## 2 Methodology
### Dip Estimation Based on Structure Tensor  
In coherence-based structural analysis, dip estimation is a critical step to ensure that reflector-parallel features are preserved during attribute calculation. Local dip vectors  
Figure 1: Schematic diagram of geological discontinuities: (**a**) channel; (**b**) fault. [PERSON] and pink arrows indicate directions perpendicular to and parallel to the discontinuities, respectively. define the dominant orientation of seismic events and are used to guide directional trace construction and alignment. In this study, we adopt a structure tensor-based approach for dip estimation, which offers robustness to noise and high precision in complex geological settings. To enhance the structural sensitivity of dip estimation and reduce the impact of amplitude variations, the real-valued seismic trace \(x(t)\) is first converted into its complex analytic form using the Hilbert transform [45]:  
\[z(t)=x(t)+i\cdot H(x(t))=A(t)e^{i\varphi(t)}, \tag{1}\]  
where \(H(x(t))\) denotes the Hilbert transform of the original signal \(x(t)\), \(i\) is imaginary unit, \(A(t)\) is the instantaneous amplitude, and \(\varphi(t)\) is the instantaneous phase. In order to ignore the amplitude, the instantaneous complex seismic trace is usually normalized as  
\[y(t)=\frac{z(t)}{|z(t)|}=e^{i\varphi(t)} \tag{2}\]  
This normalized complex trace preserves the reflector geometry while suppressing amplitude discontinuities. It also improves the stability and accuracy of subsequent gradient and tensor calculations. To estimate local reflector orientation, spatial gradients of the normalized phase-only trace \(y(t)\) are computed in three orthogonal directions: vertical (z), inline (x), and crossline (y). The partial derivatives \(G_{z}\), \(G_{x}\), \(G_{y}\) represent the rate of phase change in each respective direction. These gradient components are used to construct the local structure tensor \(\mathbf{T}\in\mathbb{R}^{3\times 3}\), defined as  
\[\mathbf{T}= \begin{bmatrix}\left\langle G_{z}^{2}\right\rangle&\left\langle G_ {z}G_{x}\right\rangle&\left\langle G_{z}G_{y}\right\rangle\\ \left\langle G_{x}G_{z}\right\rangle&\left\langle G_{x}^{2}\right\rangle& \left\langle G_{x}G_{y}\right\rangle\\ \left\langle G_{y}G_{z}\right\rangle&\left\langle G_{y}G_{x}\right\rangle& \left\langle G_{y}^{2}\right\rangle\end{bmatrix} , \tag{3}\]  
where \(\left\langle\cdot\right\rangle\) denotes local Gaussian-weighted averaging to suppress high-frequency noise. The structure tensor captures the directional variance of seismic waveform changes within a neighborhood, providing a compact and robust representation of local structural orientation. It can be seen from Equation (3) that the structure tensor \(\mathbf{T}\) is a symmetric positive semi-definite matrix, so its eigenvalues are non-negative. Then, eigenvalue decomposition is performed on the structure tensor:  
\[\mathbf{T}=\lambda_{1}\mathbf{v}_{1}\mathbf{v}_{1}^{\intercal}+\lambda_{2} \mathbf{v}_{2}\mathbf{v}_{2}^{\intercal}+\lambda_{3}\mathbf{v}_{3}\mathbf{v}_ {3}^{\intercal}, \tag{4}\]  
where \(\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq 0\) are the eigenvalues, and \(\mathbf{v}_{i}\in\mathbb{R}^{3\times 3}\) are the corresponding unit eigenvectors. The eigenvector \(\mathbf{v}_{1}=\left(u_{z},u_{x},u_{y}\right)^{\intercal}\) associated with the largest eigenvalue represents the direction of maximum local phase coherence, which is orthogonal to the seismic reflector. Based on this orientation vector, the apparent dips in inline and crossline directions are computed as  
\[\alpha=\frac{u_{x}}{u_{z}},\beta=\frac{u_{y}}{u_{z}}, \tag{5}\]  
where \(\alpha\) and \(\beta\) represent the slope (dip) of reflectors in the \(x\) and \(y\) directions, respectively. These dip fields serve as geometric constraints for directional trace alignment in the following steps. To prevent local dip vectors from becoming unstable due to noise or low signal quality, we apply a slope-clipping operation to limit extreme values.