# Global Formulation of the Synthetic Storm Technique Oriented to Satellite Link-Budget Design
## 2 The Full SST and the Global SST
### The Full SST  
A synthetic storm is obtained by converting a rain rate time series \(R(t)\), recorded at a site by a rain gauge and averaged over 1 min, to a rain rate space series \(R(x)\) along a line (\(x-\) axis), by using an estimate of the storm translation speed \(v\) to transform time to distance. Then, the specific rain attenuation \(Y(x,t)\) (dB/km), calculated from the rain rate time series, is integrated along the path to obtain the rain attenuation time series \(A_{SST}(t)\). The mathematical operation to obtain \(A_{SST}(t)\) is a convolution from which, through Fourier transforms, useful theorems and conservation properties have been derived [1].  
The SST can generate rain attenuation time series at any frequency and polarization, and for any slant path above about \(10^{\circ}\), as long as the hypothesis of isotropy of the rainfall spatial field holds, in the long term. The statistical properties of rain attenuation derived from a large sample of rain rate time series closely agree with those of the actual measured rain attenuation, and the predictions are significantly insensitive to a very precise value of \(v\). If the motion of the rain storm is along the satellite path, then the measured \(A(t)\) practically coincides with \(A_{SST}(t)\), as shown for the ITALSAT experiment [39].  
Rain attenuation time series can be simulated by using the value of \(v\) obtainable from measurements of wind speed, e.g., at 700 mbar height, extracted from the database of the European Centre for Medium-Range Weather Forecast [40]. The average horizontal rain-storm speed at Spino d'Adda is about 10.6 m/s, estimated directly from a meteorological radar [1], and in 1 min the rain storm moves a linear distance of 636 m.  
The vertical (zenith) structure of precipitation is modelled with two layers of precipitation of different depths, a model taken from [41]. Starting from the ground there is rain (hydrometeors in the form of raindrops, water temperature of 20 \({}^{\circ}\)C, called layer A), followed by a melting layer, i.e., melting hydrometeors at 0 \({}^{\circ}\)C, layer B. The vertical rain rate \(R\) (mm/h) in layer A is assumed to be uniform and given by that measured at the ground, i.e., by the rain gauge, \(R_{A}=R\). By assuming simple physical hypotheses, calculations show that the vertical precipitation rate in the melting layer, termed \"apparent rain rate\", can also be supposed to be uniform and given by \(R_{B}=3.134\times R_{A}\)[41]. The height of the precipitation (rain and melting layer, \(H_{R}\)) above sea level to be used in the simulation is the latest estimate of the ITU-R [42]. The layer B depth is 0.40 km, regardless of the latitude [41].  
Let \(H_{S}\) be the altitude of the site, \(\theta\) be the elevation angle of the slant path, then the precipitation path length \(L\) (km) is given by  
\[L=\frac{H_{R}-H_{S}}{\sin\theta}=\frac{H_{R}-0.4-H_{S}}{\sin\theta}+\frac{0.4 }{\sin\theta}=L_{A}+L_{B} \tag{1}\]  
For example, at Spino d'Adda (\(H_{S}=0.084\) km), \(H_{R}=3.341\) km (see Table 2), therefore the length of the rainy path is \(L=L_{A}+L_{B}=8.353+1.170=9.523\) km at \(\theta=20^{\circ}\), \(L=5.714+0.800=6.514\) km at \(\theta=30^{\circ}\), \(L=3.299+0.462=3.761\) km at \(\theta=60^{\circ}\), and, of course, \(L=[(3.341-0.084)-0.4]+0.4=L_{A}+L_{B}=2.857+0.4=3257\) km at \(\theta=90^{\circ}\). In the zenith path (\(\theta=90^{\circ}\)), the rain rate is uniform, see Equation (29) of [1].