## 2 Related Works
### Air Channel Model Simulating a Maritime Communications Environment  
To accurately replicate the realistic communication environment between sonobuoys and MPA, we employed an air channel model that combines free-space path loss and the Curved-Earth Two-Ray Model [23]. This model simulates multipath effects and signal attenuation by considering the curvature of the earth and ground surface reflections. Moreover, the model is advantageous for replicating the signal path characteristics that occur in maritime communication environments by incorporating various paths and surface reflection effects.  
The free-space path loss model predicts signal attenuation in an ideal linear path between the transmitter and receiver, inversely proportional to the square of the distance, thereby estimating signal loss. However, this model does not sufficiently account for practical complexities, such as ground surface reflections, which occur in marine environments. To address these limitations, we incorporated the Curved-Earth Two-Ray Model.  
The Curved-Earth Two-Ray Model divides the signal paths into two routes that account for the curvature of the earth: the LOS path and the NLOS ground reflection path. By considering the path differences caused by reflections, the model simulates multipath interference and signal attenuation. It calculates the characteristics of the signal path, including the time delays and phase differences of the two paths. The primary equations of the model are as follows:  
\[h_{2 Ray}(t,\tau)=\alpha_{0}(t)\text{exp}(j\varphi_{0}(t))\delta(\tau-\tau_{0}(t ))+\alpha_{1}(t)\text{exp}(j\varphi_{1}(t))\delta(\tau-\tau_{1}(t))\, \tag{1}\]  
where \(t\) represents the continuous time variable, describing the dynamic behavior of the signal over time, while \(\tau\) refers to the delay time, which indicates the time it takes for the signal to travel along a specific path. Specifically, the delay times of the LOS path, \(\tau_{0}(t)\) and the NLOS reflection path \(\tau_{1}(t)\) are expressed as follows:  
\[\tau_{0}(t)=\frac{R_{1}}{c}\,\ \tau_{1}(t)=\frac{R_{2}}{c}, \tag{2}\]  
where \(R_{1}\) and \(R_{2}\) represent the LOS and NLOS path distances and \(c\) is the speed of light.  
The amplitude of the LOS signal, \(\alpha_{0}(t)\) is influenced by free-space path loss and is expressed as:  
\[\alpha_{0}(t)=\frac{c}{4\pi f_{c}R_{1}}, \tag{3}\]  
where \(f_{c}\) is the carrier frequency. In contrast, the amplitude of the reflected signal \(\alpha_{1}(t)\) is influenced by a combination of the reflection coefficient \(\Gamma\), the surface roughness attenuation factor \(r_{F}\), and the divergence factor \(D_{k}\), and is expressed as:  
\[\alpha_{1}(t)=\frac{c}{4\pi f_{c}R_{2}}\cdot\Gamma\cdot r_{F}\cdot D_{k}\, \tag{4}\]  
where the reflection coefficient \(\Gamma\) accounts for the sea surface properties, specifically the relative permittivity of seawater \((\varepsilon_{r}=81)\), the conductivity \((\sigma=5\,\mathrm{S}/\mathrm{m}\), Siemens per meter) [25], and the incidence angle. Meanwhile, the surface roughness attenuation factor \(r_{F}\) quantifies the signal loss caused by the roughness of the sea surface, while the divergence factor \(D_{k}\) incorporates the geometric spreading effect of the curved reflection path [23; 26]. The phase variations of the LOS and NLOS paths are represented as \(\varphi_{0}(t)\) and \(\varphi_{1}(t)\), respectively, and are calculated based on the path lengths \(R_{1}\) and \(R_{2}\). These phase shifts are expressed as follows:  
\[\varphi_{0}(t)=-\frac{2\pi f_{c}R_{1}}{c},\ \varphi_{1}(t)=-\frac{2\pi f_{c}R_{2 }}{c}, \tag{5}\]  
Together, these parameters delay times, amplitudes, and phase variations, enabling a more realistic representation of signal attenuation and multipath interference in maritime communication environments.