## 3 Volterra Integral Equation (VIE) of the Second Kind Inversion Method  
Since the extinction coefficient \(k\) and the radar reflectivity \(\eta\) will be zero, according to Equations (12) and (16), if the precipitation rate is zero, we can easily integrate the numerous formulas in Appendix A into one:  
\[\begin{array}{l}\sigma_{SAR}(x)=\sigma_{0}exp\left[-\frac{2}{\sin\theta} \left(\int_{x-z_{l}\tan\theta}^{x-z_{0}\tan\theta}k_{s}(\delta)d\delta+\int_{ x-z_{0}\tan\theta}^{x}k_{r}(\delta)d\delta\right)\right]+\\ \tan\theta\int_{x+z_{0}/\tan\theta}^{x+z_{0}/\tan\theta}exp\left(-\frac{2}{ \sin\theta}\int_{X-[z_{l}-(X-x)\tan\theta]\tan\theta}^{x}k_{s}(\delta)d\xi \right)\eta_{s}(X)dX+\\ \tan\theta\int_{x}^{x+z_{0}/\tan\theta}exp\left[-\frac{2}{\sin\theta}\left( \int_{X-[z_{l}-(X-x)\tan\theta]\tan\theta}^{X-[z_{0}-(X-x)\tan\theta]\tan \theta}k_{s}(\xi)d\xi\right)\right]\eta_{r}(X)dX.\end{array} \tag{18}\]  
This nonlinear formula, however, is not easy to deal with. To simplify this formula, we can introduce a function:  
\[p(x)=exp\left(\frac{2}{\sin\theta}\int_{0}^{x}k_{r}(m)dm\right). \tag{19}\]  
Using \(p(x)\), the extinction coefficients of rain, the radar reflectivity of rain and snow, together with the precipitation rate can also be expressed, as is derived in Appendix B. After the substitution and transformation, also shown in Appendix B, we can obtain:  
\[\begin{array}{l}p(x)^{\frac{\delta_{t}}{\pi}}=f_{1}(x)-\lambda_{1}\int_{x+z _{l}/\tan\theta+z_{t}\tan\theta}^{x+z_{t}/\tan\theta}K_{3}(X)p(t)^{\frac{ \delta_{t}}{\pi}}dt\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad-\lambda_{2}\int_{x}^{x+z_{0}/\tan \theta+z_{0}\tan\theta}K_{4}(X)p(t)^{\frac{\delta_{t}}{\pi}}dt,\end{array} \tag{20}\]  
where  
\[f_{1}(x)=\frac{\sigma_{SAR}(x+z_{t}\tan\theta)\cdot p(x+z_{t}\tan\theta)\cdot p [x+(z_{t}-z_{0})\tan\theta]^{\frac{\delta_{t}}{\pi}-1}}{\sigma_{0}}, \tag{21}\]  
\[K_{3}(X)=K_{1}(X)\cdot p[x+(z_{t}-z_{0})\tan\theta]^{\frac{\delta_{t}}{\pi}-1} \cdot p(x+z_{t}\tan\theta), \tag{22}\]  
\[K_{4}(X)=K_{2}(X)\cdot p[x+(z_{t}-z_{0})\tan\theta]^{\frac{\delta_{t}}{\pi}-1} \cdot p(x+z_{t}\tan\theta), \tag{23}\]  
\[\lambda_{1}=\frac{G_{1}\cos^{2}\theta}{\sigma_{0}}, \tag{24}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Frequency (GHz)** & **rms Error of (17) (Percent)** & **rms Error of (16) (Percent)** \\ \hline
7.5 & 28 & 31 \\
9.4 & 30 & 36 \\
16.0 & 22 & 28 \\
34.9 & 10 & 12 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The root mean square(rms) error of the linear model in Equation (16) and the logarithmic model in (17) [15].  
\[\lambda_{2}=\frac{G_{2}\cos^{2}\theta}{\sigma_{0}}. \tag{25}\]  
Assume that for \(x\geq x_{m}\), there is no precipitation; that is \(I(x)=0\). According to Equations (16) and (19), for \(I(x)=0\), we can always obtain \(k(x)=0\) and \(p(x)=1\). Then, we can obtain the values of \(K_{3}(\mathrm{X})\), \(K_{4}(\mathrm{X})\), \(K_{1}(\mathrm{X})\), \(K_{2}(\mathrm{X})\) and \(f_{1}(x)\) for \(x_{m}-(z_{t}-z_{0})\tan\theta\leq x\leq x_{m}\). Furthermore, naturally, we can obtain \(p(x)\) for \(x_{m}-(z_{t}-z_{0})\tan\theta\leq x\leq x_{m}\). Furthermore, we can then obtain \(p(x)\) for \(x_{m}-(n+1)(z_{t}-z_{0})\tan\theta\leq x\leq x_{m}-n(z_{t}-z_{0})\tan\theta\), where n is a non-zero natural number. Through such iterations, we can obtain \(p(x)\) under ranges of interest. The iteration processes are explicitly shown in Appendix B.