## 5 Applications and Discussion
### Results of Rock Physics Analysis  
This study investigates a 3D seismic survey area (400 km\({}^{2}\)) with 23 wells distributed unevenly. To assess the validity of the waveform indicator inversion algorithm under sparse-well conditions, we conducted an inversion experiment using an uneven well network: 12 wells were selected for inversion (training set), while 11 wells served as independent validation data (Table 2, Figure 11c).  
Figure 11: Verification cross-section and sandy thickness conformity analysis of Paleogene Series Dongying Member in HHK Depression. (**a**) Inversion Validation Profile; (**b**) Comparative Analysis of Inversion-Predicted Sand Body Thickness and Logging-Interpreted Sandstone Thickness; (**c**) Profile Location Map.  
Pearson's correlation coefficient was used to separately compute correlation values between well log-derived and inversion-predicted sandstone thicknesses for both the participant and verification wells (Figure 11a). The P-wave impedance inversion results (Figure 11b) demonstrate strong consistency with both the training and validation wells, yielding high correlation coefficients (R\({}^{2}\) = 0.927 for training wells, R\({}^{2}\) = 0.821 for validation wells).  
The inversion successfully resolves thin sandstone layers (3-8 m) and accurately captures their lateral variations, as evidenced by the clear identification of layer boundaries in the impedance profile, consistency between predicted sandstone thickness (from inversion) and log-derived measurements, and effective delineation of lateral facies transitions matched by seismic reflectivity patterns. These results confirm that the waveform indicator inversion method provides reliable lateral resolution (\(\geq\)3 m) for lithological characterization in complex reservoirs with sparse well coverage.  
To further validate the precision of the waveform-based inversion method in identifying thin sandstone layers, we extracted well-site curves using well C4-3-a as an example (Figure 12). The first purple curve in Figure 12 represents the measured DT curve, the second blue curve corresponds to the measured GR curve, and the third curve (IMP) is the inversion result. The inversion-derived wave impedance profile (waveform indicator inversion result) shows high consistency with the measured curves. By applying a sandstone discrimination threshold, the lithological interpretation of the inversion results aligns well with the actual well logging interpretations, accurately identifying sandstones with thicknesses of 1.8-3.0 m (Figure 12).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Participant Well**} & \multicolumn{2}{c}{**Sandstone Thickness/m**} & \multicolumn{3}{c}{**Sandstone Thickness/m**} \\ \cline{2-6}  & **Well Log** & **Inversion** & **Verification Well** & **Well Log** & **Inversion** \\  & **Interpretation** & **Prediction** & & **Interpretation** & **Prediction** \\ \hline B7-2-b & 1.3 & 0.5 & B8-2S-b & 1.8 & 2.3 \\ \hline B7-2-d & 1.5 & 3.2 & C4-2-a & 8 & 7.6 \\ \hline B7-2-a & 2.4 & 1.1 & C-5-b & 4.5 & 3.9 \\ \hline C4-3-bD & 2.9 & 3.4 & B7-2-e & 4.6 & 2.6 \\ \hline C4-4-e & 2.9 & 3.3 & C4-1-a & 5.3 & 5.