## 3 Simulation Results  
### Simulation Comparison between Inverse Function for HG cdf and LUT Method for FF cdf  
The inverse equation of HG cdf for scatter angle \(\psi\) is  
Figure 3: Lidar geometric model for analytically estimating the probability of photon scatter back to a remote receiver.  
\[\psi=\cos^{-1}[\frac{1+g^{2}}{2g}-\frac{1}{2g}(\frac{1-g^{2}}{1+g-2g\Re})^{2}] \tag{11}\]  
Figure 4 shows the simulation comparison between the inverse cdf method for HG and the LUT method for FF. Simulations were conducted using a total of 10,000,000 photons for a given detector height of \(H=300\) m, \(FOV=50\) mrad, detector aperture of \(A=0.06\) m\({}^{2}\), sea bottom depth of \(d=40\) m, vertical layer resolution of 0.11 m, water absorption coefficient of \(\alpha=0.04\) m\({}^{-1}\), scattering coefficient of \(b=0.06\) m\({}^{-1}\), and sea bottom albedo of \(\rho_{b}=0.03\). The given \(n\) and \(\mu\) for FF spf were 1.10 and 3.62, respectively. The HG parameter was set to 0.924. It took 129 s and 140 s for the LUT method and ICDF method simulations, respectively. It shows that the simulation curve using the LUT method of FF is larger and more stable than using the ICDF method of HG under the same input condition. It may be because the forward scattering of HG spf is much less than that of the [PERSON]'s measurements (see Figure 1). Besides, the back scattering is also less; therefore, the p(\(\theta\)) value in Equation (2) is less. We can see that the black curve by LUT method looks smoother than the red curve by the inverse function method. The adjusted R-square (N = 498) between the two curves in Figure 1 is 0.948. The residual sum of squares is 3.2 \(\times\) 10\({}^{-8}\). The RMSE is 8.03 \(\times\) 10\({}^{-6}\). The min value, max value, mean value, and standard deviation of the red curve are 1.54 \(\times\) 0\({}^{-8}\), 7.70 \(\times\) 0\({}^{-4}\), 9.82 \(\times\) 0\({}^{-5}\), and 1.73 \(\times\) 0\({}^{-4}\), respectively. The min value, max value, mean value, and standard deviation of the black curve are 1.11 \(\times\) 10\({}^{-9}\), 1.63 \(\times\) 10\({}^{-4}\), 2.48 \(\times\) 10\({}^{-5}\), and 3.53 \(\times\) 10\({}^{-5}\), respectively (see Table 2). The signal intensity by LUT method appears higher than ICDF method: This is because the ICDF method to calculate the backscattering probability return to receiver was to obtain a possible certain backscattering angle, which may be less than the small receive angle range, while the LUT method to calculate the backscattering probability return to receiver was to obtain the integrals between two discrete values, which is a little larger than the small receive angle range. The standard deviation of the red curve by ICDF method is more than 5 times larger than the black curve by LUT method. It looks like there is more noise and disturbance in the red curve by ICDF method under the same preset parameters. The results also indicate the LUT method is more stable than the inverse function method. We also find that the sea bottom curve based on the LUT method for FF cdf has less pulse widening effect than the inverse function method for HG cdf.  
Figure 4: Simulation comparison between inverse equation for Henyey-Greenstein (HG) cumulative distribution function (cdf) and lookup table (LUT) method for Fournier-Forand (FF) cdf. The black curve is based on the LUT method for FF cdf, and the red curve is based on the inverse function for HG cdf.