## 3 Methodology
### Training Issues  
Precipitation is a statistical rare event. Analyzing many polar orbits, it turns out that just a few percent (or even less) of the pixels are rainy. Furthermore, considering cloudy pixels, only 10% to 30% of them are rainy as it is shown in Table 9. Using indiscriminately all the pixels of the images, the dataset would be totally unbalanced towards zero-rain \begin{table}
\begin{tabular}{c c c} \hline \hline
**Input \#** & **Data Type** & **Explanation** \\ \hline
1–6 & BITR10.8 t0... t5 & Brightness temperature of the pixel for the current IR image (t0) and the previous five images. Cloud Top Temperature, Cloud Phase, Frontal System (time). \\ \hline
7 & BITR3.9-BITR10.8 & Microphysical and optical cloud properties. \\ \hline
8 & BITR3.9-BITWV7.3 & Water Vapor Content (CWP). \\ \hline
9 & BITR8.7-BITR10.8 & Cloud Phase. \\ \hline
10 & BITR10.8-BITR12.1 & Cloud Optical Thickness, Cirrus test. \\ \hline
11–15 & MEAN3(1, 7, 8, 9, 10) & Mean value over a 3 by 3 pixel neighborhood. Minimization of geolocation and synchrony problems. Texture information about the cloud cover. \\ \hline
16–20 & STD3(1, 7, 8, 9, 10) & Standard deviation of pixel value over the 3 by 3 pixel neighborhood. Texture information about the cloud cover. \\ \hline
21–25 & MEAN5(1, 7, 8, 9, 10) & Mean value over a 5 by 5 pixel neighborhood. Minimization of geolocation and synchrony problems. Texture information about the cloud cover. \\ \hline
26–30 & STD5(1, 7, 8, 9, 10) & Standard deviation of pixel value over the 5 by 5 pixel neighborhood. Texture information about the cloud cover. \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Night time input. points, and a NN trained with such a dataset will not be able to appreciate differences in rain rate during a storm or to distinguish between a short violent thunderstorm from a long stratiform rain. To avoid this problem a neural network classifier has been first created. A subset of the total data, choosing randomly from the original dataset an equal number of rainy and not-rainy pixels, is used to train a neural network classifier. GPM data are used as output of this classifier with a threshold of 0.2 mm/h (above the threshold the pixel is considered rainy; below not rainy). Using the collocation procedure described above, with geographical areas and temporal windows provided in Table 2, the subset to create the rainy mask is obtained. The subset is randomly divided in three parts, 70% for training set, 15% for the validation set, and 15% for the test set. The training set is used for computing the network weights and biases. The error on the validation set is monitored during the training process. The validation error normally decreases during the initial phase of training, as does the training set error. However, when the network begins to overfit the data, the error on the validation set typically begins to rise. The network weights and biases are saved at the minimum of the validation set error. The test set error is not used during training, but it is used for the final evaluations. The numbers of pixels used to create the NNs are reported in Tables 5 and 6 for the day-time and night-time cases, respectively (in Table 6 the 8 July 2015 event is not reported due to night time data unavailability). As training algorithm, a Bayesian Regularization has been preferred. This algorithm typically requires more time, but provides good generalization for difficult, small, or noisy datasets like those used in this study. As a final step, the dataset composed only of rainy pixel obtained at the previous step is used to train the neural network with GPM rain rate as output. Moreover, in this case, the subset is randomly divided in three parts, 70% for training set, 15% for the validation set, and 15% for the test set as reported in Tables 7 and 8 (as for Table 6, in Table 8 the 8 July 2015 event is not reported due to night time data unavailability).