## 5 Gates Formula Improvement
### Coefficient  
The second approach for improving the accuracy of the formula involves applying coefficients. Unlike other pile driving formulas, the Gates formula shows a nonlinear trend when comparing the bearing capacity with the PDA. In such cases, applying coefficients based on segmented intervals can significantly improve the accuracy of the formula. Four intervals were defined to closely match the values calculated by the Gates formula and Gates (PHC) values with those obtained using the PDA (Pile Driving Analyzer), and coefficients were applied accordingly. These four intervals were determined using two approaches. The first approach applies coefficients at every 1000 kN, with segments of \(\leq\)3000 kN,  
Figure 4: Comparisons of bearing capacities (Qu) determined using 242 PDA data points and bearing capacities calculated using pile driving formulas. (**a**) Gates, (**b**) Gates (PHC). Figure 3: Fitting of the new Gates formula using 242 PDA data points and corresponding error variance graph, (**a**) data fitting result, (**b**) error variance graph for the new Gates formula (Gates (PHC)). 3000-4000 kN, 4000-5000 kN, and >5000 kN. The second approach calculates and applies coefficients by optimizing the segments through the iteration process. Two approaches were applied to the original Gates and the Gates (PHC) formula discussed in Section 5.1. The coefficients determined using the first approach are summarized in Table 2. The Gates formula with the applied coefficients is named Gates (Coef), and the Gates (PHC) formula with the applied coefficients is named Gates (PHC Coef). Figure 5 shows the results of applying the coefficients to each section. Figure 5 presents the results of applying the coefficients to the Gates formula. In contrast, Figure 5 presents the results of applying the coefficients to the Gates (PHC) formula, comparing them with the existing formula. Based on Figure 5, applying the coefficients to both formulas results in an overall increase in accuracy. The second approach, through an iterative process, determined not only the coefficients but also the optimized segments. These segments round to the nearest ten units. The optimized segments and coefficients are summarized in Table 3.  
The comparison of the Gates formula with the results of applying the optimal interval coefficients is shown in Figure 6. Figure 6 presents the results of using the optimal interval coefficients of the Gates formula, named Gates (Coef2). In contrast, Figure 6 presents the results of applying the optimal interval coefficients to the Gates (PHC) formula, named \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Interval (kN)** & **Gates Coefficient** & **Gates (PHC) Coefficient** \\ \hline \multirow{3}{*}{Fixed Interval} & \textless{}3000 & 1.09 & 0.80 \\  & 3000–4000 & 1.21 & 1 \\  & 4000–5000 & 1.19 & 1.04 \\  & \textgreater{}5000 & 1.10 & 0.89 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coefficients for Gates and Gates (PHC) formulas by fixed interval. Figure 5: Comparisons of bearing capacities (Qu) determined using 242 PDA data points and bearing capacities calculated using pile driving formulas. (**a**) Gates and Gates (Coef), (**b**) Gates and Gates (PHC Coef). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Interval (kN)** & **Gates Coefficient** & **Gates (PHC) Coefficient** \\ \hline \multirow{3}{*}{Optimized Interval} & \textless{}2400 & 0.97 & 0.87 \\  & 2400–3150 & 1.05 & 0.95 \\ \cline{1-1}  & 3150–3720 & 1.27 & 1.15 \\ \cline{1-1}  & \textgreater{}3720 & 1.17 & 1.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Coefficients for Gates and Gates (PHC) formulas by optimized interval. Gates (PHC Coef2). From a statistical perspective, there is an improvement in accuracy by applying the optimal coefficient. However, as shown in the two comparison graphs, a significant data gap between 3300 and 4000 kN is observed. The intervals could cause this gap, and the coefficients applied may have shifted the original Gates values away from this specific range.