## 3 Simulation Results
### Simulated-Signal vs. Airborne Lidar Measured-Signal  
Subsequently, simulation results (from both LUT and ICDF methods) were compared with airborne lidar measurements. The experimental airborne oceanic lidar was developed by the Shanghai Institute of Optics and Fine Mechanics (SIOM), Chinese Academy of Sciences (Figure 6a). The transmitting system utilized a 532 nm frequency-doubled Nd:YAG laser. Its pulse duration is 1.5 ns, and pulse repetition rate (PRF) is 1 kHz. The detector telescope's diameter is 200 mm and FOV is 40 mrad. The layer thickness employed in our simulation is 0.11 m; the phase function lookup table was kept unchanged during a simulation process. The profile optical properties of sea water measured at the station (109\({}^{\circ}\)43.808\({}^{\prime}\) E, 18\({}^{\circ}\)2.205\({}^{\prime}\) N) in Sanya Bay in the South China Sea on 12 March 2018 were input into the model, as shown in Figure 6b. The green line is the chlorophyll \(a\) concentration profile, and the black line represents water attenuation coefficient profile. We can see that a subsurface chlorophyll maximum (SCM) layer exists at about 30 m. Clearly, most current MC simulation methods based on the assumption of optically vertical homogeneous are no longer suitable. Figure 6c shows the comparison between simulation signal based on our method with airborne lidar measured-signal. The simulations were conducted using a total of 10,000,000 photons and actual lidar detector height is \(H\) = 307 m. We find that the simulated curve using the LUT method (blue line) agrees well with the lidar measured-signal (red line). The ICDF method appears to overestimate the actual signal, which is because the ICDF method used to calculate the backscattering probability return to receiver obtained a possible certain backscattering angle, which is less than small receive angle range. The LUT method appears to underestimate the actual signal at greater depths and also at shallower depths, crossing at  
Figure 5: Wind-driven roughened sea surface and multiple scattering have significant effects on simulation results in logarithmic form: (**a**) Separation of contributions by different scattering order in the total signal, (**b**) field of view (FOV) influence on return signal, (**c**) influence of wind-driven roughened sea surface various incident angles on transmittance from air to sea, and (**d**) influence of wind-driven roughened sea surface various incident angles on transmittance from sea to air. about 20 m. This is because the LUT method used to calculate the backscattering probability return to receiver obtained the integrals between two discrete values, which is a little larger than small receive angle range at deeper depths. It may be better to set the LUT increments according to the solid receiver angle. Figure 6d shows the relative errors between simulation and measurements for the LUT method within 14% mean relative error and for the ICDF method within 22% mean relative error. We can see that the relative error was less than 10% for the LUT method and less than 20% for the ICDF method within 20 m. Both methods have the maximum relative errors when the depth is between 25 and 30 m, where there was a chlorophyll maximum scattering layer. It may be due to the fact that the spfs in our study are no longer applicable when reaching the chlorophyll maximum scattering layer. Figure 7 shows the regression plots of simulated-signal vs. airborne lidar measured-signal. Figure 7a,b presents the regression plots of ICDF and LUT retrieved data vs. measured data, respectively. Figure 7c,d presents the residual plots of ICDF and LUT, respectively. It shows that the adjusted R-square between simulation and measurements (N = 347) for LUT method (R\({}^{2}\) = 0.996) is larger than that by ICDF method (R\({}^{2}\) = 0.996). The data retrieved from both methods have significant correlations with measured data, although the RMSE by LUT method (RMSE = 0.016) is relatively smaller than that by ICDF method (RMSE = 0.052).