# Hailstorm Detection by Satellite Microwave Radiometers
## 3 The Hail Detection Model
### The Hail Detection Model  
The hail detection model is based on a probability model of growth [43] that exploits the TBs at 150 and 190 GHz. The general concept is based on the inverse proportionality between the upwelling radiation and hail cross-sections. As demonstrated above, the frequencies at 150 and 190 GHz are the most sensitive to the presence of hail being subject to a modification of their radiation field as a function of hail diameter. Furthermore, although their dynamics is pretty similar, the signal at 150 GHz channel tends to decrease more steeply than at 190 GHz. Therefore, we assume as model for hail detection a modified sigmoidal function as follows:  
\[f(x,y)=\ 1-\frac{1}{1+e^{-(x-y)}} \tag{2}\]  
where \(x\) and \(y\) are the TBs at 150 and 190 GHz, respectively. The difference between these two frequencies is demonstrated to be very sensitive to the growth of ice aggregates particularly when hail size is the order of centimeters. Eq. 2 shows boundaries between 0 with clear sky conditions and 0.99 when hail is observed. The probability model described by Equation (2) was trained with the entire dataset described in Table 5 knowing that signatures from very large hail, typically correlated with \(>\)8-10 cm, can be found when \(TB_{184}^{\text{par}}\)\(\sim\)35%. Equation (2) is also reinforced by a variable inspired by the concept of carrying capacity [44] of biological species, that is the maximum population size that the environment can sustain indefinitely or on the contrary the maximal load of the environment which modulates the population equilibrium. The carrying capacity is then a sort of regulator of population dynamics typically approximated by a probability model, which controls the occurrence of an overshooting of survival conditions that in long term leads to the collapse of the species. In the case of hail detection, the population (hail) impacts on the upwelling radiation (environment) by increasing the scattering effects (environment sustainability). The growth of hail sizes increases the scattering effects and then decreases the radiation to the satellite. This process is well described by Equation (2), which approximates the expected probability distribution as a function of decreasing TBs. To govern the dynamics of the probability model by stopping the computation when the sensitivity reaches its maximum (overshooting) and no hail signatures can be recognized (saturation condition), a dynamic carrying capacity was introduced. The carrying capacity attains values in the range [0,1] and is described by the dimensionless variable K,  
\[K(x)=\frac{\alpha}{x} \tag{3}\]  
where \(\alpha=104\ K\) and \(x\) is the TB at 150 GHz, frequency chosen because it reaches saturation conditions faster. For \(K<<1\) the population load is minimum while for \(K=1\) the model reaches the overshooting and the computation stops. A realistic distribution of Equations (2) and (3) fed by data in Table 6 is reported in Figure 7. Note that both distributions converge to 1 when the minimum value is reached but largely diverge for the maximum TB. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**TB90 (K)** & **TB150 (K)** & **TB184 (K)** & **TB186 (K)** & **TB190 (K)** & \(TB_{184}^{\text{par}}\) **(\%)** & **No. Data** \\ \hline
216.40 & 181.30 & 205.27 & 191.38 & 178.09 & 25–35 & 1 \\
180.90 & 152.51 & 185.10 & 167.04 & 153.30 & \(>35\) & 2 \\
114.30 & 103.70 & 122.20 & 112.00 & 107.50 & Abs. min & 3 \\ \hline \end{tabular}
\end{table} 
Table 6: Average TBs defining the dynamic range for hail detection. Therefore, by regressing Equations (2) and (3) we derive a compact general model able to represent the hail size distribution as a function of TB depression. The result is a new function \(H(K)\) that maps the probability of hail detection as only function of TB150. \[H(K(x))=\ 0.9844\ln(K(x))+0.9072 \tag{4}\]  
The dynamics of the hail detection model when applied to the whole dataset (2882 data) is reported in Figure 8a.