## 4 Results and Discussion
However, owing to the considerable drift caused by spatial sea surfaces, the average power of the spatial model may always be lower than that of the statistical model. Owing to the distortion effects induced by the spatial model, a suitable positioning method must be adopted to enhance the energy features and anti-interference capabilities  
Figure 7: **(a-l)** Cross-sectional power distribution along the \(x\)-axis for wind speeds of (left to right) 1, 5, 10 and 15 m s\({}^{-1}\) and underwater transmission depths of (top to bottom) 20, 60 and 100 m, respectively. The shaded area shows the statistics after 50 simulated spatial sea surface topography transmissions with the mean power (solid line) in simulations. And the black dashed line is the result of a statistical sea surface, corresponding to Figure 6a. of the spot center. This study uses the square-weighted centroid positioning method to quantify the deviation of the laser spot [50]. The centroid (x\({}_{c}\), y\({}_{c}\)) can be expressed as  
\[x_{c}=\frac{\sum_{i,j}^{N_{i}N_{j}}{w_{i,j}}^{2}{x_{i,j}}}{\sum_{i,j}^{N_{i}N_{j} }{w_{i,j}}^{2}} \tag{17}\]  
and  
\[y_{c}=\frac{\sum_{i,j}^{N_{i}N_{j}}{w_{i,j}}^{2}{w_{i,j}}}{\sum_{i,j}^{N_{i}N_{ j}}{w_{i,j}}^{2}} \tag{18}\]  
where \(w_{i}\) is the energy weight at the coordinate (\(x_{i,j}\), \(y_{i,j}\)). Figure 9 shows the spatial distribution of the centroid and the offset distance calculated in this simulation experiment. As the wind speed increases, the centroid of the spot shifts further from the initial emission position. Furthermore, enhancing the underwater transmission depth can decrease the extent of laser deviation, as corroborated by the results in Figure 7. Notably, as depicted in Figure 9a-d, the distribution of the centroid of the energy of the laser spot in space presents an elliptical structure rather than an ideal circular symmetric structure. To analyze this phenomenon, the azimuth directional statistics of the two sea surface models are separately computed and depicted in Figure 10.  
Figure 8: (**a–i**) Examples of spatial distribution of beam power across simulated sea surface at various moments for a wind speed of 1 m s\({}^{-1}\) and an underwater transmission depth of 20 m. (**j**) Boxplot of power error at (x = 0, y = 0), corresponding to Figure 7a–d, for wind speeds of 1, 5, 10 and 15 m s\({}^{-1}\) and an underwater transmission depth of 20 m. And stars indicate the results from the statistical model. Due to the simulation parameter settings, the spatial model resolutions are higher at low wind speeds, resulting in sharp jittering of the simulated sea surface at 1 and 5 m s\({}^{-1}\) compared to that at 10 and 15 m s\({}^{-1}\). And the high wind speeds mainly reflect the statistical distribution trend of the sea surface angle. The spatial model exhibits an anisotropic \(W\)-shaped distribution of the azimuth, aligning with the fundamental sea surface slope model and capturing the directional variability [16; 51]. The statistical model, on the other hand, assumes an isotropic sea surface distribution. The \(W\)-shaped distribution of sea surface directions effectively corresponds to the elliptical distribution, highlighting the directional characteristics of the sea surface. Additionally, as wind speed and underwater transmission depth increase, the coordinate distribution gradually approaches isotropy, aligning with the assumptions of statistical modeling [52]. At low wind speeds, this study improves the performance of MSS estimation by refining truncation errors in the Elfouhaily spectrum. This enhancement is particularly important for optical applications, where surface roughness plays a critical role in beam scattering and reflection properties. As a result, the simulated sea surface demonstrates strong agreement with the existing empirical models and remote sensing observations under typical ocean conditions, making it suitable for simulating a wide range of realistic virtual scenarios.