## 2 Principles of Rainfall Field Retrieval by ESLs and HMLs
### Principle of Rainfall Retrieval by HML  
The principle of using HML for rainfall inversion is similar to that of ESL. It also uses the physical phenomenon that rainfall can cause attenuation of microwave signals. However, because the line-of-sight HML composed of microwave transmitter and receiver propagates near the ground, the change of meteorological elements in its path range is small. Additionally, the existing calculation models all assume that the meteorological elements at the height of HML are constant. Therefore, using HML to retrieve rainfall is  
Figure 2: Correspondence between ESL signal and rain rate from 13 June to 14 June 2020. The black line represents the SNR of ESL, and the blue line represents the rain rate.  
more convenient than using ESL to retrieve rainfall. Next, the process of retrieving rainfall by HML is described in detail [42].  
When the microwave signal sent by the microwave transmitter of HML propagates in the atmosphere near the ground, the total power attenuation \(A_{total}\) on the path can be expressed as  
\[A_{total}=A_{gas}+A_{water}+A_{FS}+A_{min} \tag{14}\]  
where \(A_{gas}\) is the attenuation caused by gases (mainly oxygen) in the atmosphere, \(A_{water}\) is the attenuation due to water vapor, \(A_{FS}\) is the loss of microwave transmission in the path of free space, and \(A_{rain}\) is the attenuation caused by rainfall. The key point in using HML to retrieve rainfall is to record rain attenuation accurately. Therefore, we rewrite Equation (14) as  
\[A_{total}=A_{rain}+A_{baseline}. \tag{15}\]  
The \(A_{baseline}\) represents the attenuation of the signal caused by other factors except rainfall, and it is generally approximated by the attenuation during non-rainfall periods after revision. Thus, as far as the receive signal level (RSL) of HML receiver is concerned, regarding \(A_{baseline}\) as \(\text{RSL}_{baseline}\), the rain attenuation can be given by  
\[A_{rain}=\text{RSL}_{baseline}-\text{RSL}. \tag{16}\]  
To further describe the relationship between rain rate and rain attenuation, a model for calculating HML rain attenuation was fitted based on the Mie scattering theory and a large amount of observational data [43]. As shown in  
\[A_{rain}=aR^{b}.l, \tag{17}\]  
where \(l\) is the length of HML, and the values of coefficients \(a\) and \(b\) are obtained by ITU-R P.838-3 [40]. Revising the coefficients \(a\) and \(b\) based on DSD data from different locations will further improve the accuracy of this calculation model and thus better match the practical situation in different locations. Therefore, according to Equations (16) and (17), the average rain rate retrieved by HML can be written as  
\[R=\left(\frac{\text{RSL}_{baseline}-\text{RSL}}{a.l}\right)^{\frac{1}{b}}. \tag{18}\]  
To visualize the correspondence between the HML signal and the rain rate, we used the HML (26 GHz, vertical polarization) set up in Jiangyin, China, to realistically record the RSL of its receiver during rainfall. Additionally, a rainfall event in June 2021 is selected to validate the correlation between RSL and rain rate as shown in Figure 3. Similar to the ESL signal, the HML signal intensity decreases significantly in the presence of rainfall. This phenomenon becomes more obvious as the rain rate increases. In addition, rainfall is the most powerful factor in the variation of the HML signal. Additionally, the CC between RSL and rain intensity was calculated to be \(-0.762\) by further analysis. The strong correlation indicates that the HML signal is highly sensitive to rain rate and is more suitable for retrieving rainfall.