On the Design of RIS–UAV Relay-Assisted Hybrid FSO/RF Satellite–Aerial–Ground Integrated Network

Satellite–aerial–ground integrated network (SAGIN) has been widely envisioned as a promising network architecture for 6G. In the SAGINs, high-altitude platform (HAP)-aided relaying satellite systems using hybrid free-space optics (FSO)/radio-frequency (RF) communications have recently attracted research efforts worldwide. Nevertheless, the main drawback of hybrid FSO/RF systems is the restricted bandwidth of the RF connection, especially when the FSO one is blocked by cloud coverage. This article explores a novel solution for the hybrid FSO/RF HAP-based SAGIN under the impact of weather and atmospheric conditions. Specifically, an additional unmanned aerial vehicle (UAV) is deployed to diverse the FSO link from the HAP-to-ground station to avoid cloud blockage while maintaining a high-speed connection of the FSO link. A mirror array constructed by reconfigurable intelligent surface (RIS), an emerging technology, is mounted on the UAV to reflect the signals from the HAP. The channel model of RIS–UAV takes into account both atmospheric turbulence and hovering-induced pointing errors. Furthermore, we present a novel link switching design with a multirate adaptation scheme for the proposed network under different weather and turbulence conditions. Numerical results quantitatively confirm the effectiveness of our proposal. Additionally, we provide insightful discussions that can be helpful for the practical system design of RIS–UAV-assisted HAP-based SAGIN using hybrid FSO/RF links. Monte Carlo simulations are also performed to validate the accuracy of theoretical derivations.


I. INTRODUCTION
Recently, with the advance of the Internet of Things, relying solely on the traditional terrestrial networks can no longer meet the exploding requirements of high-speed and reliable network access at any time and anywhere on the earth, in consideration of its limited coverage and network capacity. This has aroused widespread concern in the academia and industry on the air-ground coordination [1]. Thanks to the popularity of low Earth orbit (LEO) satellites, the satellite-aerial-ground integrated network (SAGIN) has been widely envisioned as a promising network architecture for future sixth-generation (6G) wireless communications [2].
In the SAGIN, high-altitude platform (HAP)-based relaying and free-space optical (FSO) communications are expected as the key technologies for extending the area coverage and the provision of extremely high-speed connectivities, respectively. The HAPs, which can be airships and balloons, operate at altitudes of 17-25 km, where the impact of weather is negligible. With the benefits of flexible deployment, cost-effectiveness, easy maintenance, and expansive coverage, HAPs have been widely considered as relaying nodes to enhance the scalability of satellite systems [3]. On the other hand, FSO communication has established a reputation for itself as capable of delivering extremely high-speed data services over long distances without exhausting radio frequency (RF) resources [4]. HAP-based SAGIN using FSO communications can be considered for a wide range of applications, e.g., Internet of vehicles, postdisaster emergency communications, and vertical backhaul solutions. While the FSO link can be reliably maintained for the first hop of satellite-to-HAP, it is challenging for the FSO connection on the second hop of HAP-to-ground station (GS) under the impact of weather and atmospheric conditions. The primary concerns of such links are cloud coverage and atmospheric turbulence, which pose various challenges to the design and performance of the SAGIN. To tackle this issue, an efficient solution is to use the hybrid FSO/RF scheme for the second hop of the HAP-to-GS, where the RF link serves as a backup link in case of FSO link failure [5]. This is because the RF link is less subject to atmospheric turbulence and clouds. For example, the impact of atmospheric turbulence and heavy rain on the corresponding FSO and RF links is drastically, but these factors rarely happen simultaneously. As a result, two links can function in a complementary manner.

A. Related Work and Motivation
Driven by the potential of the hybrid FSO/RF scheme, its implementation has been extensively studied in terrestrial networks [5] and recently extended to the HAP-based SAGIN [6]- [10]. Particularly, Swaminathan et al. analyzed the performance of downlink SAGIN using HAP-based relay station with hybrid FSO/RF links [6]. Swaminathan et al. [7] investigated the performance of hybrid FSO/RF uplink for the HAP-based SAGIN, assuming Gamma-Gamma fading turbulence with pointing errors for the FSO link and shadowed-Rician fading for the RF link. In [8], Shah et al. analyzed the performance of dual-hop SAGIN-based hybrid FSO/RF systems, both uplink and downlink, using an adaptive combining-based switching scheme. A novel HAP-assisted downlink SAGIN using hybrid FSO/RF communications was introduced in [9], in which the best HAP node was selected among multiple HAP nodes for relaying signals from the satellite. Different from the work in [6]- [9], where decode-and-forward (DF) relaying scheme with constant data-rate transmissions was considered, the authors presented the design of rate adaptation hybrid FSO/RF links for HAP-based SAGIN using amplify-and-forward (AF) relaying scheme [10]. The major drawbacks of HAP-based SAGIN using the hybrid FSO/RF scheme are, nonetheless, the lower data rate, even with the new millimeter-wave frequency bands, compared to that of FSO and the additional latency due to the optical/electrical and electrical/optical conversions. This fact is clearly indicated in the presence of cloud coverage, where the FSO link may be blocked, and the backup RF link is used more frequently [10]. An attractive solution for such a network scenario is to quickly deploy an additional unmanned aerial vehicle (UAV) as a relay node to diverse the FSO-based HAP-to-ground link in case of cloud blockage. Furthermore, a reconfigurable intelligent surface (RIS) array [11], an emerging technology, can be carried by the UAV to reflect the incoming light from HAP and ensure that the transmitted light points to the GS.
The combination of UAV and RIS offers an attractive solution for HAP-based SAGIN using hybrid FSO/RF in the presence of cloud coverage. Besides the flexible deployment, it requires less complex additional hardware than conventional relay nodes (i.e., using AF or DF at the UAV). Fig. 1 illustrates the HAP-based SAGIN using FSO/RF over clouds, where the RIS-UAV is temporarily deployed to a position with negligible (or without) cloud coverage. It is worth mentioning that the relevant studies are still in the early stage, where few works on FSO-based RIS-UAV relays have been recently reported in [12]- [14]. In these studies, the UAV carrying RIS array serves as a relay station for FSO-based terrestrial networks between two buildings in the presence of a blockage. To our best knowledge, the studies considering RIS-assisted UAV relay in the SAGIN, and HAP-based SAGIN using hybrid FSO/RF in this particular case, have not been available in the literature. For such network scenarios, a proper design of a link-switching scheme, especially with rate adaptation for different links, is a critical issue under different weather and atmospheric conditions. This plays an essential role in practical system deployment. As a result, it is of importance and necessity to provide a comprehensive design framework for HAP-based SAGIN using hybrid FSO/RF incorporating the FSO-based RIS-UAV relay in the presence of cloud coverage.

B. Major Contributions and Novelty
This article offers a complete design and insightful analysis of FSO-based RIS-UAV relay-assisted SAGIN under the impact of different weather and atmospheric conditions. The key contributions of this article are summarized as follows: In the HAP-based SAGIN, the primary FSO connection from HAP to the GS may be blocked by clouds containing high cloud liquid water content (CLWC) value [15]. In this case, instead of using a lower-rate backup RF link, the RIS-UAV is deployed to a position with negligible (or without) cloud coverage to diverse the FSO-based HAP-toground link. For this situation, the determination of time for deploying RIS-UAV is especially important as it is related to the tradeoff of achievable data rate and energy consumption of the UAV. This can be determined based on the predefined CLWC value. This practical issue, nonetheless, has not been addressed in the literature on FSO-based RIS-UAV, e.g., [12]- [14].
We propose a multirate system design of HAP-based SAGIN incorporating the FSO-based RIS-UAV relay, where a new link-switching scheme is presented.
In our proposal, the AF scheme is employed at HAP thanks to its cost-effectiveness and simple hardware requirements. In addition, the adaptive multirate transmission is considered for different links to reduce the frequent link switching. Different from the design in [10], a new link-switching scheme, considering the SAGIN with primary FSO-hybrid FSO/RF links and temporary FSO-based RIS-UAV link, is introduced, which is based on the specific condition of atmospheric turbulence and cloud.  For the HAP-based SAGIN using hybrid FSO/RF links, the FSO channel is modeled, taking into account the atmospheric turbulence described by the Gamma-Gamma distribution, weather effect with cloud attenuation, and beam spreading loss. Additionally, the Rician fading model is considered for the RF channel. Regarding the FSO-based RIS-UAV links, derive the statistical model that considers the combined effects of atmospheric turbulence-induced fading and the RIS-UAV hovering-induced pointing error. More importantly, capitalizing on the obtained probability density functions (pdfs), we derive the performance metrics in the closed-form expressions. Monte Carlo simulations are performed to validate the correctness of the analytical model. In addition, we also highlight the effectiveness of our proposal by comparing the existing state-of-the-art systems.
It is noteworthy that an essential novelty of the work is the complete solution for HAP-based SAGIN that we set up by jointly considering the hybrid FSO/RF techniques together with a temporarily optical RIS-UAV-based relay, and a new link-switching scheme between them to counteract the impact of cloud and atmospheric conditions. The rest of this article is organized as follows. The proposed system, link switching scheme, and rate adaptation design are described in Section II. Section III presents the channel models for transmission links. Different performance metrics, including outage probability, average transmission rate, and spectral efficiency, are analytically derived in Section IV. Simulation results are given in Section V. Finally, Section VI concludes this article.

II. SYSTEM DESCRIPTIONS
In this section, we first describe the hybrid FSO/RFbased satellite-HAP-GS system. Then, a proposal for the FSO-based RIS-UAV relay for our considered system in case of cloud blockage is introduced. Finally, we present the design of link switching and rate adaptation schemes for our proposed system. Fig. 2 illustrates the hybrid FSO/RF-based, HAPassisted satellite-to-GS system. In our considered system, the HAP is a relay node for the satellite-to-GS link, which employs the fixed-gain AF scheme. The FSO transmission is used for the link from the satellite to HAP. In addition, the hybrid FSO/RF one, using a switching scheme, is employed for the HAP-to-GS link, where FSO and RF are considered as the primary and backup links, respectively.

A. Hybrid FSO/RF-Based Satellite-HAP-GS System
At the satellite, the modulated signal using a quadrature amplitude modulation (QAM) modulation is expressed as s n (t ) = A nI g(t ) cos 2π f c t − A nQ g(t ) sin 2π f c t (1) where g(t ) is the pulse shaping function and f c is the QAM subcarrier frequency while A nI = {2n − 1 − I} I n=1 and A nQ = {2n − 1 − Q} Q n=1 are the in-phase and quadrature components of amplitude, respectively. The QAM signal is then used to modulate the laser intensity; as a result, the transmitted optical signal can be expressed as where P t is the satellite's transmitted power, m is the modulation index, and s n (t ) is given in (1). At the HAP relay, the received optical signal is amplified and forwarded to the GS. Here, hybrid FSO/RF transmission is considered for forwarding the signals, by which either the primary link of FSO or backup link of RF is selected based on the channel state information (CSI) feedback from GS. Particularly, if the primary FSO link is chosen, the received optical signal from the satellite is amplified with a fixed-gain G FSO so that the output at the HAP is given as where h SH is the channel coefficient of the FSO-based satellite-to-HAP link, s(t ) is given in (2), and n f 1 is the additive white Gaussian noise (AWGN) with variance σ 2 n f 1 .
Besides, if the backup RF link is activated, the received optical signal is first converted to the electrical by a photodetector followed by a fixed-gain (G RF ) amplifier. The output RF signal is thus expressed as r r H (t ) = G RF ηh SH P t ms n (t ) + n r 1 (4) where η is the electrical-to-optical conversion coefficient, n r 1 is the AWGN with variance σ 2 n r 1 , and s n (t ) is given in (1). At the GS destination, if the FSO transmission is selected for HAP-to-GS link, the received signal at GS is given as where h f HG is the FSO fading channel coefficient, and n f 2 is the AWGN with the variance σ 2 n f 2 . In addition, if the RF is used for HAP-to-GS link, the received signal at GS is written as y r D = h r HG G RF ηh SH P t ms n (t ) + h r HG G RF n r 1 + n r 2 (6) where h r HG is the RF fading channel coefficient, and n r 2 is the AWGN with the variance σ 2 n r 2 . From (5) and (6), the instantaneous received signal-tonoise ratios (SNRs) at GS, denoted as γ f e2e for FSO-FSO link and γ r e2e for FSO-RF link, are expressed as are, respectively, the instantaneous SNRs of satellite-to-HAP and HAP-to-GS FSO links while γ RF = ηh r HG 2 /σ 2 n r 2 is the instantaneous SNR of RF-based HAP-to-GS link. Additionally, we assume that the fixed-gain G FSO can completely compensate the noise, where we can set [16].

B. Proposal of FSO-Based RIS-UAV Relay
As reported in [15], the cloud coverage, especially with high CLWC values, may completely block the FSO-based HAP-to-GS link. To maintain the high-speed connection of the FSO link, a UAV equipped with a RIS array is deployed to the positions with negligible (or without) cloud coverage. The UAV relay can diverse the FSO-based HAP-to-GS link, as depicted in Fig. 2. The RIS module mounted on the UAV serves as a reflector, which can reflect and forward the optical signals from HAP to GS [17]. It is worth noting that the position for UAV deployment is based on the cloud forecast map and the actual monitoring data. Besides, the RIS-UAV relay is deployed when the CLWC value reaches a predefined CLWC threshold M th c , which is determined by a targeted outage level of the FSO-based direct link from HAP to GS.
From the optical output signal at HAP given in (3), the received signal at GS, which is reflected and forwarded by the RIS-UAV relay, can be expressed by where h u HG is the composite fading channel coefficient of HAP-UAV-GS FSO link. n u 1 and n u 2 are the AWGN with the variance σ 2 n u 1 and σ 2 n u 2 , respectively. From (9), the instantaneous received SNR at GS in the case of RIS-UAV relay, denoted by γ u e2e , is given as  Table I with three transmission strategies. In particular, we consider the FSO-based satellite-HAP-GS link, which When the clouds block the primary link of FSO-based HAP-to-GS, the system switches to transmission strategy 2, in which the FSO link from HAP to GS is reflected and forwarded by the RIS-UAV relay. Suppose the first two transmission strategies are not available. In that case, e.g., cloud blockage on the primary link and strong turbulence on the RIS-UAV relay link, the third transmission strategy is considered, where RF transmission is used for the direct link between HAP and GS. It is worth noting that the priority order considered in Table I is applicable in case of cloud blockage on the primary link, which is the main focus of the article. For example, in other scenarios, the outage occurs on the primary link due to the strong turbulence; the RF transmission is preferable to the RIS-UAV relay solution.
2) Multirate Adaptation Design: The objective of the multirate adaptation scheme is to maximize the data rate over turbulence fading channels while satisfying a predefined Quality of Service (QoS), i.e., a targeted bit error rate (BER 0 ). For each transmission link, we adopt the subcarrier M-array QAM schemes as in [18] with a fixed symbol rate of R * s for N possible transmission modes. The transmission bit rate changes for every mode given as R * b = R * s log 2 (M ), where M is the constellation size. Let γ * th 1 < γ * th 2 < · · · < γ * th N be the switching thresholds for different transmission modes, and γ * e2e be the instantaneous end-to-end SNR for the considered transmission link. The transmission mode ith is to avoid a high error rate, no transmission is allowed when γ * e2e < γ * th 1 . These thresholds are obtained based on the fact that the average BER for each transmission mode satisfies the targeted BER 0 as in [19]. In our study, the feedback channel carrying CSI information is supposed to be reliable, where a strong error correction code can be used [20]. Besides, the temporal coherence times of considered FSO and RF links (order of tens of milliseconds) are relatively long compared with the time slot duration, including the data transmission and feedback time (order of several milliseconds) [10]. Due to the slowly time-varying nature of fading channels, CSIs are still up-to-date information when arriving at the HAP and satellite.
For the sake of explicit clarity, the aforementioned parameters used for FSO-FSO1 link are }. An example of multirate adaptation design for different transmission links is illustrated in Table II, in which N 1 = N 2 = 3 transmission modes for FSO-FSO1 and FSO-FSO2 links, whereas N 3 = 2 transmission modes for the FSO-RF link.

III. CHANNEL MODELING
This section presents the considered channel models for each transmission hop, including FSO-based satellite-to-HAP, hybrid FSO/RF-based HAP-to-GS, and FSO-based HAP-UAV-GS.
A. FSO-Based Satellite-to-HAP Link As for the FSO-based satellite-to-HAP link, the effect of atmospheric turbulence is generally small, which can be ignored as the laser beam goes through a non-atmospheric path (above 20 km compared to the ground level) [21]. In this article, we assume that HAP remains stable at its fixed position, and a fine-tracking system [22] with perfect alignment is deployed. It is, therefore, supposed that the beam spreading loss is a major impairment for this link. Besides, as reported in [23], the maximum frequency shift between an LEO satellite and a stable HAP is the order of several gigahertz. These values are, nonetheless, within the capability of the current receiver design for LEO satellitebased FSO systems (i.e., can deal with frequency shift up to more than 10 GHz [24]). As a result, we ignore the Doppler effect in the performance analysis.
Considering the Gaussian beam profile, the fraction of collected power at HAPs circular detector with an aperture radius D HAP is approximated as [21] where corresponding to the ratio of the aperture radius and beamwidth, and w L = is the beam-waist at the distance L SH , which is given by L SH = (H SAT − H HAP ) /cos (ξ SAT ), in which H SAT is the altitude of the satellite, H HAP is the altitude of the HAP, and ξ SAT is the zenith angle of the satellite, w 0 = (2λ) / (πθ SAT ) is the beam-waist at L SH = 0 with θ SAT is the divergence angle of the satellite, = 1 + 2 [25]. Besides, the equivalent beam-waist is determined as B. Hybrid FSO/RF-Based HAP-to-GS Link 1) RF Channel: For the RF channel, we consider the major impairments, including path loss, cloud attenuation, and fading phenomenon. The channel coefficient of the RF link can be expressed as h r HG = g r h r , where h r is the channel fading coefficient, whereas g 1 is the path-loss of the RF link, where G T and G R are the transmitted and received antenna gains (in dB), respectively. L A (dB) is the gaseous atmospheric loss. L F (dB) = 92.45 + 20log f c + 20logL HG is the free space loss for HAP-to-GS RF link, in which f c (GHz) is the frequency for RF link, and L HG = (H HAP − H GS ) /cos (ξ HG ) is the transmission distance from HAP to the GS with HAPs zenith angle ξ HG and H GS is the height of the GS.
Regarding the cloud attenuation, as recommended by ITU-P840 [26], the Rayleigh scattering model is suitable for modeling the attenuation with frequencies up to 200 GHz, in which the attenuation on RF link due to cloud can be given as [26] L c = d r c α r cos (ξ HG ) (13) where d r c is the vertical extent of cloud, ξ HG is the HAPs zenith angle for HAP-GS beam, and α r = K c M c is the specific attenuation within clouds in which K c ((dB/km)/(g/m 3 )) is the specific attenuation coefficient, M c (g/m 3 ) is the CLWC [26].
As for fading model, due to the line-of-sight (LOS) of the vertical link from HAP to GS, the small-scale fading is widely described by the Rician distribution [27]. Therefore, the pdf and cumulative density function (cdf) of the instantaneous received SNR can be given as [27] where Q 1 (·, ·) is the Marcum Q 1 -function, K is the Rician factor, and γ r e2e is the average SNR of the FSO-RF link. 2) FSO Channel: We take into account the major impairments on the FSO-based HAP-to- GS  Regarding the cloud attenuation, the liquid water particles in clouds cause the scattering phenomenon of the FSO beam propagation, leading to a decrease in visibility and significant attenuation of the received signal's power. Here, we consider low cloud types (1-3 km), e.g., Stratus and Cumulus, which are the most challenging for FSO communications [15]. The attenuation due to clouds can be expressed as [19] where α f is the attenuation coefficient, which is given as [10] where V is the visibility, which is a function of the number cloud droplet concentration (N c ) and CLWC [15]. Additionally, q is the size distribution of the scattering particles, and by using the Kim model, it can be found in [28].
As for the atmospheric turbulence, this phenomenon causes the scintillation effect leading to the signal power fluctuations at the GS. As reported in [28], Gamma-Gamma (GG) is one of the most suitable models for describing a wide range of turbulence conditions in FSO-based satellite communications. The pdf of h f a is then given as [17] where (·) is the gamma function, and K v (·) is the vth-order modified Bessel function of the second kind. Additionally, α and β are, respectively, the effective number of large-scale and small-scale eddies of scattering environment found in [17] and are the functions of Rytov variance (σ 2 R ), which is given as σ 2 R = 2.25k 7/6 w sec 11/6 (ξ HG ) where k w = 2π λ is the optical wave number corresponding to the optical wavelength λ. Besides, the variation of refractive index structure parameter, C 2 n (h), according to the altitude h can be described by the most widely used Hufnagel-Valley model and is given as [28], i.e.,  For the beam spreading loss, similar to (11), the fraction of the power collected by GS with aperture radius of D GS can be approximated as [21] where ρ f is the radial displacement between centers of HAP beam footprint and GS's detector, ω f L,e(eq) is the equivalent beam waist, and Additionally, ω f L,e ≈ θ HG L HG , where θ HG is the divergence angle for the HAP-GS beam, and L HG is the propagation distance from HAP to GS, i.e., L HG = (H HAP − H GS ) sec (ξ HG ). Given and h f s are found in, respectively, (16), (18), and (20), and by using a simple transformation, the cdf of instantaneous SNR for FSObased HAP-to-GS link, denoted by γ f HG , can be expressed as whereγ HG is the average SNR for FSO-based HAP-to-GS link, and G m,n p,q (·) is the MeijerG function.
C. UAV-Assisted HAP-to-GS FSO Link Fig. 3(a) depicts the structure and perspective view of RIS-based liquid crystal (LC) materials for the UAVassisted HAP-to-GS FSO link. In this article, for the sake of simplicity, we adopt RIS-based LC structure 1 in [29] using a single-element (large size) to illustrate our proposed network scenario. For this structure, there are seven layers of thin materials, including a glass substrate layer, two indium tin oxide (ITO) layers, two alignment layers, the LC material, and a CMOS silicon backplane layer.
Particularly, the incident beam, which faces refraction on the glass substrate, propagates through the RIS-layer structure. The glass substrate layer is also in charge of generating the director, which defines the final light orientation. Here, in a cell, LCs molecules tend to point toward a predefined direction called the director. In addition, the ITO is a thin conductive coating material realized with a fair tradeoff between conductivity and transparency. Its primary role is to assist with heat generation and control of the LC cell. The alignment layers guide light arrays through the predefined direction in the LC cell. The CMOS silicon backplane, which represents the bottom layer as depicted in Fig. 3(a), is made of a material with a high reflection coefficient. The RIS depth and crystal orientation can be adjusted by a tunable element so that the light reflected on the CMOS silicon backplane is oriented in the receiver's direction.
A schematic of the considered optical RIS-UAVassisted HAP-to-GS links is illustrated in Fig. 3(b), wherein the Cartesian coordinate system [x, y, z] ∈ R 3 is considered. Without loss of generality, we assume that the RIS array lies on the x − y plane and its center is at [0, 0, 0], which is the intersection of the incident beam (from HAP) and reflected beam (at RIS). Here, θ i and θ r are the incident and reflected angles at the RIS array, respectively. Additionally, θ rl is the angle between the reflected beam and the normal vector of the lens at GS. In this article, we take into account the major impairments on the links, including 1) atmospheric attenuation h u l , 2) RIS-UAV hovering-induced misalignment h u m , and 3) atmospheric turbulence h u a . The composite channel fading coefficient is, then, formulated as h u HG = h u l h u m h u a . The attenuation h u l , which can be found in [30, (7)], is considered in clear sky conditions.
As for the RIS-UAV hovering-induced misalignment, since the RIS-UAV relay receives a sufficiently large beam footprint from HAP, the beam truncation can be ignored. As a result, the position fluctuations of the RIS array within its plane, i.e., the x − y plane, can be neglected. In other words, only the fluctuations along the z-axis, denoted by z r ∼ N (0, σ 2 ris ), are considered. The misalignment due to the UAV hovering results in the misalignment between center of reflected beam footprint from the RIS-UAV relay and the center of the GSs detector, denoted by u. Using Fig. 3(b), if the UAV position fluctuation of z r , the misalignment u at GS can be computed as u = 1 cos θ rl sin(θ i +θ r ) cos θ i z r . As z r ∼ N 0, σ 2 ris , the misalignment also follows a zeros mean Gaussian distribution, i.e., u ∼ N 0, σ 2 u with the variance of σ 2 u = 1 cos 2 θ rl sin 2 (θ i +θ r ) cos 2 θ i σ 2 ris . On the other hand, as the Gaussian beam profile is considered, the fraction of collected power at the GSs detector with aperture radius of D GS can be approximated as [13] where is the optical beam waist at the GSs detector. Here, F 0 is the radius of curvature (F 0 = ∞, for a collimated Gaussian beam), and L UG is the propagation distance from UAV to GS, i.e., L UG = (H UAV − H GS ) sec (ξ UG ) with ξ UG the UAV's zenith angle. Additionally, w u = 2λ πθ RIS with θ RIS = θ HU (1 + L HU /L UG ), where L HU = (H HAP − H UAV ) sec (ξ HU ) with ξ HU is the HAPs zenith angle for the HAP-UAV beam. It is worth noting that this article considers the power amplifying-RIS structure as depicted in Fig. 3(a), in which the geometric loss at RIS array can be compensated as reported in [29].
From (22) and by assuming that u follows a zero mean Gaussian distribution, the pdf of h u m is given as [31] Regarding the atmospheric turbulence, it can be expressed as h u a = h u a 1 h u a 2 , in which the pdf of h u a z , z ∈ {1, 2} can be modeled by the GG distribution as where α z and β z are, respectively, the effective number of large-scale and small-scale eddies of scattering environment. From (24), the pdf of h u a can be expressed With the help of [32, (9.31.2), (8.4.23.1)], [33, (8.4.23.1)], and [34, (21)], along with some mathematical manipulations, the pdf of h u a can be derived as Composite Channel Statistical Model: The pdf of composite channel coefficient, i.e., h u HG = h u l h u m h u a , is given as By using a transformation δ = ln and applying the Gauss-Hermit quadrature integration [35, (25.4.46)] along with several mathematical manipulations, the closed form of (26) can be obtained as where w i is the weighting factor and x i is the zeros of the Hermite polynomial found in [35]. It is noted that the Gauss-Hermite used in (26) quickly converges for finite values of N (e.g., N = 90 terms).

D. End-to-End Channel Statistical Model
We now analyze the end-to-end statistical models, i.e., cdf, which is then used to obtain the performance metrics. The cdfs shown at the top of the next page are derived for each link in the considered HAP-based SAGIN, including FSO-FSO1, FSO-FSO2, and FSO-RF (refer to Table I for the definition), as follows: 1) FSO-FSO1 Link: From (7) and (21), the cdf of end-to-end instantaneous SNR of the primary FSO-FSO1 link can be derived in (28) shown at the bottom of the next page. Here,γ HG is the average SNR for FSO-based HAP-to-GS link, and G m,n p,q (·) is the MeijerG function.
2) FSO-FSO2 Link: From (27), we can derive the cdf of h u HG , i.e., F h u HG (h u HG ), and then, the cdf of γ u HG , i.e., F γ u HG γ u HG , by using a variable transformation γ u HG = ηh u is derived in the Appendix A. From (10) and using the cdf of F γ u HG γ u HG , the cdf of end-to-end SNR of the FSO-FSO2 link can be computed in (29) shown at the bottom of this page.
3) FSO-RF Link: From (8) and (15), the cdf of end-toend FSO-RF link can be obtained in (30) shown at the bottom of this page.
Here, it is noted that pdfs of end-to-end instantaneous SNR for each link can be easily obtained from the derived cdfs by the following expression, i.e., f γ e2e (γ ) = ∂ ∂γ F γ e2e (γ ).

IV. PERFORMANCE ANALYSIS
This section focuses on the performance analysis of the proposed system with rate adaptation transmission over atmospheric turbulence channels. In particular, the probability of transmission mode selection for the rate adaptation scheme is investigated, then used to analytically derive the performance metrics, including outage probability, average transmission rate, and spectrum efficiency.

A. Transmission Mode Selection Probability
As mentioned in Section II-C, to determine the mode selection probability for the system's performance analysis, we need to determine the switching SNR threshold levels. As reported in [10], these threshold levels are obtained based on the condition that the average BER for each transmission mode satisfies the targeted BER 0 . As a result, the switching SNR threshold level for the transmission mode ith can be given as where M is the modulation order [10]. In addition, the mode selection for each transmission link can be formulated as Mode kth for FSO-RF if γ r th k ≤ γ r e2e < γ r th k+1 (32) where i ∈ {1, 2, . . . , N 1 − 1}, j ∈ {1, 2, . . . , N 2 − 1}, and k ∈ {1, 2, . . . , N 3 − 1} with N 1 , N 2 , and N 3 are the number of transmission modes for FSO-FSO1, FSO-FSO2, and FSO-RF links, respectively. From (31) and (32), the probability that FSO-FSO1 link using the mode ith for transmission can be given as where f f γ e2e (·) and F f γ e2e (·) are, respectively, the pdf and cdf of FSO-FSO1 link found in (28). Regarding the FSO-FSO2 link, the probability of selecting the transmission mode jth, given the outage occurs on the FSO-FSO1 link, can be expressed as where γ f th 1 is the outage threshold level of the FSO-FSO1 link while f u γ e2e (·) and F u γ e2e (·) are, respectively, the pdf and cdf of the FSO-FSO2 link found in (29).
In addition, the FSO-RF link is activated when either 1) outage of the FSO-FSO1 link due to strong turbulence (without UAV deployment) or 2) both FSO-FSO1 and FSO-FSO2 links are not available (cloud blockage at the FSO-FSO1 link and strong turbulence at the FSO-FSO2 link). Then, the probability that FSO-RF link using the mode kth If CLWC > M th c : with the UAV deployment where γ u th 1 is the outage threshold level of the FSO-FSO2 link while f r γ e2e (·) and F r γ e2e (·) are the, respectively, pdf and cdf of FSO-RF link found in (30).
An example of the mode selection probability for different transmission links over a range of CLWC values is depicted in Fig. 4. Using this figure, we can investigate the possibility of using transmission modes and links for different values of CLWC. Specifically, with low CLWC values, the primary link (FSO-FSO1) can select the highest mode for the transmission. In addition, when the CLWC value increases, lower transmission modes are chosen for the primary link to satisfy the predefined QoS. The probability of using the primary link starts decreasing at high CLWC values, e.g., CLWC = 14 mg/m 3 and higher, in which the system switches to the backup options, i.e., FSO-FSO2 and FSO-RF links.

B. Performance Metrics
Using the transmission mode probabilities derived in Section IV, we now investigate different performance metrics for our proposed system, including outage probability, average transmission rate, and spectrum efficiency.
1) Outage Probability: The outage probability, denoted as P out , is defined as the probability that the system moves to the zero-rate mode, where no data is transmitted to avoid the high error rate. It is computed as where f * γ e2e (·) and F * γ e2e (·) are the corresponding pdf and cdf while γ f th 1 , γ u th 1 , and γ r th 1 are the outage threshold levels for the FSO-FSO1, FSO-FSO2, and FSO-RF links, respectively.
2) Average Transmission Rate: The average data bit rate for each transmission link can be calculated as where p f i and p u j are given in corresponding (33) and (34) while p r k can be found in (35) and (36) [19]. Additionally, represents for total number of transmission modes and data bit rate of the FSO-FSO1, FSO-FSO2, and FSO-RF links, respectively. From (38), the average system's transmission rate can be given asR 3) Spectral Efficiency: It is supposed that a Nyquist pulse shaping filter with the bandwidth B = 1/R * s , where R * s is the symbol rate. The achievable spectral efficiency for different transmission links is, then, determined as where R f s , R u s , and R r s are the fixed symbol rates of the FSO-FSO1, FSO-FSO2, FSO-RF links, respectively [10]. From (40), the average system's spectral efficiency can be given asS

V. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we present the performance evaluation for our proposed system analyzed in Section IV with different parameter settings. We also comparatively discuss the performance of our proposed system with the existing state-of-the-art systems, including all-optical HAP-assisted SAGIN and the hybrid FSO/RF-based HAP-assisted SA-GIN proposed in [10], which highlights the effectiveness of our proposed system in the presence of cloud coverage. Regarding the rate adaptation design, N 1 = N 2 = N 3 = 3 transmission modes, i.e., BPSK, QPSK, 8-QAM. In addition, the parameters, unless otherwise noted, are shown in Table III. Monte Carlo simulations using MATLAB are also performed to validate the correctness of analytical results, and a good match can be confirmed.

A. Outage Performance
First, we investigate the time for deploying the RIS-UAV relay in our proposed system, which is determined by the CLWC threshold M th c . For that purpose, the rule of thumb here is that we want to maximize the achievable data rate while saving the energy consumption of the UAV. As a result, the UAV needs to deploy when the outage performance of the primary link (FSO-FSO1) is about to happen, which is decided by a predefined outage level. Specifically, Fig. 5 analyzes the outage performance of the primary FSO link over a range of CLWC values. Also, different satellite's powers, i.e., P t = 7 dBm, P t = 11 dBm, and P t = 13 dBm, are considered. We could find the CLWC thresholds from a predefined outage level that the UAV should be deployed using this figure. For example, with the predefined outage level of 10 −3 , the UAV should be deployed at the time that CLWC = 7.8 mg/m 3 , 10.3 mg/m 3 , and 11.7 mg/m 3 for the transmitted power levels of 7 dBm, 11 dBm, and 13 dBm, respectively. Next, we highlight the effectiveness of our proposed system in comparison with 1) all-optical HAP-assisted SAGIN (FSO-FSO Only) and 2) the hybrid FSO/RF-based HAP-assisted SAGIN (FSO-Hybrid FSO/RF Only) proposed in [10]. Particularly, Fig. 6 investigates the system's outage performance for different CLWC values, when P t = 7 dBm and CLWC threshold M th c = 7.8 mg/m 3 . As is expected, in the presence of clouds, our proposed system outperforms the existing ones in terms of outage performance. The reason is that when the CLWC increases, the outage performance of the system with FSO-FSO only is significantly deteriorated. In contrast, our proposed system can maintain a better performance than the FSO-hybrid FSO/RF system in [10] because it can attain spatial diversity with the aid of the RIS-UAV relay. For instance, to retain the outage level of 10 −6 , our proposed system can operate under the CLWC value of 12.5 mg/m 3 , are taken into account. Additionally, different divergence angles for HAP-GS beam, i.e., θ HG = 3 mrad and 4 mrad, are considered. Using this figure, we can select a proper GSs aperture size for the given satellite's transmitted power and divergence angle for HAP-GS beam to retain a specific system's outage level. For example, when P t = 7 dBm (or P t = 11 dBm), the GSs aperture size should be 14 cm (or 7 cm) to maintain an outage level of 10 −7 for the corresponding HAP-GS beam's divergence angles, i.e., θ HG = 3 mrad.

B. Average Transmission Rate and Spectral Efficiency
We are now able to analyze the other essential system performance metrics, including average transmission rate and spectral efficiency under the impact of clouds.
We further highlight the effectiveness of our proposed system in terms of average transmission rate performance. Fig. 8 analyzes the average transmission rate over a range of CLWC values. Also, P t = 11 dBm and M th c = 10.3 mg/m 3 . As is expected, our proposed system can achieve a considerable enhancement in terms of achievable rate in comparison with the systems with FSO-FSO only as well as with FSO-hybrid FSO/RF only as in [10]. At high CLWC values, while the achievable data rates significantly decrease in systems with FSO-FSO only (deteriorated by clouds) and with FSO-hybrid FSO/RF only (switching to the lower-rate mode of RF link), our proposed system can even attain a higher data rate due to the fact that the UAV-assisted FSO-FSO2 link can have more chance to select higher data-rate modes for transmission. For example, when CLWC = 18 mg/m 3 , our proposed system can maintain the achievable data rate   Fig. 9, we analyze the achievable data rate of the proposed multirate adaptation design for the backup FSO-FSO2 link in comparison with the fixed-rate schemes. Also, P t = 7 dBm, M th c = 7.8 mg/m 3 , and different fixed-rate schemes, i.e., M 2 = 2 (BPSK), M 2 = 4 (QPSK), and M 2 = 8 (8-QAM), are taken into account. As is expected, the proposed system with FSO-FSO2 link using the rate adaptation scheme offers considerably better achievable data rate than the one using the fixed-rate scheme. For instance, when CLWC = 18 mg/m 3 , the system with FSO-FSO2 link using the rate adaptation scheme can achieve 1.17 Gb/s while that using the fixed-rate scheme can maintain only 83 Mb/s, 173 Mb/s, and 0.96 Gb/s for M 2 = 2, M 2 = 4, and M 2 = 8, respectively. Fig. 10 investigates altitudes for the deployment of RIS-UAV relay in terms of achievable data rate over a  range of CLWC. We set the satellite's transmitted power P t = 11 dBm. Also, different HAPs altitudes, i.e., h HAP = 20 and 24 km, are considered. As seen, when the CLWC value increases, the system has more chance to select the UAV-assisted FSO-FSO2 link for transmission, resulting in an increase in achievable data rate. In addition, using this figure, we can decide the altitude for deploying the RIS-UAV relay and HAP. A higher achievable data rate can be achieved with lower UAV and HAP altitudes due to smaller RISs virtual divergence angle for the UAV-GS beam, defined by a ratio between HAP-UAV and UAV-GS distances. The rule of thumb here is to maintain a targeted achievable data rate while quickly deploying the RIS-UAV relay. As a result, for example, when CLWC = 19 mg/m 3 , the HAP and RIS-UAV relay should be deployed at the altitude of 20 km and 1 km, respectively, to retain a targeted achievable data rate of 1.25 Gb/s. Finally, we highlight the effectiveness of our proposed system in terms of average spectral efficiency. Fig. 11 investigates the average spectral efficiency of different systems over a range of CLWC values. This figure shows a considerable achievable spectral efficiency improvement of the proposed system compared to those with FSO-FSO only and with FSO-hybrid FSO/RF only. For instance, when CLWC = 19 mg/m 3 , the achievable spectral efficiency of our proposed system is 2.5 b/symbol. In comparison, they are 0.29 b/symbol and 0.52 b/symbol for the systems with FSO-FSO only and with FSO-hybrid FSO/RF only.

C. Design Guidelines
From the obtained insightful numerical results, we now provide the design guidelines recommended for effectively designing our proposed system in the presence of cloud coverage, as follows: 1) Given a predefined outage level of 10 −3 for the primary FSO-FSO1 link, the CLWC thresholds for deploying RIS-UAV relay are M th c = 7.8 mg/m 3 , 10.3 mg/m 3 , and 11.7 mg/m 3 for the satellite's transmitted power levels of 7 dBm, 11 dBm, and 13 dBm, respectively. 2) When P t = 7 dBm and M th c = 7.8 mg/m 3 , to maintain the outage level of 10 −6 , our proposed system can operate under the CLWC value of 12.5 mg/m 3 while they are 5 mg/m 3 and 10 mg/m 3 for, respectively, the system with FSO-FSO only and one with FSOhybrid FSO/RF only, which highlights the effectiveness of our proposed system. 3) When P t = 7 dBm (or P t = 11 dBm), the GSs aperture size should be 14 cm (or 7 cm) to maintain an outage level of 10 −7 for the corresponding HAP-GS beam's divergence angles, i.e., θ HG = 3 mrad. 4) At higher CLWC values, our proposed system can achieve a higher data rate (more chance to select higher transmission modes). In contrast, achievable data rates significantly decrease in systems with FSO-FSO only and with FSO-hybrid FSO/RF only. For example, when P t = 11 dBm and CLWC = 18 mg/m 3 , our proposed system can maintain the achievable data rate of 1.15 Gb/s while they are merely 264 Mb/s and 352 Mb/s for the systems with FSO-FSO only and with FSO-hybrid FSO/RF only, respectively. 5) The effectiveness of the multi-rate adaptation scheme is confirmed in which, for example, when P t = 7 dBm and CLWC = 18 mg/m 3 , the system with FSO-FSO2 link using the rate adaptation scheme can achieve 1.17 Gb/s while that using the fixed-rate scheme can maintain only 83 Mb/s, 173 Mb/s, and 0.96 Gb/s for M 2 = 2, 4, 8, respectively. 6) The higher achievable data rate can be achieved with higher UAV altitudes due to smaller RISs virtual divergence angle for the UAV-GS beam, defined by a ratio between HAP-UAV and UAV-GS distances. When CLWC = 19 mg/m 3 , for quick deployment, the HAP and RIS-UAV relay should be deployed at the altitude of 20 km and 1 km to retain a targeted achievable data rate of 1.25 Gb/s.

7)
The effectiveness of our proposed system in terms of achievable spectral efficiency is confirmed in which, for instance, when CLWC = 19 mg/m 3 , our proposed system can achieve 2.5 b/symbol. In comparison, they are 0.29 b/symbol and 0.52 b/symbol for, respectively, systems with FSO-FSO only and with FSO-hybrid FSO/RF only.
It is worthy that the above design guidelines are useful for practical deployment of network scenarios, e.g., spaceassisted vertical backhaul/fronthaul solutions for wireless mobile networks. Our network could be complementary to the terrestrial backhaul/fronthaul networks in some situations. For instance, it is deployed in case of failure in the terrestrial networks or temporary demand for backhaul/fronthaul during a social event such as sporting events. Moreover, our network is capable of offering backhaul/fronthaul to the small-cell base stations that are located in hard-to-reach areas where fiber or microwave links may not be readily available and expensive to deploy. Examples of these locations could be in rural/remote or urban areas.

VI. CONCLUSION
In this article, we have proposed to deploy the additional UAV equipped with a RIS array in the HAP-based SAGIN to enhance the system performance under the impact of weather conditions. We also proposed a link switching scheme and rate adaptation design. Several performance metrics, including outage probability, average transmission rate, and spectral efficiency, were analytically obtained. Numerical results illustrated the outperformance of the proposed system compared to the existing state-of-the-art ones over the turbulence fading channels. We also provided the design guidelines that can be helpful for the practical system design of RIS-UAV relay-assisted HAP-based SAGIN. Future work would be interesting to investigate the deployment of multiple RIS-UAV relays. The spatial diversity can be attained using selection combining, equal-gain combining, and maximum ratio combining schemes.