# Light Transmission in Fog: The Influence of Wavelength on the Extinction Coefficient
## 4 Extinction Coefficient Results  
### Estimation of the Extinction Coefficient  
Three devices measuring three different quantities were used during the tests carried out. In order to perform the analysis, it was necessary to link the data of the three sensors by their time stamp. A database containing the following information was created:  
* spectroradiometer/light source distance \(D\),
* fog type (advection or radiation),
* MOR (temporal resolution of one second),
* DSD (temporal resolution of one second),
* and radiance for each wavelength \(\lambda\) (temporal resolution of a few seconds). For both types of fog, the radiance as a function of wavelength \(\lambda\), MOR, and distance \(D\) could therefore be obtained. The radiance is therefore expressed in the following form:  
\[\text{Radiance}\,=\,Rad(\lambda,V,D) \tag{11}\]  
The measurements were then grouped by MOR, in increments of 5 m (plus or minus 2 m) from 10 m-200 m (10, 15, 20,..., 200 m). For a fixed spectroradiometer/light source distance and a fixed fog type, several measurement points were obtained. However, this number of measurements was not constant as a function of MOR. This was because fog dissipated slowly in the platform when MOR was low (dense fog), and then, the rate of fog dissipation increased with increasing MOR (light fog). This is shown in Table 1. A sufficient number of measurements was still obtained for all the cases tested. In order to determine the extinction coefficient \(k(\lambda)\) for each wavelength, the following procedure was applied. Owing to the fact that the spectroradiometer resolution was 8 nm for wavelengths between 1000 nm and 1900 nm, all spectral measurements were grouped every 10 nm. Furthermore, in order to reduce the database, MORs were grouped every 5 m. Each MOR class (e.g., the class for 10 m) contained the MOR value of the class, as well as those within plus or minus 2 m of this value (e.g., [8; 12 m]). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Number of Spectroradiometer**} & \multicolumn{2}{c}{**Number of PSA**} \\
**MOR, \(V\) (m)** & \multicolumn{2}{c}{**Measurements**} & \multicolumn{2}{c}{**Measurements**} \\ \cline{2-5}  & **Radiation Fog** & **Advection Fog** & **Radiation Fog** & **Advection Fog** \\ \hline
15 & 360 & 238 & 2198 & 1427 \\
20 & 240 & 206 & 1399 & 1185 \\
30 & 241 & 121 & 1354 & 688 \\
40 & 214 & 96 & 1057 & 567 \\
50 & 141 & 100 & 732 & 519 \\
70 & 76 & 90 & 320 & 480 \\
100 & 57 & 34 & 202 & 176 \\
125 & 64 & 24 & 266 & 133 \\
150 & 58 & 30 & 235 & 127 \\
175 & 54 & 27 & 250 & 112 \\ Ref (\(>\)1000 m) & 1028 & 957 & 0 & 0 \\ Total & 2533 & 1923 & 8013 & 5414 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistics on the number of measurements made during the tests. Spectroradiometer data were collected at a variable time step depending on the conditions (a few seconds). The PSA data were collected at a time step of one second. All radiance spectra measured by the spectroradiometer for a three-tuple \"source/receiver distance\", \"fog type\", and \"MOR class\" were collected and sorted. After recovering the raw spectrum, the ratios of the radiance in fog condition versus the reference radiance at the same distance \(D\) were calculated using Equation (12). As in the previous step, a spectrum ratio was obtained for a fixed distance \(D\), fog type and MOR. These ratios were then used to determine the extinction coefficient corresponding to a given MOR and wavelength. \[Ratio(\lambda,D,V)=\frac{Rad(\lambda,D,V)}{Rad_{ref}} \tag{12}\]  
where \(Rad_{ref}\) is the mean of \(Rad(\lambda,D,V)\) with \(V\geq 1000\) m.  
From the ratios previously calculated, it is now possible to go back to the value of the \(k\) coefficient for each pair \((\lambda,V)\).