## 2 Geometric Modeling Approaches
### Modeling of Dielectric Constant of the Medium
#### 2.3.2 Dielectric Property Modeling of Soil  
The dielectric properties of soil are primarily determined by its volumetric water content. Commonly used empirical or semi-empirical models include the Topp model [24], the generalized refractive mixing dielectric model (GRMDM) [25], the Wang and Schmugge model [26], and the MICRO-SWEAT model [27]. Among them, the four-component model proposed by [PERSON] demonstrates higher accuracy and applicability in estimating soil dielectric properties, as it accounts for factors such as temperature, bulk density of bare soil, and soil moisture content. Unlike traditional models that consider only soil moisture, this model incorporates soil composition and temperature, enabling a more comprehensive characterization of the soil's dielectric constant. The influencing factors of each dielectric constant of soil are shown in Table 1.  
When estimating the dielectric constant of soil using the four-component model, the soil is typically considered as a mixture of sand, clay, rock matrix, and water [28]. 5% denote the sand content and C\({}^{\circ}\)% represents the clay content in the soil, subject to the constraint \(S+C\leq 100\). The soil moisture at the wilting point satisfies the relationship given in Equation (7). \[Wp=0.06774-0.00064\times S+0.00478\times C \tag{7}\]  
The critical volumetric moisture content is empirically defined by Equation (8). \[m_{t}=0.49 Wp+0.165 \tag{8}\]  
The parameters are defined according to Equation (9). \[\beta=-0.57 Wp+0.481 \tag{9}\]  
Under typical conditions, the density of soil minerals is taken as \(\rho_{r}=2.6\) g/cm\({}^{3}\), and the bulk density of dry soil is assumed to be \(\rho_{b}\). The soil porosity can be calculated using Equation (10). \[p=1-\frac{\rho_{b}}{\rho_{r}} \tag{10}\]  
\(\rho_{b}\) is determined by Equation (11). \[\rho_{b}=3.455/R^{0.3018} \tag{11}\]  
The condition satisfied by \(R\) is given in Equation (12). \[R=25.1-0.21\times S+0.22\times C \tag{12}\]  
After determining the above parameters, the equivalent dielectric constant can be calculated based on soil moisture \(m_{v}\). When \(m_{v}\leq m_{t}\),  
\[\epsilon_{x}=\epsilon_{i}+(\epsilon_{w}-\epsilon_{i})\frac{m_{v}}{m_{t}}\beta \tag{13}\]  
When \(m_{v}>m_{t}\),  
\[ \begin{array}{l}\epsilon_{s}=m_{t}\epsilon_{x}+(m_{v}-m_{t})\epsilon_{w}+(p -m_{v})\epsilon_{a}\\ +(1-p)\epsilon_{r}\end{array}  \tag{14}\]  
The relationships satisfied by the coefficients in the above equations are given in Equation (15). \[\epsilon_{x}=\epsilon_{i}+(\epsilon_{w}-\epsilon_{i})\beta \tag{15}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**The Influencing Factors of Each Dielectric Constant of Soil**} \\ \hline
**Influencing Factors** & **Frequency of Incidence** & **Soil Moisture** & **Soil Surface Temperature** & **Soil Sediment Content** & **Soil Clay Content** \\ \hline Symbolic representation & \(f(\mathrm{GHz})\) & \(m_{v}(\mathrm{g/cm^{3}})\) & \(T(^{\circ}\mathrm{C})\) & \(S\)\% & C\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The influencing factors of each dielectric constant of soil. In the above equations, \(\varepsilon_{x}\) is the dielectric constant of the soil. \(\varepsilon_{i}\) represents the dielectric constant of ice; \(\varepsilon_{r}\) is the dielectric constant of rock; \(\varepsilon_{a}\) is the dielectric constant of air; and the dielectric constant of water, denoted as \(\varepsilon_{w}\), is calculated according to Equations (16)-(18). \[\varepsilon_{w}=4.9+\frac{\varepsilon_{w0}-4.9}{1+j2\pi f\tau_{w}} \tag{16}\]  
\[\varepsilon_{w0}(T)=88.045-0.4147T\] \[+6.295\times 10^{-4}T^{2}+1.075\times 10^{-5}T^{3} \tag{17}\]  
\[2\pi\tau_{w}(T)=1.1109\times 10^{-10}\] \[-3.824\times 10^{-12}T+6.938\times 10^{-14}T^{2} \tag{18}\]  
In the equation, \(\tau_{w}\) is the relaxation time of water, the soil Celsius temperature is denoted as \(T\). When the other parameters in the equation are determined, the effective dielectric constant of the soil can be obtained.