## 3 Laboratory Tests
### Relationship between \(k_{a}\) and \(\theta\)  
The dielectric constant measured by the PTDR is influenced by the surrounding soil, and the epoxy and MC nylon in the PTDR module. However, several researchers have shown that the dielectric constant measured by TDR probes can be related to the water content in soils, regardless of soil type [25; 40]. A calibration was thus carried out for the estimation of the water content in the soils.  
The soils sampled in the study site were used for the calibration. Note that the dielectric constant of the solid material can be changed according to the mineralogical composition of the solid material of the soil [41; 42]. Using the sampled soil, seven different specimens mixed with each water amount (\(\theta\) = 5, 10, 15, 20, 25, 30, 35 m\({}^{3}\) m\({}^{-3}\)) were prepared in a cylindrical acrylic cell with a diameter of 220 mm and a height of 200 mm. To measure the TDR waveforms, the PTDR module was driven into the center of the specimen.  
TDR waveforms measured by the PTDR module are plotted in Figure 5a for several specimens. The bump apex was determined as the first reflection point. For the determination of the second reflection point, the TDR waveforms were smoothed and then differentiated with respect to the travel time. After finding a maximum tangent line of the smoothed wave, the intersection of the tangent line with the horizontal line was selected as the second reflection point. The selection of the points was done manually. Then, the travel time (t) was determined as the time difference between the first and second reflection points. Figure 5a shows that the travel time of the reflected wave increased with the increase in the water content. The dielectric constant was determined from the travel time using Equation (1). A relationship between the dielectric constant (\(\mathrm{k_{a}}\)) obtained from the PTDR and the volumetric water content (\(\theta\)) was then established by the least-square fitting, and can be represented as follows:  
\[\theta=\mathrm{a}+\mathrm{b}\mathrm{k_{a}}+\mathrm{c}\mathrm{k_{a}}^{2}+ \mathrm{d}\mathrm{k_{a}}^{3} \tag{2}\]  
where \(\mathrm{a}\), \(\mathrm{b}\), \(\mathrm{c}\), and \(\mathrm{d}\) are constants determined by the cubic polynomial regression models of dielectric constant and volumetric water content, which corresponded to \(-33.406\), \(19.657\), \(-2.313\), and \(0.106\), respectively, as shown in Figure 5b. Note that the calibration equation suggested by [PERSON] et al. [11] was the same cubic polynomial equation as that obtained from this study, but with different constant values than those obtained from this study. In fact, the measured dielectric constant of 2.2 for the dried soil in this study represents the result of combining each dielectric constant of the soil particle, epoxy, and MC nylon of the PTDR module [43]. The values of the dielectric constant ranging from 2.2 to 9.4 in the specimens with different water contents were similar to the typical values of dielectric constants of soil particle, epoxy, and MC nylon when compared to that of water. Note that the water and soil particles can only be placed on one side of the electrodes, whereas three sides of the electrodes are enclosed with the epoxy and MC nylon.  
Figure 4: Grain size distribution with depth.