## 3 Time-Based Localization Method and Localization Performance in Numerical Examples
### Localization Performance in Numerical Examples  
Numerical simulations were conducted to verify the proposed time-based localization method. It begins with the investigation of the effects of actual pile toe depths and spatial relationships between monitoring points and driving locations on localization performance. Figure 7 shows the ground surface (i.e., _X-Y_ plane) of the simulated soil domain. A total of nine vibratory sheet pile driving locations (S1 to S9) and two pile toe depths (i.e., the ground surface and 10 m below ground surface) at each location were modeled. Linear and square layouts of monitoring points were deployed on the ground surface to model different deployment cases on construction sites, wherein the distances between two adjacent monitoring points were set as 30 m and 90 m in the linear and square layouts, respectively. A constant sampling interval was set as 0.001 s (i.e., a sampling frequency of 1 kHz) in the simulated velocity signals. After vibratory driving-induced velocity signals were collected at the deployed monitoring points, localizations of those tested source locations were performed according to the time-based localization method proposed in this study. Figure 8 displays the localization results when the vibratory sheet pile driving takes place on the ground surface. Monitoring points in the linear sensor layout were used for the localization. As observed from Figure 8, deviations exist between the estimated and actual vibratory driving locations due to the possible inconsistent wave propagation velocities among monitoring points. Such inconsistency in wave propagation velocities can be induced by the interference of body waves, given that the single type of surface wave is pre-assumed to propagate along the ground surface. The relative errors were defined as the ratio of estimation errors in the coordinates \((x,y)\) to the site dimension (i.e., 120 m in both directions in numerical examples). Table 3 shows the calculated relative errors for different vibratory driving locations (S1 to S9). The relative error ranges from 0.92% to 2.08% in the X direction and from 0.02% to 0.19% in the Y direction. Generally, the localization accuracy in the Y direction was better than that in the X direction because the used monitoring points were aligned in the Y direction. Figure 7: Two layouts of monitoring points deployed on the ground surface. (**a**) Linear layout; (**b**) square layout. Figure 9 displays the localization of the plane coordinates when the pile toe is pre-embedded 10 m below the ground surface. The localization results derived by using monitoring points in the linear and square layouts were displayed. It can be observed that a much better localization accuracy is achieved by using the monitoring points in the square layout. As also listed in Table 3, the mean relative error decreases from 2.69% in the linear layout to 1.28% in the square layout, and the maximum relative error decreases from 4.00% in the linear layout to 1.75% in the square layout. It can be concluded that spatial relationships have a great impact on localization performance. The layout of monitoring points spreading in both directions and surrounding the driving locations can improve the localization performance compared with the layout of monitoring points deployed along a single direction. Moreover, because of the more complex wave types and paths generated when the pile toe is in deep positions, a higher degree of inconsistency in wave propagation velocity was expected among the investigated monitoring points, which degraded the localization accuracy compared with those cases when the pile toe is placed on the ground surface.