# Scaling of Rain Attenuation Models: A Survey
## 3 Existing Scaling Techniques
### Frequency Scaling
#### 3.3.1 Frequency-Ratio-Based Scaling  
Several formulations have been proposed to scale long-term attenuation statistics, each with applications and limitations [12,27]. Most of these relations are intended for scaling attenuation statistics (that is, for calculating SAR), not for instantaneous (real-time) scaling of attenuation:  
I. SAR-based method:  
Instead, in real-time applications, the frequency scaling principle has been proposed to provide static scaling attenuation. Most of the statistical scaling methods consider only frequency as the parameter. The SAR method considers the percent of time exceeded with a rain rate. The statistical attenuation due to clear air is required to be calculated similar to (1). A significant disadvantage of such a model is that these apply only to the same pair of frequencies, for which the related coefficients are determined through the investigated datasets. Electromagnetic propagation becomes disturbed by many environmental factors such as gases, raindrops, water vapor, and similar factors. Many researchers tried to establish empirical relationships of a satellite link for two channels by approximating the ratio /\({}_{2}\)/\({}_{1}\) (\(f_{2}\), \(f_{1}\) is used to mean \(f_{2}>f_{1}\)) for different GHz frequency bands. The most straightforward approach to present the SAR through mathematical means is [28]:  
\[SAR=\
icefrac{{A(f_{2})}}{{A(f_{1})}}=\
icefrac{{(f_{2}/f_{1})}}{{n}} \tag{12}\]  
The SAR-based method implementation was proposed in [12] as:  
\[SAR_{CCIR}=\
icefrac{{A(f_{2})}}{{A(f_{1})}}=\
icefrac{{\Phi(f_{2})}}{{\Phi( f_{1})}} \tag{13}\]  
\[\Phi(f)=\
icefrac{{f^{1.72}}}{{(1+3\times 10^{-7}}}\
icefrac{{\beta^{4.4}}}{{ \Phi(f)}}) \tag{14}\]  
where \(A(f_{2})\) and \(A(f_{1})\) are the attenuations in dB at frequencies \(f_{2}\) (GHz) and \(f_{1}\) (GHz), respectively;II. SAR from models:  
These categorized models do not include an explicit dependence on frequency, but stem from attenuation models. These attenuation models yield attenuation values that can form a ratio to predict SAR for a given pair of frequencies. \[SAR(f_{1},f_{2},p)=SACA(f_{2},p)/SACA(f_{1},p) \tag{15}\]  
where SACA is the statistical attenuation in reference to clear air; \(p\) is the percent of time exceedance of the occurring rain event for a considered pair of frequencies \(f_{1}\) and \(f_{2}\);  
(a) DSD-based path-averaged rain rate:  
The rain attenuation is significantly influenced by the distribution of the raindrop size rather than the rain rate. The size of raindrops varies in a single rain event. Therefore, the chance of scattering, refraction, or absorption varies greatly. For this reason, the analytical raindrop-based attenuation technique has good accuracy compared to the rain-rate-based model. The specific attenuation is calculated from the rain rate, and the raindrop size impacts the specific attenuation [29]. Different types of statistical and physical raindrop size distribution models were presented in [30; 31]. Based on the experimental measurements, an exponential distribution was used to represent the DSD in [32]:  
\[N(D)=N_{0}\cdot exp(-\Gamma D) \tag{16}\]  
where \(N_{0}=8000\) is a constant proposed in [33] and \(\Gamma\) is a coefficient parameter in millimeters depending on the rainfall rate (\(\Gamma=4.1R\) where \(R\) is the rainfall rate measured in \(mm/h\)). The DSD can also be used to calculate a path-averaged rain rate; \(R(z)\) is a function of distance in the horizontal direction; \(z\) is related to \(N(D,z)\); \(v(D)\) is the raindrop falling velocity; the diameter of the raindrop is \(D\) as in [34].