## 2 Principle
### Noise Modeling  
The UOWC system noises can be categorized into two parts, electrical noise and solar noise arising from the Rx components and ambient sunlight respectively. The electrical noise sources mainly consist of photocurrent shot noise, dark current noise, and thermal noise [25; 26]. The shot noise is caused by random incident photons and the generated current flowing through the device, which is generally modeled as a Poisson distribution random process. Nevertheless, when the received optical signal power maintains a high level, that is to say, there is a significant increase in the incident photons, the Poisson distribution-based shot noise can be approximated to a Gaussian distribution model. Thus, its variance is expressed as:  
\[\sigma_{sn}^{2}=2 eBMF(M)(I_{os}+I_{sr}) \tag{5}\]  
where \(e\) denotes the electron charge, \(B\) is the bandwidth, \(M\) represents the multiplication factor, \(F(M)\) is the excess noise figure related to \(M\) and the excess noise index, \(I_{os}\) and \(I_{sr}\) indicate the photocurrent generated at Rx from the optical signal and solar radiation, respectively. Similarly, the dark noise variance is given by:  
\[\left\{ \begin{array}{c}\sigma_{dn}^{2}=2 eBM^{2}F(M)I_{db}+2 eBI_{ds}\\ I_{dark}=I_{ds}+MI_{db}\end{array} \right. \tag{6}\]  
where \(I_{dark}\) is the total dark current, \(I_{db}\) and \(I_{ds}\) represent the dark bulk and surface leakage currents, respectively. For a PIN-Rx, the multiplication factor \(M_{PIN}\) and noise figure are always equal to one. In terms of an APD-Rx, these two factors are dependent on the reverse voltage, we commonly have \(10<M_{APD}<200\), \(F(M)_{APD}=\vartheta M_{APD}(1-\vartheta)(2-1/M_{APD})\), in which \(\vartheta\) is the holes/electrons ionization rate. The thermal noise mainly arises from the trans-impedance \(R_{L}\) and can be classically modeled as a zero-mean Gaussian random process given by:  
\[\sigma_{tn}^{2}=4 KTB/R_{L} \tag{7}\]  
in which \(K\) and \(T\) denote the Boltzmann constant and the temperature in Kelvin, separately. To evaluate the ambient light noise impact, we should first quantify the solar radiation level at a certain depth below the sea surface. In ocean optics, the ambient light becomes cumulatively diffusive and attenuates approximately exponentially with the depth after permeating into the seawater [27]. The downwelling plane irradiance is assumed as follows using the K function:  
Figure 3: Multiple scattering path of each photon. \[E_{d}(z,\lambda)=E_{d}(0,\lambda)exp\left[-\int_{0}^{z}K_{d}(z^{\prime},\lambda) dz^{\prime}\right] \tag{8}\]  
where \(E_{d}(0,\lambda)\) denotes the sea surface direct irradiance in \(W\cdot m^{-2}\cdot nm^{-1}\), \(K_{d}(z,\lambda)\) is the diffuse attenuation coefficient in \(m^{-1}\), \(\lambda\) is the wavelength in \(nm\). Although \(K_{d}(z,\lambda)\) is always related to depth and the wavelength, it is pretty complicated to derive the precise value at each level of depth and the accurate theoretical modeling of \(K_{d}(z,\lambda)\) is out of the scope of this work. In addition, \(K_{d}(z,\lambda)\) is independent of depth under several typical conditions such as in homogeneous water or far enough above the bottom in optically shallow water where the incident lighting mainly comes from the sun and sky [28]. Thus, for the convenience of estimation, the \(K_{d}(z,\lambda)\) is assumed to be a constant at a certain depth range and Equation (8) can be simplified as \(E_{d}(z,\lambda)=E_{d}(0,\lambda)exp[-K_{d}(\lambda)z]\). The experimental measurement of \(K_{d}(\lambda)\) and \(E_{d}(0,\lambda)\) has been conducted in several works [28; 29; 30]. The exact value of \(E_{d}(0,\lambda)\) exhibits a huge difference under different climate conditions (i.e., clear or hazy) and the sun locations (i.e., zenith and azimuth angle). To present a worst-case analysis, we consider the clear weather and sun at its zenith based on the measured data in [24; 28].