## 2 Methodology
### Optimally Oriented Coherence Calculation with Multi-Frequency Fusion
The weight \(w_{i}\) is determined by its relative energy contribution, defined as  
\[w_{i}=\frac{E_{i}}{\sum_{j=1}^{N}E_{j}}, \tag{14}\]  
where the energy \(E_{i}\) is calculated as the sum of squared magnitudes of the Gabor responses along both the \(l\) and \(l^{*}\) directions:  
\[E_{i}=\sum_{t}\lvert G_{l}(t,f_{i})\rvert^{2}+\sum_{t}\lvert G_{l^{*}}(t,f_{i })\rvert^{2}. \tag{15}\]  
After computing the fused coherence coefficients across all orthogonal direction pairs, we adopt a minimum-coherence selection strategy to enhance sensitivity to structural discontinuities. Specifically, let the fused coherence values corresponding to all directional pairs be defined as  
\[coh(t)=\min\Bigl{\{}\rho_{1}{}^{fused}(t),\rho_{2}{}^{fused}(t),\dots,\rho_{L }{}_{2}{}^{fused}(t)\Bigr{\}}, \tag{16}\]  
where \(L/2\) denotes the number of orthogonal directional pairs, and \(coh(t)\) represents the final coherence attribute extracted by selecting the minimum fused response across all directions at time sample \(t\). Conventional directional coherence methods typically rely on a set of predefined orientations, compute coherence values based on the similarity between windowed traces on either side of a central trace, and select the direction with the minimum coherence as the final result. While this approach is relatively robust, it lacks geological context, is sensitive to noise, and often incurs high computational costs. More importantly, it does not guarantee alignment with the true structural orientation, making it difficult to reliably identify the optimal direction. Figure 3 compares the differences between the published geosteering coherence method [47] and the method proposed in this paper. In the published method, for each direction, two model traces (left and right) are constructed (magenta and cyan circles shown in Figure 3a). Then, a cross-correlation operation is performed on each direction to obtain the correlation coefficient. Finally, the minimum is considered as the final coherence. While, in this paper, only one model trace is built for each direction, and the center trace is removed. Also, the model traces in the vertical direction are subjected to cross-correlation operations. Finally, in line with the first method, the minimum is taken as the coherence. The new method has two main advantages: higher accuracy to identify minor discontinuities and higher computational efficiency. Firstly, the method proposed by [PERSON] (2018) [47] essentially reflects the differences between the leftmost and the rightmost seismic trace in each direction, as shown in Figure 3a. During model construction, other seismic traces, including their weights, are completely the same. Small differences between the leftmost and rightmost seismic traces caused the two model traces to be similar, yielding a high correlation coefficient. Therefore, it is difficult to identify the minor discontinuities within the analysis window. However, in the proposed method, the same seismic traces are not involved in the model trace construction. From another perspective, it is equivalent to predicting the center trace in two vertical directions. Throughout the scanning, the model traces constructed perpendicular and parallel to the discontinuities will differ significantly, thus lowering the correlation coefficient, which is beneficial to identify minor discontinuities. Secondly, unlike traditional approaches that compute symmetric cross-correlations within a moving window, OOCA constructs only one structure-oriented model trace per direction and excludes the central trace, reducing the number of cross-correlation operations by approximately 50%. Moreover, the new method, which employs the gradient structure tensor method for dip estimation, is more robust to noise than the method of estimating time shift via cross-correlation used in the first method. Therefore, the proposed method is more stable and reliable.