## 2 Methodology
### Optimally Oriented Coherence Calculation with Multi-Frequency Fusion  
To further enhance the multi-scale responsiveness of the optimally oriented coherence attribute in complex structural interpretation, this study introduces a Gabor-based multi-frequency decomposition strategy into the directional coherence framework. The Gabor transform combines the temporal localization of a Gaussian window with the spectral resolution of the Fourier transform, enabling effective extraction of localized frequency features at different scales. This strategy allows us to obtain structural responses across multiple frequency channels from seismic signals, thereby improving the imaging capability for discontinuities of varying scales, such as small faults and narrow channels. Specifically, given a directionally guided model trace \(\hat{\textbf{y}}_{l}\) constructed under local dip constraints, the complex response of the \(l\)th direction at frequency \(f_{i}\) obtained via the Gabor transform can be expressed as [46]  
\[G_{l}(t,f_{i})=\int y_{l}(\tau)\cdot e^{-\frac{(t-\tau)^{2}}{2\sigma^{2}}}.e^{- i2\pi f_{l}\tau}d\tau, \tag{11}\]  
where \(y_{l}\) denotes the normalized complex-valued model trace along the \(l\) direction. The parameter \(\sigma\) controls the width of the Gaussian window function, and \(f_{l}\) represents the center frequency of the Gabor kernel. By specifying a set of discrete frequency points \(\{f_{1},f_{2},f_{3},\ldots,f_{N}\}\), each directional model trace can be decomposed into multiple frequency scales, resulting in a sequence of complex-band responses \(G_{l}(t,f_{i})\). Here, \(\tau\) is the integration variable corresponding to the time axis. Following the Gabor-based multi-frequency decomposition of the directional model traces, complex-valued cross-correlation is performed between each pair of orthogonal directions to evaluate structural coherence at each frequency scale. Let \(G_{l}(t,f_{i})\) and \(G_{l^{*}}(t,f_{i})\) denote the complex Gabor responses at frequency \(f_{i}\) along the \(l\) direction and its orthogonal counterpart \(l^{*}\), respectively. The normalized complex cross-correlation coefficient between these two directional responses at frequency \(f_{i}\) is then defined as  
\[\rho_{l}{}^{(f_{i})}=\frac{\sum_{t=-K}^{K}\left(G_{l}(t,f_{i})-\overline{G_{l }}{}^{(f_{i})}\right)\cdot\left(G_{l^{*}}(t,f_{i})-\overline{G_{l^{*}}}{}^{(f_ {i})}\right)^{*}}{(2K+1)\cdot\sigma_{G_{l}}^{(f_{i})}\sigma_{G_{l^{*}}}^{(f_{ i})}}, \tag{12}\]where \(\widehat{G_{l}}^{(f_{i})}\) and \(\widehat{G_{l^{*}}}^{(f_{i})}\) denote the mean values of the complex Gabor responses along the \(l\) and \(l^{*}\) directions at frequency \(f_{i}\), respectively. The terms \(\sigma_{G_{l}}^{(f_{i})}\) and \(\sigma_{G_{l^{*}}}^{(f_{i})}\) represent the corresponding standard deviations, and the superscript asterisk \((\cdot)^{*}\) indicates the complex conjugate operation. After obtaining the cross-correlation coefficients across all frequency components, a multi-band weighted fusion strategy is introduced to construct a comprehensive directional coherence indicator. Let the total number of frequency bands be \(N\); the fused result can be expressed as  
\[\rho_{l}fused(t)=\sum_{i=1}^{N}w_{i}\cdot\rho_{l}{}^{(f_{i})}(t). \tag{13}\]  
To integrate directional coherence responses across different frequency scales, we adopt an energy-based weighting scheme. The fused coherence \(\rho_{l}fused(t)\) is computed as the weighted sum of normalized cross-correlation coefficients \(\rho_{l}{}^{(f_{i})}(t)\) over all frequency bands.