## 4 Results and Discussion
### Experimental Validation of Current Injection Models
#### 4.2.3 Gaussian Function  
Case 1: Current Flows in and out of the Same Boundary Surface  
Figure 16 illustrates the boundary conditions when the Gaussian function was applied to the current injection model. Here, boundary condition function \(g(y)=0\) because the boundary surface at \(x=0\) has no current flow, while \(f(y)\) can be expressed as follows:  
\[f(y)=\left(\frac{I}{\sigma}\right)\cdot\left(\frac{1}{\sigma_{s}\sqrt{2\pi}} \right)\left[e^{-0.5\left(\frac{y-y_{1}}{\sigma_{s}}\right)^{2}}-e^{-0.5\left( \frac{y-y_{2}}{\sigma_{s}}\right)^{2}}\right] \tag{40}\]  
where, \(I\) is the injected current, \(\sigma\) is the electrical conductivity of concrete, \(\sigma_{s}\) is a standard deviation for the Gaussian function, and \(y_{1}\) and \(y_{2}\) are the electrode positions where the current flows in and out, respectively. Standard deviation \(\sigma_{s}\) is an important constant. The electric potential field function corresponding to the Gaussian function has not yet been derived, but constant \(\sigma_{s}\) will first discussed using the electric potential function, which will be derived later. Figure 16: Boundary illustration of the current injection model using Gaussian function for Case 1. Figure 15: Comparison between a normal distribution and residuals obtained from the Heaviside step function for Case 2: (a) histogram of residuals (the label of x axe: residuals (differences) between calculations and measurements in \(10^{-3}\) volt unit, the label of y axe: the number of residuals in frequency) (b) Q-Q plot. Figure 17 shows the plot of the RAE as a function of \(\sigma_{s}\) using Gaussian function in Case 2, which had in a low RAE. In this case, the voltage data from V1 of Sets #6 to #8 were excluded as outliers because the fitting process was mainly dominated by these three sets, whose residual values were much higher than those of the others. Figure 18 shows histograms with various \(\sigma_{s}\) values. As shown in Figure 17, a \(\sigma_{s}\) value between 2 and 3 had the lowest RAE. Figure 18 shows that the histogram of residuals followed the normal distribution most closely with \(\sigma_{s}\) = 2.5. Therefore, considering that the electrode width was 5 mm, half that width is the best selection for constant \(\sigma_{s}\). Equation (41) provides the coefficients of series \(A_{n}\) using Equation (40). By applying Equation (41) to Equation (27), it is possible to obtain the electric potential equation \(u(x,y)\) for Case 1. \[ \begin{split} A_{n}&=\frac{2}{n\pi\sinh\left(\frac {n\pi y}{b}\right)}\cdot\left(\int_{0}^{b}f(y)\cdot\cos\left(\frac{n\pi y}{b} \right)dy\right)\\ &=\frac{2\cdot(1/\sigma)}{n\pi\sinh\left(\frac{n\pi y}{b}\right) }\cdot\left(\frac{1}{\sigma_{s}\sqrt{2\pi}}\right)\int_{0}^{b}\left[e^{-0.5 \left(\frac{y-y_{1}}{\sigma_{s}}\right)^{2}}-e^{-0.5\left(\frac{y-y_{2}}{ \sigma_{s}}\right)^{2}}\right]\cdot\cos\left(\frac{n\pi y}{b}\right)dy\end{split}  \tag{41}\]  
Table 7 shows a comparison between the measured and calculated voltages. The fitted electrical conductivity was 15.915 mS\(\cdot\)m\({}^{-1}\), and the RAE was 21.955%. The RAE was marginally reduced compared to the Heaviside step function. Since the residual rates of the Gaussian function shown in Table 7 were similar to those of the Heaviside step function, the histograms and Q-Q plots of two functions were also strikingly similar. For this reason, the histogram and Q-Q plot of the Gaussian function are not displayed here. Figure 17: Relative absolute error as a function of constant \(\sigma_{s}\) of the Gaussian function for Case 2.