# Scaling of Rain Attenuation Models: A Survey
## 3 Existing Scaling Techniques
### Frequency Scaling
#### 3.3.1 Frequency-Ratio-Based Scaling
The path-averaged rain rate can be defined as:  
\[\bar{R}=\int_{0}^{D_{m}}4\pi/3(D/2)^{3}v(D)\bar{N}(D)dD \tag{17}\]  
The DSD-derived attenuation values can be used to produce a ratio, yielding a \(SAR\) model, as follows:  
\[SAR_{DSD}=A_{DSD}(f_{2})/A_{DSD}(f_{1}) \tag{18}\]  
The problem with this approach of frequency scaling is that it needs a sophisticated device to record the drop size, and most rain attenuation-related datasets do not include the DSD-based information;  
(b) DSD-based specific attenuation ratio:  
The specific attenuation (\(\gamma(\mathrm{dB/km})\)) is given by the following integral equation [35]:  
\[\gamma=4.343\times 10^{3}\int\limits_{0}^{D_{\max}}\sigma_{t}(D).N(D)dD \tag{19}\]  
where \(\sigma_{t}\) is the total cross-section for a spherical raindrop, \(N(D)dD\) is the number density of raindrops with an equivalent radius in the interval \(dD\), \(N(D)\) represents raindrop size distribution, and \(D_{\max}\) is the maximum mean raindrop radius. The integral equation can be changed to summation for numerical calculation as in [36; 37]:  
\[\gamma(f)=4.343\times\left(10^{3}\lambda^{2}/\pi\right)\sum\limits_{p=1}^{p} \sigma_{t}(p\Delta D)N(p\Delta D)\Delta D \tag{20}\]where \(\Delta D\) is the incremental radius, \(P\) is the integer number of \(D_{\text{max}}/\Delta D\), and \(\lambda\) is the wavelength corresponding to the frequency \(f\). The specific attenuation factor can be defined for the frequency \(f_{1}\) (GHz) and \(f_{2}\) (GHz) as \(\gamma_{f_{1}}\) and \(\gamma_{f_{2}}\), respectively. Now, the ratio of these two specific attenuations can be calculated as:  
\[SAR_{f_{1}f_{2}}=\gamma_{f_{1}}/\gamma_{f_{2}} \tag{21}\]  
where the notation \(SAR_{f_{1}f_{2}}\) is constant. Thus, the constant \(SAR_{f_{1}f_{2}}\) and the specific attenuation of a baseline link can be used to estimate a target links' attenuation. There is a popular application of the specific attenuation to predict the attenuation of a satellites' downlink channel using the specific attenuation of the uplink channel;  
(c) The use of the spatial rain rate profile [38]:  
A real-time method for frequency scaling that was demonstrated in [13] has addressed the spatial rain distribution by adopting the exponential rain cell distribution known as the SAM, as in [38]. The fundamental concept is to provide the geographical rain rate profile for the rainfall rates, where the rain rate can be recorded. The SAM considers the rainfall rate's spatial profile as exponential and varies as:  
\[R(t,h)= \begin{cases}R_{0}(t),&\text{for }R_{0}\leq 10\,\text{mm}/\text{h}\\ R_{0}(t)exp[-\gamma\text{ln}(R_{0}/10)h],&\text{for }R_{0}>10\,\text{mm}/\text{h} \end{cases}  \tag{22}\]  
where \(R(t,h)\) is the rain rate at a range of \(h\) (the projection of the slant link up to the rainy region) from the receiver at any time and \(R_{0}(t)\) is the point rain rate at the receiver located at the corresponding time. Frequency scaling has to do with the variation of the propagation effects as a function of frequency. The concept of frequency scaling was proposed to yield the scaling attenuation statics instead of real-time applications;  
III. Instantaneous (ratio-based):  
The instantaneous scaling technique is defined as the ratio of clear air attenuation \((A_{CA})\) at a high and low frequency, respectively. In the experiment [6], additional parameter time was considered representing the sample timings in (1). Usually, the rain attenuation value given by (1) is noisy, so a moving average filter is used to smooth the samples where the window duration is 30 s [6]. However, a larger window duration of 60 s was reported in [39].