Performance of Integrated Ground-Air-Space FSO Links Over Various Turbulent Environments

We analyze the outage probability of integrated ground-air-space free-space optical (FSO) communication links for different atmospheric turbulence channel models. The performance of ground to high altitude platforms (HAPS), HAPS to geostationay Earth orbit (GEO) satellite, HAPS to HAPS and ground to GEO satellite is investigated for various link configurations (downlink, horizontal, and uplink) and parameters such as zenith angle, channel state, horizontal and vertical deviations, altitude, beam waist, receiver aperture diameter, wind speed, and visibility. The atmospheric attenuation (the effects of fog, cloud and volcanic activities), atmospheric turbulence, angle of arrival (AOA) fluctuations and pointing error are included in the considered model. Closed-form expressions for the probability density function (PDF) and cumulative distribution function (CDF) are obtained for Lognormal, Gamma and exponentiated Weibull distributed channel models. The benefit of using HAPS is shown by comparison of outage probabilities for direct ground to GEO satellite and HAPS assisted ground to GEO satellite links.


I. INTRODUCTION
A IR and space based applications are increasingly emerging in various areas and the demand for communication links of these applications having high data rates is rising. Since optical beam significantly suffers from atmospheric turbulence and environmental conditions, free-space optical (FSO) communication has found distance and weather limited usage in general. However, researchers are working for FSO based communication solutions to meet the need for high data rates. Besides some atmospheric turbulence mitigation techniques such as aperture averaging, adaptive optics (AO) correction and using multipleinput multiple-output (MIMO) systems; high altitude platforms (HAPS) relayed both hybrid radio frequency (RF)/FSO and only FSO links have become a prominent alternative in terms of increasing the performance of the wireless communication systems. In this frame, the concept of using HAPS to enhance the performance of terrestrial based and satellite wireless communication systems was discussed and system architectures for some applications were presented [1]. The advantegous the HAPS relayed satellite RF and FSO link was examined for the link budget and it was shown that the improvement in the performance can be up to 30 dB for HAPS relayed geostationary Earth orbit (GEO) satellite link [2]. Moreover, the cost effectivity of using HAPS instead of using many terrestrial base stations in terms of covering larger area was discussed [3]. A multi-hop HAPS assisted ground to ground network communication was analyzed and it was shown that the average bit-error-rate (BER), channel capacity and link availability of the communication system improve with the HAPS application and increased number of hops [4]. In [5], an unmanned aerial vehicle (UAV)-enabled mobile relaying for ground to ground communication was investigated and it was concluded that the larger link gains (or less path losses) are obtained for mobile UAV rather than static UAV. Link budget analysis of vertical backhaul and fronthaul for 5G+ application was performed in [6] depending on the different weather conditions. In addition to the significant influence of the some weather conditions, the key role of the divergence angle was shown with the multi Gb/s data rates with the reduced divergence angle. It was shown that the impact of atmospheric turbulence becomes negligible for downlink and uplink communication between GEO satellite and UAV at a certain altitude [7]. The effects of the angle of arrival (AOA) fluctuations and pointing error on the performance of the high altitude HAPS to HAPS link was studied in [8] and an adaptive beam control algorithm was proposed to define the optimum beam size and avoid the link loss. In [9], the performance analysis for ground to HAPS link including AOA fluctuations, pointing error and atmospheric turbulence channel with Lognormal and Gamma-Gamma distribution was studied. The misalignment errors in [9], which are assumed to be identical in both vertical and horizontal axes, are represented by Rayleigh distribution. A similar approach was used for a UAV-relayed FSO communication system in the Lognormal distributed turbulent channel in [10]. A compherensive survey was published to emphasize the opportunities, challenges, and unexplored potentials of the HAPS and their future deployments [11]. In [12], we analysed the performance of ground to HAPS downlink and uplink FSO communication systems in Gamma-Gamma distributed turbulent channel model and mitigation of the turbulence effect by applying adaptive optics correction. In this study, we make contribution by using different channel models (Lognormal, Gamma and exponentiated Weibull) and extending the communication link from ground station (GS) to GEO satellite both directly and using HAPS relaying. We also did not include adaptive optics correction effect but accounted for aperture averaging in this study.
Regarding integrated ground-air-space network application, the error probability of a HAPS assisted relaying using hybrid RF/FSO communication system has been analyzed recently [13]. The better performance with the HAPS relaying ground to satellite link was obtained compared with the direct ground to satellite link. A novel hybrid RF/FSO satellite communication system was offered to improve the usability of FSO link and the importance of the zenith-angle dependency for system design was concluded [14].
All above mentioned studies show that the performance of the integrated ground to air or satellite FSO communication systems is degraded considerably with the weather conditions (especially foggy weather), atmopsheric turbulence, pointing error and AOA fluctuations due to the hovering aerial vehicle. However, HAPS assisted links improve the performance of communication systems up to certain level. It is obvious that more detailed analyzes are needed from different perspectives for accurate modeling and optimum deployment of different communication link configurations. Our motivation is to investigate the performance of slant path and horizontal links in both direct and HAPS assisted applications in various turbulence environments modeled by different channel distributions. There are several channel models which are used to characterize the turbulent channel statistics such as Lognormal, Gamma, Gamma-Gamma, exponentiated Weibull and K-distribution. Among these models, Lognormal and Gamma-Gamma channel models are widely adopted to model turbulent channels. In our previous study [12], we investigated the Gamma-Gamma distribution model for ground to HAPS communication link in different aspects. In this study, our motivation is to extend our previous study and reflect the behaviour of the ground to satellite communication links and relay applications in different turbulence channel models. Although the Lognormal channel model is partially effective in stronger turbulence with a sufficiently large receiver, it generally yields quite effective results for a point receiver operating in weak turbulence conditions [15], [16], [17]. Gamma distribution is presented to have highly effective approximation when the propagation is analyzed in terms of intensity and amplitude variations [18]. Exponentiated Weibull distribution, that is the generalized version of the Weibull distribution with a shape parameter, is shown to be yielding accurate results from weak-to-strong turbulence conditions with all aperture averaging variations [15]. However, it is obvious that each model has limitations under certain conditions and parameters. Due to these limitations, in direct and/or HAPS assisted ground-air-space applications, it becomes very important to reveal the performances of the above-mentioned models and validate their accuracy in the turbulent environment, and to compare the suitability and performance of these models to each other. For this purpose, we derived closed-form expressions for probability density function (PDF), cumulative distribution function (CDF) and probability of outage for FSO communication systems operating in Lognormal, Gamma and exponentiated Weibull distributed turbulent From here on the paper is formed as follows. In Section II, the system model is given. Atmospheric attenuation is defined and channel model for Lognormal, Gamma and exponentiated Weibull distribution is given in Section III using turbulence parameters for various links. The effects of pointing error with Hoyt distribution and AOA fluctuations with Rayleigh model are also given in Section III. The outage performance analysis is performed in Section IV for all these channel models. Results are presented and discussed in Section V. Conclusion of our study is given in Section VII. Finally, the procedures used to obtain analytical expressions for channel models are offered in Appendix A, B, and C. Table I lists the used symbols and their definitions.

II. SYSTEM MODEL
The block diagram of the integrated ground-air-space communication systems is given in Fig. 1. All link configurations between ground to GEO satellite, HAPS to GEO satellite and ground to HAPS are assumed to be FSO based and, single transmitter and single receiver are used on both sides. The receiver is assumed to use aperture averaging [18]. The atmospheric turbulence is taken into account for three different models; Lognormal, Gamma and exponentiated Weibull distributed

A. Atmospheric Attenuation
The contribution of absorption and scattering to the optical beam attenuation in the atmosphere can be expressed by Beer-Lambert law as where σ is the attenuation coefficient and L is the link length. The attenuation coefficient σ in (1) is obtained as follows depending on the visibility [19] where V is the visibility (in Km), λ is the wavelength (in nm), and the parameter q is expressed using the Kim model as The visibility and attenuation coefficient of various fog conditions for λ = 1550 nm is given in Table II [20].Clouds also cause attenuation in atmosphere. The attenuation due to the clouds is modeled by using liquid water concentration (LWC) and cloud number concentration (CNC) and, the visibility is obtained by [21] V = 1.002 The visibility and attenuation for various cloud types are given in Table III [20].
Another attenuation factor affecting the optical beam propagation is the stratospheric attenuation that is mainly caused by in Hufnagel-Valley (HV) model, where l is the parameter of height, w denotes the root mean square (rms) of the wind speed in m/s, A is the turbulence structure constant at the ground level (C 2 n (0)) in m −2/3 varying with the wind speed in the HV5/7 model. Then, the power spectrum of the turbulence based on the Kolmogorov theory becomes where κ is the spatial wave number. Being valid for weak, moderate and strong turbulence conditions; the scintillation index is [18] (p. 420) where D G is the receiver aperture diameter, σ 2 lnX and σ 2 lnY are the large-and small-scale log variances respectively and they are defined as where index i is i ∈ d, u, h depending on the downlink, uplink and horizontal link communication, is the non-dimensional Fresnel parameter, W G is the radius of Gaussian lens and W 2 is the Fresnel ratio of Gaussian beam at receiver, F 0 is the phase front radius of curvature, are the beam curvature parameters at the transmitter and transmitter, Θ 1 = 1 − Θ 1 is the complementary parameter. In (8) and (9), σ 2 Bi and σ 2 Ri are the Rytov variances for Gaussian beam and plane wave. a) Slant Path Links: The Rytov variance of Gaussian beam propagating in slant path link is [18] where k = 2π/λ is the wavenumber, ζ is the zenith angle, ξ is the normalized distance parameter, h 0 is the height of the ground station, H is the altitude of air or space platform, and Re denotes the real part. The normalized distance parameter of the downlink is ξ = (l−h 0 ) (H−h 0 ) . Using (10), the Rytov variances for a Gaussian beam and plane wave in downlink are obtained as [18], . Then, the Rytov variance for uplink is [18] (13) On setting Θ 1 = 1 and Λ 1 = 0 in (13), the Rytov variance of plane wave for uplink can be obtained as b) Horizontal Path Link: The Rytov variance of a Gaussian beam for horizontal link is [18] Since the height is constant for horizontal FSO link, the refractive index structure parameter C 2 n (l) given in (5) will also keep its constant value for fixed l over link length. Then, the Rytov variance of propagating Gaussian beam can be obtained by using (13) as And, the Rytov variance of plane wave for horizontal link will be σ 2 Rh = 1.23C 2 n (l)k 7/6 L 11/6 . 2) Different Turbulence Channel Models: Here, the different types of channel models are given to characterize the atmospheric turbulence-induced fluctuations.

C. Pointing Error
Assuming that the jitter variances over the two dimensions are not identical, the PDF of the misalignment by Hoyt distribution for the orthogonal directions s and z is [24] where q H = σ z /σ s varying in interval (0,1], σ s and σ z are the jitter variances in both directions. Using the relationships s = r cos(ϕ) and z = r sin(ϕ) for polar coordinates, PDF takes the form [24] where Then the PDF of the pointing error given in (21) is found to be [24] where , ω b is the beamwaist, erf(.) is error function, υ = π/2r a /ω b , r a = D G /2 is the receiver aperture radius, and A 0 = erf 2 (υ). Including the deviation angle θ d then, the conditional probability for the PDF of pointing error h pl can be found We note that (25)

D. AOA Fluctuations
For small deviation angle in comparison to the field of view (FOV) (θ d ≤ θ F OV ), the fading due to AOA fluctuations is expressed by [9] where J n (.) is the n th order Bessel function of the first kind. The PDF of random variable θ d modeled by Rayleigh distribution is [9] where σ 2 0 is the variance of random variable θ d .

IV. OUTAGE PROBABILITY ANALYSIS
The outage probability of slant path (ground to HAPS, ground to GEO satellite, HAPS to GEO satellite) and horizontal (HAPS to HAPS) links is analyzed in this section. Taking into account all the effects mentioned above, the channel state for downlink, uplink and horizontal link can be written as where h al , h at , h pl , and h af represent the effects of the attenuation loss, atmospheric turbulence, pointing error, and AOA fluctuations, respectively. The conditional channel state on θ d can be represented as

A. Lognormal Distributed Channel Model
Inserting the PDF of the Lognormal channel model and the conditional probability of PDF of the pointing error given in (17) and (25) into (29); the PDF of h ag conditioned on the deviation angle θ d is found by (30) After derivations given in Appendix-A (A.1)-(A.8), the PDF of the channel state for Lognormal distribution will be obtained as The CDF of the channel state h is expressed as follows depending on the PDF of the Lognormal distributed channel model Substituting (31) into (32), one can find the CDF of the Lognormal distributed channel as Following the derivations given in Appendix-A (A.9)-(A.14), the CDF of the channel state h will be obtained as The probability of outage can be found by using the CDF as Substituting (34) in (35), the probability of outage will be For the special case q H = 1, the closed-form expression of outage probability in (36) takes the form of

B. Gamma Distributed Channel Model
Inserting (18) and (25) into (29), we find the conditioned PDF of h ag as Then, following the procedures given in Appendix-B (B.1),(B.2), the PDF of the channel state is found as Inserting (39) into (32), the CDF of the channel state for Gamma distributed channel model is To solve x dependent integration in (40), we will use (26) of [25]. The, the CDF will be Using the relationship given in (35), the outage probability is For the special case q H = 1, the closed-form expression of outage probability in (42) becomes

C. Exponentiated Weibull Distributed Channel Model
Inserting (19) and (25) into (29), we find the conditioned PDF of h ag as Then, following the procedures given in Appendix-C (C.1)-(C.6), the PDF of the channel state is found as (45) Using (32), the CDF of exponentiated Weibull distributed channel is (46) To solve x dependent integration in (32), first changing the variable as t = x β and then using (26) of [25] Then, the outage probability will be obtained as (48) For the special case q H = 1, the closed-form expression of outage probability for exponentiated Weibull distribution will It can be seen from (36), (42) and (48) that, showing the outage probabilities for Lognormal, Gamma, and exponentiated Weibull distributions, an integral term depending on the polar angle ϕ brings a complexity and it is not possible to solve analytically. The results are obtained numerically by evaluating ϕ dependent integral. Also, a series with infinite summation is involved in the equations related to the exponentiated Weibull distribution resulting from expansion of the Newton's generalized-binomial theorem. We verified that the series converges in the first 10 terms or more terms. However, we took into account 20 terms in our numerical computation to make sure that we present the exact result of the converging sum.

V. RESULTS AND DISCUSSION
In this section, the analytical results obtained from the derivated expressions are presented depending on the various parameters and conditions. The values of used parameters are chosen as listed in Table V. Any difference in the parameter from those in the table are mentioned in either figure captions or in the plots. Also, the used channel model for turbulence-induced fading is given on each plot. The performance of various links (slant path and horizontal) are illustrated and evaluated in terms of outage probability. Moreover, the comparisons of direct and HAPS assisted ground to GEO satellite links and, channel models are shown.

A. Ground to HAPS Communication Link
In Fig. 2, the probability of outage for a downlink between ground to HAPS is depicted versus zenith angle ζ and, ω b /r a . As it can be observed from Fig. 2 that the probability of outage degrades with the increase of the zenith angle and keeping the zenith angle smaller yields better performance. For example, the outage probability varies from P out ≈ 3.5 × 10 −4 to P out ≈ 1.1 × 10 −2 when the zenith angle varies from ζ = 0 • to ζ = 60 • . In [18], it is expressed that the atmospheric turbulence remains in weak regime for slant path link while the zenith angle is ζ < 45 • or ζ < 60 • . We also observe similar variation in Fig. 2. When zenith angle exceeds ≈ ζ > 70 • , the outage probability takes its highest values and the performance of the communication link becomes worst. We also observe some reversed decrease trends in outage probability when ζ > 80 • which is the result of scintillation saturation in the strong turbulence regime. Again, we see from Fig. 2 that the outage probability takes smaller values when the beam waist is on the higher order of receiver aperture radius. The outage probability of downlink between ground to HAPS for zenith angle ζ = 50 • drops from P out ≈ 2 × 10 −1 to P out ≈ 2.7 × 10 −4 when the ratio of beam waist to aperture radius increases from ω b /r a = 5 to ω b /r a = 15. This is due to the increasing possibility of collecting optical beam by the receiver aperture with the increase of beam waist that occurs for the long range of the ground to HAPS communication link. This shows that the performance of FSO communication link can be improved significantly with the appropriate level of the beam waist. Fig. 3 demonstrates the variation of the outage probability depending on the outage threshold for various displacement values. The outage probability increases with increasing of the channel state threshold for a FSO downlink. Keeping the beam deviation as σ S = 6 × r a in Fig. 3, the outage probability rises from P out ≈ 3 × 10 −3 to P out ≈ 5.9 × 10 −1 when the threshold of the channel state increases from h th = 1 × 10 −6   to h th = 1 × 10 −3 . The performance degrading effect of the beam displacement is also clearly seen in Fig. 3. Increasing σ S from σ S = 4 × r a to σ S = 12 × r a causes significant performance degradation yielding outage probabilities from P out ≈ 1.5 × 10 −4 to P out ≈ 3.5 × 10 −1 while the outage threshold is kept as h th = 1 × 10 −5 . Figs. 4 and 5 illustrate the outage performance of an FSO uplink between ground to HAPS using the same parameters given in Figs. 2 and 3, respectively. Similar to downlink, the performance degradation with the larger zenith angle, channel state threshold, beam displacement and smaller ω b /r a is observed for uplink communication. The benefit of smaller zenith angle, beam displacement and larger beam waist is clearly seen from Figs. 2-5 for both downlink and uplink.
We note that although the downlink and uplink yield similar performances (e.g., Figs. 2-4, Figs. 3-5), the outage probability of downlink remains lower very slightly. This slight difference  tends to increase for higher turbulence effects. To reflect the difference between outage performances of downlink and uplink, the comparison of uplink and downlink is given in Fig. 6 for two different receiver aperture diameter values (D G = 5 cm and D G = 0 point receiver). It is seen from Fig. 6 that the outage performance of downlink maintains at lower level than uplink for all cases. The reason why the performance of downlink is better may be that the optical beam encounters the higher turbulence effect in the last phase of its propagation (optical beam first propagates and then is distorted in short duration) while this happens in the initial phase for uplink (optical beam is distorted and then distorted beam propagates). However, the level of difference between the outage probabilities of downlink and uplink can be assumed to be negligible when receiver with large aperture is used. It is also observed from Fig. 6 that outage performances of downlink and uplink are very close to each other for larger aperture sizes and the difference between downlink and uplink performances becomes more visible when receiver aperture size takes smaller sizes. This shows that the use of aperture average is also minimizing the difference in performance between uplink and downlink scenarios.

B. Ground to Satellite Communication Link
We evaluate the performance of FSO downlink and uplink communication between ground to GEO satellite for different parameters in Figs. 7-10. Fig. 7 shows the probability of outage variation for the FSO downlink communication versus zenith angle forvarious q H values. When the beam deviations in both   directions are identical (q H = 1), the outage probability starts from P out ≈ 4.3 × 10 −4 and reachs to P out ≈ 3.5 × 10 −1 with the rise of the zenith angle from ζ = 0 • to ζ = 70 • . When the zenith angle exceeds the value of ζ ≈ 70 • , the link is almost lost and the outage performance takes the highest values. It can be inferred from Fig. 7 the FSO downlink benefits from the asymmetrical behaviour of the beam deviations while q H < 1. This is due to the σ Z remains smaller than σ S and leads to a decrease in the probability of outage compared with the symmetrical case σ Z = σ S . For example, keeping the zenith angle as ζ = 30 • , the probability of outage changes from P out ≈ 1.4 × 10 −3 to P out ≈ 6 × 10 −3 when the beam deviatons change from Hoyt distribution (q H = 0.1) to Rayleigh distribution (q H = 1). In Fig. 8, a monotonic logarithmic increase in the outage probability is seen depending on the increase in the threshold of the channel state. The impact of the height of the GS is also plotted in Fig. 8. When ground station is deployed at the ground level In Fig. 9, we observe a degradation in the probability of outage with the rise of the zenith angle for FSO uplink similar to the variation for downlink given in Fig. 7. For Hoyt distributed deviation case q H = 0.1, the performance of the uplink FSO communication system degrades by taking the values of P out ≈ 4.2 × 10 −3 and P out ≈ 3.5 × 10 −1 when zenith angle is ζ = 0 • and ζ = 70 • . The decrease in the ratio of vertical and horizontal deviations yields an improvement in the performance of the uplink FSO communication systems as it can be seen from Fig. 9.
The benefit of using LAPS and HAPS compared to the ground based uplink can be inferred from Fig. 10 for uplink FSO communication. The outage probability falls from P out ≈ 4.7 × 10 −2 to P out ≈ 2.4 × 10 −3 when the height of the lower station increases from h 0 = 0 (ground level) to h 0 = 20 km (HAPS assisted) for threshold of the channel state as h th = 1 × 10 −4 . The results for both downlink and uplink FSO communication show that keeping the lower station as much as higher altitude will lead of avoidance of the atmospheric attenuation and turbulence effects. It is also seen that increasing the receiver aperture size causes performance improvement for FSO link. For example, the outage probability decreases from P out ≈ 1.7 × 10 −5 to P out ≈ 8.7 × 10 −6 with the increase of receiver aperture diameter from D G = 1 cm to D G = 20 cm while the zenith angle is fixed to ζ = 20 • . The increase in the receiver aperture diameter starts to change the outage performance slightly because of the larger  receiver collects all the intensity and saturates showing that the receiver aperture size should be optimized to provide the optimum performance improvement with the aperture averaging. The performance improvement with enlargement of the receiver aperture diameter becomes smaller when zenith angle increases. However, it is observed that aperture averaging remains a useful technique in terms of mitigating the degradation effects caused by turbulence.

C. HAPS to Satellite Communication Link
In Fig. 12, the radius of receiver aperture is fixed to r a = 5 cm and only the beam waist is changed to observe the performance of HAPS to satellite FSO uplink. Keeping the outage threshold as h th = 1 × 10 −5 then changing the beam waist from ω b = 0.1 m to ω b = 0.5 m yields a remarkable performance improvement by pulling down the outage probability from P out ≈ 1.3 × 10 −1 to P out ≈ 1.4 × 10 −5 .

D. HAPS to HAPS Communication Link
The ground to satellite communication link may be provided via first ground to HAPS later HAPS to HAPS network and the last HAPS to satellite links. That's why we show the performance results of the horizontal FSO link in the following Figs. 13 and 14. In Fig. 13, it is observed that the probability of outage rises with the increase of altitude up to a certain level (here H ≈ 10 km) and then outage probability starts to decrease with the increase of altitude until taking almost its constant value H ≈ 20 km. The outage probability maintains its almost constant trend after H ≈ 20 km altitude. Another parameter, the wind speed, shows its influence on the outage probability until H ≈ 20 km the outage probability almost does not change with the wind speed change.  In Fig. 14, the performance of a horizontal FSO link is plotted versus of link distance and the visibility parameter. The performance degradation with the link distance is seen as expected. The high impact of the visibility that depends on the atmospheric conditions is also observed from Fig. 14. Outage probability takes the worst values at L ≈ 14 km and L ≈ 30 km for the values of visibility as V = 3 km and V = 5 km, respectively. Given that the V = 1.9 km for thin foggy weather in Table II, the drastic performance degradation of the FSO links with fog still remains challenging.

E. Comparison of Direct Link Versus Relayed Link
The outage probability of HAPS assisted ground to satellite link with decoding and forwarding capabilities at the HAPS can be found as where P GH out (h ≤ h th ) and P HS out (h ≤ h th ) are the outage probabilities of ground-to-HAPS and HAPS-to-satellite links.
To characterize the benefit of using HAPS relaying between ground to GEO satellite communications, the outage probability variation versus channel state threshold is given for various HAPS deployment configurations (depending on the zenith angle) in Fig. 15. Here, the zenith angle for ground to GEO satellite link is set to ζ GS = 50 • , the zenith angle for ground to HAPS ζ GH is kept smaller than ζ GS and the zenith angle for HAPS to satellite ζ HS is calculated by using geometric relations depending on the ζ GH , ζ GS and altitudes. The receiver aperture diameter D G values are chosen as 5 cm for ground to HAPS link, 10 cm for HAPS to satellite link and 10 cm for ground to satellite link. Keeping the outage threshold as h th = 1 × 10 −5 , the probability of outage takes the value of P out ≈ 1 × 10 −2 . When HAPS relaying is used, the outage probability values become P out ≈ 5.5 × 10 −3 , P out ≈ 3.2 × 10 −3 and P out ≈ 2.7 × 10 −3 for zenith angles ζ GH = 40 • , ζ GH = 20 • and ζ GH = 0 • . It is obvious that the best performance can be obtained for the vertical alignment of the GS and HAPS (ζ GH = 0 • ). Finally, to verify the accuracy of our derivations and compare the turbulent channel models, we present the comparison of Lognormal, Gamma and exponentiated Weibull distributions in Fig. 16. We see that three models match perfectly. The important point is the Lognormal distribution gives the results until zenith angle reaches ζ = 62 • then it does not perform calculations. This behavior is consistent with the Lognormal distribution that yields effective results when weak atmospheric turbulence conditions are available. The Gamma and exponentiated Weibull distributions yield the results in moderate to strong turbulence regimes and then they lose their sensitivity when zenith angle exceeds ζ = 72 • which corresponds to the boundary of saturated strong turbulence regime.
Results show that the selection of appropriate channel model should be based on the turbulence strength rather than application type (HAPS to HAPS, ground to HAPS, HAPS to satellite). Lognormal, Gamma and exponentiated Weibull channel models can be used for the performance analysis in weak turbulence conditions. When strong turbulence conditions are available, Lognormal tends not to perform then Gamma and exponentiated Weibull models remain as accurate solutions providing almost the same performances. For a slant path communication link having small zenith angle, Lognormal, Gamma and exponentiated Weibull models can be used with similar performances. However, Lognormal model do not provide any results when zenith angle is high due to strong turbulence effect.
In terms of application types, HAPS to satellite communication link, that is exposed almost no turbulence effect, can be modeled by Lognormal, Gamma and exponentiated Weibull distributed channel models. However, a horizontal or slant path link experiencing the turbulence conditions in troposphere and lower stratosphere layers (approximately < 20 km altitude), including ground to HAPS, HAPS to HAPS and ground to satellite communication links, need to be modeled carefully by taking into account the strength of the turbulence.

VI. FUTURE INVESTIGATIONS
In this study, the performance of integrated ground-air-space communication links is presented for single-input single-output (SISO) systems. The benefit of using HAPS as relaying is seen from the obtained results up to a certain level. To overcome turbulence-induced degradation effects and increase the performance of integrated ground-air-space based communication systems, mitigation techniques (spatial diversity techniques and adaptive optics correction methods are analyzed extensively for various applications) can be other considerable solutions in terms of performance improvement. We have already investigated the performance improvement with adaptive optics correction methods for ground to HAPS communication links in Gamma-Gamma distributed turbulent channel model [12]. Diversity techniques, using multiple apertures at transmitter and/or receiver side, are worthy of investigation in terms of additional improvement in the performance of HAPS assisted applications. The power gain and reduced scintillation with the help of statistically independent multiple apertures for single-input multiple-output (SIMO), multiple-input single-output (MISO) and MIMO systems have the potential to be the extension of this work.
In addition, the inclusion of the round-trip time (RTT) latency is seen as an important figure of merit to ensure the continuity of the stream of data, especially comparing the cases of HAPS relaying ground to satellite communication and direct ground to satellite link. Including RTT latency is another investigation parameter to characterize the performance of HAPS assisted communication links.

VII. CONCLUSION
We investigated the performance of integrated ground-airspace FSO communication links between ground and GEO satellite, ground and HAPS, HAPS and HAPS and, HAPS and satellite. The joint effect of atmospheric attanuation, pointing error, AOA fluctuations and atmospheric turbulence is included in proposed model. Three models, namely Lognormal, Gamma and exponentiated Weibull distributions, are used for atmospheric turbulence characterization. Closed-form expressions for channel PDF, CDF and probability of outage are extracted using Lognormal, Gamma and exponentiated Weibull distributions separately. The pointing error effect is modeled by Hoyt distribution. The performance of slant path and horizontal links is affected from beam waist, beam deviation, channel state, the ratio of vertical and horizontal deviations, the altitude of upper stations, the height of lower stations, receiver aperture diameter, wind speed and zenith angle. The HAPS assisted ground to satellite link yields better performance than the direct ground to satellite link. The best performance is obtained with HAPS assisted link with zenith angle ζ GH = 0 • between ground to HAPS. The results given in this study may be useful for characterizing different slant path and horizontal FSO links for integrated ground-air-space applications operating in various turbulence environment.