## 3 Results and Discussion
### Verification of the Fast Calculation Model with PSF Convolution
#### 3.3.3 Results Verification of Uplink Transmission Scenario Results  
In order to further verify the performance of the convolution fast calculation method, we further study the scene of transmission at a certain elevation angle. The laser is transmitted 10 km obliquely upward at an elevation angle of 30 degrees from the ground. For a horizontal transmission path, the turbulent refractive index structure parameter is usually assumed to be a constant. For a slant transmission path, a turbulence profile model is required. There are already many atmospheric turbulence profile models [35; 36; 37; 38]. We chose the Hufnagel-Valley model to begin our simulation because it is well known and often used and has two parameters to adapt to atmospheric conditions; one is the surface refractive index structure parameter \(C_{n}^{2}(0)\) (m\({}^{-2/3}\)), and the other is root mean square wind speed \(U\) (m/s). This model calculates \(C_{n}^{2}\) using the following equation [35; 36]:  
Figure 11: The relative error between the fast calculation results and 4D wave optics simulation results for \(Enc_{axis}\) and \(Enc_{peak}\) under S5, S6, and S7 scenarios. Figure 10: The change curve comparison between the fast calculation results and 4D wave optics simulation results of (**a**) \(Enc_{axis}\) and (**b**) \(Enc_{peak}\) with facula expansion multiple \(\beta\) under S5, S6, and S7 scenarios. \[C_{n}^{2}(h)=8.148\times 10^{-26}U^{2}h^{10}\exp(-h)+2.700\times 10^{-16}\exp(-h/1.5)+C_{n}^{2}(0)\exp(-10h) \tag{15}\]  
where \(h\) is height in kilometers. For the H-V5/7 model, \(C_{n}^{2}(0)=1.7\times 10^{-14}\) m\({}^{-2/3}\) and \(U=21\) m/s. The wind-speed profile we adopt is the Bufton wind model [39; 40], that is:  
\[V(h)=V_{g}+30\exp(-((h-9.4)/4.8)^{2}) \tag{16}\]  
where 30 is wind speed at the 200 hPa level, 9.4 is the height of the upper boundary of the tropospehr, 4.8 is the thickness of the tropopause, and the ground wind speed parameter \(V_{g}\) is usually taken to be 5 m/s. The absorption coefficient and extinction coefficient is shown in Figure 12. \(PSF_{\text{FC}}\) is obtained by convoluting the \(PSF\) with different effects, as seen in Table 5.  
We consider the combined effects of turbulence and thermal blooming. Figure 13 shows the comparison between the theoretical calculation results of \(PSF_{\text{FC}}\) and the 4D wave optics simulation program calculation results under different thermal distortion parameters for S8, S9, and S10 scenarios. In Figure 13, the left column is the calculation result of the 4D wave optics simulation program, and the right column is the fast calculation result of \(PSF_{\text{FC}}\). It can be seen from the figures that, when the thermal blooming effect is weak, the calculation results of the two methods are in good agreement. With the increase of thermal distortion parameters, the light intensity distribution calculated by the two methods is different. The light facula energy calculated by the OTF method is more dispersed. Again, we get the same conclusion. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Scenario** & **Parameters** & **Fast Calculation** \\ \hline S8 & \(\beta_{0}\)\(=5, jitter=5\mu rad,\text{D}/r_{0}=3.37,N_{D}=37.7\) & \(PSF_{\text{FC}}=PSF_{b0-5}\otimes PSF_{intr-5}\otimes PSF_{D/r_{0}=3.37}\) \\ S9 & \(\beta_{0}\)\(=5, jitter=5\mu rad,\text{D}/r_{0}=3.37,N_{D}=149.4\) & \(PSF_{\text{FC}}=PSF_{b0-5}\otimes PSF_{intr-5}\otimes PSF_{D/r_{0}=3.37} \otimes PSF_{N_{D}=149.4}\) \\ S10 & \(\beta_{0}\)\(=5, jitter=5\mu rad,\text{D}/r_{0}=3.37,N_{D}=261.4\) & \(PSF_{\text{FC}}=PSF_{b0-5}\otimes PSF_{intr-5}\otimes PSF_{D/r_{0}=3.37} \otimes PSF_{N_{D}=261.4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Fast calculation with PSF convolution process. Figure 12: The absorption coefficient and extinction coefficient vary with height.