## 4 Proposed Empirical Models for Weather Factors
### Dust Model  
Due to the fact that no suitable model for the coefficients could be found in the literature, a simplified model is proposed based on the literature available in different research areas.  
Using a visibility-related model for the number of particles, ref. [46] have calculated the coefficients by assuming a fixed effective particle diameter. Here, it is not taken into account that particularly small particles with a diameter in the order of the laser wavelength have a large influence on both coefficients, which is also depicted in Figure 3. The particle size distribution of sand storms can be very different [34], as the it is highly influenced by the soil of the storm's origin. The ISO 12103-1 [47] standard describes the parameters of different types of dust. Here, the dust type \"A4 - coarse test dust\" can be considered similar to the dust referred to in the literature [34]. Based on this definition of dust, the weight \(m\) of a fixed number of particles \(N\) can be calculated using the dust's bulk density \(\rho\) of 1.2 g/cm\({}^{3}\)[47] and the particle density distribution \(PDF\):  
Figure 9: Comparison of the scattered quantities of light in regards to the two coefficients and the number of particles per droplet radius.  
\[m=\int_{0}^{\infty}\rho_{bulk}\cdot\frac{4}{3}\pi r^{3}\cdot N\cdot PDF(r)\,dr \tag{13}\]  
This result can then be used to estimate the number of particles in relation to the visibility \(V\) and the total suspended particle mass \(M\). To calculate the mass, the visibility model of [33] was used, as this model showed a reasonable empirical determination of \(R^{2}=0.79\)[30]:  
\[N(V)=\frac{M}{m}=\frac{1}{m}\cdot(\frac{3553.405}{V})\frac{127}{125} \tag{14}\]  
Plugging Equation (14) into Equation (7) makes it possible to calculate the extinction coefficient \(\alpha\) depending on visibility \(V\). The backscatter coefficient can be obtained in a similar way, by replacing Equation (14) in Equation (7):  
\[\alpha(V) =\frac{\pi}{8}\cdot\int_{0}^{\infty}D^{2}\cdot Q_{ext}(D)\cdot N( V)\cdot PDF(D)\,dD \tag{15}\] \[\beta(V) =\frac{\pi}{8}\cdot\int_{0}^{\infty}D^{2}\cdot Q_{back}(D)\cdot N( V)\cdot PDF(D)\,dD \tag{16}\]  
Figure 10 shows the coefficients in relation to the visibility during dust storms by calculating 1000 data points with different visibility for Equations (15) and (16). To enable an easy calculation of both coefficients, a curve fit of an inverse power law was performed on the data point sets, which shows a coefficient of determination of \(R^{2}=1\) for both fits:  
\[\alpha =5.26\,V^{1.016} \tag{17}\] \[\beta =5.38\,V^{1.016} \tag{18}\]  
This model yields a reasonable value for the extinction coefficient, as the overall scattering anisotropy of SiO\({}_{2}\) particles is smaller than for water (see Figure 3), which results in a higher extinction coefficient. Below is a comparison of the extinction coefficient \(\alpha\)  
Figure 10: Extinction coefficient \(\alpha\) and backscattering coefficient \(\beta\) in relation to the visibility during dust storms.  
of dust storms and fog at an exemplary visibility of 100 m. As expected, the extinction coefficient of dust storms is higher than water:  
\[\alpha_{Dust}|_{V=100m} =0.0488 \tag{19}\] \[\alpha_{Fog}|_{V=100m} =0.0391 \tag{20}\]