## 5 System Model
### Ambient Noise  
The downwelling irradiance (\(E_{t}\)) is the integral of all spectral irradiances in W m\({}^{-2}\) (radiometric), expressed in (19). The background power experienced by a receiver of surface area \(A_{r}\) is derived using (20).  
\[E_{t}=\int E(\lambda)\ d\lambda \tag{19}\]  
\[P_{BG}(S,\lambda,z)=\frac{\pi A_{r}\Omega^{2}T_{f}\int_{\Delta}\left(L_{sol}( S,\lambda,z)+L_{bb}(S,\lambda,z)\right)\ d\lambda}{4} \tag{20}\]  
where \(\Omega\) is the plane angle of the field-of-view (FOV) of the receiver in radians, \(T_{f}\) and \(\Delta\) are the optical filter transmissivity and bandwidth, respectively, \(L_{sol}\) is the upwelling solar spectral radiance, and \(L_{bb}\) is the blackbody radiation of the water body. \(c_{d}\) is the diffuse attenuation coefficient, an AOP that describes how a background light field may attenuate in water expressed from IOP using (21) [31]. Equation (4), which corresponds to the light  
Figure 7: Change in attenuation coefficient with depth and wavelength in (**a**) clear ocean profile S2 and (**b**) coastal profile S8. Amber is preferred at the surface, shifting to blue-green with depth in S8.  
intensity at a location, may be thereafter adapted for the attenuation of downwelling irradiance in a stratified environment using (22), where \(n\) is a layer of unit depth in meters and \(z\) is the receiver depth. Therefore, \(L_{sol}\) may be derived for each profile with (23), where \(R_{f}\) is the underwater reflectance that expresses a ratio between downwelling and upwelling irradiances correlated to the backscattering coefficient with (24) [59]. \(L_{f}\) is the radiance factor, expressed as the ratio between the radiance observed at an angle of interest (receiver orientation) with respect to the nadir, or looking vertically down. The corresponding values for a horizontal link and downlink (receiver looking up) were extracted from the sources for oceanic (S1-S4) and coastal (S5-S8) waters from Duntley [48], Hoejerlev and Aas [50], and Aas and Hoejerslev [60].  
\[c_{d}(S,\lambda,z)\approx\ \frac{c(S,\lambda,z)}{3} \tag{21}\]  
\[E_{t}=\int_{\Delta}E_{0}(\lambda)\prod_{n=1}^{z}e^{-c_{d}(S,\lambda,z_{n})}\ d\lambda \tag{22}\]  
\[L_{sol}(S,\lambda,z)=\frac{E_{0}(\lambda)R_{f}(S,\lambda,z)L_{f}\prod_{n=1}^{ z}e^{-c_{d}(S,\lambda,z_{n})}}{\pi} \tag{23}\]  
\[R_{f}(S,\lambda,z)\approx\ \frac{b_{B}(S,\lambda,z)}{3a(S,\lambda,z)} \tag{24}\]  
Blackbody radiation is described as the EM radiation emitted by an object at a particular temperature on the Kelvin scale, idealized for a perfect black object. It is expressed by (25), where \(h\) is [PERSON]'s constant (6.626 \(\times\) 10\({}^{-34}\) J s), \(v\) is the speed of light in water (2.253 \(\times\) 10\({}^{8}\) m s\({}^{-1}\)), \(k_{B}\) is the Boltzmann constant (1.381 \(\times\) 10\({}^{-23}\) J K\({}^{-1}\)), \(T\) is the temperature of the body in Kelvin, and \(\alpha\) is the radiance absorption factor (0.5) [19]. Given that the underwater environments are about 300 K or lower, much of this radiation available is in the infrared spectrum.  
\[L_{bb}(S,\lambda,z)=\frac{2 hv^{2}\alpha}{\lambda^{5}\left(e^{(\frac{hv}{k_{B }T(S,z)})}-1\right)} \tag{25}\]