## 2 Materials and Methods
### Methods  
If instantaneous RSL/TSL measurements are available (as in the Active approach), the total attenuation (in dB) of the CML's signal can be extracted by subtracting one from the other, and it can be described by [10]:  
\[A_{T}(t,l)=A_{0}(l)+A_{P}(t,l)+A_{H}(t,l)+A_{C}(t)+N(t) \tag{2}\]  
where \(t\) is the time index and, as in Equation (1), \(l\) is the length of the link. \(A_{0}\) is the frequency-dependent propagation loss which is constant over time. \(A_{P}\) is the attenuation caused by precipitation, if existing. For rain, for example, \(A_{P}=A\) of Equation (1). \(A_{H}\) is the attenuation due to hydrometeors other than precipitation, e.g., fog or water vapor. This component changes slowly over time, while being small compared with \(A_{P}\), if existing. \(N(t)\) represents the measurements noise,  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Characteristics**} & **Passive** & **Active** \\ \hline \multirow{2}{*}{Source of measurements} & Existing records from network & \\  & management systems (NMS) & \\ \hline \multirow{2}{*}{Temporal resolution} & Minutes-days & Seconds-minutes \\  & Typical-15 min & Typical-10 s \\ \hline \multirow{2}{*}{Non-linear preprocessing} & Typically min/max values over & \\  & a given interval & Non \\ \hline \multirow{2}{*}{Quantization} & Yes & Yes \\ \hline \multirow{2}{*}{Major advantage} & Simple access, no risk for cellular & \\  & operators & Real time \\ \hline \multirow{2}{*}{Major disadvantage} & Not available in real time & Hard to get \\ \hline \multirow{2}{*}{Summary} & Recommended for research & \\  & purposes and for historic studies & Essential for real time applications \\ \hline \hline \end{tabular}
\end{table}
Table 1: Available commercial microwave link (CML) measurements.  
and \(A_{C}(t)\) is a component which represents the disruption of the signal by other objects, if existing. Note that Equation (2) is a simplified description of the heavily studied physics of the attenuation of a signal propagating in the atmosphere [11; 12], which has a great effect on the performance of communication systems [13].  
Depending on the application, the first stage is to isolate \(A_{P}(t,l)\) (for precipitation monitoring), or \(A_{H}(t,l)\) (for monitoring other-than-rain atmospheric phenomena). This is commonly done by using side information, or by taking advantage of the built-in diversity of CML technology, as there are usually multiple CMLs in a given area of different lengths and frequencies (e.g., Reference [14]). The next step is to estimate the parameter of interest (e.g., rain-rate) from the corresponding term, using the known relation between the signal attenuation and the phenomenon of interest. A flowchart of this possible process is presented in Figure 2.  
Obviously, measurements collected by the passive approach are far from being _ready to use_ for this analysis. The instantaneous total attenuation \(A_{T}(t,l)\) required by Equation (2) cannot be extracted from the \(\min/\max\) indicators of the TSL/RSL. A systematic approach, for example, for extracting rain-rate estimates from extreme measurements provided using the passive approach is described in Reference [15] and is depicted in Figure 3.