## Appendix B  
This appendix is the mathematical details on VIE analytic formulations.  
Using the function \(p(x)\) defined in Section 3, the extinction coefficients of rain, the radar reflectivity of rain and snow, together with the precipitation rate can also be expressed as:  
\[k_{r}(x)=\frac{\sin\theta}{2}\frac{p_{x}{}^{\prime}(x)}{p(x)},\] (A40)  
\[\eta_{r}(x)=a_{1r}\left[\frac{\sin\theta}{2a_{r}}\frac{p_{x}{}^{\prime}(x)}{p( x)}\right]^{b_{1r}},\] (A41)  
\[\eta_{s}(x)=a_{1s}\left[\frac{\sin\theta}{2a_{r}}\frac{p_{x}{}^{\prime}(x)}{p( x)}\right]^{b_{1s}},\] (A42)  
\[I(x)=\frac{\sin\theta}{2a_{r}}\frac{p_{x}{}^{\prime}(x)}{p(x)},\] (A43)  
where \(p_{x}{}^{\prime}(x)\) is the derivative of the function \(p(x)\).  
\[K_{3}(X) = K_{1}(X)\cdot p[x+(z_{t}-z_{0})\tan\theta]^{\frac{\theta_{x}}{\theta}-1} \cdot p(x+z_{t}\tan\theta), \tag{100}\] \[K_{4}(X) = K_{2}(X)\cdot p[x+(z_{t}-z_{0})\tan\theta]^{\frac{\theta_{x}}{ \theta}-1}\cdot p(x+z_{t}\tan\theta),\] (101) \[\lambda_{1} = \frac{G_{1}\cos^{2}\theta}{\sigma_{0}},\] (102) \[\lambda_{2} = \frac{G_{2}\cos^{2}\theta}{\sigma_{0}}. \tag{103}\]  
The iteration processes are explicitly shown as follows.  
For \(x\geq x_{m}\), we have  
\[p(x)=1. \tag{104}\]  
For \(x\geq x_{m}-(z_{t}-z_{0})\tan\theta\), we have  
\[I(x+(z_{t}-z_{0})\tan\theta)=0,p(x+(z_{t}-z_{0})\tan\theta)=1. \tag{105}\]  
We can obtain:  
\[p(x)=\left[\frac{\sigma_{SAR}(x+z_{t}\tan\theta)}{\sigma_{0}}\right]^{\frac{ \theta_{x}}{\theta_{x}}}. \tag{106}\]  
For \(x\geq x_{m}-z_{t}\tan\theta\), we have \(I(x+z_{t}\tan\theta)=0,p(x+z_{t}\tan\theta)=1\). Assume that \(z_{t}\geq 2z_{0}\), then we have:  
\[p[x+(z_{t}-z_{0})\tan\theta]=\left[\frac{\sigma_{SAR}(x+2z_{t}\tan\theta-z_{0} \tan\theta)}{\sigma_{0}}\right]^{\frac{\theta_{x}}{\theta_{x}}}. \tag{107}\]  
We can obtain:  
\[p(x)=\left\{\frac{\sigma_{SAR}(x+z_{t}\tan\theta)}{\sigma_{0}}\left[\frac{ \sigma_{SAR}(x+2z_{t}\tan\theta-z_{0}\tan\theta)}{\sigma_{0}}\right]^{1-\frac {\theta_{x}}{\theta_{x}}}\right\}^{\frac{\theta_{x}}{\theta_{x}}}. \tag{108}\]  
Continuing such iteration, \(p(x)\) within the range of interest can be obtained. Furthermore, through Equation (104), we can then determine \(I(x)\) within range of interest.  
According to the geometric characteristics of rectangles, triangles and trapezoids, once we know the precipitation rate, we can obtain the precipitation distribution. A rough method to distinguish the horizontal variation can be used here [11]. We first calculate two parameters:  
\[dI_{1}(x)=\frac{I(x+\Delta x)-I(x)}{\Delta x}, \tag{109}\]  
\[dI_{2}(x)=\frac{I(x+2\cdot\Delta x)-I(x+\Delta x)}{\Delta x}. \tag{110}\]  
where \(\Delta x\) is a small increment of \(x\), \(dI_{1}(x)\) and \(dI_{2}(x)\) are the slopes at two successive points, respectively.  
When the slope is considerably large, taking the typical value 30 as an example:  
\[dI_{1}(x)>30, \tag{111}\]  
the horizontal variation can be seen as rectangular. The \(x\)-coordinate of the left point of the precipitation area \(\hat{x}_{L}=x\). Similarly, the \(x\)-coordinate of the right point of the precipitation area \(\hat{x}_{R}\) is the point at which \(dI_{1}(x)\) is considerably small. The width of the precipitation area \(\hat{w}\) can be consequently calculated through:  
\[\hat{w}=\hat{x}_{R}-\hat{x}_{L}. \tag{112}\]  
When  
\[dI_{1}(x)\geq 2\cdot dI_{2}(x), \tag{113}\]\[dI_{2}(x)\geq 0, \tag{11}\]  
the horizontal variation can be seen as trapezoidal.  
When  
\[dI_{1}(x)\geq 2\cdot dI_{2}(x), \tag{12}\]  
\[dI_{2}(x)\leq 0, \tag{13}\]  
the horizontal variation can be seen as triangular. For trapezoidal rain-cloud cells and triangular rain-cloud cells, the \(x\)-coordinate of the left edge of the precipitation area \(\hat{x}_{L}\), the \(x\)-coordinate of the right edge of the precipitation area \(\hat{x}_{R}\) and the width of the precipitation area \(\hat{w}\) can be calculated similar to those of the rectangular rain-cloud cells.