## 2. Saturated-Unsaturated Seepage Control Equation  
It has been proven that [PERSON]'s law is not only suitable for saturated soils but also for unsaturated soils. Compared to saturated soils, the seepage parameters for unsaturated soils are not constant but a function of soil saturation. [PERSON]'s law of unsaturated soils is as follows:  
\[v=k(\theta)J \tag{1}\]  
\[v=\frac{K(\theta)\gamma}{\mu}J=K\frac{K_{r}(\theta)\gamma}{\mu}J \tag{2}\]  
where \(k(\theta)\) is the permeability coefficient, \(K(\theta)\) is the penetration of unsaturated soils, which is a function of the volumetric moisture content (the volume of water per unit volume), \(K\) is the penetration of saturated soils, \(K_{r}(\theta)\) is relative penetration, and \(\mu\) is the dynamic viscosity of water. The two-dimensional saturated-unsaturated control equation is as follows:  
\[-\frac{\partial(\rho v_{i})}{\partial x_{i}}+Q=\frac{\partial(\rho\theta)}{ \partial t} \tag{3}\]  
where \(\rho\) is the density of water and \(Q\) is the flow. According to [PERSON]'s law, Equation (3) can be rewritten as  
\[\frac{\partial}{\partial x_{i}}[k_{ij}^{s}k_{r}(h_{c})\frac{\partial h_{c}}{ \partial x_{j}}+k_{ij}^{s}k_{r}(h_{c})]+Q=[C(h_{c})+\beta S_{s}]\frac{\partial h _{c}}{\partial t} \tag{4}\]  
where \(k_{ij}^{s}\) is the saturated permeability tensor, \(k_{r}\) is the relative permeability coefficient in a saturated area, \(k_{r}=1\), \(C(h_{c})=\frac{\partial\theta}{\partial h_{c}}\) is the water capacity, indicating changes in the moisture content as the water head changes. \(\beta\) is 0 in an unsaturated area and 1 in a saturated area. \(S_{s}\) is the unit storage factor, it can be seen as 0 in most instances. In a stable seepage field, Equation (4) can be rewritten as  
\[\frac{\partial}{\partial x_{i}}[k_{ij}^{s}k_{r}(h_{c})\frac{\partial h_{c}}{ \partial x_{j}}+k_{ij}^{s}k_{r}(h_{c})]+Q=0 \tag{5}\]  
There are two common models for obtaining \(k_{r}\), the van Genuchten model and Brooks-Corey model. The van Genuchten model is as follows:  
\[k_{r}(\theta_{w})=\theta_{w}{}^{1/2}[1-(1-\theta_{w}\frac{1}{n})^{m}]^{2} \tag{6}\]  
where \(\theta_{w}=\frac{\theta_{w}-\theta_{r}}{\theta_{s}-\theta_{r}}\) is effective saturation, \(\theta_{r}\) is the saturated moisture content, \(\theta_{s}\) is the remaining moisture content, and \(\theta\) is the moisture content. \[\theta_{w}=[\frac{1}{1+  
The Brooks-Corey model is as follows:  
\[\theta_{w}=\left(\frac{P_{d}}{P_{c}}\right)^{-\lambda} \tag{9}\]  
where \(P_{d}\) is the initial pressure, \(P_{c}\) is the pore water pressure, and \(\lambda\) is the shape parameter. **3. Inversion Analysis Principle**  
_3.1. BP Network_  
_3.1.1. Forward Propagation  
In this step, raw input data can be converted to output data in hidden layers. The theory is shown in Figure 1.  
Each input datum is multiplied by the weight and the activation function to obtain the hidden layer data. The sigmoid function is a common activation function, which is as follows:  
\[S(x)=\frac{1}{1+e^{-x}} \tag{10}\]  
Figure 2 shows the drawing of the sigmoid function; each datum is within 0 to 1. Use some matrix to represent the content of Figure 1, and define the input matrix \(X\), hidden layer1 matrix \(H1\), hidden layer2 matrix \(H2\), and output matrix \(Y\) as  
\[X=\left(in1\quad in2\quad in3\quad\cdots\quad in8\right) \tag{11}\]  
\[v=\frac{K(\theta)\gamma}{\mu}J=K\frac{K_{r}(\theta)\gamma}{\mu}J \tag{2}\]0  
\[v=\frac{K(\theta)\gamma}{\mu}J=K\frac{K_{r}(\theta)\gamma}{\mu}J \tag{2}\]1  
\[v=\frac{K(\theta)\gamma}{\mu}J=K\frac{K_{r}(\theta)\gamma}{\mu}J \tag{2}\]2  
\[v=\frac{K(\theta)\gamma}{\mu}J=K\frac{K_{r}(\theta)\gamma}{\mu}J \tag{2}\]3 Theory of a BP network. Define the weight matrix and activation function matrix as  
\[U= \begin{pmatrix}u_{11}&u_{12}&u_{13}\\ u_{21}&u_{22}&u_{3}\\ \vdots&\ddots&\vdots\\ u_{81}&\cdots&a_{83}\\ \end{pmatrix}  \tag{15}\]  
\[H_{0}=\mathit{sigmoid}(H1) \tag{16}\]  
\[W= \begin{pmatrix}w_{11}&w_{12}\\ w_{21}&w_{22}\\ \end{pmatrix}