# Influence of Soil Parameters on Detecting Voids behind a Tunnel Lining Using an Impact Echo Method
## 3 Experimental Study
### Soil Density Alteration
The regular geometric space includes the point, line, plane and volume, with corresponding dimensions of 0, 1, 2 and 3; nevertheless, the counting dimension is not integral. Within fractal theory, the box-counting dimension method is widely used because it is easier to be calculated. It can recognize the difference among various signals quantitatively. The box-counting dimension is obtained by covering the object using square boxes with side length \(\delta\). To cover the unit square using a box-counting dimension, \(N\) small square boxes with a side length \(\delta\) are needed. The number of \(N\) is calculated using:  
\[N=\left(\frac{1}{\delta}\right)^{2}. \tag{1}\]  
For a unit square, the dimension \(D\) is 2, which can be calculated using:  
\[D=\frac{\log N}{-\log\delta}. \tag{2}\]  
If the fractal object has an irregular shape, the box-counting dimension can be calculated in the same way. However, it should be noted that the dimension may not be an integer. The acceleration vibration signal discussed in this study was an irregular curve that varied with time (Figure 6). In this case, the box-counting dimension is defined as:  
\[D= \begin{array}{c}\lim\\ \delta_{1}\to 0\\ \delta_{2}\to 0\end{array} \frac{\log N}{-\log\delta_{i}}, \tag{3}\]  
where \(i=1\) or 2. The small boxes used to cover the signal have the dimensions of \(\delta_{1}\times\delta_{2}\), of which, \(\delta_{1}\) is the sampling time interval and \(\delta_{2}\) is the smallest difference of the accelerations between two sampling points (non-zero). Because the vibration signals were discrete, the counting dimension \(D\) was obtained as the slope of the linear regression of a series data (\(-\)log\(k_{j}\delta_{i}\), log\(N_{j}\) (\(j=1,2,\dots\) )), instead of calculating the limit. \(k_{j}\) is a constant that controls the size of the boxes \(k_{j}\) (\(\delta_{1}\times\delta_{2}\)). However, in order to make the covering process meaningful, it should be noted that the constant \(k_{j}\) is limited according to [29]:  
\[\left\{ \begin{array}{c}\delta_{1}<k_{j}\delta_{1}<\frac{\Delta T}{2}\\ \delta_{2}<k_{j}\delta_{2}<\frac{a_{\max}}{2}\end{array} \right., \tag{4}\]  
where \(\Delta T\) is the time internal between the peak value point and the adjacent zero value point (Figure 6), and \(a_{\max}\) is the peak value of the acceleration. The box-counting dimension expresses the covering level of the vibration signal and is between 1 and 2. In the present study, if the box-counting dimension is larger, the vibration signal may attenuate more slowly or dominate at a higher frequency range, and vice versa. Figure 7 illustrates the box-counting dimensions for different soil density cases. Each value was averaged using 10 useful data points. The counting dimension value increased with a decrease in soil density. For case D3, with the largest density, the existence of the void induced a sharp increase in the dimension, from 1.2601 to 1.3416. Thus, the compactness between the soil and concrete plate played an important role in the counting dimension. Case D3 had the densest soil, and subsequently, the best compactness, while case D3v had the worst compactness due to the void's influence. The box-counting dimension can reflect the contact condition between the soil and concrete, and evaluate the void quantitatively. If the density is significantly altered, the void inspection results may be affected. The frequency spectrum can be calculated using a Fourier transform. Figure 8 illustrates the Fourier spectra for three different density cases, each of which was averaged using multiple signals. It can be observed that the dominant frequency was between 1640 and 1690 Hz for all cases. This frequency can be explained by the local resonance theory of a plate. Suppose the plate displacement \(u(r)\) at location \(r\) is [30]:  
\[u(r)=u_{0}\bigg{(}1-\frac{r^{2}}{a^{2}}\bigg{)}^{2}, \tag{5}\]  
Figure 6: A vibration signal covered with small boxes. Figure 7: Fractal box-counting dimensions for different soil density cases.