## Appendix A  
This appendix is the analysis of radar backscattering echo path. The radar backscattering echo path is shown in Figure 11. Let the x-coordinate of the point where the radar beam reaches the ground be x. When the microwaves emitted by the radar have not entered the precipitation area, that is, \(\mathrm{x}\in[-\infty,0]\), there is surface backscattering echoes caused by the ground, without attenuation by the precipitation. The x-coordinates of the left and right limits of the attenuation path of scattering echo along l-direction and the left and right limits of the attenuation path of reflecting along t-direction caused by meteorological particles are:  
\[x_{s_{1}}=0, \tag{1}\]  
\[x_{s_{2}}=0, \tag{2}\]  
\[x_{v_{1}}=0, \tag{3}\]  
\[x_{v_{2}}=0. \tag{4}\]  
The surface backscattering cross section and the volume backscattering cross section can be expressed as:  
\[\sigma_{sf}(x)=\sigma_{0}, \tag{5}\]  
\[\sigma_{vol}(x)=0. \tag{6}\]  
When the reflected microwaves begin to enter the snowfall area, that is, \(x\in[0,x_{1}]\), the surface backscattering echoes caused by the ground are still unattenuated:  
\[\sigma_{sf}(x)=\sigma_{0}. \tag{7}\]  
However, the volume backscattering echoes caused by snowfall appear:  
\[x_{v_{1}}=x_{L}, \tag{8}\]  
\[x_{v_{2}}=x+z_{t}/\tan\theta, \tag{9}\]  
\[x_{s_{1}}=0, \tag{1}\]0\[x_{s_{1}}=0, \tag{1}\]1 (A11)  
Since the volume backscattering echoes are only attenuated by the snowfall, the extinction coefficient is \(k_{s}\) and the radar reflectivity is \(\eta_{s}\). The volume backscattering cross section can be expressed as:  
\[x_{s_{1}}=0, \tag{1}\]2 (A12)  
For \(x\in[x_{1},x_{1}]\), the surface backscattering echoes caused by the ground are still unattenuated, there are snowfall backscattering echoes, and the rainfall backscattering echoes start to occur. The Equations (A8)-(A11) still hold, but the extinction coefficient k and the radar reflectivity \(\eta\) differ:  
\[k(\xi)=\left\{ \begin{array}{l}k_{r}(\xi),\,\xi\in[x_{v_{3}},X-[z_{0}-(X-x) \tan\theta]\tan\theta]\\ k_{s(\xi)},\xi\in[X-[z_{0}-(X-x)\tan\theta]\tan\theta,x_{v_{4}}]\end{array}  \right.,\] (A13)  
\[\eta(X)=\left\{ \begin{array}{l}\eta_{r}(X),X\in[x_{v_{1}},x+z_{0}/\tan \theta]\\ \eta_{s}(X),X\in[x+z_{0}/\tan\theta,x_{v_{2}}]\end{array} \right..\] (A14)  
The volume backscattering cross section can be expressed as:  
\[\sigma_{vol}(x)=\tan\theta\int_{x_{v_{2}}/\tan\theta}^{x_{v_{2}}}exp\left(- \frac{2}{\sin\theta}\int_{x_{v_{3}}}^{x_{v_{2}}}k_{s}(\xi)d\xi\right)\eta_{s} (X)dX+\] (A15)  
When the microwaves emitted by the radar begin to enter the snowfall area, that is, \(x\in[x_{L},x_{R}]\), the surface backscattering echoes caused by the ground begin to be attenuated, and there are precipitation backscattering echoes:  
\[x_{s_{1}}=max\{x_{L},x-z_{t}\tan\theta\},\] (A16)  
\[x_{s_{2}}=x.\] (A17)  
\[x_{v_{1}}=x,\] (A18)  
\[x_{v_{2}}=min\{x_{R},x+z_{t}/\tan\theta\}.\] (A19)  
\[x_{v_{3}}=max\{x_{L},X-[z_{t}-(X-x)\tan\theta]\tan\theta\},\] (A20)  
\[x_{v_{4}}=X.\] (A21)  
The surface backscattering cross section can be expressed as:  
\[\sigma_{srf}(x)=\sigma_{0}\cdot exp\left[-2/\sin\theta\left(\int_{x_{s_{1}}}^ {x-z_{0}\tan\theta}k_{s}(\delta)d\delta+\int_{x-z_{0}\tan\theta}^{x_{v_{2}}}k _{r}(\delta)d\delta\right)\right].\] (A22)  
The volume backscattering cross section can be expressed by Equation (A15).