### Modified Rel Model and Proposed Piecewise Approach  
This paper proposes a Modified REL model to accurately model reflections over ground and antenna gains based on the simplified REL model. While the REL model generally provides a structure that considers surface roughness and reflections, the inclusion of parameters, such as antenna gains and best Rx height, into the model significantly increases the estimation accuracy. The prediction of the reflections and losses of a signal on a rough surface considering the curvature of the Earth is the main purpose of the REL model. The most general expression for the pattern propagation (path gain) factor (\(F\)) in the REL model is as follows:  
\[F=\left|1+\Gamma_{eff}\cdot Div\cdot S\cdot\frac{G_{ref}D}{G_{dir}R}e^{-jk \Delta d}\right| \tag{1}\]  
where \(D=\sqrt{(h_{t}-h_{r})^{2}+d^{2}}\) and \(R=\sqrt{(h_{t}+h_{r})^{2}+d^{2}}\) express direct and reflected wave paths, respectively. \(h_{t}\) and \(h_{r}\) are the Tx and Rx antenna heights, respectively, and the \(\Delta d\) parameter represents the distance difference between the directly transmitted signal and the reflected signal (\(R-D\)), which plays a decisive role in the phase shifts and interference of the signals. \(G_{ref}\) and \(G_{dir}\) represent the angle-dependent gains of reflected and direct waves, respectively. The angle-dependent gains are expressed as \(G_{ref}=G_{tr}G_{rr}\) and \(G_{dir}=G_{td}G_{rd}\), where \(G_{tr}\) and \(G_{rr}\) represent the angle-dependent gains of the Tx and Rx antennas, respectively, for the reflected wave, and \(G_{td}\) and \(G_{rd}\) represent the angle-dependent gains for the direct wave. \(\Gamma_{eff}\) refers to the effective (modified) reflection coefficient obtained by multiplying the Fresnel reflection coefficient (\(\Gamma\)) by the roughness coefficient (\(\rho_{s}\)). \(\rho_{s}\) is calculated by using the zero-order modified Bessel function (\(I_{0}\)) and the surface roughness (\(g\)) as follows:  
\[\rho_{s}=\exp\left[-2(2\pi g)^{2}\right]\cdot I_{0}\left(2(2\pi g)^{2}\right),\quad g=\frac{\sigma_{h}\cdot\sin\psi}{\lambda} \tag{2}\]  
where \(\sigma_{h}\) shows Root Mean Square (RMS) surface height, \(\psi\) is grazing angle, and \(\lambda\) is the wavelength of the signal. Divergence (Div.) considers the effects of the curvature of the Earth during the propagation of the signal over the surface, and helps model the deviations that the signal undergoes until it reaches the Rx as follows:  
Figure 2: The Rx routes and sample coastal photos for all scenarios. \[Div=\frac{1}{\sqrt{1+\frac{2G_{1}G_{2}}{a_{e}C\sin\zeta}}},\quad\xi= \sin^{-1}\Bigg{(}\frac{2a_{e}h_{t}+h_{t}^{2}-R_{1}^{2}}{2a_{e}R_{1}}\Bigg{)}, \quad\phi_{i}=\frac{G_{i}}{a_{e}},i=1,2 \tag{3}\] \[R_{1}=\sqrt{h_{t}^{2}+4a_{e}(a_{e}+h_{t})\sin^{2}\!\left(\frac{ \phi_{1}}{2}\right)},\quad R_{2}=\sqrt{h_{t}^{2}+4a_{e}(a_{e}+h_{r})\sin^{2}\! \left(\frac{\phi_{2}}{2}\right)}\]  
Here, \(a_{e}\) = 6370 km represents the effective Earth radius, \(G\) represents the distance between the Rx and the Tx considering the curvature of the Earth, and \(G_{1}=G/2+p\cos((\pi+\alpha)/3)\) and \(G_{2}=G-G_{1}\) represent the distances between the reflection point and the Tx and Rx, respectively. Here, the propagation distance was taken as \(G\cong d\) since it is relatively short.