## 2 System Model
Assuming further that the transmitter uses the simple on-off keying (OOK) intensity modulation to transmit useful signals while the receiver adopts direct detection to make decisions, the received current signal of an APD detector after considering the channel path loss, the photo-current shot noise and thermal noise [13; 15], has the following expression as  
\[Y=GL\tau x+\sqrt{GL\tau x}n^{\prime}+n, \tag{1}\]  
where \(G\) is the average gain of APD; \(L\) is the channel path loss in clear ocean water; \(\tau=\eta e/\hbar v\) is the photoelectric transformation coefficient, in which \(\eta\) is the quantum efficiency, \(h\) is the [PERSON]'s constant, \(v\) is the frequency of light wave in seawater, and \(e\) is the electron charge; \(x\) denotes the transmitted light-intensity signal, i.e., the OOK symbol, which is assumed to be drawn equal-probably from an OOK modulation constellation, that is, \(x\in\{0,P_{\mathrm{s}}\}\), where \(P_{\mathrm{s}}\) is the average transmitting power; \(\sqrt{GL\tau x}n^{\prime}\) is the so-called photo-current shot noise term [13], which is proportional to the received average current signal value, and \(n^{\prime}\) is a Gaussian white noise random variable with mean zero and variance \(2 eBGF_{1}\), in which \(B\) is the filter bandwidth, \(F_{1}=\zeta G+(2-1/G)(1-\zeta)\) is the excess noise factor, and \(\zeta\) is the ionization ratio; \(n\) represents the background noise of APD, which is independent of noise \(n^{\prime}\) and can be modeled by the thermal noise of the load resistance of amplifier circuit as a Gaussian white noise term with mean zero and variance \(\sigma^{2}\), i.e., reference [13]  
\[\sigma^{2}=\frac{4 KTBF}{R}, \tag{2}\]  
Figure 1: Geometric configuration model for UWOC system. where \(K\) is the Boltzmann constant, \(T\) is the temperature in Kelvin, \(B\) is the filter bandwidth mentioned above, \(R\) is the resistance value of the load, and \(F\) is the noise factor of the system. For simplicity of description below, we further record the ratio of the variance of input-dependent noise term \(n^{\prime}\) to that of the background noise term \(n\) as \(\varsigma^{2}\), where \(\varsigma^{2}=2 eBGF_{1}\), so the variance of input-dependent noise term \(n^{\prime}\) can be simply expressed as \(\sigma^{2}\varsigma^{2}\). Please note that although the spectral efficiency of OOK modulation is only \(1\text{bit/s}/\text{Hz}\) compared with other higher order modulation schemes [20], for example, quadrature phase shift keying (QPSK), \(M\)-ary quadrature and amplitude modulation (\(M\)-QAM), etc., however, its simple detection method and easy implementation for intensity modulation make it widely adopted and used in the research of optical communication community. In fact, due to the very high carrier frequency of light beam, the transmission data rate of UWOC system even with OOK modulation can reach Giga bits/second with quite low BER detection performance [3; 6]. In (1), the channel path loss \(L\) can be described as [4]  
\[L=\tau_{channel}\cdot\tau_{point}, \tag{3}\]  
where \(\tau_{channel}\) is the channel loss, which is from the absorption and scattering due to seawater. Compared with the conventional [PERSON]'s law, a double-exponential channel loss model can more accurately depict the channel fading in a clear ocean water type [14; 21]. Therefore, it can be modified and depicted as  
\[\tau_{channel}=C_{1}e^{C_{2}d}+C_{3}e^{C_{4}d}, \tag{4}\]  
where \(C_{1}\), \(C_{2}\), \(C_{3}\), and \(C_{4}\) are the fitting coefficients obtained by Monte Carlo simulations; \(d\) is the propagation path from the light source to receiver depicted in Figure 1. However, when misalignment deployment of the receiver and transmitter occurs, the pointing loss must be considered.