# Cross-Attention U-Net for Elastic Wavefield Decomposition in Anisotropic Media
## 4 Discussion  
While the proposed cross-attention U-Net shows promising results in elastic wavefield decomposition, several aspects require further consideration. One of the main challenges is the computational complexity that is introduced by the cross-attention mechanism. Since attention operations scale quadratically with input size, this can create a bottleneck for applications involving large seismic datasets. This limitation necessitates the exploration of optimization strategies such as sparse attention techniques, hierarchical attention implementations, or alternative attention mechanisms that can maintain performance while reducing the computational overhead.  
Another area of concern is the model's dependence on labeled training data. Unlike physics-based approaches, the proposed approach requires a considerable amount of labeled data for effective generalization. Although the use of low-rank approximations for generating labels has been effective in the current setting, this method may introduce potential bias due to the assumptions involved in the approximation. In particular, it is unclear how these labels generalize to complex real data, where assumptions may not hold. Investigating alternative label generation strategies, such as weak supervision or self-supervised learning, could help reduce such bias while improving model robustness.  
Extending the method to three-dimensional wavefield decomposition is an important next step for real data applications. However, this also increases the computational and memory demands of the model. Future work should focus on improving the scalability for 3D domains, possibly by introducing hybrid attention architectures that combine local and global representations or by leveraging the structural characteristics of seismic data to reduce the processing overhead. In addition, inference with the trained network is considerably faster than conventional decomposition methods, such as low-rank approximations, which require intensive computation for each snapshot. Once training is complete, the model yields predictions almost instantly through a single forward pass, making it well suited for large-scale or time-sensitive applications.  
Figure 8: Wavefield separation results on the noisy Hess VTI model using the sliding-patch approach. Input displacement fields (top row: \(u_{x}\) and \(u_{z}\)) with added Gaussian noise, ground truth qP and qS components (second row: \(u_{x,\_0}\), \(u_{x,\_0}\), \(u_{z,\_0}\), \(u_{z,\_0}\)), predicted components using the cross-attention U-Net (third row), and difference maps between the predicted and ground truth results (bottom row).  
In addition, when the model is applied to larger domains using a sliding-patch inference approach, some reduction in accuracy can be observed near patch boundaries. This appears to stem from the limited receptive field of the network, which is trained on relatively small patches (200 \(\times\) 200) and may not capture long-range spatial dependencies effectively. This limitation becomes more noticeable when processing extended domains, as illustrated in Figures 7 and 8. Potential improvements could include overlapping patch strategies with smooth blending, or the integration of global context modeling techniques such as attention modules that span wider spatial extents.  
While the model performs well across different velocity structures and anisotropic characteristics, further development is needed to improve its performance in more complex geological settings. This could include incorporating physical constraints directly into the network architecture or loss function, helping the model align more closely with the underlying physics. Additionally, interpretability tools tailored to wavefields decomposition could offer useful insights into how the model makes decisions and support validation in practical scenarios.