## 2 The Improved Model for Non-Ponding Infiltration  
### The Infiltration Model with the Hypothesis of Uniform Initial Moisture Content  
For non-ponding infiltration, that is, when the rainfall intensity is less than the saturated permeability coefficient of soil, which means an unsaturated infiltration process, all rainwater will keep into soil and will not accumulate on the ground. The whole process can be divided into the following two stages: the transitional stage and stable stage. During the initial transition stage, the moist area formed in soil expands continuously, and the surface moisture content and moisture depth increase simultaneously, while the soil moisture content decreases along the depth direction. Based on the experimental results, some scholars assumed that the moisture content profile in the unsaturated zone is a \(1/4\) elliptic curve, and achieved a good result [14]. Therefore, this paper also adopted the same hypothesis and considered the profile of soil moisture content in the transitional stage as elliptical, supposing the surface moisture content of soil is \(\theta_{i}\), the initial moisture content is \(\theta_{0}\), as shown in Figure 1.  
If the the rainfall intensity is \(q\), the infiltration rate is \(i\), then for a one-dimensional infiltration situation perpendicular to the ground, the infiltration process can be expressed using the Green-Ampt model:  
\[i=K(\theta)\Bigg{(}\frac{S_{(\theta_{0})}-S_{(\theta_{i})}}{z_{f}}+1\Bigg{)} \tag{1}\]  
where \(K(\theta)\) is the unsaturated permeability coefficient corresponding to moisture content \(\theta\), \(z_{f}\) is the wetting front depth and \(S_{(\theta_{i})}\) and \(S_{(\theta_{0})}\) are the matric suction at the soil surface and wetting front, respectively. It should be noted that [PERSON] et al. [3] believed that even without ponding on the ground, the moisture content still reaches saturation, and proposed the following formula to calculate the wetting front depth:  
\[z_{f}=\frac{qT}{\theta_{s}-\theta_{0}} \tag{2}\]  
where \(\theta_{s}\) is saturated moisture content, \(T\) is the rainfall time. However, some scholars [9] found that the soil moisture content could not reach saturation during unsaturated infiltration, so it could not be directly applied to the infiltration calculation. Meanwhile, the unsaturated permeability coefficient \(K(\theta)\) in Equation (1) generally corresponds to the state of a stable and uniform moisture content. However, the moisture content is not uniform at the initial stage, so it cannot be expressed by the conventional permeability coefficient. [PERSON] et al. [15] tried to use the permeability coefficient that corresponds to the average moisture content in the moist zone to represent it, but the result was not accurate. Therefore, the equivalent permeability coefficient in the moist zone was adopted to represent the permeability coefficient [17], which can be expressed as  
\[\widetilde{K}=\frac{K(\theta_{i})-K(\theta_{0})}{\ln K(\theta_{i})-\ln K( \theta_{0})} \tag{3}\]  
Meanwhile, the Van Genuchten model [18] was introduced to represent the soil matrix potential and unsaturated permeability coefficient corresponding to moisture content, as shown in Equations (4) and (5). \[K(\theta)=K_{s}\left(\frac{\theta-\theta_{r}}{\theta_{s}-\theta_{r}}\right)^{ 1/2}\left\{1-\left[1-\left(\frac{\theta-\theta_{r}}{\theta_{s}-\theta_{r}} \right)^{\frac{n}{n-1}}\right]^{\frac{n-1}{n}}\right\}^{2} \tag{4}\]  
\[S(\theta)=\frac{1}{\alpha}\left[\left(\frac{\theta-\theta_{r}}{\theta_{s}- \theta_{r}}\right)^{\frac{n}{1-n}}-1\right]^{\frac{1}{2}} \tag{5}\]  
where \(\alpha\) and \(n\) are constants and \(\theta_{r}\) is the residual moisture content. Is the rainwater is fully infiltrated, the infiltration rate \(i\) equals the rainfall intensity \(q\).