# Enhancing Density Prediction of Agricultural Land Soil through Void Area Curve Analysis
## 2 Materials and Methods
### Void Area in the Soil Image  
Soil mass comprises three phases: solids (soil particles), water, and air. The composition of the volume of the soil mass is shown in Equation (1) [39].  
\[V_{t}=V_{s}+V_{w}+V_{a}=V_{s}+V_{v} \tag{1}\]  
where \(V_{t}\) is the total volume, \(V_{s}\) is the volume of the solids, \(V_{w}\) is the volume of the water, \(V_{a}\) is the volume of the air, and \(V_{v}\) is the volume of the void. The dry density of the soil is defined as the ratio of the total volume of soil (\(V_{t}\)) to the mass of soil particles (\(W_{s}\)) and represents the degree of compaction. When the soil is compacted, the dry density increases because the void and total volume of the soil decreases. Therefore, if the void of the soil can be quantified, the dry density can be predicted. The voids in the soil can be quantified as the void ratio (\(e\)) or porosity (\(n\)), which are expressed by Equations (2) and (3), respectively [39].  
\[e=V_{v}/V_{s}=V_{v}/(V_{t}-V_{v})=n/(1-n) \tag{2}\]  
\[n=V_{v}/V_{t}=V_{v}/(V_{v}+V_{s})=e/(1+e) \tag{3}\]  
\(V_{t}\) is a constant of the sum of \(V_{s}\) and \(V_{v}\). Figure 1 shows the change in \(e\) and \(n\) according to \(V_{v}\) when \(V_{t}\) is 1.  
\begin{table}
\begin{tabular}{c c c c c} \hline \multicolumn{5}{c}{**Soil Sample Condition**} & \multirow{2}{*}{**Repetition**} & \multirow{2}{*}{**Total Images**} \\ \hline
**Texture** & **Dry Density (t/m\({}^{3}\))** & **Water Content (\%)** & \\ \hline SL, L, SiL, SiCL & 1.1, 1.2, 1.3, 1.4 & 5, 10, 15, 20, 25 & 3 times & 240 \\ \hline \end{tabular}
\end{table}
Table 1: Condition of the soil samples for image analysis.  
As \(V_{v}\) increases, both \(e\) and \(n\) tend to increase. \(n\) increases linearly with \(V_{v}\), ranging from a minimum of 0 to a maximum of 1. \(e\) increases exponentially with respect to \(V_{v}\) and diverges as \(V_{v}\) approaches 1. Equations (2) and (3) can be rewritten by the specific gravity of the soil (\(\rho_{s}\)) and dry density (\(\rho_{d}\)) as Equations (4) and (5).  
\[e=\frac{n}{1-n}=\frac{1-\frac{\rho_{d}}{\rho_{s}}}{1-(1-\frac{\rho_{d}}{\rho_{ s}})}=\frac{\frac{\rho_{s}-\rho_{d}}{\rho_{s}}}{\frac{\rho_{d}}{\rho_{s}}}= \frac{\rho_{s}-\rho_{d}}{\rho_{d}}=\frac{\rho_{s}}{\rho_{d}}-1 \tag{4}\]  
\[n=\frac{V_{v}}{V_{t}}=\frac{V_{t}-V_{s}}{V_{t}}=1-\frac{V_{s}}{V_{t}}=1-\frac{ W_{s}V_{s}}{V_{t}W_{s}}=1-\frac{\rho_{d}}{\rho_{s}} \tag{5}\]  
The changes of \(e\) and \(n\) according to \(\rho_{d}\) are shown in Figure 2.  
\(n\) is linearly proportional to \(\rho_{d}\) and \(V_{v}\). Therefore, if \(n\) can be quantified using soil \(V_{v}\), \(\rho_{d}\) can be predicted. However, because the volume cannot be obtained from a two-dimensional soil image, \(n\) must be calculated indirectly through an area rather than a  
Figure 1: Changes in the void ratio and porosity according to the volume composition of the soil mass.  
Figure 2: Changes in the void ratio and porosity according to dry density.  
volume [40]. The porosity in the digital image (\(n^{\prime}\)) based on the area of the void in the digital image rather than the volume is expressed in Equation (6).  
\[n^{\prime}=A_{\triangledown}/A_{t} \tag{6}\]  
\(n^{\prime}\) is the porosity in the digital image, \(A_{\triangledown}\) is the area of the void in the digital image, and \(A_{t}\) is the total area of the digital image. The basic unit of area in a digital image is the pixel. Accordingly, Equation (6) can be rewritten as Equation (7).  
\[n^{\prime}=A_{\triangledown}/A_{t}=N_{void}/N_{total} \tag{7}\]  
where \(N_{void}\) is the number of pixels of the void and \(N_{total}\) is the total number of pixels in the digital image.