## 1 Introduction
### The Principle and Development of Rainfall Estimation by CMLs  
CMLs typically provide line-of-sight wireless backhaul connectivity for cellular communication networks, as well as serving as infrastructure for services such as smart cities. As shown in Figure 1, CMLs transmit microwave signals which carry data information between two cell towers to achieve message communication at two locations. Currently, the operating frequencies of CMLs mainly range from 15 to 40 GHz and are extending to the E-band due to the demand for higher bandwidth. In these frequencies, where the wavelengths are comparable to the size of the raindrop particles, the microwave signals are significantly attenuated by scattering and absorption. Considering the rainfall at location \(x\) on the path, the specific rain-induced attenuation \(\gamma(x)\) (dB\(\cdot\)km\({}^{-1}\)) at that location can be calculated as:  
\[\gamma(x)=4.343\times 10^{3}\int_{D}Q_{ext}(D,f,T)\cdot N_{x}(D)dD \tag{1}\]  
where \(Q_{ext}(D,f,T)\) (m\({}^{2}\)) is the extinction cross section of a raindrop particle of size \(D\) (mm), which is related to the signal frequency \(f\) (GHz) and raindrop temperature \(T\) (K). \(N_{x}(D)\) (m\({}^{-3}\cdot\)mm\({}^{-1}\)), i.e., the DSD function, is the number concentration of raindrop particles of size \(D\) at location \(x\). Meanwhile, according to the definition of rain rate, the rain rate \(R(x)\) (mm\(\cdot\)h\({}^{-1}\)) at this location can be expressed as:  
\[R(x)=6\times 10^{-4}\pi\int_{D}V(D)N_{x}(D)D^{3}dD \tag{2}\]  
where \(V(D)\) (m\(\cdot\)s\({}^{-1}\)) is the falling velocity of a raindrop of size \(D\). Since both \(\gamma(x)\) and \(R(x)\) are functions of \(N_{x}(D)\), the relationship between the two can be deduced. The theoretical calculation results show that the relationship between \(\gamma\) and \(R\) approximates a power-law form [17]:  
\[\gamma(x)=aR(x)^{b} \tag{3}\]  
where \(a\) (mm\({}^{-b}\cdot\)h\({}^{b}\cdot\)dB\(\cdot\)km\({}^{-1}\)) and \(b\) (dimensionless) are the power-law parameters, which are associated with frequency, polarization and, to a lesser extent, with DSD [18; 19] (compared to the \(Z\)-\(R\) relationship). The total attenuation \(A\) (dB) of a CML of length \(L\) (km) due to rainfall on the path can be obtained by integrating \(\gamma(x)\) over the path:  
\[A=\int_{0}^{L}\gamma(x)dx=a\int_{0}^{L}R(x)^{b}dx \tag{4}\]Then, the specific rain-induced attenuation over the entire CML path can be expressed as:  
\[\gamma=\frac{A}{L}=\frac{a}{L}\int_{0}^{L}R(x)^{b}dx \tag{5}\]  
Substitute Equation (3) into (5) to obtain an estimate \(\hat{R}\) of the path-averaged rain rate, with the real path-averaged rain rate \(\overline{R}\) shown by Equation (7):  
\[\hat{R}=\left(\frac{\int_{0}^{L}R(x)^{b}dx}{L}\right)^{1/b} \tag{6}\]  
\[\overline{R}=\frac{1}{L}\int_{0}^{L}R(x)dx \tag{7}\]  
Figure 2 presents the scatter plot of CML rainfall estimates versus real rain rates at different frequencies and rainfall types. It can be seen that the deviation between the CML rain rate and the real rain rate is much larger for frequencies outside of 22 to 38 GHz. Comparing the path-averaged rain rate, \(\overline{R}\), with \(\hat{R}\), it is straightforward to realize that the two are equal only if \(b\) approximates unity or \(R(x)\) is nearly constant. This means that taking into account the spatial variability of rainfall, only CMLs operating within 20 to 40 GHz are suitable for rainfall estimation. However, in the last century, the low frequency of prevalent CMLs was not applicable for monitoring rainfall. This is reflected in the insensitivity of the microwave signal to rainfall on the one hand, and due to the inherent error between the rainfall estimate \(\hat{R}\) and the path-averaged rain rate \(\overline{R}\) on the other. In addition, it is too expensive to deploy a dedicated microwave link rainfall monitoring network compared to radar-based regional measurements. As a result, studies 20 years ago were mainly limited to dedicated experimental microwave links.