# Real-Time Rain Rate Evaluation via Satellite Downlink Signal Attenuation Measurement
## 4 Direct Method for the Estimation of the Rain Rate
### Experimental Derivation of the Analytical Regression Formula  
More than 6 years of DSDs collected each minute by using a Parsivel disdrometer made by OTT Hydromet Deutschland (85,207 samples) in the area of Rome (the disdrometer is installed on the roof of the Institute of Atmospheric Sciences and Climate of the National Research Council, CNR) have been used as input of the T-matrix electromagnetic simulation method to compute the specific attenuation, according to the integral in (1), at different frequencies and for different polarizations. The used dataset is one of the longest time-series of measured DSD in Italy and include a huge amount of different types of precipitation events. The choice of using relationships derived by DSDs, instead of others available from the literature (in particular, those obtainable from the ITU-R P.838-3 recommendation [29]), is supposed to produce \(R\)-\(k\) relations that are more representative of the climate of the area in which the  
Figure 7: Daily height of the 0 \({}^{\circ}\)C isotherm over the city of Florence, Italy, obtained by averaging 35-year observation data from the modern-era retrospective analysis for research and applications (MERRA) database [48]. -disdrometer measurements are collected. Since such a database is unique in Italy, it is not currently possible to quantify at what extent such relationships are representative for different regions. However, we can reasonably assume that they are valid at least for the climatic region in which Rome is included, which actually include almost all the regions of West Italy, like Tuscany (49; 50). The following assumptions have been considered in the T-matrix method: The environmental temperature of 20 \({}^{\circ}\)C (51); fall velocity has been applied to compute \(N(D)\); drop shapes have been modelled with the (52) relationship; and the variability of the canting angle has been modelled with a Gaussian function with zero mean and 10\({}^{\circ}\) standard deviation. Furthermore, in the simulation, an elevation angle of 40\({}^{\circ}\) was assumed. The values of the rain rate can be straightforwardly obtained from measured DSDs. Once the values of \(k\) (in dB/km) and \(R\) (in mm/h) are known for a given DSD, the coefficients \(\alpha\) and \(\beta\) of the following relation have been determined by performing a nonlinear regression  
\[R=\alpha\ k^{\beta}. \tag{5}\]  
Figure 8 shows the regression fits between \(R\) and \(k\) for the forward link (i.e., 11,345.6 MHz with vertical polarization) and the return link (i.e., 14,216.6 MHz with horizontal polarization), both in Ku-band, but with different polarizations. Finally, Table 2 shows the values of the obtained parameters and those relative to the statistics computed, for both the polarizations, between disdrometer-based rain rate and the rain rate estimates through the \(R\)-\(k\) algorithms, namely the normalized mean absolute error (NMAE), normalized bias (NB), root mean square error (RMSE), and correlation coefficient (CC). NB is an index of the systematic error: Negative values means that the \(R\)-\(k\) relation slightly underestimates the rainfall rate with respect to disdrometer measured values. This statement is particularly valid for high rain rates (\(R\) > 80 mm/h). RMSE is a measure of the accuracy of the proposed algorithm; the obtained low values indicate an overall good performance of the \(R\)-\(k\) relation. NMAE values indicate the possible influence of the DSD variability on the accuracy of the rain rate estimates. The values of 17.9% and 20.6% indicates a limited impact of DSD variability. Figure 8 and the statistics mentioned above point out a fundamental advantage of choosing experimentally-derived relationships. In fact, in this way, we have a way to assess the influence of DSD variability on the selected relationship, which will be essential information when the performance of the NEFOCAST system will be characterized with respect to the different sources of error.