## 1 Introduction
Assuming a consistent particle size distribution changes with increasing or decreasing visibility, [PERSON] and [PERSON] [3] empirically fitted the measurement data with \(\gamma\), thus relating \(M\) and \(V\) as demonstrated in Equation (5). The relationship given by Equation (5) is often used to solve for \(N_{T}\), assuming a case of polydispersive particle distribution. To illustrate, the particle size distribution term [5] is introduced into Equation (3), which becomes:  
\[A=\frac{8.19*10^{4}}{\lambda}*\Im\left(\frac{\varepsilon-1}{\varepsilon+2} \right)*\int_{0}^{\infty}n(r)\;r^{3}dr\;dB/km, \tag{7}\]  
where \(n(r)\;dr\) is the number of sand/dust particles in the radii range of \(r\to r+dr\). A cubic meter of all dust particles per cubic meter of air, i.e., the total relative volume (\(v_{r}\)), is given by  
\[v_{r}=\frac{4}{3}\pi\int_{0}^{\infty}r^{3}n(r)dr\;\text{m}^{3}\text{ of dust/m}^{3}\text{ of air}. \tag{8}\]  
At the same time, \(v_{r}\) has the following relation to the dust mass concentration (\(M\)) and the mass particle density (\(\rho\)) in g/m\({}^{3}\)  
\[v_{r}=\frac{M}{\rho}\;\text{m}^{3}\text{ of dust/m}^{3}\text{ of air} \tag{9}\]  
Using Equations (8) and (9), \(v_{r}\) may alternately be expressed as  
\[v_{r}=\frac{M}{\rho}=\frac{C}{\rho\;V^{\prime}}\;\text{m}^{3}\text{ of dust/m}^{3}\text{ of air}. \tag{10}\]  
Considering the form of Equation (8) and the relation given in Equation (10) for the size distribution in Equation (7), we get  
\[N_{T}=3.67*10^{-5}\frac{\alpha_{0}}{r^{2}}\ (\text{\# of particles}/\text{m}^{3}), \tag{2}\]0  
Taking advantage of the established relationships and the equations mentioned above, the various authors that have attempted modifying [PERSON]'s [1] attenuation model adopted the visibility parameter, and below are some of the known attenuation models constructed by them:  
1. [PERSON] et al. 's [6] attenuation model: \[N_{T}=3.67*10^{-5}\frac{\alpha_{0}}{r^{2}}\ (\text{\# of particles}/\text{m}^{3}), \tag{2}\]1 (12)
2. [PERSON]'s [7] attenuation model: \[N_{T}=3.67*10^{-5}\frac{\alpha_{0}}{r^{2}}\ (\text{\# of particles}/\text{m}^{3}), \tag{2}\]2 (13)
3. [PERSON] et al. 's [2] attenuation model: \[N_{T}=3.67*10^{-5}\frac{\alpha_{0}}{r^{2}}\ (\text{\# of particles}/\text{m}^{3}), \tag{2}\]3 (14)
4. [PERSON]'s [8] attenuation model: \[N_{T}=3.67*10^{-5}\frac{\alpha_{0}}{r^{2}}\ (\text{\# of particles}/\text{m}^{3}), \tag{2}\]4 (15)
5. [PERSON]'s [9] attenuation model: \[N_{T}=3.67*10^{-5}\frac{\alpha_{0}}{r^{2}}\ (\text{\# of particles}/\text{m}^{3}), \tag{2}\]5 (16)* [PERSON] et al. 's [10] attenuation model: \[N_{T}=3.67*10^{-5}\frac{\alpha_{0}}{r^{2}}\ (\text{\# of particles}/\text{m}^{3}), \tag{2}\]6 (17) where \(F\) is the frequency in GHz in (12), (15), (16), and (17), \(V\) is the visibility in km in (12) to (17), \(r_{e}\) is the effective particle radius in \(\upmu\)m in (14) and (15), \(r\) is the particle radius in \(\upmu\)m in (16), and \(\uplambda\) is the wavelength in meters in (13) and (14). It is evidently clear that the attempts are very persistent in designating visibility as a critical factor in signal attenuation calculations and as indicative of sand/dust storms intensity. However, while statistical information on dust storm visibility can be retrieved from weather stations, other commonly encountered sources may offer visibility in different terms. For instance, [PERSON] and [PERSON]'s [3] work, as briefly hinted at earlier, attempted to deduce visibility from dust concentration in the air based on an empirical relationship. In fact, several additional empirical relationships have been established (Table 1) to calculate visibility (\(V\)) in km from a known dust concentration (\(M\); occasionally, referred to as total suspended particles (\(TSP\)) in \(\upmu\)g/m\({}^{3}\). However, there are significant accompanying uncertainties, one of which is the tendency to over- or under-estimate visibility [11], which is this paper's interest.