# Scaling of Rain Attenuation Models: A Survey
## 3 Existing Scaling Techniques
### Elevation Angle Scaling  
If \(\theta_{1}\) and \(\theta_{2}\) are elevation angles and \(A(\theta_{1})\), \(A(\theta_{2})\) are the attenuations due to rain attenuation, then for a settled operating frequency, the attenuation scales with the elevation angle, which can be analytically written as:  
\[A(\theta_{1})\sin\theta_{1}=A(\theta_{2})\sin\theta_{2} \tag{7}\]  
The implementation of (7) can be performed through various means such as power-law-based techniques, cosecant-rule-based techniques, and synthetic time-series-based techniques. All these different implementation techniques are briefly explained below:  
I. Power-law based technique:  
Elevation angle scaling is defined as the ratio of rain attenuation at the zenith and the slant paths (\(\theta\) from \(1^{\circ}\) to \(89^{\circ}\)) in [11]. The following simple formulation presents a competent model as the scaling factor \(SF(\theta)\):  
\[SF(\theta)=1.741\theta^{-0.6015}+0.8931 \tag{8}\]  
It was shown in [11] that this method can predict slant link attenuation at a satisfactory level of accuracy from \(5^{\circ}\leq\theta\leq 90^{\circ}\). According to the CCIR model [23], it is noteworthy that the elevation angle scaling is defined for \(\theta>10^{\circ}\);  
II. Cosecant-rule-based technique [11]:  
The scaling factor (\(SF[P,\theta]\)) is defined as the proportion of the attenuation across \(\theta\) and the one anticipated from the zenith path using the simple cosecant law:  
\[SF(P,\theta)=\
icefrac{{A_{B}(P,\theta)}}{{\rm{osc}}(\theta)A_{B}(P,90^{ \circ})} \tag{9}\]  
The scaling factor \(SF\) can be calculated as the approximated fourth-order polynomial as:  
\[SF(p,\theta)=\alpha_{1}(\theta)P^{3}+\alpha_{2}(\theta)P^{2}+\alpha_{3}( \theta)P+\alpha_{4}(\theta) \tag{10}\]  
where \(\alpha_{1},\alpha_{2},\alpha_{3}\), and \(\alpha_{4}\) are the coefficients of (10);  
III. Time-series-based technique:  
There is a strong impact of the elevation angle on the design of telecommunication facilities, particularly the low Earth orbit (LEO) satellites. At a low elevation angle for the LEO satellite, if a single rain cell intercepts the propagated signal from the Earth station antenna, it is possible to scale the elevation angle with the rainy time series generation technique [24] as per (11). \[F=\left[\
icefrac{{(h_{r}-h_{a})}}{{\
icefrac{{f}}{{(h_{r,1}-h_{a,1})}}}} \right]\cdot\left[\
icefrac{{\sin(\theta_{1})}}{{\sin(\theta_{2})}}\right] \cdot\left(\
icefrac{{f_{2}}}{{f_{1}}}\right)^{1.72} \tag{11}\]  
where the subscripts \"1\" and \"2\" are used for the base (reference) measured and operating (target) datasets. Here, the rain height and the antenna height are, respectively, \(h_{r}\)(km) and \(h_{a}\)(km), and both are with reference to the sea level, \(\theta\) is the elevation angle, and \(f\) is the frequency in GHz. However, if multiple rain cells exist, it is impossible to scale the attenuation, and (11) will not be valid. It is assumed that by picking a collection of episodes from the ITALSAT database and scaling it to the factor indicated in (11), it is possible to recreate rain reduction statistics at a different frequency, height, and elevation for any other location. This technique needs several data processing steps, and among these steps, the most important things to check are: the original elevation angle needs to be mapped (e.g., from 37.7\({}^{\circ}\) to 107\({}^{\circ}\)); 20 terms are then expanded to result in a database to investigate whether there exist multiple rain cells (as in the case of multiple rain cells, the assumption made by (11) will not be valid). In [25], the effect of rain attenuation on the elevation angle was assessed using the synthetic storm approach to create rain attenuation time series. The study showed that if the elevation angle decreases to 20\({}^{\circ}\), the impact of rain attenuation increases significantly, and at higher elevation angles (i.e., 40\({}^{\circ}\) to 80\({}^{\circ}\)), the influence of rain attenuation is negligible.