# Cross-Attention U-Net for Elastic Wavefield Decomposition in Anisotropic Media
###### Abstract  
Elastic wavefield separation in anisotropic media is essential for seismic imaging but remains challenging due to complex interactions among multiple wave modes. Traditional methods often rely on solving the Christoffel equation, which is computationally expensive, particularly in heterogeneous models. This study proposes a deep learning-based approach using a cross-attention U-Net architecture to achieve efficient vector decomposition of elastic wavefields. The model employs a dual-branch encoder with cross-attention mechanisms to preserve and exploit inter-component relationships among wavefield components. The network was trained on patches extracted from the BP (British Petroleum) 2007 anisotropic benchmark model, with ground truth labels being generated via low-rank approximation methods. Quantitative evaluations show that the cross-attention U-Net outperforms a baseline U-Net, improving the peak signal-to-noise ratio(PSNR) by 1.25 dB (44.10 dB vs. 42.85 dB) and structural similarity index (SSIM) by 0.014 (0.904 vs. 0.890). The model demonstrates effective generalization to larger domains and different geological settings, validated on both the extended BP model and the Hess vertically transversely isotropic (VTI) model. Overall, this approach provides a computationally efficient alternative to traditional separation methods while maintaining physical consistency in the separated wavefields. e +
Footnote †: journal: Applied sciences  
e +
Footnote †: journal: Applied sciences  
are calculated from the Christoffel equation in the wavenumber domain. While this method preserves the amplitude, phase, and vector characteristics of the separated waves, it shares a common challenge with wave mode separation methods: both approaches require solving the Christoffel equation, which is computationally demanding and needs to be repeated for each spatial location and propagation direction, creating challenges for large-scale or heterogeneous models. To address the computational limitations of solving the Christoffel equation, several advancements have been proposed. [PERSON] and [PERSON] (2009) [1] introduced nonstationary spatial filters that adapt to local anisotropic properties. These filters still use polarization vectors from the Christoffel equation but represent them as localized filtering operators in the space domain, reducing the computational overhead while maintaining separation accuracy. Building on this, [PERSON] and [PERSON] (2011) [6] developed a mixed-domain algorithm that integrates space and wavenumber domain operations. By approximating the medium's heterogeneity with reference models, this approach reduces the calculations that are needed for polarization vectors. [PERSON] and [PERSON] (2014) [2] proposed using Fourier integral operators with low-rank approximations for both wave mode separation and vector decomposition methods. This modification enables more efficient wavefield separation without directly solving the Christoffel equation at every spatial point, making these methods more practical for large, heterogeneous models. [PERSON] and [PERSON] (2016) [7] developed an approach that simplifies the separation operators to forms resembling divergence and curl operators in isotropic media. Their method uses rotated wave vectors and Poynting vectors to approximate polarization directions, reducing the computational complexity in complex geological settings. Further research has extended these methods to low-symmetry anisotropic media. [PERSON] et al. (2017) [8] adapted vector decomposition for orthorhombic, monoclinic, and triclinic models, addressing challenges like polarization discontinuities and singularities through smoothing filters and local signal-noise orthogonalization. These improvements enhance the computational efficiency and broaden the application of elastic wavefield separation methods to more complex geological models.