## 4 Comparative Analysis of Pile Driving Formulas  
The pile driving formula, or dynamic formula, has limitations in reliably estimating the ultimate bearing capacity of piles. These issues arise from comparing the formula, which does not account for time factors, with static load test results that do consider time factors [24]. Thus, it should be used for quality control purposes immediately after the End of Initial Driving (E.O.I.D). For ensuring high reliability in quality control, existing pile driving formulas were reviewed. This review utilized a total of 258 data points from a PDA (Pile Driving Analyzer) and various other data collected from 11 different sites. Only PHC piles with a diameter of 600 mm were used, with the piles driven into weathered rock. The ground above the weathered rock mainly consisted of silty sand and weathered soil. The dataset used in this analysis includes measured bearing capacities from a PDA, calculated using CAPWAP analysis and calculated bearing capacities from four different formulas: the Hiley, Gates, Danish, and original Gates formulas. Among these, the Hiley and Danish formulas utilize the EMX value obtained through PDA, reflecting the actual energy. Therefore, versions of these formulas that include the EMX value for higher accuracy were used. Additionally, the Hiley formula specifically extracts the well-estimated rebound value using the DMX obtained from PDA, resulting in even greater accuracy. Figure 2 compares the bearing capacity derived from CAPWAP with the results calculated using various pile driving formulas. Figure 2a shows the comparison with the Hiley formula, Figure 2b with the Danish formula, Figure 2c with the Gates formula, and Figure 2d with the original Gates formula. The closer the data points lie to the 1:1 diagonal line, the higher the accuracy of the pile driving formula. Data points below the diagonal line indicate that the formula underestimates the bearing capacity, while points above it indicate an overestimation. The comparison reveals that the Hiley formula produces results most similar to the bearing capacity obtained from the dynamic load test. The Danish and Gates formulas also show comparable accuracy, though the Danish formula exhibits a relatively higher degree of variance. In contrast, the Gates formula tends to have less dispersion. Despite not utilizing highly accurate variables obtained from a PDA like the Hiley or Danish formulas, the Gates formula demonstrated creditable accuracy. The original Gates formula consistently produces lower values than those obtained from PDA, indicating very low reliability. Two Gates (original Gates and Gates) formulas exhibit a nonlinear shape, likely due to the inclusion of root and logarithmic terms in their equations, reflecting inherent model limitations. Further metrics for evaluating the performance of these formulas, such as the mean error, absolute mean error rate, and standard deviation of absolute mean error rate, were calculated (Table 1). These metrics provide insights into both the accuracy (how close predictions are to actual values) and the consistency (how variable the predictions are) of each formula. Firstly, the mean error can be used to determine whether a formula generally underestimates or overestimates the bearing capacity. The absolute mean error is a measure used to quantify the accuracy of predictions. It is calculated as the average of the absolute differences between predicted values and actual values. A lower absolute mean error indicates that the predictions are, on average, closer to the actual values, suggesting higher accuracy. The standard deviation of errors measures the variability or dispersion of prediction errors around the mean error. A lower standard deviation indicates that the prediction errors are more consistent and closely clustered around the mean error, suggesting more reliable predictions.