# Secrecy Performance Analysis of Integrated RF-UWOC IoT Networks Enabled by
UAV and Underwater-RIS

Abrar Bin Sarawar, A. S. M. Badrudduza, Member, IEEE, Md. Ibrahim, Member,
IEEE, Imran Shafique Ansari, Senior Member, IEEE, Heejung Yu, Senior Member,
IEEE

###### Abstract

## Abstract

In the sixth-generation (6G) Internet of Things (IoT) networks, the use of
UAV-mounted base stations and reconfigurable intelligent surfaces (RIS) has
been considered to enhance coverage, flexibility, and security in non-
terrestrial networks (NTNs). In addition to aerial networks enabled by NTN
technologies, the integration of underwater networks with 6G IoT can be
considered one of the most innovative challenges in future IoT. Along with
such trends in IoT, this study investigates the secrecy performance of IoT
networks that integrate radio frequency (RF) UAV-based NTNs and underwater
optical wireless communication (UOWC) links with an RIS. Considering three
potential eavesdropping scenarios (RF signal, UOWC signal, and both), we
derive closed-form expressions for secrecy performance metrics, including
average secrecy capacity, secrecy outage probability, probability of strictly
positive secrecy capacity, and effective secrecy throughput. Extensive
numerical analyses and Monte Carlo simulations elucidate the impact of system
parameters such as fading severity, the number of RIS reflecting elements,
underwater turbulence, pointing errors, and detection techniques on system
security. The findings offer comprehensive design guidelines for developing
such a network aiming to enhance secrecy performance and ensure secure
communication in diverse and challenging environments.

## Keywords

Reconfigurable intelligent surface, unmanned aerial vehicle, non-terrestrial
network, underwater optical wireless communication, physical layer security

##  1 Introduction

###  1.1 Background

Advancements in machine-type communications, particularly in Internet of
Things (IoT), have led to the proliferation of connecting devices with diverse
communication needs which poses significant challenges in network design. In
non-terrestrial networks (NTNs), mobile base stations (BSs), mounted on
unmanned aerial vehicles (UAVs) (a.w.a. drones), offer a flexible network
infrastructure for uncovered areas by ground BSs [1]. Bypassing the need for
extensive terrestrial infrastructure deployment, UAVs enhance network coverage
and performance across various IoT applications and use cases. UAV-mounted BSs
offer network flexibility and scalability for future networks by flying on-
demand to locations, bypassing terrestrial constraints, and ensuring
connectivity with both ground users and BSs [2]. In addition to aerial
networks, the integration of underwater networks into 6G IoT networks has been
considered an essential research area, especially in IoT applications such as
underwater monitoring with unmanned underwater vehicles (UUVs). This makes
them essential components of an integrated network spanning space, air, land,
and sea, and optimal for serving predictable traffic demands of massive
machine-type communication (MMTC) and IoT networks, where nodes are typically
static and traffic demands are predictable [3].

Nowadays, the reconfigurable intelligent surface (RIS) plays a pivotal role in
the landscape of IoT-enabled 6G wireless applications, adeptly addressing
challenges by modifying phase shifts and reflecting signals to intended
locations [4]. An RIS is now a promising solution to address localized
coverage gaps in crowded city instances and difficult residential propagation
environments and can easily be mounted on indoor or outdoor walls. Beyond
traditional NTN and/or underwater optical wireless communication (UOWC)
networks, the integration of UAV-based NTN and UOWC technologies demonstrates
valuable impacts in achieving wider coverage and higher data rate [5]. The
integration of RISs into an NTN-UOWC dual-hop link has become essential,
offering cutting-edge research avenues for seamlessly connecting terrestrial
and underwater networks. Furthermore, the signal modification technique of an
RIS can be leveraged to minimize propagated signals to eavesdroppers,
introducing a new dimension to physical layer security (PLS) [6].

###  1.2 Literature Review

RISs comprised of cost-effective meta-surfaces have been recognized as a
prospective technology in wireless communication, offering creative ways to
enhance performance across diverse dimensions. In relay systems, radio signals
are amplified to extend transmission range. On the other hand, an RIS acts
passively to reflect signals by utilizing the meta-surfaces as reflective and
refractive radio mirrors and an RIS is regarded as a highly promising
technology for optimizing data transmission efficiency. A comparative analysis
of coverage extension of RIS-aided systems compared to direct link was
unveiled in [7]. In [8] presented a reliable analytical basis for evaluating
the maximum diversity order and outage probability (OP) in RIS-aided systems.
When compared to other active RIS designs, the recommended design in [9] was
proficient in both affordability and energy efficiency. For RIS-aided
millimeter wave (mmWave) and sub-THz systems, channel estimation techniques
were studied in [10]. In [11], the design approaches of RIS phase shift were
investigated. The concept of underwater RISs was introduced in [12]. However,
the detailed performance analysis of underwater links including RISs has been
rarely investigated.

Nowadays, dual-hop networks, especially those integrating RF and optical
communication technologies, have the ability to broaden coverage and enable
higher data rates. Recently, there has been growing interest in using UAVs to
enhance the connectivity and coverage of dual-hop networks, especially in
areas where traditional infrastructure is unavailable or impractical. In
particular, 6G NTN enabled by UAVs as well as satellites have gathered
research interests to realize three-dimensional (3D) global coverage. A
notable UAV relay method was approached for a hybrid RF-FSO network in [13]
emphasizing the critical role of minimizing pointing errors to reduce OP. In
[14], ensuring the collaboration with IoT, the UAV-aided RF-FSO model was
considered in terms of OP and bit error rate (BER). The authors demonstrated
the closed-form expressions in the presence of atmospheric turbulence.
However, for ensuring line-of-sight (LOS) communication, a UAV-aided triple-
hop RF-FSO-UOWC network was demonstrated in [15] whereby the end-to-end OP and
average BER were evaluated.

In wireless communication, ensuring the confidentiality of transmitted data is
paramount for safeguarding sensitive information, and guaranteeing the success
of strategic operations to uphold national security, especially in military
defense applications. A series of studies on the intricacies of secrecy
evaluation shed light on innovative techniques e.g., introducing artificial
noise (AN), using RISs [16], incorporating UAVs, etc., to enhance the security
of communication systems [17, 18, 19, 20, 21, 22, 23, 24]. Beginning with
[25], the authors explored PLS mechanisms in simultaneous power and data
transfer scenarios by introducing AN-aided transmission systems. In [26], a
secrecy analysis of RF-UOWC networks was conducted, revealing that a decrease
in fading and underwater turbulence (UWT) could fortify the security of such
communication systems. In parallel efforts, the authors of [27, 28] focused on
the secrecy performance of similar types of networks, employing sophisticated
statistical distributions to model RF and UOWC links along with antenna
selection approaches. The authors of [29] conducted a comprehensive
investigation into the secrecy outage probability (SOP) of hybrid RF-FSO
models, providing valuable insights into system behavior at high SNR regimes.
The nuances of RF and FSO wiretapping were analyzed in [30], uncovering the
superior performance of FSO eavesdropping in weak turbulence conditions.
Furthermore, the authors in [31] had provided an RIS-aided hybrid RF-UOWC
network, emphasizing the critical role of reflective elements in ensuring
secure communication for users and thwarting potential eavesdroppers. Besides,
researchers in [32] looked into how temperature gradients and air bubble
levels affected the UOWC channel within UAV-based RF (NTN)-UOWC networks.

###  1.3 Motivation and Contribution

It is noteworthy that the existing research predominantly explores secrecy
performance in scenarios where the source remains stationary [29] limiting the
applicability in dynamic environments. This gap in literature underscores the
need to investigate new models that accommodate dynamic sources, particularly
UAVs and low earth orbit (LEO) satellites in NTNs. Moreover, while some recent
studies have considered secrecy frameworks within the hybrid RF-FSO paradigm,
these investigations have primarily focused on terrestrial network (TN)
scenarios, neglecting the unique challenges posed by UOWC. Additionally,
although an RIS has shown promise in enhancing security in RF and FSO
communications [33] its potential in UOWC scenarios remains largely untapped.
Therefore, there exists a critical research gap in exploring the integration
of an RIS for enhancing both performance and security in UOWC environments.
Furthermore, while previous studies have examined eavesdropping scenarios in
either RF or UOWC paths individually [26, 31], simultaneous eavesdropping in
both domains remains unexplored. This limitation hampers the development of
comprehensive security strategies for hybrid RF-UOWC networks, necessitating
novel approaches to address potential threats effectively. In light of these
considerations, we address the research gap by introducing innovative models
that incorporate UAVs as dynamic sources and leverage an RIS in the UOWC link
for enhanced performance and security. Taking into account the small-scale
fluctuations of the RF links, the generalized $\kappa-\mu$ distribution is
considered, which yields the widely-used Rice and Nakagami-$m$ distributions
as special cases. All the underwater links are followed by a mixture
Exponential-Generalized Gamma model (mEGG) provides a vivid representation of
the random nature of irradiance fluctuations on the laser beams. In the end,
the following outline holds the main contribution of this study:

  * •

Our primary contribution lies in addressing the gap in research by proposing a
UAV-based RF NTN-UOWC model that encompasses three potential eavesdropping
scenarios: RF, UOWC, and simultaneous RF-UOWC eavesdropping. Unlike previous
studies that primarily focused on stationary sources, our model accounts for
the dynamic nature of UAVs, enabling a more accurate representation of real-
world scenarios. Furthermore, apart from the existing secrecy analysis on
hybrid RF NTN-UOWC networks, we extend our analysis to include the deployment
of a RIS in UOWC environments offering a more comprehensive security analysis.

  * •

We derive the cumulative distribution function (CDF) for the dual-hop RF-UOWC
link, followed by the derivation of analytical expressions for secrecy metrics
such as average secrecy capacity (ASC), SOP, probability of strictly positive
secrecy capacity (SPSC), and effective secrecy throughput (EST), validated
through Monte Carlo (MC) simulations. We also provide asymptotic expressions
of SOP that facilitate a deeper understanding of the system’s behavior at a
high SNR regime. To the best of the authors’ knowledge, the derived
expressions offer a level of generality due to the incorporation of
statistical distributions that are themselves generalized, and have not been
previously reported in the literature, representing a novel contribution to
the field.

  * •

Through the derived closed-form expressions, we investigate the influence of
fading parameters, number of reflecting elements, UWT, pointing error,
detection techniques, etc., on the overall behavior of the system.
Additionally, we offer a graphical comparison between RIS-aided and non-RIS
systems, providing valuable insights into the efficacy of an RIS deployment in
UOWC environments.

###  1.4 Organization

The paper is organized into several key sections to systematically address
different aspects of the proposed system. Section II provides a pictorial
representation of the entire system. In Section III, we detail the channel
models utilized in both RF NTN and UOWC links. Performance evaluation metrics
are thoroughly examined in Sections IV through VII. Section VIII presents the
results of our analytical analyses and simulation studies, offering insights
into the system’s behavior and performance under various conditions. Finally,
Section IX outlines the conclusions drawn from our findings.

##  2 System Model

Figure 1: System model incorporating a source ($\mathcal{S}$), a relay
($\mathcal{R}$), an RIS ($\mathcal{I}$), a destination ($\mathcal{D}$) and two
eavesdroppers ($\mathcal{E}$ and $\mathcal{\tilde{E}}$).

The suggested system model illustrates a dual-hop UAV-based RF NTN and an RIS-
assisted UOWC wireless communication setup. The initial stage involves a UAV
as the source $\mathcal{S}$, a relay $\mathcal{R}$, and an eavesdropper
$\mathcal{E}$ in the first hop. The subsequent phase includes an RIS
$\mathcal{I}$, a destination $\mathcal{D}$, and an additional eavesdropper
$\mathcal{\tilde{E}}$ situated below the water’s surface in the second hop.
Such a type of UAV-based NTN communication scenario is particularly beneficial
in remote areas with diverse IoT applications, where UAV-based BSs can
efficiently serve a multitude of devices across different zones during a
single flight, maximizing network coverage and performance. In this system, a
UAV hovering in the air transmits information during two distinct time slots.
During the initial slot, it transfers data to $\mathcal{R}$ positioned on a
ship. Conversely, in the subsequent slot, $\mathcal{R}$ receives the RF
signal, transforms it into its optical counterpart, and then sends it to
$\mathcal{\tilde{E}}$ via $\mathcal{I}$. Here, to maintain uninterrupted
underwater communication, $\mathcal{I}$ is deployed on an underwater drone.
Unwanted entities, identified as eavesdroppers, can intercept private data
through the $\mathcal{S}$-$\mathcal{E}$ and
$\mathcal{R}$-$\mathcal{I}$-$\mathcal{\tilde{E}}$ links and monitoring these
eavesdropping activities, the three scenarios are considered as follows.

  * •

In Scenario-I, the RF NTN communication link between $\mathcal{S}$ and
$\mathcal{R}$ is intercepted by $\mathcal{E}$. Consequently, both
$\mathcal{R}$ and $\mathcal{\tilde{E}}$ intercept and receive the exact
signals transmitted from $\mathcal{S}$.

  * •

In Scenario-II, both $\mathcal{D}$ and $\mathcal{\tilde{E}}$ capture the
identical UOWC signal transmitted from $\mathcal{R}$ through $\mathcal{I}$.

  * •

Scenario-III incorporates the consequence of conjointly eavesdropping action
at both the RF NTN and UOWC links.

However, within the proposed framework, $\mathcal{R}$ operates as a
transceiver with $L_{\mathcal{R}}$ receive antennas and a transmit aperture,
$\mathcal{S}$ is equipped with a single antenna, whereas $\mathcal{E}$ is
equipped with $L_{\mathcal{E}}$ receive antennas. Moreover, $\mathcal{N}$
denotes the count of reflective meta-surfaces within $\mathcal{I}$, wherein
$\mathcal{D}$ and $\mathcal{\tilde{E}}$ solely possess a single photodetector
for receiving optical signals. All the RF links adhere to the $\kappa-\mu$
fading model, while the UOWC network, encompassing the
$\mathcal{R}$-$\mathcal{I}$-$\mathcal{\tilde{E}}$ and
$\mathcal{R}$-$\mathcal{I}$-$\mathcal{D}$ links, encounters the mEGG
distribution coupled with pointing error impairments.

##  3 Problem Formulation

###  3.1 SNRs of UAV-aided RF NTN Links

The received signal at $l$th antenna ($l=1,2,\ldots,L_{j}$) of $\mathcal{R}$
and $\mathcal{E}$ can be expressed as

$\displaystyle
y_{jl}=\sqrt{\frac{\varrho_{jl}\mathcal{P}_{T}}{q_{j}^{\alpha_{jl}}}}h_{jl}x_{s}+z_{jl},$
(1)

where $j\in\\{\mathcal{R},\mathcal{E}\\}$, $h_{jl}$ is the channel gain,
$\varrho_{jl}=G_{s}G_{q_{jl}}\left(\frac{\Upsilon_{s}}{4\pi}\right)^{2}$,
$G_{s}$ is the gain of the antenna at $\mathcal{S}$, $G_{q_{jl}}$ is the gain
of each antenna at $\mathcal{R}$ and $\mathcal{E}$, $\Upsilon_{s}$ outlines
the wavelength, the symbol for the path loss components are $\alpha_{jl}$, the
transmitted power is denoted by $\mathcal{P}_{T}$ wherein $x_{s}$ is expressed
as the transmitted signal and complex Gaussian noise with zero mean value and
fixed standard deviation $\sigma_{z_{jl}}^{2}$ is symbolized by $z_{jl}$.
According to (1), the received instantaneous SNRs can be accomplished as
follows.

$\displaystyle\gamma_{jl}=\varrho_{jl}\left|h_{jl}\right|^{2}q_{j}^{-\alpha_{jl}}\bar{\gamma}_{jl},$
(2)

where the average SNRs are expressed by
$\bar{\gamma}_{jl}=\frac{\mathcal{P}_{T}}{\sigma_{z_{jl}}^{2}}$. Is it
noteworthy that due to the deployment of $L_{j}$-branch MRC receivers at
$\mathcal{R}$ and $\mathcal{E}$, the output SNRs at the respective combiners
can be defined as $\gamma_{j}=\sum_{l=0}^{L_{j}}\gamma_{jl}$.

###  3.2 SNRs of RIS-aided UOWC Links

In the second hop, the channel coefficient of the $\mathcal{R}$-$\mathcal{I}$
link is specified by $h_{u}$ $(u=1,2,\ldots,\mathcal{N})$ and in the similar
manner, $g_{u,n}$ indicates the channel gain of the
$\mathcal{I}$-$\mathcal{D}$ and $\mathcal{I}$-$\mathcal{\tilde{E}}$ links.
Thus expressions of the received signal at $\mathcal{D}$ and
$\mathcal{\tilde{E}}$ is given by

$\displaystyle
y_{n}=\left[\sum_{u=1}^{\mathcal{N}}h_{u}e^{j\theta_{u,n}}g_{u,n}\right]y_{R}+w_{n},$
(3)

where $n\in\\{\mathcal{D},\mathcal{\tilde{E}}\\}$ and $y_{R}$ is the re-
transmitted signal from $\mathcal{R}$. Here,
$h_{u}=\alpha_{u}e^{j\varphi_{u}}$, $g_{u,n}=\beta_{u,n}e^{j\varrho_{u,n}}$,
$\alpha_{u}$ and $\beta_{u,n}$ are mEGG random variables (RVs), $\varphi_{u}$
and $\varrho_{u,n}$ outline the uniformly distributed phases of the
corresponding links with ranges $[0,2\pi)$, $\theta_{u,n}$ stands for the
phase shift induced by $u$th reflecting element of $\mathcal{I}$ and
$w_{n}\sim\widetilde{\mathcal{N}}(0,N_{n})$ is the AWGN samples with a power
of $N_{n}$. Equation (3) can be expressed in an alternative form as

$\displaystyle y_{n}=\mathbf{g}_{n}^{T}\theta_{n}\mathbf{h}y_{R}+w_{n},$ (4)

where $\mathbf{h}=[h_{1}h_{2}\ldots h_{\mathcal{N}}]^{T}$ and
$\mathbf{g}_{n}=[g_{1,n}g_{2,n}\ldots g_{\mathcal{N},n}]^{T}$ are the channel
coefficient vectors and the diagonal matrices for phase shifts are given by
$\theta_{n}=\text{diag}([e^{j\theta_{1,n}}e^{j\theta_{2,n}}\ldots
e^{j\theta_{\mathcal{N},n}}])$ which involves the phase shifts carried out by
the RIS elements. The instantaneous SNRs at $\mathcal{D}$ and
$\mathcal{\tilde{E}}$ can now be presented as

$\displaystyle\gamma_{n}=\left[\sum_{u=1}^{\mathcal{N}}\alpha_{u}\beta_{u,n}e^{j(\theta_{u,n}-\varphi_{u}-\varrho_{u,n})}\right]^{2}\bar{\gamma}_{n},$
(5)

where $\bar{\gamma}_{n}=\frac{\mathcal{P}_{R}}{N_{n}}$ indicate the average
SNR of $\mathcal{R}$-$\mathcal{I}$-$\mathcal{D}$ and
$\mathcal{R}$-$\mathcal{I}$-$\mathcal{\tilde{E}}$ links. It is true that the
RIS elements serve as intelligent reflecting surfaces, manipulating wireless
signals to optimize reception quality while thwarting eavesdropping attempts
by adjusting reflection coefficients, beamforming, power allocation, and phase
shifts. Achievement of this dual optimization enhances both secure
communication and overall system performance against malicious interception.
Similar to [34, 31], considering the worst-case scenario (i.e., maximum
eavesdropper’s SNR), the optimal selection of $\theta_{u_{n}}$ that maximizes
the instantaneous SNR is $\theta_{u_{n}}=\varphi_{u}+\varrho_{u_{n}}$. Hence,
the maximized SNR can be found at $\mathcal{D}$ and $\mathcal{\tilde{E}}$ is

$\displaystyle\gamma_{n}=\left(\sum_{u=1}^{\mathcal{N}}\alpha_{u}\beta_{u,n}\right)^{2}\bar{\gamma}_{n}.$
(6)

###  3.3 SNR of Dual-hop Link

The received SNR at $\mathcal{D}$ employing variable gain relay, is given by

$\displaystyle\gamma_{eq}$
$\displaystyle=\frac{\gamma_{R}\gamma_{\mathcal{D}}}{\gamma_{R}+\gamma_{\mathcal{D}}+1}$
(7a)
$\displaystyle\cong\text{min}\left\\{\gamma_{R},\gamma_{\mathcal{D}}\right\\}.$
(7b)

######  Remark 3.1.

The analytical modeling of the proposed system, incorporating SNR statistics
as described in (7a), is notably challenging due to mathematical
intractability. Consequently, we resort to the approximation provided in (7b),
which remains valid even in the context of DF relay, as testified in vast
existing literature, such as [35].

###  3.4 Statistics of UAV-aided RF NTN Links

Since the UAV is hovering in the air, the distance between $\mathcal{S}$ to
$\mathcal{R}$ and $\mathcal{S}$ to $\mathcal{E}$ can be modeled as a 3D random
waypoint model (RWP) and the corresponding random distances are denoted by
$q_{j}$. Following this 3D model, the probability distribution function (PDF)
of $q_{j}$ is given by

$\displaystyle
f_{q_{j}}(q)=\sum_{i_{j}=1}^{m}\frac{C_{i_{j}}}{D_{j}^{\beta_{i_{j}}+1}}q^{\beta_{i_{j}}},\quad
0\leq q\leq D_{j}$ (8)

where $m$ = 3, $C_{i_{j}}=\frac{1}{72}[735,-1190,455]$ and
$\beta_{i_{j}}=[2,4,6]$ [36]. The $\mathcal{R}$ is presumed to be placed at
the center of a sphere with a maximum radius of $D_{j}$ in the 3D network
topology of the RWP model.

######  Remark 3.2.

It is noteworthy that the RWP is widely used for modeling UAV links due to its
capability to capture the dynamic nature of UAV movement since the
trajectories of UAVs are not predetermined. Furthermore, the RWP allows for
the simulation of realistic UAV mobility patterns, which is essential for
accurately assessing network dynamics in communication applications.
Additionally, the parameters of RWP can be adjusted to mimic various UAV
behaviors, proving it a versatile tool for UAV link modeling [36].

Since the $\mathcal{S}$-$\mathcal{R}$ and $\mathcal{S}$-$\mathcal{E}$ links
experience $\kappa-\mu$ fading distribution, the PDF of $h_{jl}$ can be
written as [37]

$\displaystyle
f_{h_{jl}}(x)=A_{j}x^{\mu_{j}}e^{-B_{j}x^{2}}I_{\mu_{j}-1}[M_{j}x],$ (9)

where
$A_{j}=\frac{2\mu_{j}(1+\kappa_{j})^{\frac{\mu_{j}+1}{2}}}{\kappa_{j}^{\frac{\mu_{j}-1}{2}}e^{\mu_{j}\kappa_{j}}}$,
$B_{j}=\mu_{j}(1+\kappa_{j})$,
$M_{j}=2\mu_{j}\sqrt{\kappa_{j}(1+\kappa_{j})}$. Herein, $\kappa_{j}$ and
$\mu_{j}$ are known as the fading parameters, and $I_{v}(.)$ is the first kind
of modified Bessel function of order $v$ [38].

######  Remark 3.3.

The $\kappa-\mu$ fading channel model serves as a valuable tool enabling
accurate channel modeling and aiding in the optimization of communication
systems for diverse and realistic wireless application scenarios, e.g.,
wireless sensor network, satellite communication, underwater acoustic
networks, mmWave communication for 5G and beyond networks, and vehicular
networks where signals encounter rapid variations due to vehicle mobility.
Furthermore, based on the values of $\kappa$ and $\mu$, the $\kappa-\mu$
distribution exhibits enormous generality since it yields various multipath
channels [39] as special cases.

######  Lemma 3.4.

The PDF of $\gamma_{j}$ can be expressed as

$\displaystyle
f_{\gamma_{j}}(\gamma)=\sum_{i_{j}=1}^{m}\sum_{k_{j}=0}^{p}K_{1_{j}}\gamma^{-1}\text{G}_{1,2}^{1,1}\left[K_{2_{j}}\gamma\middle|\begin{matrix}1-\Psi_{i_{j}}\\\
L_{j}\mu_{j}+k_{j},-\Psi_{i_{j}}\end{matrix}\right],$ (10)

where
$K_{1_{j}}=\frac{1}{\alpha_{j}e^{L_{j}\mu_{j}\kappa_{j}}}C_{i_{j}}V_{j}(k_{j},p,L_{j}\mu_{j}-1)\left(L_{j}\mu_{j}\kappa_{j}\right)^{k_{j}}$,
$K_{2_{j}}=\frac{B_{j}D_{j}^{\alpha_{j}}}{\varrho_{j}L_{j}\bar{\gamma_{j}}}$,
and $\Psi_{i_{j}}=\frac{1}{\alpha_{j}}\left(\beta_{i_{j}}+1\right)$.

######  Proof 3.5.

See Appendix 10.

######  Lemma 3.6.

The CDF of $\gamma_{j}$ can be expressed as

$\displaystyle
F_{\gamma_{j}}(\gamma)=\sum_{i_{j}=1}^{m}\sum_{k_{j}=0}^{p}K_{1_{j}}\text{G}_{2,3}^{1,2}\left[K_{2_{j}}\gamma\middle|\begin{matrix}1-\Psi_{i_{j}},1\\\
L_{j}\mu_{j}+k_{j},0,-\Psi_{i_{j}}\end{matrix}\right].$ (11)

######  Proof 3.7.

See Appendix 11.

######  Lemma 3.8.

The asymptotic CDF of $\gamma_{j}$ can be expressed as

$\displaystyle F_{\gamma_{j}}^{\infty}(\gamma)$
$\displaystyle=\sum_{i_{j}=1}^{m}\sum_{k_{j}=0}^{p}\frac{K_{1_{j}}}{(K_{2_{j}}\gamma)^{-L_{j}\mu_{j}-k_{j}}}$
$\displaystyle\times\frac{\prod_{l_{1}=1}^{2}\Gamma(T_{1,l_{1}}+L_{j}\mu_{j}+k_{j})}{\prod_{l_{1}=2}^{3}\Gamma(T_{2,l_{1}}+L_{j}\mu_{j}+k_{j})},$
(12)

where $T_{1}=(\Psi_{i_{j}},0)$, $T_{2}=(1,1+\Psi_{i_{j}})$, and the $y^{th}$
term of $T_{x}$ can be represented as $T_{x,y}$.

######  Proof 3.9.

See Appendix 12.

$\displaystyle\aleph_{1}$
$\displaystyle=\frac{(w_{I}w_{n}\xi_{I}^{2}\xi_{n}^{2})(J_{I}J_{n}\nu_{I}\nu_{n})^{p}\left\\{\Gamma(1+p)\right\\}^{2}}{\left(\xi_{n}^{2}+p\right)\left(\xi_{I}^{2}+p\right)}.$
(14a) $\displaystyle\aleph_{2}$
$\displaystyle=\frac{(1-w_{n})w_{I}\xi_{I}^{2}\xi_{n}^{2}(\nu_{I}b_{n}c_{n}J_{I}J_{n})^{c_{n}p}\Gamma(a_{n}+p)\Gamma(1-\Delta\left(c_{n},0\right)+p)\Gamma(1-\Delta\left(c_{n},1-\xi_{I}^{2}\right)+p)}{c_{n}^{\frac{1}{2}}(2\pi)^{\frac{1}{2}(c_{n}-1)}\left(\frac{\xi_{n}^{2}}{c_{n}}+p\right)\Gamma(1-\Delta\left(c_{n},-\xi_{I}^{2}\right)+p)}.$
(14b) $\displaystyle\aleph_{3}$
$\displaystyle=\frac{(1-w_{I})w_{n}\xi_{I}^{2}\xi_{n}^{2}(\nu_{n}b_{I}c_{I}J_{n}J_{I})^{c_{I}p}\Gamma(a_{I}+p)\Gamma(\Delta\left(c_{I},1\right)+p)\Gamma(\Delta\left(c_{I},\xi_{n}^{2}\right)+p)}{c_{I}^{\frac{1}{2}}(2\pi)^{\frac{1}{2}(c_{I}-1)}\left(\frac{\xi_{I}^{2}}{c_{I}}+p\right)\Gamma(a_{I})\Gamma(\Delta\left(c_{I},1+\xi_{n}^{2}\right)+p)}.$
(14c) $\displaystyle\aleph_{4}$
$\displaystyle=\frac{(1-w_{I})(1-w_{n})\xi_{I}^{2}\xi_{n}^{2}(b_{I}J_{I})^{c_{I}p}(b_{n}J_{n})^{c_{n}p}\Gamma(a_{n}+p)\Gamma(a_{I}+p)}{\left(\frac{\xi_{n}^{2}}{c_{n}}+p\right)\left(\frac{\xi_{I}^{2}}{c_{I}}+p\right)\Gamma(a_{I})\Gamma(a_{n})}.$
(14d)

###  3.5 Statistics of RIS-aided UOWC Links

The optical beam transmitted from $\mathcal{R}$ reaches $\mathcal{D}$ after
reflection from $\mathcal{I}$. The UWT due to air bubbles, temperature
gradients, and water salinity (fresh and salty water) causes irradiance
fluctuations which can be modeled effectively with mEGG distribution [40].
Besides, as $\mathcal{I}$ is placed on an underwater drone in water, due to
the waves in the water, $\mathcal{I}$ moves slightly in the water resulting in
a pointing error that can be modeled using [41]. We assume $\alpha_{u}$ and
$\beta_{u,n}$ are mEGG distributed with pointing error impairments. Hence, the
resultant RV associated with an RIS-aided
$\mathcal{R}-\mathcal{I}-\mathcal{D}$ link for $u$th element of $\mathcal{I}$
is obtained as $Y_{u,n}=\alpha_{u}\beta_{u,n}$. Due to $\mathcal{N}$
reflecting elements, we obtain ‘$\mathcal{N}$’ number of RVs as
$\chi_{n}=\sum_{u=1}^{\mathcal{N}}Y_{u,n}$, which can be easily approximated
by the gamma distribution that results from expanding the first term of a
Laguerre series. The corresponding shape and scale parameters are given by
$\rho_{n}=\frac{(\mathbb{E}(\chi_{n}))^{2}}{\text{Var}(\chi_{n})}$, and
$w_{n}=\frac{\text{Var}(\chi_{n})}{\mathbb{E}(\chi_{n})}$, respectively. Since
the expectation operator is a linear one,
$\mathbb{E}(\chi_{n})=\mathcal{N}\mathbb{E}(Y_{u,n})$,
$\text{Var}(\chi_{n})=\mathcal{N}\text{Var}(Y_{u,n})$. Here, $p$th moment of
$Y_{u,n}$ is given by

$\displaystyle\mathbb{E}(Y_{u,n}^{p})=\aleph_{1}+\aleph_{2}+\aleph_{3}+\aleph_{4},$
(13)

where $\aleph_{1}$, $\aleph_{2}$, $\aleph_{3}$, and $\aleph_{4}$ are given in
(14a), (14b), (14c), and (14d), respectively. Finally, using the RV transform,
the PDF of $\gamma_{n}$ is expressed as follows [42].

$\displaystyle
f_{\gamma_{n}}(\gamma)=\Upsilon_{1_{n}}\gamma^{\frac{\rho_{n}}{r_{n}}-1}\text{G}_{0,1}^{1,0}\left[\Upsilon_{2_{n}}\gamma^{\frac{1}{r_{n}}}\middle|\begin{matrix}-\\\
0\end{matrix}\right],$ (15)

where
$\Upsilon_{1_{n}}=\frac{(\mathbb{E}(Y_{u,n}))^{\rho_{n}}}{r_{n}\Gamma(\rho_{n})w_{n}^{\rho_{n}}\tau_{r_{n},n}^{\frac{\rho_{n}}{r_{n}}}}$,
$\Upsilon_{2_{n}}=\frac{\mathbb{E}(Y_{u,n})}{w_{n}\tau_{r_{n},n}^{\frac{1}{r_{n}}}}$,
$r_{n}$ designates the detection method (i.e. $r_{n}$ = 1 implies the
heterodyne detection (HD) method and $r_{n}$ = 2 represents the intensity
modulation/direct detection (IM/DD) method),
$\tau_{r_{n},n}=\frac{(\eta\mathbb{E}(Y_{u,n}))^{r_{n}}}{N_{n}}$ symbolizes
electrical SNRs, $\xi_{I}$ and $\xi_{n}$ denote the pointing errors and
$J_{I}$ and $J_{n}$ indicate the pointing loss. For $\mathcal{R}-\mathcal{I}$
link, $\nu_{I}$ stands for the exponential distribution parameter, $0<w_{I}<1$
indicates the mixture weight, and the GG distribution parameters are specified
by $a_{I}$, $b_{I}$ and $c_{I}$, respectively whereas for
$\mathcal{I}-\mathcal{D}$ and $\mathcal{I}-\mathcal{\tilde{E}}$ links, those
parameters are defined as $\nu_{n}$, $w_{n}$, $a_{n}$, $b_{n}$ and $c_{n}$.
The values of those parameters are considered from [40] by taking into account
different air bubble levels, and temperature gradients in fresh and salty
water that induce diverse turbulence situations. In consequence, the CDF of
$\gamma_{n}$ can be presented as

$\displaystyle
F_{\gamma_{n}}(\gamma)=\Upsilon_{3_{n}}\gamma^{\frac{\rho_{n}}{r_{n}}}\text{G}_{1,r_{n}+1}^{r_{n},1}\left[\Upsilon_{4_{n}}\gamma\middle|\begin{matrix}1-\frac{\rho_{n}}{r_{n}}\\\
0,\Upsilon_{5_{n}},-\frac{\rho_{n}}{r_{n}}\end{matrix}\right],$ (16)

where
$\Upsilon_{3_{n}}=\frac{(\mathbb{E}(Y_{u,n}))^{\rho_{n}}}{\sqrt{r_{n}}\Gamma(\rho_{n})w_{n}^{\rho_{n}}\tau_{r_{n},n}^{\frac{\rho_{n}}{r_{n}}}(2\pi)^{\frac{r_{n}-1}{2}}}$,
$\Upsilon_{4_{n}}=\frac{(\mathbb{E}(Y_{u,n}))^{r_{n}}}{r^{r_{n}}w_{n}^{r_{n}}\tau_{r_{n},n}}$
and $\Upsilon_{5_{n}}=\frac{r_{n}-1}{r_{n}}$. Similar to (12),
$F_{\gamma_{n}}(\gamma)$ can be asymptotically expressed as

$\displaystyle F^{\infty}_{\gamma_{n}}(\gamma)$
$\displaystyle=\Upsilon_{3_{n}}\gamma^{\frac{\rho_{n}}{r_{n}}}\sum_{k_{2}=1}^{r_{n}}\left(\frac{1}{\Upsilon_{4_{n}}\gamma}\right)^{T_{3,k_{2}}-1}$
$\displaystyle\times\frac{\prod_{l_{2}=1;l_{2}\neq
k_{2}}^{r_{n}}\Gamma(T_{3,k_{2}}-T_{3,l_{2}})\Gamma(1+\frac{\rho_{n}}{r_{n}}-T_{3,k_{2}})}{\prod_{l_{2}=r_{n}+1}^{r_{n}+1}\Gamma(1+T_{3,l_{2}}-T_{3,k_{2}})},$
(17)

where $T_{3}=(1,1-\Upsilon_{5_{n}},1+\frac{\rho_{n}}{r_{n}})$.

######  Remark 3.10.

The unified mEGG distribution emerges as the optimal probability distribution
for characterizing fluctuations in underwater optical signal irradiance,
attributed to different levels of air bubbles, temperature, and salinity
gradients. This distribution not only demonstrates exceptional agreement with
measured data across varying turbulence conditions but also offers a more
manageable model, facilitating the derivation of straightforward expressions
for various performance metrics in UOWC systems. Furthermore, in the case of
uniform temperature, the intensity of the incident laser beam can be described
by the exponential-Gamma (EG) distribution which is a special case of the mEGG
model [40].

###  3.6 Statistics of Dual-hop Link

The expression for the CDF of end-to-end SNR $\gamma_{eq}$ of dual-hop RF NTN-
UOWC system is given by

$\displaystyle
F_{\gamma_{eq}}(\gamma)=F_{\gamma_{R}}(\gamma)+F_{\gamma_{\mathcal{D}}}(\gamma)-F_{\gamma_{R}}(\gamma)F_{\gamma_{\mathcal{D}}}(\gamma).$
(18)

Replacing (11) and (16) into (18), the CDF of $\gamma_{eq}$ is derived as

$\displaystyle F_{\gamma_{eq}}(\gamma)$
$\displaystyle=\Upsilon_{3_{\mathcal{D}}}\gamma^{\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}}\text{G}_{1,r_{D}+1}^{r_{D},1}\left[\Upsilon_{4_{\mathcal{D}}}\gamma\middle|\begin{matrix}1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\\\
0,\Upsilon_{5_{\mathcal{D}}},-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\end{matrix}\right]$
$\displaystyle+\sum_{i_{R}=1}^{m}\sum_{k_{R}=0}^{p}K_{1_{R}}\text{G}_{2,3}^{1,2}\left[K_{2_{R}}\gamma\middle|\begin{matrix}1-\Psi_{i_{R}},1\\\
L_{R}\mu_{R}+k_{R},0,-\Psi_{i_{R}}\end{matrix}\right]$
$\displaystyle\times\biggl{(}1-\Upsilon_{3_{\mathcal{D}}}\gamma^{\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}}G_{1,r_{\mathcal{D}}+1}^{r_{\mathcal{D}},1}\left[\Upsilon_{4_{\mathcal{D}}}\gamma\middle|\begin{matrix}1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\\\
0,\Upsilon_{5_{\mathcal{D}}},-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\end{matrix}\right]\biggl{)}.$
(19)

$F_{\gamma_{eq}}(\gamma)$ can be written asymptotically as

$\displaystyle F^{\infty}_{\gamma_{eq}}(\gamma)\approx
F^{\infty}_{\gamma_{R}}(\gamma)+F^{\infty}_{\gamma_{\mathcal{D}}}(\gamma).$
(20)

Substituting (12) and (17) into (20), the asymptotic equivalent CDF of the
dual-hop link can be obtained as

$\displaystyle F^{\infty}_{\gamma_{eq}}(\gamma)$
$\displaystyle=\sum_{i_{R}=1}^{m}\sum_{k_{R}=0}^{p}\frac{K_{1_{R}}}{(K_{2_{R}}\gamma)^{-L_{R}\mu_{R}-k_{R}}}$
$\displaystyle\times\frac{\prod_{l_{1}=1}^{2}\Gamma(T_{1,l_{1}}+L_{R}\mu_{R}+k_{R})}{\prod_{l_{1}=2}^{3}\Gamma(T_{2,l_{1}}+L_{R}\mu_{R}+k_{R})}$
$\displaystyle+\Upsilon_{3_{\mathcal{D}}}\gamma^{\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}}\sum_{k_{2}=1}^{r_{\mathcal{D}}}\left(\frac{1}{\Upsilon_{4_{\mathcal{D}}}\gamma}\right)^{T_{3,k_{2}}-1}$
$\displaystyle\times\frac{\prod_{l_{2}=1;l_{2}\neq
k_{2}}^{r_{\mathcal{D}}}\Gamma(T_{3,k_{2}}-T_{3,l_{2}})\Gamma(1+\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}-T_{3,k_{2}})}{\prod_{l_{2}=r_{\mathcal{D}}+1}^{r_{\mathcal{D}}+1}\Gamma(1+T_{3,l_{2}}-T_{3,k_{2}})}.$
(21)

##  4 Average Secrecy Capacity

Mathematically, for Scenario-I, ASC is expressed as [43, Eq. (15)]

$\displaystyle
ASC=\int_{0}^{\infty}\frac{1}{1+\gamma}F_{\gamma_{\mathcal{E}}}(\gamma)\left[1-F_{\gamma_{eq}}(\gamma)\right]\text{d}\gamma.$
(22)

Upon substituting (11) and (19) into (22), and then performing integration,
ASC is expressed as

$\displaystyle ASC$
$\displaystyle=\sum_{i_{\mathcal{E}}=1}^{m}\sum_{k_{\mathcal{E}}=0}^{p}K_{1_{\mathcal{E}}}\left(\mathfrak{R}_{1}-\sum_{i_{R}=1}^{m}\sum_{k_{R}=0}^{p}K_{1_{R}}\mathfrak{R}_{2}\right.$
$\displaystyle-\Upsilon_{3_{\mathcal{D}}}\mathfrak{R}_{3}+\left.\sum_{i_{R}=1}^{m}\sum_{k_{R}=0}^{p}K_{1_{R}}\Upsilon_{3_{\mathcal{D}}}\mathfrak{R}_{4}\right),$
(23)

where the four integral parts $\mathfrak{R}_{1}$, $\mathfrak{R}_{2}$,
$\mathfrak{R}_{3}$ and $\mathfrak{R}_{4}$ are derived as follows.

######  Remark 4.1.

The ASC expression in (23) represents the average rate at which secure
information can be transmitted over a communication channel. It is a measure
of the system’s overall ability to maintain security, considering variations
in channel conditions. A higher average secrecy capacity indicates a more
consistently secure communication system.

####  4.0.1 Derivation of $\mathfrak{R}_{1}$

$\mathfrak{R}_{1}$ is expressed as

$\displaystyle\mathfrak{R}_{1}=\int_{0}^{\infty}(1+\gamma)^{-1}\text{G}_{2,3}^{1,2}\left[K_{2_{\mathcal{E}}}\gamma\middle|\begin{matrix}1-\Psi_{i_{\mathcal{E}}},1\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},0,-\Psi_{i_{\mathcal{E}}}\end{matrix}\right]\text{d}\gamma.$

To convert $(1+\gamma)^{-1}$ into Meijer’s $G$ functions and then to do
integration, we use the identities from [44, Eq. (8.4.2.5)] and [44, Eq.
(2.24.1.2)]. Finally, we obtain $\mathfrak{R}_{1}$ as the following.

$\displaystyle\mathfrak{R}_{1}$
$\displaystyle=\int_{0}^{\infty}\text{G}_{1,1}^{1,1}\left[\gamma\middle|\begin{matrix}0\\\
0\end{matrix}\right]\text{G}_{2,3}^{1,2}\left[K_{2_{\mathcal{E}}}\gamma\middle|\begin{matrix}1-\Psi_{i_{\mathcal{E}}},1\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},0,-\Psi_{i_{\mathcal{E}}}\end{matrix}\right]\text{d}\gamma$
$\displaystyle=\text{G}_{3,4}^{2,3}\left[K_{2_{\mathcal{E}}}\middle|\begin{matrix}1,0\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},0,0,-\Psi_{i_{\mathcal{E}}}\end{matrix}\right].$
(24)

####  4.0.2 Derivation of $\mathfrak{R}_{2}$

$\mathfrak{R}_{2}$ is expressed as

$\displaystyle\mathfrak{R}_{2}$
$\displaystyle=\int_{0}^{\infty}(1+\gamma)^{-1}\text{G}_{2,3}^{1,2}\left[K_{2_{\mathcal{E}}}\gamma\middle|\begin{matrix}1-\Psi_{i_{\mathcal{E}}},1\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},0,-\Psi_{i_{\mathcal{E}}}\end{matrix}\right]$
$\displaystyle\times\text{G}_{2,3}^{1,2}\left[K_{2_{R}}\gamma\middle|\begin{matrix}1-\Psi_{i_{R}},1\\\
L_{R}\mu_{R}+k_{R},0,-\Psi_{i_{R}}\end{matrix}\right]\text{d}\gamma.$

Utilizing [38, Eq. (1.112.1)] and carrying out integration with the aid of
[44, Eq. (2.24.1.1)], $\mathfrak{R}_{2}$ is obtained as

$\displaystyle\mathfrak{R}_{2}$
$\displaystyle=\sum_{z=1}^{\infty}(-1)^{z-1}\int_{0}^{\infty}\gamma^{z-1}\text{G}_{2,3}^{1,2}\left[K_{2_{\mathcal{E}}}\gamma\middle|\begin{matrix}1-\Psi_{i_{\mathcal{E}}},1\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},0,-\Psi_{i_{\mathcal{E}}}\end{matrix}\right]$
$\displaystyle\times\text{G}_{2,3}^{1,2}\left[K_{2_{R}}\gamma\middle|\begin{matrix}1-\Psi_{i_{R}},1\\\
L_{R}\mu_{R}+k_{R},0,-\Psi_{i_{R}}\end{matrix}\right]\text{d}\gamma$
$\displaystyle=\sum_{z=1}^{\infty}\frac{(-1)^{z-1}}{K_{2_{\mathcal{E}}}^{z}}\text{G}_{5,5}^{3,3}\left[\frac{K_{2_{R}}}{K_{2_{\mathcal{E}}}}\middle|\begin{matrix}1-\Psi_{i_{R}},1,\Omega_{1},\Omega_{2},\Omega_{3}\\\
L_{R}\mu_{R}+k_{R},\Omega_{4},-z,0,-\Psi_{i_{R}}\end{matrix}\right]$ (25)

where $\Omega_{1}=1-z-L_{\mathcal{E}}\mu_{\mathcal{E}}-k_{\mathcal{E}}$,
$\Omega_{2}=1-z$, $\Omega_{3}=1-z+\Psi_{i_{\mathcal{E}}}$ and
$\Omega_{4}=\Psi_{i_{\mathcal{E}}}-z$.

####  4.0.3 Derivation of $\mathfrak{R}_{3}$

$\mathfrak{R}_{3}$ is expressed as

$\displaystyle\mathfrak{R}_{3}$
$\displaystyle=\int_{0}^{\infty}(1+\gamma)^{-1}\gamma^{\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}}\text{G}_{2,3}^{1,2}\left[K_{2_{\mathcal{E}}}\gamma\middle|\begin{matrix}1-\Psi_{i_{\mathcal{E}}},1\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},0,-\Psi_{i_{\mathcal{E}}}\end{matrix}\right]$
$\displaystyle\times\text{G}_{1,r_{\mathcal{D}}+1}^{r_{\mathcal{D}},1}\left[\Upsilon_{4_{\mathcal{D}}}\gamma\middle|\begin{matrix}1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\\\
0,\Upsilon_{5_{\mathcal{D}}},-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\end{matrix}\right]\text{d}\gamma.$
(26)

Making use of [38, Eq. (1.112.1)] and implementing integration via [44, Eq.
(2.24.1.1)], $\mathfrak{R}_{2}$ is written as

$\displaystyle\mathfrak{R}_{3}$
$\displaystyle=\sum_{z=1}^{\infty}(-1)^{z-1}\int_{0}^{\infty}\gamma^{\varpi-1}\text{G}_{2,3}^{1,2}\left[K_{2_{\mathcal{E}}}\gamma\middle|\begin{matrix}1-\Psi_{i_{\mathcal{E}}},1\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},0,-\Psi_{i_{\mathcal{E}}}\end{matrix}\right]$
$\displaystyle\times\text{G}_{1,r_{\mathcal{D}}+1}^{r_{\mathcal{D}},1}\left[\Upsilon_{4_{\mathcal{D}}}\gamma\middle|\begin{matrix}1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\\\
0,\Upsilon_{5_{\mathcal{D}}},-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\end{matrix}\right]\text{d}\gamma$
$\displaystyle=\sum_{z=1}^{\infty}\frac{(-1)^{z-1}}{K_{2_{\mathcal{E}}}^{\varpi}}\text{G}_{4,r_{\mathcal{D}}+3}^{r_{\mathcal{D}}+2,2}\left[\frac{\Upsilon_{4_{\mathcal{D}}}}{K_{2_{\mathcal{E}}}}\middle|\begin{matrix}\Omega_{5},\Omega_{6},\Omega_{7},\Omega_{8}\\\
S_{1r_{\mathcal{D}}},\Omega_{9},-\varpi,S_{2r_{\mathcal{D}}}\end{matrix}\right],$
(27)

where $\varpi=\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}+z$,
$\Omega_{5}=1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}$,
$\Omega_{6}=1-\varpi-L_{\mathcal{E}}\mu_{\mathcal{E}}-k_{\mathcal{E}}$,
$\Omega_{7}=1-\varpi$, $\Omega_{8}=1-\varpi+\Psi_{i_{\mathcal{E}}}$,
$\Omega_{9}=\Psi_{i_{\mathcal{E}}}-\varpi$, $S_{11}=0$,
$S_{12}=(0,\Upsilon_{5_{\mathcal{D}}})$, $S_{21}=\Upsilon_{5_{\mathcal{D}}}$
and $S_{22}=-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}$.

####  4.0.4 Derivation of $\mathfrak{R}_{4}$

$\mathfrak{R}_{4}$ is expressed as

$\displaystyle\mathfrak{R}_{4}$
$\displaystyle=\int_{0}^{\infty}(1+\gamma)^{-1}\gamma^{\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}}\text{G}_{2,3}^{1,2}\left[K_{2_{\mathcal{E}}}\gamma\middle|\begin{matrix}1-\Psi_{i_{\mathcal{E}}},1\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},0,-\Psi_{i_{\mathcal{E}}}\end{matrix}\right]$
$\displaystyle\times\text{G}_{1,r_{\mathcal{D}}+1}^{r_{\mathcal{D}},1}\left[\Upsilon_{4_{\mathcal{D}}}\gamma\middle|\begin{matrix}1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\\\
0,\Upsilon_{5_{\mathcal{D}}},-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\end{matrix}\right]$
$\displaystyle\times\text{G}_{2,3}^{1,2}\left[K_{2_{R}}\gamma\middle|\begin{matrix}1-\Psi_{i_{R}},1\\\
L_{R}\mu_{R}+k_{R},0,-\Psi_{i_{R}}\end{matrix}\right]\text{d}\gamma.$ (28)

Applying [38, Eq. (1.112.1)] and implementing integration with the aid of [45,
Eq. (07.34.21.0081.01)], $\mathfrak{R}_{4}$ is expressed as

$\displaystyle\mathfrak{R}_{4}$
$\displaystyle=\sum_{z=1}^{\infty}(-1)^{z-1}\int_{0}^{\infty}\gamma^{\varpi-1}\text{G}_{2,3}^{1,2}\left[K_{2_{\mathcal{E}}}\gamma\middle|\begin{matrix}1-\Psi_{i_{\mathcal{E}}},1\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},0,-\Psi_{i_{\mathcal{E}}}\end{matrix}\right]$
$\displaystyle\times\text{G}_{1,r_{\mathcal{D}}+1}^{r_{\mathcal{D}},1}\left[\Upsilon_{4_{\mathcal{D}}}\gamma\middle|\begin{matrix}1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\\\
0,\Upsilon_{5_{\mathcal{D}}},-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\end{matrix}\right]$
$\displaystyle\times\text{G}_{2,3}^{1,2}\left[K_{2_{R}}\gamma\middle|\begin{matrix}1-\Psi_{i_{R}},1\\\
L_{R}\mu_{R}+k_{R},0,-\Psi_{i_{R}}\end{matrix}\right]\text{d}\gamma$
$\displaystyle=\sum_{z=1}^{\infty}\frac{(-1)^{z-1}}{K_{2_{\mathcal{E}}}^{\varpi}}\times$
(29) $\displaystyle\text{G}_{3,2\colon 1,r_{\mathcal{D}}+1\colon
2,3}^{2,1\colon r_{\mathcal{D}},1\colon
1,2}\left[\begin{matrix}\Omega_{6},\Omega_{7},\Omega_{8}\\\
\Omega_{10},-\varpi\end{matrix}\middle|\begin{matrix}\Omega_{5}\\\
S_{3r_{D}}\end{matrix}\middle|\begin{matrix}1-\Psi_{i_{R}},1\\\
L_{R}\mu_{R}+k_{R},0,-\Psi_{i_{R}}\end{matrix}\middle|\Omega_{11}\right],$

where $\Omega_{10}=\Psi_{i_{\mathcal{E}}}-\varpi$,
$\Omega_{11}=\left(\frac{\Upsilon_{4_{\mathcal{D}}}}{K_{2_{\mathcal{E}}}},\frac{K_{2_{R}}}{K_{2_{\mathcal{E}}}}\right),$
$S_{31}=(0,\Upsilon_{5_{\mathcal{D}}})$,
$S_{32}=(0,\Upsilon_{5_{\mathcal{D}}},-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}})$,
and
$\text{G}_{-,-\colon-,-\colon-,-}^{-,-\colon-,-\colon-,-}\left[\cdot\mid\cdot\mid\cdot\mid\cdot\right]$
denotes the Bivariate Meiger’s $G$-function [46].

##  5 Secrecy Outage Probability

Mathematically, the SOP can be defined as the probability that the
instantaneous secrecy capacity drops below a certain threshold chosen to
represent an acceptable level of security. In the following subsections, we
derive the expression of SOP considering the three eavesdropping scenarios.

###  5.1 Scenario-I

The SOP is defined in the case of RF eavesdropping as

$\displaystyle SOP^{I}$
$\displaystyle=\Pr(\gamma_{eq}\leq\phi+\phi\gamma_{\mathcal{E}}-1)$
$\displaystyle=\int_{0}^{\infty}F_{\gamma_{eq}}(\phi+\phi\gamma-1)f_{\gamma_{\mathcal{E}}}(\gamma)d\gamma,$
(30)

$\phi=2^{\mathcal{R}_{s}}$ and $\mathcal{R}_{s}>0$. Obtaining the exact
expression for the SOP is challenging due to the involvement of shifts in
Meiger’s $G$ function. Consequently, researchers often resort to deriving
lower bounds for SOP. Drawing on the formulation presented in [47, Eq. (21)],
the lower bound of SOP can be expressed as

$\displaystyle SOP^{I}\geq SOP_{L}^{I}$
$\displaystyle=\Pr\left\\{\gamma_{eq}\leq\phi\gamma_{\mathcal{E}}\right\\}$
$\displaystyle=\int_{0}^{\infty}F_{\gamma_{eq}}(\phi\gamma)f_{\gamma_{\mathcal{E}}}(\gamma)\text{d}\gamma.$
(31)

######  Remark 5.1.

It is noted that comparing (5.1) with (31), the tightness between the exact
SOP and lower bound for SOP can easily be found which depends on
$\phi=2^{\mathcal{R}_{s}}$ and $\gamma_{\mathcal{E}}$. It can be easily
observed that a smaller $\mathcal{R}_{s}$ and/or a larger
$\bar{\gamma}_{\mathcal{E}}$ is responsible for a tighter bound.

Finally, SOP${}^{I}_{L}$ is obtained after putting (10) and (19) into (31) and
then performing integration as the following.

$\displaystyle SOP^{I}_{L}$
$\displaystyle=\sum_{i_{\mathcal{E}}=1}^{m}\sum_{k_{\mathcal{E}}=0}^{p}K_{1_{\mathcal{E}}}\biggl{(}\sum_{i_{R}=1}^{m}\sum_{k_{R}=0}^{p}K_{1_{R}}\mathfrak{Z}_{1}$
$\displaystyle+{\Upsilon}^{{}^{\prime}}_{3_{\mathcal{D}}}\mathfrak{Z}_{2}-\sum_{i_{R}=1}^{m}\sum_{k_{R}=0}^{p}K_{1_{R}}{\Upsilon}^{{}^{\prime}}_{3_{\mathcal{D}}}\mathfrak{Z}_{3}\biggl{)},$
(32)

where
${\Upsilon}^{{}^{\prime}}_{3_{\mathcal{D}}}=\Upsilon_{3_{\mathcal{D}}}\phi^{\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}}$
and $\mathfrak{Z}_{1}$, $\mathfrak{Z}_{2}$ and $\mathfrak{Z}_{3}$ are three
integral terms.

######  Remark 5.2.

The derived expression in (32) can be used to quantify the probability of a
communication link failing to maintain the secrecy of the transmitted
confidential information. It mainly assesses the likelihood that this secrecy
is compromised or outed during the communication process.

The three integral parts $\mathfrak{Z}_{1}$, $\mathfrak{Z}_{2}$ and
$\mathfrak{Z}_{3}$ are derived in the following manner.

#### Derivation of $\mathfrak{Z}_{1}$

$\mathfrak{Z}_{1}$ is derived as

$\displaystyle\mathfrak{Z}_{1}$
$\displaystyle=\int_{0}^{\infty}\gamma^{-1}\text{G}_{2,3}^{1,2}\left[{K}^{\prime}_{2_{R}}\gamma\middle|\begin{matrix}1-\Psi_{i_{R}},1\\\
L_{R}\mu_{R}+k_{R},0,-\Psi_{i_{R}}\end{matrix}\right]$
$\displaystyle\times\text{G}_{1,2}^{1,1}\left[K_{2_{\mathcal{E}}}\gamma\middle|\begin{matrix}1-\Psi_{i_{\mathcal{E}}}\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},-\Psi_{i_{\mathcal{E}}}\end{matrix}\right]d\gamma.$
(33)

Performing the integration through [44, Eq. (2.24.1.1)], $\mathfrak{Z}_{1}$ is
expressed as

$\displaystyle\mathfrak{Z}_{1}=\text{G}_{4,4}^{3,2}\left[\frac{K_{2_{\mathcal{E}}}}{{K}^{\prime}_{2_{R}}}\middle|\begin{matrix}1-\Psi_{i_{\mathcal{E}}},1-L_{R}\mu_{R}-k_{R},1,1+\Psi_{i_{R}}\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},\Psi_{i_{R}},0,-\Psi_{i_{\mathcal{E}}}\end{matrix}\right],$
(34)

where ${K}^{\prime}_{2_{R}}=\phi K_{2_{R}}$.

#### Derivation of $\mathfrak{Z}_{2}$

$\mathfrak{Z}_{2}$ is expressed as

$\displaystyle\mathfrak{Z}_{2}$
$\displaystyle=\int_{0}^{\infty}\gamma^{\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}-1}\text{G}_{1,r_{\mathcal{D}}+1}^{r_{\mathcal{D}},1}\left[{\Upsilon}^{\prime}_{4_{\mathcal{D}}}\gamma\middle|\begin{matrix}1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\\\
0,\Upsilon_{5_{\mathcal{D}}},-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\end{matrix}\right]$
$\displaystyle\times\text{G}_{1,2}^{1,1}\left[K_{2_{\mathcal{E}}}\gamma\middle|\begin{matrix}1-\Psi_{i_{\mathcal{E}}}\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},-\Psi_{i_{\mathcal{E}}}\end{matrix}\right]\text{d}\gamma.$
(35)

After implementing the integration identity of [44, Eq. (2.24.1.1)],
$\mathfrak{Z}_{2}$ is expressed as

$\displaystyle\mathfrak{Z}_{2}=({\Upsilon}^{\prime}_{4_{\mathcal{D}}})^{-\left(\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}+1\right)}\text{G}_{2+r_{\mathcal{D}},3}^{2,1+r_{\mathcal{D}}}\left[\frac{K_{2_{\mathcal{E}}}}{{\Upsilon}^{\prime}_{4_{\mathcal{D}}}}\middle|\begin{matrix}1-\Psi_{i_{\mathcal{E}}},S_{4r_{\mathcal{D}}}\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},0,-\Psi_{i_{\mathcal{E}}}\end{matrix}\right],$
(36)

where ${\Upsilon}^{\prime}_{4_{\mathcal{D}}}=\phi\Upsilon_{4_{\mathcal{D}}}$,
$S_{41}=(1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}},1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}-\Upsilon_{5_{\mathcal{D}}})$
and
$S_{42}=(1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}},1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}-\Upsilon_{5_{\mathcal{D}}},1)$.

#### Derivation of $\mathfrak{Z}_{3}$

$\mathfrak{Z}_{3}$ is expressed as

$\displaystyle\mathfrak{Z}_{3}$
$\displaystyle=\int_{0}^{\infty}\gamma^{\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}-1}\text{G}_{2,3}^{1,2}\left[{K}^{\prime}_{2_{R}}\gamma\middle|\begin{matrix}1-\Psi_{i_{R}},1\\\
L_{R}\mu_{R}+k_{R},0,-\Psi_{i_{R}}\end{matrix}\right]$
$\displaystyle\times\text{G}_{1,r_{\mathcal{D}}+1}^{r_{\mathcal{D}},1}\left[{\Upsilon}^{\prime}_{4_{\mathcal{D}}}\gamma\middle|\begin{matrix}1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\\\
0,\Upsilon_{5_{\mathcal{D}}},-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\end{matrix}\right]$
$\displaystyle\times\text{G}_{1,2}^{1,1}\left[K_{2_{\mathcal{E}}}\gamma\middle|\begin{matrix}1-\Psi_{i_{\mathcal{E}}}\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},-\Psi_{i_{\mathcal{E}}}\end{matrix}\right]\text{d}\gamma.$
(37)

$\mathfrak{Z}_{3}$ is represented as after integrating with the aid of [45,
Eq. (07.34.21.0081.01)] as

$\displaystyle\mathfrak{Z}_{3}$
$\displaystyle=\left({K}^{\prime}_{2_{R}}\right)^{-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}}\times$
$\displaystyle\text{G}_{3,2\colon 1,r_{\mathcal{D}}+1\colon 1,2}^{2,1\colon
r_{\mathcal{D}},1\colon
1,1}\left[\begin{matrix}\Omega_{12},\Omega_{13},\Omega_{14}\\\
\Omega_{15},-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}\end{matrix}\middle|\begin{matrix}\Omega_{16}\\\
S_{3r_{\mathcal{D}}}\end{matrix}\middle|\begin{matrix}1-\Psi_{i_{\mathcal{E}}},1\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},-\Psi_{i_{\mathcal{E}}}\end{matrix}\middle|\Omega_{17}\right],$
(38)

where
$\Omega_{12}=1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}-L_{R}\mu_{R}-k_{R}$,
$\Omega_{13}=1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}$,
$\Omega_{14}=1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}+\Psi_{i_{R}}$,
$\Omega_{15}=\Psi_{i_{R}}-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}$,
$\Omega_{16}=1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}$, and
$\Omega_{17}=\left(\frac{{\Upsilon}^{\prime}_{4_{\mathcal{D}}}}{{K}^{\prime}_{2_{R}}},\frac{K_{2_{\mathcal{E}}}}{{K}^{\prime}_{2_{R}}}\right)$.

#### Asymptotic Analysis

The asymptotic expression of $SOP^{I}_{L}$ can be obtained as

$\displaystyle
SOP^{I}_{\infty}=\int_{0}^{\infty}F^{\infty}_{\gamma_{eq}}(\phi\gamma)f_{\gamma_{\mathcal{E}}}(\gamma)d\gamma.$
(39)

Substituting (10) and (21) into (39), and performing integration through [38,
Eq. (7.811.4)], $SOP^{I}_{\infty}$ is obtained as shown in (49).

$\displaystyle SOP^{I}_{\infty}$
$\displaystyle=\sum_{i_{\mathcal{E}}=1}^{m}\sum_{k_{\mathcal{E}}=0}^{p}K_{1_{\mathcal{E}}}\left(\sum_{i_{R}=1}^{m}\sum_{k_{R}=0}^{p}\frac{K_{1_{R}}}{(K_{2_{R}}\phi)^{-L_{R}\mu_{R}-k_{R}}}\frac{\prod_{l_{1}=1}^{2}\Gamma(T_{1,l_{1}}+L_{R}\mu_{R}+k_{R})}{\prod_{l_{1}=2}^{3}\Gamma(T_{2,l_{1}}+L_{R}\mu_{R}+k_{R})}\frac{\Gamma\left(L_{R}\mu_{R}+k_{R}+L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}}\right)}{k_{2_{\mathcal{E}}}^{L_{R}\mu_{R}+k_{R}}\left(\Psi_{i_{\mathcal{E}}}-L_{R}\mu_{R}-k_{R}\right)}\right.$
$\displaystyle+\left.\sum_{k_{2}=1}^{r_{\mathcal{D}}}\frac{\Upsilon_{3_{\mathcal{D}}}\phi^{\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}}}{\Upsilon_{4_{\mathcal{D}}}^{T_{3,k_{2}}-1}}\frac{\prod_{l_{2}=1;l_{2}\neq
k_{2}}^{r_{\mathcal{D}}}\Gamma(T_{3,k_{2}}-T_{3,l_{2}})\Gamma(1+\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}-T_{3,k_{2}})}{\prod_{l_{2}=r_{\mathcal{D}}+1}^{r_{\mathcal{D}}+1}\Gamma(1+T_{3,l_{2}}-T_{3,k_{2}})}\right.$
$\displaystyle\times\left.\frac{\Gamma\left(L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}}+\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}-T_{3,k_{2}}+1\right)\Gamma\left(\Psi_{i_{\mathcal{E}}}-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}+T_{3,k_{2}}-1\right)}{K_{2_{\mathcal{E}}}^{\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}-T_{3,k_{2}}+1}\Gamma\left(\Psi_{i_{\mathcal{E}}}-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}+T_{3,k_{2}}\right)}\right)$
(49) $\displaystyle SOP_{L}^{II}$
$\displaystyle={\Upsilon}^{\prime}_{3_{\mathcal{D}}}\Upsilon_{1_{\mathcal{\tilde{E}}}}\frac{r_{\mathcal{\tilde{E}}}^{\frac{1}{2}}({\Upsilon}^{\prime}_{4_{\mathcal{D}}})^{-\zeta}}{(2\pi)^{\frac{1}{2}(r_{\mathcal{\tilde{E}}}-1)}}\text{G}_{r_{\mathcal{D}}+1,r_{\mathcal{\tilde{E}}}+1}^{r_{\mathcal{\tilde{E}}}+1,r_{\mathcal{D}}}\left[\Omega_{18}\middle|\begin{matrix}S_{5r_{\mathcal{D}}}\\\
0,\Omega_{19}\end{matrix}\right]\left(1-\sum_{i_{R}=1}^{m}\sum_{k_{R}=0}^{p}K_{1_{R}}\text{G}_{2,3}^{1,2}\left[\Omega_{20}\middle|\begin{matrix}1-\Psi_{i_{R}},1\\\
L_{R}\mu_{R}+k_{R},0,-\Psi_{i_{R}}\end{matrix}\right]\right)$
$\displaystyle+\sum_{i_{R}=1}^{m}\sum_{k_{R}=0}^{p}K_{1_{R}}\text{G}_{2,3}^{1,2}\left[\Omega_{20}\middle|\begin{matrix}1-\Psi_{i_{R}},1\\\
L_{R}\mu_{R}+k_{R},0,-\Psi_{i_{R}}\end{matrix}\right].$ (52)

###  5.2 Scenario-II

Whenever the eavesdropper is present at the UOWC link, then SOP is represented
as [48, Eq. (13)]

$\displaystyle SOP^{II}$
$\displaystyle=\int_{0}^{\infty}F_{\gamma_{\mathcal{D}}}(\phi\gamma+\phi-1)f_{\gamma_{\mathcal{\tilde{E}}}}(\gamma)d\gamma$
$\displaystyle\times(1-F_{\gamma_{R}}(\phi-1))+F_{\gamma_{R}}(\phi-1).$ (50)

Since deriving the exact SOP is difficult, we define the SOP at the lower
bound as

$\displaystyle SOP^{II}\geq SOP_{L}^{II}$
$\displaystyle=\int_{0}^{\infty}F_{\gamma_{\mathcal{D}}}(\phi\gamma)f_{\gamma_{\mathcal{\tilde{E}}}}(\gamma)d\gamma$
$\displaystyle\times(1-F_{\gamma_{R}}(\phi-1))+F_{\gamma_{R}}(\phi-1).$ (51)

Substituting (15) and (16) into (5.2) and utilizing [44, Eq. (2.24.1.1)], the
expression of $SOP_{L}^{II}$ is obtained as shown in (52), where
${\Upsilon}^{\prime}_{3_{\mathcal{D}}}=\phi\Upsilon_{3_{\mathcal{D}}}$,
$\zeta=\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}}+\frac{\rho_{\mathcal{\tilde{E}}}}{r_{\mathcal{\tilde{E}}}}$,
$\Omega_{18}=\frac{\Upsilon_{2_{\mathcal{\tilde{E}}}}^{r_{\mathcal{\tilde{E}}}}r_{\mathcal{\tilde{E}}}^{-r_{\mathcal{\tilde{E}}}}}{{\Upsilon}^{\prime}_{4_{\mathcal{D}}}}$,
$\Omega_{19}=-\frac{\rho_{\mathcal{\tilde{E}}}}{r_{\mathcal{\tilde{E}}}}$,
$\Omega_{20}=K_{2_{R}}(\phi-1)$,
$S_{51}=(1-\zeta,1-\zeta-\Upsilon_{5_{\mathcal{D}}})$ and
$S_{52}=(1-\zeta,1-\zeta-\Upsilon_{5_{\mathcal{D}}},1-\frac{\rho_{\mathcal{D}}}{r_{\mathcal{D}}})$.

$\displaystyle SOP^{II}_{\infty}$
$\displaystyle={\Upsilon}^{\prime}_{3_{\mathcal{D}}}\Upsilon_{1_{\mathcal{\tilde{E}}}}\frac{r_{\mathcal{\tilde{E}}}^{\frac{1}{2}}({\Upsilon}^{\prime}_{4_{\mathcal{D}}})^{-\zeta}}{2\pi^{\frac{1}{2}(r_{\mathcal{\tilde{E}}}-1)}}\sum_{k_{3}=1}^{r_{\mathcal{D}}}\Omega_{18}^{T_{4,k_{3}}-1}\frac{\prod_{l_{3}=1;l_{3}\neq
k_{3}}^{r_{\mathcal{D}}}\Gamma\left(T_{4,k_{3}}-T_{4,l_{3}}\right)\prod_{l_{3}=1}^{r_{\mathcal{\tilde{E}}}+1}\Gamma\left(1+T_{5,l_{3}}-T_{4,k_{3}}\right)}{\prod_{l_{3}=r_{\mathcal{D}}+1}^{r_{\mathcal{D}}+1}\Gamma\left(1+T_{4,l_{3}}-T_{4,k_{3}}\right)}$
$\displaystyle\times\left(1-\sum_{i_{R}=1}^{m}\sum_{k_{R}=0}^{p}\frac{K_{1_{R}}}{\Omega_{20}^{-L_{R}\mu_{R}-k_{R}}}\frac{\prod_{l_{1}=1}^{2}\Gamma(T_{1,l_{1}}+L_{R}\mu_{R}+k_{R})}{\prod_{l_{1}=2}^{3}\Gamma(T_{2,l_{1}}+L_{R}\mu_{R}+k_{R})}\right)+\sum_{i_{R}=1}^{m}\sum_{k_{R}=0}^{p}\frac{K_{1_{R}}}{\Omega_{20}^{-L_{R}\mu_{R}-k_{R}}}$
$\displaystyle\times\frac{\prod_{l_{1}=1}^{2}\Gamma(T_{1,l_{1}}+L_{R}\mu_{R}+k_{R})}{\prod_{l_{1}=2}^{3}\Gamma(T_{2,l_{1}}+L_{R}\mu_{R}+k_{R})}.$
(53)

#### Asymptotic Analysis

Upon utilizing the identity provided in [49, Eq. (41)], the asymptotic
expression of $SOP_{L}^{II}$ can be obtained as shown in (53), where
$T_{4}=S_{5r_{\mathcal{D}}}$ and $T_{5}=(0,\Omega_{14})$.

###  5.3 Scenario-III

The lower bound of SOP for the RIS-aided hybrid RF-UOWC architecture with
simultaneous eavesdropping attack over the RF and UOWC links is given by [33]

$\displaystyle SOP^{III}_{L}=1-(SOP_{1}\times SOP_{2}),$ (54)

where

$\displaystyle
SOP_{1}=1-\int_{0}^{\infty}F_{\gamma_{R}}(\phi\gamma)f_{\gamma_{\mathcal{E}}}(\gamma)d\gamma,$
(55) $\displaystyle
SOP_{2}=1-\int_{0}^{\infty}F_{\gamma_{\mathcal{D}}}(\phi\gamma)f_{\gamma_{\mathcal{\tilde{E}}}}(\gamma)d\gamma.$
(56)

Substituting (10) and (11) into (55) and exploiting [44, Eq. (2.24.1.1)],
$SOP_{1}$ is expressed as

$\displaystyle SOP_{1}$
$\displaystyle=1-\sum_{i_{R}=1}^{m}\sum_{k_{R}=0}^{p}K_{1_{R}}\sum_{i_{\mathcal{E}}=1}^{m}\sum_{k_{\mathcal{E}}=0}^{p}K_{1_{\mathcal{E}}}$
$\displaystyle\times\text{G}_{4,4}^{3,2}\left[\frac{K_{2_{\mathcal{E}}}}{{K}^{\prime}_{2_{R}}}\middle|\begin{matrix}1-\Psi_{i_{\mathcal{E}}},1-L_{R}\mu_{R}-k_{R},1,1+\Psi_{i_{R}}\\\
L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},\Psi_{i_{R}},0,-\Psi_{i_{\mathcal{E}}}\end{matrix}\right].$
(57)

Putting (15) and (16) into (56) and completing the integration via [44, Eq.
(2.24.1.1)], $SOP_{2}$ is given by

$\displaystyle SOP_{2}$
$\displaystyle=1-{\Upsilon}^{\prime}_{3_{\mathcal{D}}}\Upsilon_{1_{\mathcal{\tilde{E}}}}\frac{r_{\mathcal{\tilde{E}}}^{\frac{1}{2}}({\Upsilon}^{\prime}_{4_{\mathcal{D}}})^{-\zeta}}{(2\pi)^{\frac{1}{2}(r_{\mathcal{\tilde{E}}}-1)}}\text{G}_{r_{\mathcal{D}}+1,r_{\mathcal{\tilde{E}}}+1}^{r_{\mathcal{\tilde{E}}}+1,r_{\mathcal{D}}}\left[\Omega_{18}\middle|\begin{matrix}S_{5r_{\mathcal{D}}}\\\
0,\Omega_{19}\end{matrix}\right].$ (58)

#### Asymptotic Analysis

The asymptotic expression of $SOP^{III}_{L}$ can be written as

$\displaystyle SOP^{\infty}_{III}=1-(SOP^{\infty}_{1}\times
SOP^{\infty}_{2}).$ (59)

Applying [49, Eq. (41)], $SOP^{\infty}_{1}$ can be expressed as shown in (59),
and $SOP^{\infty}_{2}$ can be obtained as shown in (60), where
$T_{6}=(1-\Psi_{i_{\mathcal{E}}},1-L_{R}\mu_{R}-k_{R})$ and
$T_{7}=(L_{\mathcal{E}}\mu_{\mathcal{E}}+k_{\mathcal{E}},\Psi_{i_{\mathcal{E}}},0)$.

$\displaystyle
SOP^{\infty}_{1}=1-\sum_{i_{R}=1}^{m}\sum_{k_{R}=0}^{p}K_{1_{R}}\sum_{i_{\mathcal{E}}=1}^{m}\sum_{k_{\mathcal{E}}=0}^{p}K_{1_{\mathcal{E}}}\sum_{k_{4}=1}^{2}\left(\frac{K_{2_{\mathcal{E}}}}{{K}^{\prime}_{2_{R}}}\right)^{T_{6,k_{4}}-1}\frac{\prod_{l_{4}=1;l_{4}\neq
k_{4}}^{2}\Gamma\left(T_{6,k_{4}}-T_{6,l_{4}}\right)\prod_{l_{4}=1}^{3}\Gamma\left(1+T_{7,l_{4}}-T_{6,k_{4}}\right)}{\prod_{l_{4}=3}^{4}\Gamma\left(1+T_{6,l_{4}}-T_{6,k_{4}}\right)\Gamma\left(T_{6,k_{4}}+\Psi_{i_{\mathcal{E}}}\right)}$
(59) $\displaystyle
SOP^{\infty}_{2}=1-{\Upsilon}^{\prime}_{3_{\mathcal{D}}}\Upsilon_{1_{\mathcal{\tilde{E}}}}\frac{r_{\mathcal{\tilde{E}}}^{\frac{1}{2}}({\Upsilon}^{\prime}_{4_{\mathcal{D}}})^{-\zeta}}{2\pi^{\frac{1}{2}(r_{\mathcal{\tilde{E}}}-1)}}\sum_{k_{3}=1}^{r_{\mathcal{D}}}\Omega_{18}^{T_{4,k_{3}}-1}\frac{\prod_{l_{3}=1;l_{3}\neq
k_{3}}^{r_{\mathcal{D}}}\Gamma\left(T_{4,k_{3}}-T_{4,l_{3}}\right)\prod_{l_{3}=1}^{r_{\mathcal{\tilde{E}}}+1}\Gamma\left(1+T_{5,l_{3}}-T_{4,k_{3}}\right)}{\prod_{l_{3}=r_{\mathcal{D}}+1}^{r_{\mathcal{D}}+1}\Gamma\left(1+T_{4,l_{3}}-T_{4,k_{3}}\right)}$
(60)

##  6 Strictly Positive Secrecy Capacity

The probability of SPSC can be mathematically expressed in terms of the SOP
expressions in (32), (52), and (54) as

$\displaystyle SPSC^{I}$ $\displaystyle=1-SOP^{I}\big{|}_{\mathcal{R}_{s}=0},$
(60a) $\displaystyle SPSC^{II}$
$\displaystyle=1-SOP^{II}\big{|}_{\mathcal{R}_{s}=0},$ (60b) $\displaystyle
SPSC^{III}$ $\displaystyle=1-SOP^{III}\big{|}_{\mathcal{R}_{s}=0}.$ (60c)

######  Remark 6.1.

If the secrecy capacity is strictly positive, it means that there is a
positive rate at which covert communication can be maintained securely. The
expressions in (60a), (60b), and (60c) can be used to indicate the probability
of the existence of such secrecy capacity in scenarios, where it is difficult
to achieve perfect secrecy.

##  7  Effective Secrecy Throughput

When the source transmits information at a constant rate $\mathcal{R}_{s}$,
then EST can be defined for the three considered scenarios as [50]

$\displaystyle EST^{I}$ $\displaystyle=\mathcal{R}_{s}(1-SOP^{I}),$ (61a)
$\displaystyle EST^{II}$ $\displaystyle=\mathcal{R}_{s}(1-SOP^{II}),$ (61b)
$\displaystyle EST^{III}$ $\displaystyle=\mathcal{R}_{s}(1-SOP^{III}).$ (61c)

######  Remark 7.1.

EST is a parameter employed in the field of information-theoretic security to
measure the rate at which secret information may be transmitted over a channel
while preserving a particular level of security. It is evident from (61a),
(61b), and (61c) that an increase in $\mathcal{R}_{s}$ results in a
proportional increase in the SOP, and conversely, a decrease in
$\mathcal{R}_{s}$ leads to a reduced SOP. This implies the existence of an
optimal value for $\mathcal{R}_{s}$ that maximizes the EST under the given
circumstances. Making use of (61a), (61b), and (61c), one can easily obtain
the optimal $\mathcal{R}_{s}$ and maximum EST.

##  8 Numerical Results

This section investigates the impact of various parameters on the security
performance of an RIS-assisted UAV-NTN communication within the RF NTN-UOWC
system. The parameters under consideration encompass fading parameters, the
number of reflecting elements, RF link distance, thermal gradients, detection
methods, UWT, pointing errors, and more. The objective is to comprehensively
understand the intricate interplay between these factors and the overall
security of the communication system. The examination involves figurative
illustrations of numerical instances making use of the derived expressions, as
represented in (23), (32), (52), (54), (60a)-(60c), and (61b) which offer a
visual representation, aiding in the interpretation of the numerical impact of
each parameter on the security performance. To validate the accuracy of the
newly derived expressions, MC simulations are conducted using MATLAB. In our
general case assumptions, specific values are assigned to the parameters:
$\alpha_{j}=2$, $\mu_{j}=2$, $\kappa_{j}=1$, $D_{j}=50$m, $L_{j}\geq 1$,
$\mathcal{N}=2$, $\bar{\gamma}_{j}\geq 0$, $\bar{\gamma}_{n}\geq 0$,
$r_{n}={1,2}$, $\mathcal{R}_{s}=0.01$, and $\xi_{n}=1$. Furthermore, the UWT
parameters crucial to our analysis are extracted from Table 3 in [26].

###  8.1 Impacts of UAV-NTN Link Parameters

Figure 2: The $SOP^{I}$ versus $\bar{\gamma}_{R}$ for selected values of
$\mu_{R}$, $\mu_{\mathcal{E}}$ and $\bar{\gamma}_{\mathcal{D}}$.

Figure 2 depicts the impact of the fading severity parameter on SOP by
graphing the $SOP^{I}$ versus the average SNR of the main UAV-NTN link. A
notable observation from the figure emerges, demonstrating that as both
$\mu_{R}$ and $\mu_{\mathcal{E}}$ are concurrently increased, there is a
corresponding decrease in the secrecy outage. This reduction stems from a more
challenging environment for an eavesdropper to intercept the transmitted
signal effectively due to the diminished channel fading resulting from the
augmentation of both $\mu_{R}$ and $\mu_{\mathcal{E}}$, indicating increased
resilience of the communication system against potential security breaches.

Figure 3: The $ASC^{I}$ versus $\bar{\gamma}_{R}$ for selected values of
$\kappa_{\mathcal{E}}$.

The influence of small-scale fading effects due to multipath propagation on
the $ASC^{I}$ is depicted in Fig. 3. It has been observed that the ASC
experiences a decline when $\kappa_{\mathcal{E}}$ is increased from $1$ to
$5$. This outcome is expected, as the total fading of the eavesdropper channel
diminishes with the escalation of $\kappa_{\mathcal{E}}$ towards infinity,
leading to a reduction in ASC.

Figure 4: The $ASC^{I}$ versus $\bar{\gamma}_{R}$ for selected values of
$D_{R}$, $D_{\mathcal{E}}$ and $\bar{\gamma}_{\mathcal{E}}$.

The impact of link distance in wireless communication is a pivotal factor, as
demonstrated in Figure 4, where $ASC^{I}$ is plotted against varying
$D_{\mathcal{R}}$ and $D_{\mathcal{E}}$. When $D_{\mathcal{E}}$ is maintained
at a constant value, the ASC is observed to decrease with an increase in
$D_{\mathcal{R}}$. Conversely, when $D_{\mathcal{R}}$ is fixed, the ASC
exhibits an increase with a rise in $D_{\mathcal{E}}$. This behavior is
expected, considering that the signal strength at the legitimate receiver
($\mathcal{R}$) diminishes with an increase in $D_{\mathcal{R}}$ due to
factors such as path loss, fading, diffraction, and scattering in free-space
propagation. Moreover, heightened path loss in the eavesdropper link reduces
the probability of successful eavesdropping, thereby contributing to an
enhancement in secrecy. In addition to the link distance, the influence of
$\bar{\gamma}_{\mathcal{E}}$ is also examined in the figure, revealing a
noteworthy trend of decreased ASC with an increased value of
$\bar{\gamma}_{\mathcal{E}}$. This phenomenon arises due to the higher values
of $\bar{\gamma}_{\mathcal{E}}$ indicating an improved wiretap link,
signifying enhanced eavesdropping capability for $\mathcal{E}$.

The impact of diversity on secrecy capacity is demonstrably positive, as
illustrated in Figure 5 by plotting $SOP^{I}$ against the UOWC link’s average
SNR. The plot shows that, compared to a single-branch receiver
($L_{R}=L_{\mathcal{E}}=1$), incorporating additional branches
($L_{R}=L_{\mathcal{E}}\geq 1$) at both $\mathcal{R}$ and $\mathcal{E}$
significantly improves the overall signal reception quality at $\mathcal{R}$
by reducing the probability of an outage. This improvement is due to the
diversity reception technique, which effectively mitigates the fading effects
and enhances the robustness of the main link. Moreover, it is evident from the
figure that the impact of simultaneously increasing the number of antennas at
both $\mathcal{R}$ and $\mathcal{E}$ is more advantageous for the main channel
than for the eavesdropper channel. This indicates that the diversity reception
technique is more critical for the main link, as it provides a more
substantial improvement in signal quality and secrecy capacity compared to the
eavesdropper link. Thus, the addition of multiple branches at the receiver end
not only enhances the main link’s performance but also contributes to a higher
level of security by lowering the secrecy outage probability.

###  8.2 Impacts of RIS-aided UOWC Link Parameters

Figure 5: The $SOP^{I}$ versus $\bar{\gamma}_{\mathcal{D}}$ for selected
values of $L_{R}$ and $L_{\mathcal{E}}$.

Figure 6: The $SOP^{I}$ versus $\bar{\gamma}_{R}$ for selected values of
$\mathcal{N}$ and $r_{n}$.

The effect of the number of reflecting elements is demonstrated in Fig. 6 in
terms of $SOP^{I}$. It is evident that the SOP experiences a significant
decrease with an increase in $\mathcal{N}$, as corroborated by [33]. This
reduction is attributed to the enhanced capabilities provided by a larger
value of $\mathcal{N},$ allowing for more sophisticated beamforming and
directional control of reflected signals. The increased $\mathcal{N}$ provides
opportunities to exploit diversity and multipath effects in the wireless
channel, resulting in a higher SNR at $\mathcal{D}$. Additionally, adjusting
the phase shifts and amplitudes of individual elements enables the RIS to
steer the transmitted signal in a specific direction, thereby reducing the
probability of signal leakage to the intruders. Figure 6 also depicts a
comparative analysis between HD and IM/DD techniques in terms of the SOP.
Remarkably, the HD method outperforms the IM/DD method in achieving lower
outage behavior more effectively, resulting in a significant enhancement in
secrecy performance. The superiority of the HD method can be attributed to its
integration of a local oscillator, facilitating the conversion of optical
frequency to an intermediate frequency for ease of detection. This integration
results in a higher SNR at the receiver compared to the IM/DD method.
Additionally, the HD technique generally offers higher sensitivity due to its
capability to recover both amplitude and phase information for which it is
always favoured for high-speed and long distance communication.

Figure 7: The $SOP^{II}$ versus $\bar{\gamma}_{\mathcal{D}}$ for selected
values of $h_{I}$, $h_{\mathcal{D}}$ and $\bar{\gamma}_{\mathcal{\tilde{E}}}$.

Figure 8: The $SOP^{II}$ versus $\bar{\gamma}_{\mathcal{D}}$ for selected
values of $h_{I}$, $h_{\mathcal{D}}$, $h_{\mathcal{\tilde{E}}}$ and
$\bar{\gamma}_{\mathcal{\tilde{E}}}$.

The investigation, centering on analyzing the impact of various UWT scenarios
within both a thermal gradient and a thermally uniform system, is carried out
by plotting the $SOP^{II}$ against $\gamma_{\mathcal{D}}$ in Figs. 7 and 8. In
Fig. 7, a fixed $h_{\mathcal{\tilde{E}}}$ and $l_{\mathcal{\tilde{E}}}$,
valued at 2.4 (L/min) and 0.10 (C $cm^{-1}$) respectively, are assumed.
Notably, as the temperature gradient increases from 0.15 to 0.22, the SOP
exhibits an increase, leading to a degradation in system performance. This is
attributed to the amplification of irradiance variations with an increase in
the temperature gradient [26]. The incorporation of asymptotic analysis
provides further insights, and remarkably, the outcomes from simulations,
asymptotic approaches, and analytical methods closely align, especially at
high SNR. In Fig. 8, the impact of UWT in two different types of water, fresh
and salty, is demonstrated. It is evident that the increased salinity of water
results in a higher SOP, thereby deteriorating channel performance. The
findings indicate that freshwater environments are more resilient during
outages compared to saltwater environments.

Figure 9: The $EST^{II}$ versus $\bar{\gamma}_{\mathcal{D}}$ for selected
values of $\bar{\gamma}_{\mathcal{\tilde{E}}}$.

The impact of eavesdropper’s average SNR on $EST^{II}$ against the average SNR
of the UOWC link is illustrated in Fig. 9. As observed, the EST experiences a
reduction when the average SNR of the eavesdropper increases from 5 dB to 25
dB. This outcome indicates that higher values of the eavesdropper’s average
SNR enhance the corresponding link’s overhearing capability, subsequently
declining the overall secrecy performance.

Figure 10: The $SOP^{III}$ versus $\bar{\gamma}_{\mathcal{D}}$ for selected
values of $h_{\mathcal{R}}$, $h_{\mathcal{D}}$, $h_{\mathcal{\tilde{E}}}$,
$\xi_{R}$ and $\xi_{\mathcal{D}}$.

In optical wireless communication systems, pointing error refers to the
deviation between the intended target and the actual direction in which a
system is pointing. Figure 10 illustrates the $SOP^{III}$ against the average
SNR of the UOWC link for a scenario involving simultaneous eavesdropping,
showcasing the impacts of pointing errors at $D$ and $\mathcal{\tilde{E}}$.
When $\xi_{\mathcal{\tilde{E}}}$ is held constant at 2, it is evident that the
SOP decreases as the value of $\xi_{\mathcal{D}}$ increases from 1 to 2,
thereby enhancing system performance. Similar results were also shown in [35].
Conversely, when $\xi_{\mathcal{D}}$ is fixed at 2, the SOP decreases with the
growing value of $\xi_{\mathcal{\tilde{E}}}$ from 1 to 2, indicating declining
misalignment error between the transmitter and the eavesdropper, thereby
degrading the system performance. The asymptotic SOP is presented alongside
analytical and simulation results, revealing a close alignment, particularly
in the high SNR regime. This observation signifies a robust agreement with the
analytical findings, emphasizing the reliability and accuracy of the
asymptotic analysis in capturing system behavior, especially under conditions
of high SNR.

###  8.3 Comparative Analysis

Figure 11: The $SPSC$ versus $\bar{\gamma}_{\mathcal{D}}$ for three different
scenario.

Figure 12: The $SPSC^{I}$ versus $\bar{\gamma}_{\mathcal{R}}$ for selected
values of $\bar{\gamma}_{\mathcal{D}}$.

The SPSC is plotted against $\bar{\gamma}_{\mathcal{D}}$ in Figure 11 and
$\bar{\gamma}_{\mathcal{R}}$ in Figure 12. A comparative analysis among three
eavesdropping scenarios is presented in Figure 11, highlighting that Scenario
III exhibits the poorest secrecy performance, suggesting that simultaneous
eavesdropping poses the most significant threat. Additionally, it is observed
that the RF eavesdropper is more detrimental than the FSO eavesdropper,
implying that the FSO link is more susceptible to eavesdropping than the RF
link. Figure 12 presents a comparison between Scenario I and the same scenario
without the inclusion of the RIS, revealing that better secrecy performance
can be achieved when communication occurs with the aid of the RIS.

###  8.4 Design Guideline

Based on the results of the investigation into the secrecy performance of the
proposed system, here are some design guidelines for consideration:

####  8.4.1 RF UAV-NTN Link Modeling

The security of the proposed system can be ensured by choosing higher values
of fading severity parameters, understanding small-scale fading effects to
make a balance between the total fading and the multipath exploitation, and
finally strategically selecting optimal link distances.

####  8.4.2 UOWC Link Modeling

In the case of the UOWC system, an increase in the number of reflecting
elements in the RIS for sophisticated beamforming, considering the HD method
for improved secrecy, and mitigating UWT effects to ensure robust performance
in different underwater environments, considering factors such as thermal
gradients and water salinity can significantly enhance secrecy. Besides, the
smaller pointing errors must be addressed to obtain a better alignment between
the transmitter and receiver, and the scenarios with better secrecy
performance must be prioritized in the system design.

These recommendations, which address various system parameters for three
considered scenarios to obtain a balance between the complexities faced by the
system, utilization of limited resources, and overall security will certainly
help the design engineers optimize the secrecy performance of the proposed
system.

##  9 Conclusions

This study addresses a RF NTN-UOWC model that considers dynamic sources, e.g.,
UAVs, and incorporates an RIS in UOWC environments. By meticulously examining
three potential eavesdropping scenarios, e.g., RF NTN, UOWC, and simultaneous,
we have provided valuable insights into the vulnerabilities and mitigation
strategies of such networks in terms of closed-form expressions for some
secrecy metrics, i.e., ASC, SOP, EST, and SPSC. Through numerical analyses and
simulations, our findings highlight the significance of strategic parameter
manipulation in enhancing system security, considering factors such as fading
severity, number of reflecting elements, UWT, pointing errors, and detection
techniques. It is observed that by exploiting an RIS technology, we can
manipulate signal propagation and minimize signal leakage to potential
eavesdroppers, thereby enhancing the security of the communication system. A
comparison between RIS-enabled and non-RIS systems provides remarkable
insights into the efficacy of an RIS deployment in bolstering security.
Furthermore, the study offers valuable design guidelines for the
implementation of secure communication technologies, which can pave the way
for safer and more reliable wireless communication systems in the future.

##  10 Proof of Lemma 1

In order to obtain the PDF of $\gamma_{j}$, we have to first find the PDF of
$Z_{jl}=\left|h_{jl}\right|^{2}q_{j}^{-\alpha_{j}}$. Let’s assume that
$W_{jl}=\left|h_{jl}\right|^{2}$ and $Y_{j}=q_{j}^{-\alpha_{j}}$. Utilizing
the transformation of RV, the PDF of $W_{jl}$ and $Y_{j}$ can be determined,
respectively as follows.

$\displaystyle f_{W_{jl}}(w)$
$\displaystyle=\frac{A_{j}}{2}(\sqrt{w})^{\mu_{j}-1}e^{-B_{j}w}I_{\mu_{j}-1}(M_{j}\sqrt{w})$
(62) $\displaystyle f_{Y_{j}}(y)$
$\displaystyle=\frac{1}{\alpha_{j}}\sum_{i_{j}=1}^{m}\frac{C_{i_{j}}}{D_{j}^{\beta_{i_{j}}+1}}y^{-\frac{1}{\alpha_{j}}(\beta_{i_{j}}+1)-1},\quad
D_{j}^{-\alpha_{j}}\leq q\leq\infty$ (63)

Hence, the PDF of $Z_{jl}$ can be obtained as

$\displaystyle
f_{Z_{jl}}(z)=\int_{D_{j}^{-\alpha_{j}}}^{\infty}\frac{1}{y}f_{W_{jl}}\left(\frac{z}{y}\right)f_{Y_{j}}(y)dy.$
(64)

Putting (62) and (63) to (64) results in

$\displaystyle f_{Z_{jl}}(z)$
$\displaystyle=\frac{A_{j}z^{\frac{1}{2}(\mu_{j}-1)}}{2\alpha_{j}}\sum_{i_{j}=1}^{m}\frac{C_{i_{j}}}{D_{j}^{\beta_{i_{j}}+1}}$
$\displaystyle\times\int_{D_{j}^{-\alpha_{j}}}^{\infty}y^{-\frac{1}{2}(\mu_{j}-1)-\frac{1}{\alpha_{j}}(\beta_{i_{j}}+1)-2}$
$\displaystyle\times
e^{-B_{j}zy^{-1}}I_{\mu_{j}-1}\left(M_{j}\sqrt{z}y^{-\frac{1}{2}}\right).$
(65)

Using the approximation of the Bessel function
$I_{v_{j}}(x)=\sum_{k_{j}=0}^{p}V_{j}(k_{j},p,v_{j})\left(\frac{x}{2}\right)^{v_{j}+2k_{j}}$,
where
$V_{j}(k_{j},p,v_{j})=\frac{\Gamma(p+k_{j})p^{1-2k_{j}}}{\Gamma(k_{j}+1)\Gamma(p-k_{j}+1)\Gamma(v_{j}+k_{j}+1)}$
[51], and assuming $u=y^{-1}$, (65) can be translated as follow.

$\displaystyle f_{Z_{jl}}(z)$
$\displaystyle=\frac{A_{j}}{2\alpha_{j}}\sum_{i_{j}=1}^{m}\sum_{k_{j}=0}^{p}\frac{C_{i_{j}}}{D_{j}^{\beta_{i_{j}}+1}}V_{j}(k_{j},p,\mu_{j}-1)z^{\mu_{j}+k_{j}-1}$
$\displaystyle\times\left(\frac{M_{j}}{2}\right)^{\mu_{j}+2k_{j}-1}\int_{0}^{D_{j}^{\alpha_{j}}}u^{\mu_{j}+k_{j}+\frac{1}{\alpha_{j}}(\beta_{i_{j}}+1)-1}e^{-B_{j}zu}du.$
(66)

Now, the PDF of $Z_{j}$ can be deduced by utilizing [38, Eq. (3.381.1)] as

$\displaystyle f_{Z_{jl}}(z)$
$\displaystyle=\frac{A_{j}}{2\alpha_{j}}\sum_{i_{j}=1}^{m}\sum_{k_{j}=0}^{p}\frac{C_{i_{j}}}{D_{j}^{\beta_{i_{j}}+1}}V_{j}(k_{j},p,\mu_{j}-1)$
$\displaystyle\times\left(\frac{M_{j}}{2}\right)^{\mu_{j}+2k_{j}-1}z^{-\frac{1}{\alpha_{j}}(\beta_{i_{j}}+1)-1}B_{j}^{-\mu_{j}-k_{j}-\frac{1}{\alpha_{j}}(\beta_{i_{j}}+1)}$
$\displaystyle\times\gamma\left(\mu_{j}+k_{j}+\frac{1}{\alpha_{j}}(\beta_{i_{j}}+1),B_{j}zD_{j}^{\alpha_{j}}\right),$
(67)

where $\gamma\left(.,.\right)$ indicates the lower incomplete Gamma function
[38, Eq. (8.350.1)]. For further simplification, employing [44, Eq.
(8.4.16.1)] and [44, Eq. (8.2.2.15)] to (67) and following the transformation
of RVs considering $\mu_{MRC_{j}}=L_{j}\mu_{j}$,
$\kappa_{MRC_{j}}=\kappa_{j}$, $\bar{\gamma}_{MRC_{j}}=L_{j}\bar{\gamma}_{j}$
as stated in [52], the PDF of $\gamma_{j}$ can be expressed as shown in (10),
where $\text{G}_{p,q}^{m,n}\left[.\right]$ indicates the Meiger’s $G$-function
[38].

##  11 Proof of Lemma 2

$F_{\gamma_{j}}(\gamma)$ can be determined through the integration of (10)
over the range [0, $\gamma_{j}$], employing the expression provided in [44,
Eq. (2.24.2.2)] as illustrated in (11).

##  12 Proof of Lemma 3

Making use of the identity of [44, Eq. (8.2.2.14)] and applying [49, Eq. (41)]
for expanding the Meijer’s $G$ function, the asymptotic CDF of $\gamma_{j}$
can be derived as shown in (12).

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