## 4 Computational Conditions and Generating Arrival Time Observation Errors  
The numerical model that is used for numerical experiments is shown in Figure 3. Table 1 shows the configuration of the numerical model that is illustrated in Figure 3 and the computational conditions of the numerical experiments. The model simulates a structure in which a local damaged part exists. The model of the analysis area is 0.400 m in both width and height. The sources of the events are installed at 8 cm intervals, and 16 events are generated in total. In addition, eight sensors are installed. One event is defined as the generation of one elastic wave from one source. The black squares in Figure 3 show the sensor location used in the numerical experiments. The observed arrival times are computed by assuming the straight path from the generated sources to the sensors.  
Figure 3: Analysis model schematic of AET: (**a**) locations of installed events and sensors in the model and (**b**) locations of installed candidates points in the model.  
Tables 2 and 3 show the coordinates of the sensor and source location, respectively. The cell size is 10 cm in height and width. The number of the candidate points is set to 1681, and the number of cells is 16. The initial elastic wave velocity is set to a homogeneous distribution of 4000 m/s, which is that of concrete in the soundness condition.  
In this study, the observation error is generated on the basis of truncated normal distribution. Figure 4 is an example of observation arrival time error distribution from the truncated normal distribution that is introduced in Figure 5. It is noteworthy that Figure 5 is created on the basis of the maximum observed arrival time and the error ratio. The mean error value is set to 0 for truncated normal distribution, and the absolute value \(\alpha\) of the error generated is obtained from the observation error ratio and the observed maximum  
\begin{table}
\begin{tabular}{c c} \hline Number of elements & 16 \\ Number of nodes & 25 \\ Model size (m \(\times\) m) & 0.40 \(\times\) 0.40 \\ Interval of candidate points (cm) & 1.0 \\ Elastic wave velocity in soundness condition & 4000 \\ (m/s) & 3000 \\ Damaged part elastic wave velocity (m/s) & 16 \\ Observation error ratio & 0.05–0.15 \\ Number of analyses with different errors at & 100 \\ each observed error ratio & 8 \\ Number of sensors & 8 \\ \hline \end{tabular}
\end{table}
Table 1: Numerical model conditions.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Sensors** & **x (m)** & **y (m)** \\ \hline S1 & 0.0 & 0.0 \\ S2 & 0.2 & 0.0 \\ S3 & 0.4 & 0.0 \\ S4 & 0.0 & 0.2 \\ S5 & 0.4 & 0.2 \\ S6 & 0.0 & 0.4 \\ S7 & 0.2 & 0.4 \\ S8 & 0.4 & 0.4 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Locations of installed sensors in the model.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Events** & **x (m)** & **y (m)** \\ \hline E1 & 0.08 & 0.08 \\ E2 & 0.16 & 0.08 \\ E3 & 0.24 & 0.08 \\ E4 & 0.32 & 0.08 \\ E5 & 0.08 & 0.16 \\ E6 & 0.16 & 0.16 \\ E7 & 0.24 & 0.16 \\ E8 & 0.32 & 0.16 \\ E9 & 0.08 & 0.24 \\ E10 & 0.16 & 0.24 \\ E11 & 0.24 & 0.24 \\ E12 & 0.32 & 0.24 \\ E13 & 0.08 & 0.32 \\ E14 & 0.16 & 0.32 \\ E15 & 0.24 & 0.32 \\ E16 & 0.32 & 0.32 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Locations of installed events in the model.  
arrival time, as shown in Equation (15). The range of the error distribution is from \(-\alpha\) to \(+\alpha\). The standard deviation given to the truncated normal distribution is determined separately as described in Equation (16). Random numbers generated in this distribution are used as the observation error values. The total number of random numbers generated is the same as the total number of observed values.  
\[\alpha=\pm A_{max}E \tag{15}\]  
\[\sigma=0.5\ A_{max}E \tag{16}\]  
where \(A_{max}\) is the maximum value of all observed arrival times to generate errors with a uniform distribution, \(E\) is the observation error ratio, and \(\sigma\) is the standard deviation of the truncated normal distribution.