## 3 Results  
### Physical Modeling Data Example  
To evaluate the effectiveness and robustness of the proposed optimal direction-guided coherence method in delineating subtle structural discontinuities, we first apply it to a physical modeling data set [47]. This model was designed to simulate typical subsurface complexities in deep formations, such as reverse faults, fracture systems, and lateral heterogeneity. It consists of six geological layers (Figure 5a), among which the fourth layer is selected as the target formation due to its richness in structural features. The southern part of this layer contains four nearly east-west trending reverse faults (f\({}_{1}\)-f\({}_{4}\)) with varying displacements, cutting through multiple horizons. These faults are accompanied by several groups of intersecting fractures with variable lengths and dips, forming a tectonically intense deformation zone characterized by high-angle disruptions and discontinuous reflectors. In contrast, the northern part of the layer includes systematically configured fracture sets (marked as the black numbers 1-8) with controlled variations in density, spacing, orientation, and dip angle (Figure 5b). This design allows us to test coherence attribute responses across both complex and semi-regular discontinuity patterns. The seismic dataset was generated via convolution between layer-based reflectivity and a 40 Hz Ricker wavelet, followed by conventional post-stack processing. This controlled experiment offers an ideal benchmark for assessing the multi-scale and multi-orientation sensitivity of the proposed coherence technique under deep exploration conditions. Figure 4: Workflow of the proposed OCCA method. As an initial validation, the proposed coherence enhancement method is first applied to three high-dimensional phase coherence volumes derived from Gabor decomposition at 10 Hz, 30 Hz, and 45 Hz, as illustrated in Figure 6a-c. The low-frequency 10 Hz coherence slice (Figure 6a) clearly delineates the large-scale fault structures in the southern part of the model, particularly capturing the overall fault trajectories (indicated by green arrows). However, it shows limited ability to detect subtle discontinuities within the northern fracture zones, as reflected by the weak responses inside the blue dashed box. The 30 Hz coherence slice (Figure 6b) achieves a balanced performance between fault clarity and noise suppression, enhancing the visibility of the northern fracture sets (blue box) while maintaining relatively low background noise. The 45 Hz coherence result (Figure 6c) further improves the resolution of small-scale features, offering sharper imaging of Fractures 1, 2, and 4 in the northern section, as well as fracture clusters adjacent to the southern main faults. Nonetheless, it introduces a considerable amount of high-frequency noise across the section. In contrast, the fused coherence result (Figure 6d) demonstrates better overall structural imaging performance. It successfully retains the low-frequency response to large-scale fault outlines while integrating the high-frequency fracture details, thereby enhancing both structural continuity and spatial resolution. However, this fusion strategy is not without limitations. For example, the southern fault boundary becomes partially blurred due to interference from adjacent high-frequency fracture responses. This highlights a typical trade-off in multi-frequency fusion: while improving structural completeness, it may also amplify local artifacts in regions with dense discontinuities. Figure 5: (**a**) The designed 3D physical model proposed by [PERSON] (2018) [47] and (**b**) the horizon sketch of the target layer. The physical model used in this study, consisting of six layers. The fourth layer (shaded in color) is selected as the target due to its abundant structural features. The southern region includes four east–west trending reverse faults (F\({}_{1}\)–F\({}_{4}\)) and multiple intersecting fracture groups (marked as the red numbers 1–7).