# Cross-Attention U-Net for Elastic Wavefield Decomposition in Anisotropic Media
## 3 Results  
### Comparative Evaluation of the U-Net and Cross-Attention U-Net  
Table 1 compares the overall performance of the baseline U-Net model and the proposed cross-attention U-Net on the test set. The evaluated metrics include the peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and channel-wise SSIM values for each separated component (\(u_{x,\text{q}p}\), \(u_{x,\text{q}s}\), \(u_{z,\text{q}p}\), \(u_{z,\text{q}s}\)). Higher PSNR and SSIM  
Figure 3: Example of the patch extraction process from the modified BP 2007 anisotropic benchmark model. The full model domain is divided into 200 \(\times\) 200 subregions, yielding 60 distinct patches for training. Additionally, 30 augmented patches are generated using random flipping and minor scaling. Red and blue denote high and low values of the physical property, respectively. scores indicate better decompositions of qP and qS energy from the original displacement fields. Across all quantitative metrics, the cross-attention U-Net achieves lower reconstruction errors and higher fidelity compared to the U-Net. The mean PSNR and SSIM values, for instance, increase from 42.8360 dB to 44.1008 dB and from 0.8897 to 0.9043, respectively, indicating that incorporating cross-attention enables the network to better capture the relationships between the horizontal and vertical displacement fields. The channel-wise SSIM values also improve consistently, suggesting that each separated wave mode (\(u_{x,\text{qP}}\), \(u_{x,\text{qP}}\), \(u_{z,\text{qP}}\), \(u_{z,\text{qP}}\)) benefits from the mutual exchange of complementary features across different displacement components. Figure 4 shows an example input wavefield (\(u_{x}\) and \(u_{z}\); first row), along with the corresponding ground truth snapshots (\(u_{x,\text{qP}}\), \(u_{x,\text{qP}}\), \(u_{z,\text{qP}}\), \(u_{z,\text{qP}}\)) in the second row. These ground truth snapshots were generated using the low-rank approximation method (Section 2.2) to ensure a physically consistent reference for vector decomposition. Figures 5 and 6 compare the results of the U-Net and the cross-attention U-Net for the same snapshot. In each figure, the top row shows the predicted components, and the bottom row displays the difference from the ground truth. These difference (residual) plots are scaled by a factor of three (\(\times\)3) to enhance the visibility of local errors and artifacts. In Figure 5, the U-Net successfully separates most of the P- and S-wave energy but leaves some pronounced artifacts near complex geological boundaries and areas of strong reflection. In contrast, Figure 6 demonstrates that the cross-attention U-Net yields a cleaner separation with fewer artifacts and lower overall residuals, especially in more challenging zones. This indicates that, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**PSNR**} & \multicolumn{6}{c}{**SSIM**} \\ \cline{3-6}  & & **Total** & \(u_{x,\text{qP}}\) & \(u_{x,\text{qS}}\) & \(u_{x,\text{qP}}\) & \(u_{x,\text{qS}}\) \\ \hline U-net & 42.8460 & 0.8897 & 0.8008 & 0.8519 & 0.9565 & 0.9456 \\ CA U-net & 44.1008 & 0.9043 & 0.8325 & 0.8719 & 0.9612 & 0.9515 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the baseline U-Net and the cross-attention U-Net in terms of peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and channel-wise SSIM (\(u_{x,\text{qP}}\), \(u_{x,\text{qP}}\), \(u_{z,\text{qP}}\), \(u_{z,\text{qP}}\), \(u_{z,\text{qP}}\)). Higher PSNR and SSIM values indicate better overall decomposition performance. Figure 4: Example of an input snapshot (top row: \(u_{x}\) and \(u_{z}\)) and the corresponding ground truth qP and qS components (bottom row: \(u_{x,\text{qP}}\), \(u_{x,\text{qP}}\), \(u_{z,\text{qP}}\), \(u_{z,\text{qP}}\)). Ground truth labels are generated via the low-rank approximation method.