## 5 Gates Formula Improvement
### Summary of Improved Gates Formula  
In this section, two different methods were introduced to improve the Gates formula. The first approach utilizes 242 PDA data points from this study to modify the constants of the Gates formula through data curve fitting. The second approach to enhancing the accuracy of the formula involves applying coefficients. The improved results obtained from both methods, along with the statistical results of the existing pile driving formulas, are summarized in Table 4.  
The mean error analysis indicates that Gates (PHC) significantly addresses the previous tendency to underestimate, as demonstrated by the improvement. The absolute mean error decreased from 17.52% to 14.71%, indicating enhanced accuracy. Although the variance of the error slightly increased, the absolute value itself remained small.  
Optimized coefficients resulted in a [PERSON] (Coef) value with an absolute mean error of 13.41% compared to the bearing capacity derived from the PDA values. All values moved  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Formula**} & **Mean Error (\%)** & **Absolute Mean Error (\%)** & **Absolute Error** \\  & **(\%)** & **Error (\%)** & **Std. (\%)** \\ \hline [PERSON] & 8.24 & 10.9 & 10.72 \\ [PERSON] & 8.56 & 13.97 & 12.9 \\ [PERSON] & \(-\)13.13 & 17.52 & 9.77 \\ [PERSON] (PHC) & 3.25 & 14.71 & 11.09 \\ [PERSON] (Coef) & 2.73 & 13.41 & 11.02 \\ [PERSON] (PHC Coef) & 2.56 & 13.35 & 10.34 \\ [PERSON] (Coef2) & \(-\)0.04 & 11.76 & 9.36 \\ [PERSON] (PHC Coef2) & 0.40 & 12.09 & 9.56 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of bearing capacity estimates using nine different pile driving formulas and results from 242 PDA data points (mean error, absolute mean error, and standard deviation of absolute mean error).  
Figure 6: Comparisons of bearing capacities (Qu) determined using 242 PDA data points and bearing capacities calculated using pile driving formulas. (**a**) [PERSON] and Gates (Coef2), (**b**) [PERSON] and Gates (PHC Coef2).  
closer to zero, confirming a significant improvement in accuracy. Applying the optimized coefficients to the Gates (PHC) formula resulted in an absolute mean error of 13.35% compared to the bearing capacity from the PDA values, showing a slight improvement over the [PERSON] (Coef) formula.  
The results of applying the optimal interval coefficients for both cases ([PERSON] (Coef2) and [PERSON] (PHC Coef2)) demonstrated the most significant statistical improvement across all values compared to previous examples. However, as observed in Figure 6, this method induces a data gap in specific intervals, making its use less advisable. A more detailed analysis with finer intervals than the four used in this study could mitigate this issue, but this would require the collection of more data under diverse conditions. Comprehensive analysis of all results indicates that the Hiley formula exhibits the lowest mean absolute error and generally has the smallest mean absolute standard deviation, demonstrating the highest accuracy. Nonetheless, various improvement measures attempted for the Gates formula have also significantly enhanced its accuracy. Consequently, the newly fitted Gates (PHC) and the Gates (PHC) with interval coefficients (Gates (PHC Coef)) have proven suitable for improving the Gates formula. However, as Gates (PHC Coef) lacks sufficient justification for the interval range, caution or further research is recommended.  
Given that this study employed the highly accurate EMX values derived from the PDA measurements, the improved Gates formulas are expected to have similar or higher accuracy in field applications where a PDA cannot always be used, thus demonstrating high practical utility.