. To avoid the extensive computational cost of solving the Christoffel equation across large datasets, low-rank approximation was used to generate ground truth labels for training the deep learning model applied in this study.