## 5 Numerical Results
The capacity outage probabilities for the IM/DD technique under strong turbulence levels are \(3.37\times 10^{-7}\) and \(2.35\times 10^{-8}\) at a total transmit power of 35 dBW and 40 dBW. By taking the logarithm of capacity outage probabilities, the diversity order is obtained as \(2.32\approx G_{d}=2.37\). Then we obtain \(2.22\times 10^{-7}\) and \(3.13\times 10^{-9}\) for HD technique under strong turbulence level at transmit power of 20 dBW and 25 dBW. Therefore, the diversity order is calculated as \(3.70\approx G_{d}=3.73\). Similarly, the diversity order equals to \(3.17\approx G_{d}=3.24\) for IM/DD technique under moderate turbulence level. We also obtain the diversity as \(5.40\approx G_{d}=5.48\) for the HD technique under moderate turbulence levels. The above analysis further verifies the accuracy and rigor of asymptotic analysis in (46). As the system diversity order is mainly determined by the smallest values of \(\alpha_{3}\), \(\beta_{3}\), and \(\xi_{3}\), we should flexibly set the values of beam waist in different turbulence levels to obtain the maximum returns. Figure 4: Capacity outage probability under different detection techniques with different turbulence levels. Figure 3: Capacity outage probability of different schemes under different weather conditions. The weighted average BER under different weather conditions and turbulence levels is presented in Figure 6. We assume that the average SNR of the OGS-to-satellite and satellite-to-HAP FSO link is 40 dB, and the total transmit power of HAP is 14 dBW. As can be seen, rainy weather and foggy weather can deteriorate the system-weighted average BER performance, and there exists an optimal value of \(\rho\) to minimize the system-weighted average BER. Specifically, considering clear weather and moderate rain weather, the values of \(\rho^{*}\) increases in heavy turbulence compared to moderate turbulence. It can be inferred that the average BER of the FSO link becomes the main factor that affects the weighted average BER of parallel FSO/RF links. Hence, more power should be allocated to the FSO link to obtain the optimal weighted average BER. In addition, considering the light fog case, the weighted average BER first increases and then decreases with the increase of \(\rho\). Therefore, considering the optimal \(\rho\) in Figure 2, a trade-off should be made between weighted average BER and capacity outage probability to select the optimal \(\rho\). Moreover, the effect of turbulence on the weighted average BER becomes smaller when \(\rho\) is larger than 0.75 in clear weather and moderate rain. In this case, the system-weighted average BER may be mainly dominated by the worse average BER performance of the RF link. In Figure 7, the weighted average BER under different transmission schemes is depicted where the weak turbulence is assumed with an optimal \(\rho\). It can be observed that the  
Figure 5: Asymptotic capacity outage probability under both IM/DD and HD techniques with different turbulence levels. Figure 6: Weighted average BER under different weather conditions with different power allocation coefficients. parallel FSO/RF scheme owns a worse weighted average BER performance compared with only the FSO scheme and only the RF scheme, especially in moderate rain weather. This is because the parallel FSO/RF links have a smaller average SNR in both FSO and RF links, which is caused by the power allocation between FSO and RF links. The only RF scheme owns a better average BER performance in both clear and rainy weather. Moreover, with the increase in average SNR, the average BER performance gap between only the FSO and only the RF scheme gradually decreases, and only the FSO scheme outperforms only the RF scheme in clear weather. Considering the advantages in capacity outage performance and service flexibility, the parallel FSO/RF links can be a better choice with an acceptable average BER performance loss. Figure 8 depicts the weighted average BER under different turbulence levels, in which both IM/DD and HD techniques are considered.