## 1 Introduction  
Over the past few years, lidar was widely used in ocean exploration, including plankton layer detecting [1; 2; 3], optical profiling [4; 5; 6; 7], bathymetric survey [8; 9], and so on. This is largely because blue-green lidar is the only technology that can currently produce vertical profiles of the ocean from above the sea surface [10; 11]. Meanwhile, many simulation models and inversion methods were developed such as Monte Carlo radiative transfer models [12; 13; 14] and the Wa-Lid model based on the lidar equation [15; 16]. Owing to the high polarization degree of lidar, the depolarization characteristics of return signals can provide extra information for marine particles [17]. In order to determine the depolarization induced in water, depolarization of the air-sea interface needs to be separated first. Therefore, it is necessary to analyze the depolarization of the sea surface. In fact, vector radiative transfer (VRT) over the sea surface was widely discussed by researchers in the passive remote sensing field [18; 19; 20], and an inelastic radiative process code was also achieved [21]. These models often expand the azimuthal dependence of the VRT equation into the Fourier series to reduce the computational cost [22], and the magnitude of depolarization by the rough sea surface is not calculated directly. Considering the small view angle of the lidar receiver, the reflection and transmission of the sea surface at a certain zenith and azimuth are indispensable. In this case, the methods of simulating the wind-blown ocean surface in passive remote sensing are not compatible with the propagation of lidar across the air-sea interface. In previous studies [21; 23; 24] on the polarization properties of sea surfaces, researchers usually focused on reflectance, while depolarization was not analyzed systematically. In general, simulation models for lidar hardly take rough surfaces into account or simplify the surface model; furthermore, VRT in passive remote sensing cannot be directly applied for oceanographic lidar.  
In this study, we use the Stokes vectors to denote the lidar incident light, and we calculate the full \(4\times 4\) reflection and transmission matrices for lidar interacting with a rough sea surface. Firstly, we approximate the ocean surface roughness with the isotropic Cox-Munk model [25; 26]. After obtaining the wave slope probability distribution, we calculate the reflection and transmission matrices of a small wave facet. Subsequently, we integrate azimuth and zenith to obtain the Muller matrix at a certain view angle. The reflected light and transmission light of lidar interacting with a rough sea surface could then be derived by multiplying the Muller matrix and Stokes vectors of the incident light. In particular, sphere energy patterns are plotted, making it convenient to analyze the polarized reflection and transmission properties of a wind-blown sea surface for oceanographic lidar. To quantify the magnitude of depolarization by the rough sea surface, the depolarization rate of reflection and transmission is calculated from Stokes vector parameters. The results show that the depolarization caused by sea surface is not negligible.