## 3 Time-Based Localization Method and Localization Performance in Numerical Examples
### Localization Performance in Numerical Examples
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Actual Vibratory**} & \multicolumn{2}{c}{**Pile Toe on the**} & \multirow{2}{*}{**Pile Toe 10 m below the Ground Surface**} \\ \cline{2-2} \cline{4-6}  & & **Ground Surface** & & & \\ \cline{2-2} \cline{4-6}  & \multicolumn{2}{c}{**Linear Layout**} & \multicolumn{2}{c}{**Linear Layout**} & \multicolumn{2}{c}{**Square Layout**} \\ \cline{2-6}  & **X (\%)** & **Y (\%)** & **X (\%)** & **Y (\%)** & **X (\%)** & **Y (\%)** \\ \hline S1 & 1.85 & 0.02 & 2.67 & 0.02 & 1.00 & 0.02 \\ S2 & 1.70 & 0.02  
The sensitivity studies were conducted to investigate the localization performance due to the changes in the ground shear wave velocity, the vibratory driving frequency, and the friction coefficient at the pile-soil interface when the pile toe is pre-embedded 10 m below the ground surface and the monitoring points in the linear layout are utilized. Compared with the localization results shown in Figure 9a for the simulation case: the shear wave velocity \(v_{s}\) = 120 m/s, driving frequency \(f_{0}\) = 20 Hz, and friction coefficient \(\mu\) = 0.25, Figure 10 presents the localization results of plane coordinates for two other shear wave velocities (\(v_{s}\) = 60 m/s and 250 m/s), Figure 11 shows two other driving frequencies (\(f_{0}\) = 40 Hz and 60 Hz), and Figure 12 displays two other friction coefficients (\(\mu\) = 0.1 and 0.4). Overall, the localized driving source is reasonably close to the actual driving locations in these figures. The localized deviations are mainly reflected in the X direction due to the monitoring points aligned in the Y direction. Figure 10: Localization of horizontal coordinates when pile driving takes place in different ground conditions. (**a**) \(v_{s}\) = 60 m/s; (**b**) \(v_{s}\) = 250 m/s. (Note: MP—monitoring points, AVDL—actual vibratory driving location, EVDL—estimated vibratory driving location). Figure 9: Localization of horizontal coordinates when pile toe is pre-embedded 10 m below the ground surface. (**a**) Linear layout; (**b**) square layout. (Note: MP—monitoring points, AVDL—actual vibratory driving location, EVDL—estimated vibratory driving location). Table 4 presents the mean and maximum relative errors in sensitivity studies. When considering the larger driving frequency and shear wave velocity, the adopted constant sampling rate (i.e., 1 kHz) cannot guarantee enough accuracy in AlToAs among monitoring points. As shown in Table 4, either a larger shear wave velocity or higher driving frequency can induce a slight increase in the relative errors of the estimated coordinates. For example, the mean relative error (X direction) increases by 0.87% when \(v_{s}\) changes from 120 m/s to 250 m/s, and it increases by 1.04% when \(f_{0}\) changes from 20 Hz to 60 Hz. The relative errors can also be affected by the change in the pile-soil friction coefficient, which might be explained by the generated wave contents during the pile-soil interactions. As discussed before, the body waves generated from the pile toe influence the localization accuracy by complicating the collected velocity signals. As the friction coefficient decreases, the percentage of body waves generated from the pile toe increases, and their adverse effects on the localization performance are magnified. For example, the mean relative error (X direction) increases by 0.66% when the friction coefficient \(\mu\) decreases from 0.25 to 0.1. Figure 11: Localization of horizontal coordinates when pile driving force operates at different frequencies. (**a**) \(f_{0}\) = 40 Hz; (**b**) \(f_{0}\) = 60 Hz. (Note: MP—monitoring points, AVDL—actual vibratory driving location, EVDL—estimated vibratory driving location). Figure 12: Localization of horizontal coordinates when the pile–soil interface is defined with different pile–soil friction coefficients. (**a**) \(\mu\) = 0.1; (**b**) \(\mu\) = 0.4. (Note: MP—monitoring points, AVDL—actual vibratory driving location, EVDL—estimated vibratory driving location).