## 2 Monte Carlo-Based Channel Parameter Characteristics Analysis  
### Underwater Channel Modeling Based on Monte Carlo  
Figure 1 shows the schematic of a Gaussian modulated coherent state (GMCS) CVQKD system over an underwater link, which was proposed by [PERSON] and [PERSON] in 2002 [13] and has been proven to be secure against general collective attacks and coherent attacks [24; 25; 26]. In this system, the sender Alice separates coherent light pulses generated by a laser diode into weak signal pulses and strong local oscillator (LO) pulses. Then the signal pulses are loaded with key information by using an amplitude modulator and a phase modulator, so that the \(X\) and \(P\) quadratures of the coherent state respectively follow a centred Gaussian distribution with a variance of \(V_{A}\). The LO pulses are delayed by a delay line so as to be transmitted together with the signal pulses through time multiplexing. At the receiving end, [PERSON] first separate the LO and signal pulses by using a beam splitter and then randomly measures one of the quadratures of the signal states by homodyne detection, with the help of the LO as a phase reference. The phase modulator in [PERSON] is used to randomly select the quadrature by imposing a \(\pi/2\) phase shift to the LO. During this process, the pulses propagating in seawater are affected by two main factors: absorption and scattering. Absorption depends on the water's index of refraction and leads to energy loss of the light, while scattering changes the direction of photons and leads to broadening of a pulse [27]. Both absorption and scattering are concerned with the turbidity of the water and the type of particles in water. The total effects of these two process can be described by beam extinction coefficient \(c\), which is defined as \(c(\lambda)=a(\lambda)+b(\lambda)\), where \(\lambda\) is the wavelength of the light, \(a(\lambda)\) is the absorption coefficient and \(b(\lambda)\) is the scattering coefficient. As a common practice, the chlorophyll concentration \(C\) (in mg.m\({}^{-3}\)) is employed as the free parameter to compute the values of \(a(\lambda)\) and \(b(\lambda)\), based on bio-optical models proposed in [28; 29]. The behaviour of light propagation in a medium is described by the radiative transfer equation (RTE), which can be written as [27]  
\[\frac{dL(z,\theta,\phi,\lambda)}{dr}=-c(\lambda)L(z,\theta,\phi,\lambda)+L^{E} +L^{I}, \tag{1}\]where \(L(z,\theta,\phi,\lambda)\) denotes the light radiance at a point in units of \(\text{Wm}^{-2}\text{sr}^{-1}\text{nm}^{-1}\), \(z\) is the distance of the point from the transmitter and \(r=z/\text{cos}\theta\) represents the radial distance to the source, \(\theta\) and \(\phi\) are the polar and azimuthal angles, respectively. The first term in Equation (1) accounts for the annihilation of photons from the beam, the second and third terms denote the elastic scattering and inelastic scattering process, respectively. The effect of inelastic scattering is generally neglected because of its relatively low contribution to the solution of the RTE. Many previous works also neglect the contribution of elastic scattering and consider a straight-line propagation, describing the RTE by the simple [PERSON]'s law, as  
\[L(z)=L(0)exp(-cz). \tag{2}\]  
Nevertheless, the received optical power may be severely underestimated in this way because it assumes that all the photons undergoing scattering are lost. In practice, the scattered photons may still be received, especially when the water has a high scattering coefficient. Therefore, in this paper we take the scattered photons into account and use a Monte Carlo method to solve the RTE. Monte Carlo simulation is a rigorous and flexible method for modeling the underwater light propagation and has been adopted in many underwater optical communication research [30; 31; 32; 33].