# Scaling of Rain Attenuation Models: A Survey
## 3 Existing Scaling Techniques
### Frequency Scaling
#### 3.3.2 Nonratio-Based Frequency Scaling  
In the nonratio-based scaling rule formulation, the scaling rule is not formed through a ratio-based rule; instead, the relationship is represented through an empirical formula that conveys the relation of the physical influence of scaling. The nonratio-based models can be classified into an empirical formula and instantaneous-based method, briefly discussed below:  
I. Empirical formula-based scaling method:  
The link between the upper and base frequency attenuation cannot be represented in a precise ratio form in a nonratio model. Nonratio models do not predict SAR, but quantify the relationship between \(A_{f_{1}}\) and \(A_{f_{2}}\) through empirical relationships;  
(a) Single frequency:  
The frequency scaling ratio can be implemented on the level of the attenuation of the primary frequency as explained in [41]. The primary frequency-level-based attenuation can be defined as [42]:  
\[A_{2}=4k_{2}\Big{[}A_{1}/4k_{1}\Big{]}^{\alpha_{2}/\alpha_{1}} \tag{24}\]  
where \(k\) and \(\alpha\) are constants that depend on the frequency and drop size distribution and \(A_{1}\), \(A_{2}\) are the attenuations corresponding to the frequencies \(f_{1}\) (GHz) and \(f_{2}\) (GHz). In (24), the considered pathlength was developed at about 4 km, and this scaling formula can be used at the frequency of 60 GHz. The drawback of this technique is that it needs the coefficients \(k\) and \(\alpha\) derived through the frequency and the DSD. This procedure was simplified in [27], and through the empirical formula, ITU-R (25) can be adopted. The polynomial-based frequency scaling technique was proposed in [43,44];  
i. ITU-R model:  
ITU-R has adopted a model to predict frequency scaling as [45]:  
\[A_{2}=A_{1}(\Phi_{2}/\Phi_{1})^{1-H(\Phi_{1},\Phi_{2},A_{1})} \tag{25}\]  
where the values of \(\Phi(f)\) and \(H(\Phi_{1},\Phi_{2},A_{1})\) are given by:  
\[\Phi(f) =f^{2}/(1+10^{-4}f^{2}) \tag{26}\] \[H(\Phi_{1},\Phi_{2},A_{1}) =1.12(\Phi_{2}/\Phi_{1})^{0.5}(\Phi_{1}A_{1})^{0.55} \tag{27}\]  
The applicable frequency of this model [45] is limited to 7-55 GHz;  
ii. Use of the specific attenuation relation:  
Using the power-law-based specific attenuation relationship, we can write:  
\[A(p\%)=kR^{\alpha}(p_{\%})l(p_{\%}) \tag{28}\]  
where \(p_{\%}\) is the percentage of time exceedance, \(A\) is the attenuation, \(R\) is the point rain rate with \(p_{\%}\) percentage of time exceedance and \(l\) is the corresponding effective path length. The scaling relationship assumes that both frequencies are coincidental (that is, seen along the same path, i.e., we presume that \(R\) and \(l\) are the same). \[A_{2}\approx k_{2}(A_{1}/k_{1})^{\alpha_{2}/\alpha_{1}}; \tag{29}\]  
iii. Rain rate depending on a Gaussian distribution:  
In [46], it was assumed that the point rain rate along the propagation path could be expressed as a Gaussian function of position. Therefore, the instantaneous attenuation can be formulated as:  
\[A=\int_{0}^{L}kR(x)dx\quad\text{ where }R(x)=R_{o}exp[-(x/l_{0})^{2}] \tag{30}\]  
\(R_{0}\) is the peak rain rate along the path \(L\). The distance x is measured from the position of maximum rain rate intensity, and \(l_{0}>>L\) is a measure of the cellular rain structure. Under these conditions, the attenuation ratio for frequencies \(f_{1}\) and \(f_{2}\) was given by [46] as follows:\[A_{2}/A_{1}=k_{2}/k_{1}\sqrt{\
icefrac{{{a_{1}}}}{{{a_{2}}}}}\Big{[}A_{1}/k_{1}\sqrt{ \
icefrac{{{a_{1}}}}{{{a_{1}}}}-1}\Big{]}^{\
icefrac{{{({e_{2}}/{{a_{1}}}-1)}}}{{1}}} \tag{31}\]  
where \(\alpha_{1},k_{1},\alpha_{2}\), and \(k_{2}\) are parameters associated with frequencies \(f_{1}\) and \(f_{2}\), respectively. This method does not establish a strict ratio relationship because \(A_{2}/A_{1}\) is a function of \(A_{1}\);  
(b) Dual-frequency:  
Two frequency scaling models can usually be either linear or nonlinear. The ratio of attenuation at two frequencies is used in the majority of the two frequency models.