## 4 Results
### Lithology Classification Feature Extraction
These metrics underscore UMAP's capacity to effectively capture both internal physical variability (e.g., heterogeneity in coarse sand containing gravel inclusions) and the depositional logic between lithologies (e.g., gradual energy transitions between silt and fine sandstone). As a nonlinear dimensionality reduction technique, UMAP successfully addresses lithological overlap issues caused by the linear projections of PCA--most notably enhancing the separability between silt and fine sandstone by 214% compared with PCA--and mitigates the abnormal dispersion and cluster fragmentation associated with t-SNE's local overfitting. The DBI value of 2.83 further underscores UMAP's capability for fine-grained boundary segmentation, especially in complex interbedded formations such as thin layers of clay shale and silt, surpassing both t-SNE (DBI 7.22) and PCA (DBI 3.12). Visualization outcomes demonstrate that limestone forms a distinct and compact cluster (compactness 5.37) in the UMAP-reduced space, with negligible mixing with clastic lithologies. The spatial separation between coarse- and fine-grained sandstone is geologically rational, closely aligning with the grain-size variation patterns typical of fluvial sedimentary environments. In summary, UMAP delivers a high-fidelity, interpretable low-dimensional representation of LWD data via topological manifold learning, facilitating real-time recognition of highly heterogeneous subsurface formations during drilling operations. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Lithologic Type** & **Silt** & **Fine Sandstone** & **Limestone** & **Clay Shale** & **Fine Sand** & **Coarse Sand** \\ \hline PCA & 2.06 & 1.39 & 2.03 & 2.05 & 1.40 & 1.85 \\ t-SNE & 9.36 & 9.51 & 11.31 & 9.56 & 10.15 & 13.28 \\ UMAP & 5.85 & 6.57 & 5.37 & 4.55 & 7.49 & 6.19 \\ \hline \end{tabular}
\end{table} 
Table 3: Within-cluster scatter for each lithology class after using different dimensionality reduction algorithms. Figure 9: Separation quality of each lithology class after using different dimensionality reduction algorithms. Cluster processing  
Clustering algorithms leverage the inherent structure of data to define rules for grouping observations with similar characteristics. Among these algorithms, K-means is widely employed in machine learning cluster analysis due to its ability to identify natural groupings or clusters in dimensionality-reduced data. The algorithm assigns data points to clusters with equal variance, minimizing the within-cluster sum of squared deviations, also known as inertia. The accuracy of the clustering outcome is influenced by the selection of the optimal number of clusters and the relevant variables that capture the essential patterns and information in the dataset. In this study, we applied the K-means algorithm to a dataset reduced in dimensionality using UMAP. To assess the clustering quality, we plotted the Davies-Bouldin Index (DBI) as a function of the number of clusters ranging from 2 to 24 (Figure 10). The experimental results revealed that the DBI reached its minimum value of 0.71 when K = 15, indicating the optimal clustering configuration. This result suggests that, at K = 15, there is an ideal balance between cluster compactness and separation, leading to the best overall clustering quality. The K-means algorithm was applied with K = 15 to the UMAP-reduced dataset (Figure 11), and the resulting clusters were visualized on top of the UMAP dimensionality reduction projections (Figure 12). In cluster 0, fine sand predominates, making up 86.8% of the cluster, clearly identifying it as a fine sand-dominated group. Cluster 1 contains 47.3% coarse sand and 41.4% clay shale, which likely represents a mixed layer of coarse sand and clay shale. In cluster 2, fine sandstone and limestone constitute 56.3% and 23.7%, respectively, suggesting an interaction zone between limestone and fine sandstone.