## 2 Propagation Mechanism and Particle Distribution
### Fog Droplet Distribution  
The fog is a weather phenomenon where tiny droplets or ice crystals are suspended in the near-ground air, which is the product of the condensation of water vapor in the air near the ground. According to the conditions of formation, fog can be divided into advection fog and radiation fog. Advection fog is formed by the cooling of warm and moist air as it moves to cooler land or over the colder surface of water. It generally appears in areas near the sea, with a wide range and high concentration. Radiation fog is the countless small droplets formed by the cooling of the radiation on the ground at night. It mostly occurs at night and early morning when the weather is clear, the water vapor near the surface is sufficient and stable, or there is a temperature inversion.  
Attenuation of terahertz waves in fog is caused by the absorption and scattering of waves by particles. The radius of these particles typically ranges between 1 \(\upmu\)m and 60 \(\upmu\)m. Table 1 presents the levels of fog and their corresponding horizontal visibilities (\(V\)).  
The size and distribution of fog droplets can have a significant impact on terahertz wave attenuation based on Mie theory. The generalized gamma distribution is the most commonly used empirical model for describing fog droplet distribution, which is reported as [29]:  
\[n(r)=ar^{\alpha}\exp(-br^{\beta}), \tag{1}\]  
where \(r\) is the radius of droplet particles, and \(n\) is the number of droplets per unit volume. If the droplet radius is in microns (\(\upmu\)m), the unit of \(n\) is m\({}^{-3}\upmu\)m\({}^{-1}\); if the droplet radius is in meters (m), then \(n\) is in m\({}^{-4}\). Other parameters \(a\) (\(a=\frac{9.781}{V^{0.8}M^{5}}\times 10^{15}\)) and \(b\) (\(b=\frac{1.304}{VM}\times 10^{4}\)), describing droplet size and shape, are determined by visibility and water content; \(V\) is the fog visibility in kilometers; and \(M\) is the water content of the fog in grams per meter cumbed. When \(\alpha=2\) and \(\beta=1\), the gamma distribution can accurately describe the particle size distribution:  
\[n(r)=ar^{2}\exp(-br). \tag{2}\]  
To calculate the density of liquid water \(M\) (g/m\({}^{3}\)) in advection fog and radiation fog, an empirical expression based on the fog visibility can be represented [17]:  
\[\left\{\begin{array}{ll}M=(18.35V)^{-1.43}=0.0156V^{-1.43},&\text{Advection fog},\\ M=(42V)^{-1.54}=0.00316V^{-1.54},&\text{Radiation fog}.\end{array}\right. \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Fog Level** & **0** & **1** & **2** & **3** & **4** & **5** \\ \hline
**Horizontal Visibility (km)** & \textgreater{}10 & 1–10 & 0.5–1 & 0.2–0.5 & 0.05–0.2 & \textless{}0.05 \\ \hline
**Designation** & Clear day & Mist & Fog & Heavy fog & Dense fog & Strong fog \\ \hline \hline \end{tabular}
\end{table}
Table 1: Relationship between fog level and horizontal visibility.  
Figure 1: Obstacle absorption and scattering of incident waves.  
Combining Equations (2) and (3), it can be found that:  
\[\left\{\begin{array}{rl}n(r)=1.059\cdot 10^{7}V^{1.15}r^{2}\cdot\exp(-0.8359\cdot V ^{0.43}r)(\mathrm{m}^{-3}\mathrm{\upmu m}^{-1}),&\text{Advection fog},\\ n(r)=3.104\cdot 10^{10}V^{1.7}r^{2}\cdot\exp(-4.122\cdot V^{0.54}r)(\mathrm{m}^{-3} \mathrm{\upmu m}^{-1}),&\text{Radiation fog}.\end{array}\right. \tag{4}\]