## 2 Principles and Model Establishment
### The Precipitation Spatial Distribution  
The precipitation spatial precipitation rate is simplified as the factorization of the vertical distribution variation and its horizontal variation [9]:  
\[R(x,z)=I(x)\cdot V(z), \tag{2}\]  
where \(R(x,z)\) is the precipitation rate in mm/h, \(V(z)\) is the vertical distribution of the precipitation and \(I(x)\) is the horizontal variation of \(V(z)\). The along-track resolution \(\cdot y\) is estimated to be 264 m [10]. This degraded along-track resolution is adequate for hydrological purposes. This paper will focus on the solution to the horizontal variation \(I(x)\). Thus, the vertical distribution \(V(z)\) will be considered uniform for simplicity:  
\[V(z)=V(0),\ \ 0\leq z\leq z_{t}, \tag{3}\]  
where \(V(0)\) is the near-surface precipitation rate in mm/h. \(I(x)\) representing a single rain-cloud cell can be expressed as:  
\[I(x)=\left\{\begin{array}{rl}0,\ \ 0\leq x<z_{t}/\tan\theta.\\ x/d-z_{t}/(\text{dtan}\theta),\ z_{t}/\tan\theta\leq x<d+z_{t}/\tan\theta.\\ 1,\ d+z_{t}/\tan\theta\leq x<w-d+z_{t}/\tan\theta.\\ (w-x)/d+z_{t}/(\text{dtan}\theta),\ w-d+z_{t}/\tan\theta\leq x<w+z_{t}/\tan \theta.\\ 0,\ x\geq w+z_{t}/\tan\theta.\end{array}\right. \tag{4}\]  
where \(d\) is the shape parameter of the horizontal variation of the precipitation, \(w\) is the width of the precipitation area in km.  
Figure 1: Simplified cross sectional model of precipitation. The \(x\)-axis is the cross-track direction, the \(y\)-axis is the along-track direction, the \(z\)-axis is the altitude direction, the \(l\)-direction is the propagation direction of microwaves emitted by the radar, and the \(t\)-direction is the direction of the reflected microwaves, perpendicular to the \(l\)-direction. \(z_{t}\) is the height of the rain-cloud cell in km and \(z_{0}\) is the boundary height of the rainfall and snowfall in km. \(\theta\) is the slant off-nadir angle. The microwave pulses emitted by the radar are considered as plane wave front slices, shown as a pair of lines with a width of \(r_{R}\). \(\delta\) is the \(x\)-coordinate of the point on the path of the backscattering echo caused by ground. \(\xi\) is the \(x\)-coordinate of the point on the path of the backscattering echo caused by hydrometeors. \(X\) is the \(x\)-coordinate of the point on the microwave front slices. Rectangular rain-cloud cell is chosen just for simplicity and more shapes will be used in the simulation section.  
In this paper, three common cases for the horizontal variation with a single rain-cloud cell are used. For \(d=0\):  
\[I(x)=\begin{cases}0,\ \ 0\leq x<z_{t}/\tan\theta.\\ 1,\ \ z_{t}/\tan\theta\leq x<w+z_{t}/\tan\theta.\\ 0,\ \ x\geq w+z_{t}/\tan\theta.\\ \end{cases} \tag{5}\]  
representing the rectangular variation.  
For \(d=w/2\):  
\[I(x)=\begin{cases}0,\ \ 0\leq x<z_{t}/\tan\theta.\\ (x-z_{t}tan\theta)/(w/2),\ \ z_{t}/\tan\theta\leq x<w/2+z_{t}/\tan\theta.\\ (w-x+z_{t}tan\theta)/(w/2),\ \ w/2+z_{t}/\tan\theta\leq x<w+z_{t}/\tan\theta.\\ 0,\ \ x\geq w+z_{t}/\tan\theta.\\ \end{cases} \tag{6}\]  
representing the triangular variation.  
For \(0<d<w/2\), \(I(x)\) represents the trapezoidal variation.  
To specify, \(I(x)\) representing two or more rain-cloud cells can be realized by superimposing two or more single rain-cloud cells accordingly. In this paper, the rectangular variation with two separate rain-cloud cells is also considered in the simulation and it can be expressed as:  
\[I(x)=\begin{cases}0,\ \ 0\leq x<z_{t}/\tan\theta\\ 1,\ \ z_{t}/\tan\theta\leq x<d+z_{t}/\tan\theta\\ 0,\ \ \ d+z_{t}/\tan\theta\leq x<w-d+z_{t}/\tan\theta\\ 1,\ \ w-d+z_{t}/\tan\theta\leq x<w+z_{t}/\tan\theta\\ 0,\ \ \ x\geq w+z_{t}/\tan\theta\\ \end{cases}. \tag{7}\]