# Beyond the Horizon, Backhaul Connectivity for Offshore IoT Devices
## 2 Methodology
### Additional Random Slow Fading \(\psi_{dB}\)  
The slow fading in ED is climate-induced and relates to the percent of ED height occurrence. It is the combination of slow path loss fading and fast-instantaneous fading. The slow path loss fading distributions can be determined by understanding the changes in the average path loss value. With the change in ED height, the average path loss value also changes. The ED height variations are measured in a new climatology of South China Sea showing ED height profile in [15] as shown in Figure 4a. It is also evident that the changes in average path loss values due to slow fading are not related to distance. Instead, these are temporal variations. The transmitter and receiver are fixed and the change in average pathloss is only due to changes in ED characteristics, mainly its height. The fast-instantaneous fading on top of the slow fading is because of multipath effect within the channel created by sea wave movement. To determine complete fading characteristics in ED, one needs to know the average path loss values with respect to distance along with the additional weather-dependent slow and fast fading due to multipath. The average path loss measurements help to determine the path loss exponent and type of fading. PE-tool based simulations using known ED climatology of the South China Sea, as shown in Figure 4b, can determine mean path loss and complete path loss variations. These slow path loss variations can be calculated using the best-fit distribution. The data acquired from simulations can be approximated with three-parameter (3-P) lognormal distribution. The probability density function for 3-P lognormal distribution is given in Equation (2),  
\[f(x)=\frac{(x-\gamma)^{-1}}{\sigma\sqrt{2\pi}}exp\biggl{(}-\frac{1}{2\sigma^{ 2}}\left[\ln(x-\gamma)-\mu\right]^{2}\biggr{)} \tag{2}\]  
Figure 4: (**a**). Annual average evaporation duct height EDH variations and percentage of occurrence in the South China Sea [15] (**b**). Path loss in dB for varying distance due to EDH variations causing additional slow fading. where \(x\) is in dB and \(\sigma=\) continuous shape parameter (\(\sigma>0\)), representing swaying of the peak on the left. \(\mu=\) scale parameter, represents width of peak that stretch or shrink the distribution. \(\gamma=\) location parameter, is the starting location of the data (\(\gamma\equiv 0\) yields the 2-P). The shape of the pathloss distribution depends primarily on two parameters: \(\mu\) and \(\sigma\). We have calculated values of both parameters at each distance. The probability distribution of 3-P lognormal distribution is same as the 2-P lognormal distribution except the third location parameter, which depends on the distance between the transmitter and the receiver. For demonstration, a 2-P lognormal distribution at 40 km distance is shown in Figure 5. The average path loss varies up to 42 dB due to climate-induced changes in the ED. To calculate fade margin in addition to the average path loss value, the third parameter \(\gamma\) can be neglected as in,  
\[x^{\prime}=x-\gamma \tag{3}\]  
Therefore, the equation can be represented as 2-P lognormal distribution,  
\[f(x)=\left\{\frac{1}{x^{\prime}\sigma\sqrt{2\pi}}exp\left(-\frac{1}{2}\left[ \frac{ln(x^{\prime})-\mu}{\sigma}\right]^{2}\right)\right\} \tag{4}\]  
The equation depends on the lognormal values of \(\mu\) and \(\sigma\).