## 3 Results and Discussion
### Verification of the Fast Calculation Model with PSF Convolution
#### 3.3.2 Results for Turbulence and Thermal Blooming Effects  
We then consider the combined effects of turbulence and thermal blooming. Figure 9 shows the comparison between the theoretical calculation results of \(PSF_{\text{FC}}\) and the 4D wave optics simulation program calculation results under different initial aberration and thermal distortion parameters for three propagation scenarios (S5, S6, and S7). Similarly, \(PSF_{\text{FC}}\) is obtained by convoluting the \(PSF\) with different effects, as seen in Table 4. In Figure 9, the left column shows the calculation result of the 4D wave optics simulation program, and the right column shows 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. This difference is mainly due to the assumption that the turbulent phase and the thermal blooming phase are independent in the transfer function calculation method. In actual conditions, the turbulent thermal blooming effect is interactive. However, from the point of view of rapid evaluation, the results of the two methods are in good agreement with each other, both from the far-field irradiance pattern and the energy distribution around the ring. With the increase of thermal blooming effect, the difference in the spot shape becomes larger.  
Figure 8: The relative error between the fast calculation results and 4D wave optics simulation results for \(Enc_{axis}\) and \(Enc_{peak}\) under S1, S2, S3, and S4 scenarios.  
Figure 7: 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 S1, S2, S3, and S4 scenarios.  
Figure 9: 4D wave optics simulation results (**a**–**c**) and the far-field intensity distribution for fast calculation results (**d**–**f**) with S5, S6, and S7 scenario pairs presented top to bottom, respectively.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Scenario** & **Parameters** & **Fast Calculation** \\ \hline S5 & \(\beta_{0}\)= 5, \(\beta_{0}\)= 5, \(\beta_{itter}=5\
u rad,D/r_{0}=\) & \(PSF_{FC}=PSF_{\beta_{0}=5}\otimes PSF_{\beta_{itter}=5}\otimes\) \\  & 6.13, \(N_{D}=25.5\) & \(PSF_{D/r_{0}=13}\otimes PSF_{N_{D}=25.5}\) \\ S6 & \(\beta_{0}\)= 5, \(\beta_{itter}=5\
u rad,D/r_{0}=\) & \(PSF_{FC}=PSF_{\beta_{0}=5}\otimes PSF_{\beta_{itter}=5}\otimes\) \\  & 6.13, \(N_{D}=127.3\) & \(PSF_{D/r_{0}=6.13}\otimes PSF_{\beta_{itter}=127.3}\) \\ S7 & \(\beta_{0}\)= 5, \(\beta_{itter}=5\
u rad,D/r_{0}=\) & \(PSF_{FC}=PSF_{\beta_{0}=5}\otimes PSF_{\beta_{itter}=5}\otimes\) \\  & 6.13, \(N_{D}=254.3\) & \(PSF_{D/r_{0}=6.13}\otimes PSF_{N_{D}=254.3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Fast calculation with PSF convolution process.  
Figure 10 shows the change curve comparison between the fast calculation results and 4D wave optics simulation results of \(Enc_{axis}\) and \(Enc_{peak}\) with facula expansion multiple \(\beta\) under S5, S7, and S7 scenarios. Figure 11 gives the relative error between the fast calculation results and the 4D wave optics simulation results for \(Enc_{axis}\) and \(Enc_{peak}\). With the increase of thermal distortion parameter, the difference of \(Enc_{axis}\) increases, but the difference in \(Enc_{peak}\) is not significant. For \(\beta=5,10,15\), the errors of \(Enc_{axis}\) are within \(-50\)-\(10\%\), and the errors of \(Enc_{peak}\) are within \(-25\)-\(5\%\).