## 2 Attenuation Prediction Models  
Fog, clouds, and haze are the main challenges in the implementation of a FSO link. Atmospheric attenuation caused by different environmental factors are mainly dominated by fog as the particle size in this case is comparable to the wavelengths of interest. The fog (1-20 \(\upmu\)m) and clouds (3-6 \(\upmu\)m) consist of very fine droplets of water, which may cause absorption and scattering of transmitted optical signals. Due to their small refractive index and particle size, ice crystals do not contribute much to attenuation due to absorption [18]. Thus, absorption can generally be neglected at the wavelength of interest for free space communication [19]. Haze (0.01-1 \(\upmu\)m) consists of small aerosol particles suspended in the air, dust, pollutants, etc. As the size of particles is comparable to the wavelengths in visible and IR bands,it may cause attenuation at FSO wavelengths. In order to estimate the attenuation, visibility statistics and the relationship between visibility and propagation loss has to be known. For characterization of fog and haze attenuation, the term \"specific attenuation\" is used, which represents attenuation in dB/km and is expressed as [20]:  
\[\beta(\lambda)=\frac{10}{R}\text{lg}\Big{(}\frac{P0}{Pr}\Big{)}=\frac{10}{R} \text{lg}(e^{\tau(\lambda)R}\Big{)} \tag{1}\]  
where \(P_{o}\) and \(P_{r}\) represent the transmitted and received optical power at distance \(R\), respectively. \(R\) and \(\gamma(\lambda)\) represent the link length and attenuation coefficient of the atmosphere, respectively. \(\lambda\) represents the wavelength. Mie scattering theory can be applied to predict attenuation, which involves complex computations requiring details of fog parameters. Using empirical models and taking 550 nm as the visibility reference wavelength, specific attenuation due to fog and haze may be expressed as:  
\[\beta_{f}(\lambda)=\frac{3.91(\lambda/550)^{-m}}{v} \tag{2}\]  
where \(v\) and \(\lambda\) define the visibility range in km and operating wavelength in nm, respectively. \(m\) is defined as the scattering distribution coefficient and its value can be specified by the [PERSON] and Kruse model. The [PERSON] model was originally proposed for haze particles and is one of the widely used models for defining the scattering distribution coefficient [21]:  
\[m= \begin{cases}\begin{array}{cc}1.6&v>50\\ 1.3&6<v<50\\ 0.585v^{\frac{1}{2}}&v<6\end{array}\end{cases}  \tag{3}\]  
The model has uncertainty about visibility less than 1 km and estimates less attenuation for longer wavelengths. [PERSON] suggested a modified model based on theoretical assumptions and indicates wavelength independent of attenuation for \(v\) < 0.5 km. \(m\) values as per the [PERSON] model are defined as:  
\[m= \begin{cases}\begin{array}{cc}1.6&v>50\\ 1.3&6<v<50\\ 0.16v+0.34&1<v<6\\ v-0.5&0.5<v<1\\ 0&v<0.5\end{array}\end{cases}  \tag{4}\]  
Clouds are formed when water condenses or deposits above the surface of the earth and are characterized by their properties, like cloud height, thickness, water content, phase, droplet sizes, horizontal extent, and variability. Attenuation due to clouds can be as high as several tens of decibels with water droplets ranging from 1 to 30 \(\upmu\)m. The visibility in this case may be expressed as [22]:  
\[v=\frac{1.002}{\Big{(}L\times Nc\Big{)}^{0.6473}} \tag{5}\]  
where \(L\) (g/m\({}^{3}\)) and \(Nc\) (cm\({}^{-3}\)) represent the average cloud liquid water content (CLWC) and cloud droplet number concentration, respectively. \(Nc\) may be further expressed as:  
\[\text{Nc}=\frac{L}{\frac{3}{3}\pi v^{3}\rho\times 10^{-6}} \tag{6}\]where \(r\) (\(\upmu\)m) and \(\rho\) (1 g/cm\({}^{3}\)) represent the mean of the cloud droplet radius and density of liquid water, respectively. \(r\) is 3.33, 4.7, and 6.0 (\(\upmu\)m) in the case of stratus, nimobstratus, and cumulus clouds, respectively [22].