# Rainfall Estimation from Tropospheric Attenuation Affecting Satellite Links
## 6 Performance Evaluation
### Comparison with Other Rain Rate Estimation Models: Brief Description
#### 6.1.2 Centralized Method for Real-Time Rainfall Estimation  
The model in [8] proposes a real-time and centralized rain rate estimation technique that employs Carrier-to-Noise (\(C/N\)) power ratio measurements every five minutes (exploiting both satellite forward  
Figure 15: Trend of the equivalent rain height for the events represented in Figures 11 and 13.  
and return-link data), fed back from a network of Ka-band satellite ground terminals to a central gateway, and successively stored in a database. The collected data are then processed while using an Artificial Neural Network (ANN) to detect rain events, and successively rain attenuation is extracted. The authors propose two different solutions to estimate rainfall rate from the inferred rain attenuation: the former takes the Earth-satellite link geometry into account and makes use of the equivalent rain height; the latter is instead a completely empirical relationship. The validity of both approaches is verified while using on-site rain gauges, which report the average rainfall intensity every minute.  
The lower case \(n\) denotes the \(n^{th}\) discrete-time sample of the quantities described below. The estimation procedure can be summarized, as follows:  
* estimate the baseline \(C/N\) level in dry conditions, namely \(\overline{C}(n)\);
* detect the presence of a rain event using the ANN described in [8]; and,
* once a rain event is detected, determine the corresponding rain attenuation \(A_{R}(n)\) as the decrease in the measured \(C/N\) (during the rain event) with respect to the last value of \(\overline{C}(n)\) before the beginning of precipitation;
* estimate the rainfall rate from \(A_{R}(n)\) as: \[R(n)=\frac{1}{k}\bigg{(}\frac{A_{R}(n)\sin\theta}{h_{R}-h_{s}}\bigg{)}^{1/ \alpha}\;[\text{mm}/\text{h}],\] (21) where the values of \(k\) and \(\alpha\) can be found in [18], \(\theta\) is the elevation angle of the Earth-satellite link, \(h_{s}\) is the altitude of the ground terminal above mean sea level, and \(h_{R}\) is the mean annual equivalent rain height, expressed as a function of the mean annual zero-degree isotherm height of the site \(h_{0}\): \[h_{R}=h_{0}+0.36\;[\text{km}].\] (22)  
An alternative solution proposed as part of the model is to build an empirical relationship between the observed rain attenuation and rainfall rate, exploiting the training data that were fed to the ANN. The result is an empirical power law of the following kind:  
\[R(n)=\ a(A_{R}(n))^{b}\;[\text{mm}/\text{h}], \tag{23}\]  
where the coefficients \(a\) and \(b\) are obtained by curve-fitting the used training set to the rain rate measurements produced by the rain gauge.  
1.3 Model Developed by the Department of Broadband Infocommunications and Electromagnetic Theory at Budapest University of Technology and Economics (BME-HVT)  
The model [10] allows for estimating rain intensity from rain attenuation affecting satellite links. No algorithm is provided for retrieving rain attenuation from the received downlink signal: as a matter of fact, the authors have obtained \(A_{R}\) by manually identifying the presence of rain events along the satellite link, a process that requires the visual inspection of every received power level time series.  
Once rain attenuation is derived, rainfall rate can be estimated by applying the following procedure:  
1. Compute the effective path length \(L_{E}\) of the Earth-satellite link, as described in the Recommendation ITU-R P.618 [19];
2. Obtain the values of the power-law coefficients \(k\) and \(\alpha\), which are available in [18];
3. Estimate rain rate as: \[R=10^{[\log_{10}{(A_{R}/L_{E})}-\log_{10}k]/\alpha}\;[\text{mm}/\text{h}].\] (24)