Security Performance Analysis of a NOMA-Assisted Underwater VLC System Under Imprecise Channel Estimations

The accurate estimation of underwater Visible Light Communication (VLC) channel conditions is challenging due to its widespread attenuation and scattering effects. The channel attenuation is a linear function of frequency and causes exponential signal power loss whereas due to the scattering effect, numerous photons are statistically generated as light beams strike water molecules and there arise security concerns. Assuming realistic underwater conditions, this paper investigates the security performance of a typical Non-Orthogonal Multiple Access (NOMA)-assisted underwater VLC system. It consists of a Floating Vehicle Transmitter (FVT), equipped with multiple Light Emitting Diodes (LEDs) to transmit the signal to two legitimate near-end and far-end Underwater Vehicles (UVs) in presence of an active/passive eavesdropper. The Channel State Information (CSI) of each transmitting link is estimated with the use of a Minimum Mean Square Error (MMSE) technique. Furthermore, we propose a LED selection mechanism to select an LED that can achieve the highest secrecy rate defined under the constraints of known and unknown CSI of legitimate and/or eavesdropping links. Using the Successive Interference Cancellation (SIC) technique, a novel closed-form secrecy outage probability expressions for the conventional single-LED and multi-LED NOMA-VLC links for both known and unknown CSI scenarios is derived. The security performance of the proposed multi-LED NOMA-VLC system is compared with the conventional single-LED NOMA-VLC system under the effect of air bubbles for both fresh and salty water. Finally, we verify the validity of the numerical results through Monte-carlo simulation analysis.


I. INTRODUCTION
Underwater Optical Wireless Communication (OWC) is rapidly gaining popularity among both academia and industry due to the expansion of underwater activities, such as under- The associate editor coordinating the review of this manuscript and approving it for publication was Hassan Omar .
water scientific data collection, underwater sensor networks, unmanned underwater vehicles, etc. [1]. Unlike traditional acoustic communications, underwater Visible Light Communication (VLC) is expected to provide a high-speed data transmission rate with reliable, secure, and low-latency metrics. However, attenuation and scattering are two widespread concerns the OWC networks, which affect the overall VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ performance of the underwater VLC system. Therefore, in [3], a modified Beer-Lambert formula was derived especially for the underwater VLC link that incorporates both attenuation and scattering effects unlike the conventional Beer-Lambert formula derived in [4] which did not include the scattering effect. Indeed, the attenuation effect causes signal power loss whereas due to the scattering effect, when a light beam (i.e. operating at blue/green light wavelength band 450 nm to 550 nm) originated at VLC transmitter strikes a water molecule, numerous photons are statistically generated and start traveling in random directions [1], [2], [3], [4], [5], [6]. This as a consequence may cause an insecurity issue in the underwater VLC system because a scattered photon may received by an illegitimate receiver (or eavesdropper). Considering the security as a key metric, it is important to track the scattered photon location, which indeed requires the channel estimations of realistic underwater conditions. The overall attenuation effect in the realistic underwater channel conditions, not only causes signal power loss but also moderates the VLC channel connectivity with mobile endusers [7]. In order to improve the user connectivity between transceivers, Multiple Input Multiple Output (MIMO) technology that utilizes multiple Light Emitting Diodes (LEDs) can be used [8]. However, due to small-area transmission coverage of VLC technology [9], each transmitting LED cannot be accessed by a mobile user. Then, for further improvement in the underwater VLC network connectivity, the transmitter can select a number of LEDs that are perfectly aligned with the receiver terminal. However, it again depends on the channel estimation analysis to know the exact user location [10].
Recently, Non-Orthogonal Multiple Access (NOMA) has emerged as another potential technology that can serve multiple users at the same time/frequency/code by minimizing the attenuation effect [4]. The fundamental concept of NOMA is based on superposition coding and successive decoding where the superposition coding of multiple users' signals can be done over the same subcarrier with different power levels. As a consequence, it enables the receiver to decode the signal by using the Successive Interference Cancellation (SIC) technique [11]. At the receiver terminal, the SIC starts by decoding the data of the user that have the strongest channel connectivity with the transmitter. Once the strongest user's signal is decoded, then the detected data is passed through to an iterative SIC method. Then, the other user's signal is reconstructed based on the prior knowledge of the CSI of this next user after the first user's signal is subtracted from the superimposed signal [12]. This indeed avoids the interference caused by the first user's signal and increases accuracy in decoding another user's signal. In particular, in the NOMA scheme, the power coefficients among multiple users can either be arbitrarily chosen or power can be intentionally allocated to optimize a preferable performance e.g. minimizing the secrecy outage probability and/or maximizing the capacity, secrecy capacity, and other objectives [13], [14].
On the other hand, due to the scattering effect, less number of scattered photons enter the Field-of-View (FOV) of the authenticated (legitimate) user while there is a high probability that some of the scattered photons might be incident on the FOV of the unauthenticated user (eavesdropper) [15], [16], [17], [18]. Thus, the underwater VLC systems have a requirement for security performance analysis. The security in OWC systems can be provided through either the conventional Cryptographic technique that can be applied at the network layer or the Physical Layer Security (PLS) technique, which is applicable at the physical layer itself [19]. More specifically, the PLS technique provides Information Theoretic Security (ITS) at a quantum level by exploiting the properties of the physical layer whereas the Cryptographic technique is based on encryption and decryption methods following a secrete code. Therefore, Wyner [20] first time used the PLS technique and proposed a wiretapper channel model by defining a positive secrecy rate metric that can provide the optimum guarantee to prevent eavesdropping attacks up at this particular rate [21], [22]. However, the major limitation of the Wyner model was the assumption that the legitimate channel conditions are always stronger than the eavesdropping channel conditions and the transmitter can easily obtain a positive secrecy rate. In fact, this assumption is not reliable in underwater communication scenarios because here scattering is a major cause of insecurity occurrence and hence, eavesdroppers can also realize the stronger channel conditions. Therefore, in PLS analysis instantaneous CSI estimation analysis is a key requirement in order to track the light propagation paths that are approaching the legitimate and/or eavesdropper's FOVs.
In the OWC systems, due to the fact that the user node which might be either a legitimate node or an eavesdropper one, can estimate the CSI of its receiving link during the pilot symbol transmission process. In this process, some known pilot symbols are been sent by the transmitter over each transmitting link [23]. Then, whenever the transmitter demands, the user may or may not send its estimated CSI knowledge to it. In practice, it can be assumed that a legitimate user always sends its CSI to the transmitter [24]. Whereas, an eavesdropper will send its CSI to the transmitter depending on whether it is an active user (i.e. a legitimate node of the adjacent network that sends its CSI to the transmitter) or a passive user (i.e. a malicious node that never shares its CSI with transmitter) [25]. Recently, in [26], a secure NOMAassisted multi-LED underwater VLC network is presented where the secrecy rate metric is defined against both active and passive eavesdropping attacks with an assumption that an error-free CSI estimation is possible at the receiver node. However, due to the variable underwater channel conditions and the occurrence of Gaussian distribution errors, the precise error-free CSI estimation at the receiver node is practically impossible [27]. Therefore, in the PLS, it is important to formulate the secrecy rate according to imprecise channel estimations so that the transmitter can prevent both active and passive eavesdropping attacks in a strategic manner. To the best of our knowledge, no prior research has been conducted that address the adverse scattering effects from a security perspective, specifically for underwater VLC system.
In this paper, we assume that due to the scattering phenomenon, the underwater VLC system is vulnerable in the presence of an active/passive eavesdropper. In order to estimate the exact knowledge of the eavesdropper's availability, imprecise channel estimations are needed in each transmission time interval. The main contributions of this paper can be summarized as follows: • Present the geometry of the light propagation mechanism of the underwater VLC model and derive the approximate channel gain expressions for both direct and scattered paths, respectively.
• Derive the Probability Density Function (PDF) and Cumulative Distribution Function (CDF) expressions of the received end-to-end signal-to-interference noise ratio (SINR) of each legitimate and eavesdropper links.
• A Minimum Mean Square Error (MMSE) technique is used to estimate the CSI knowledge of CSI of both legitimate and eavesdropping links. Then, according to the availability or non-availability of CSI of both legitimate and eavesdropping links, a LED selection mechanism is defined to select the LED with the highest secrecy rate.
• This paper considers the Exponential Generalized Gamma (EGG) distribution to model to measure the received signal fluctuations while considering the underwater turbulence effects because it can provide the excellent agreement between the theoretical and experimental results.
• Derive novel closed-form secrecy outage probability expressions for conventional single-LED as well as multi-LED transmission strategies, for both eavesdropping scenarios.
• Finally, computer-based simulation analysis is performed to validate each plotted numerical result.
The remainder of the paper is organized as follows. The system and channel model are presented in Section II. Additionally, a detailed description of the light propagation underwater VLC model and LED selection mechanism are presented in the same section. The performance metrics of the conventional single-LED and the proposed multi-LED transmission strategies are defined in Section III. In Section IV, the numerical results are demonstrated for the security performance of the proposed NOMA-assisted underwater VLC system and Section V concludes the findings of this paper.

II. SYSTEM AND CHANNEL MODEL
Suppose a floating vehicle transmitter (T x ) transmits signals towards legitimate near-end (M 1 ) and far-end (M 2 ) underwater vehicles (UVs), in the presence of an active and/or passive eavesdropping UV (E k ). Let T x be equipped with N transmitting LEDs, whereas M 1 and M 2 are equipped with a photo-detector (PD) node, as shown in Fig. 1. In NOMA scheme, two signals X 1 and X 2 overlap with different power levels. Each signal is encoded as (R b,k , R s,k ), when R b,k is defined as a fixed transmission rate parameter and R s,k is defined as a desired secrecy rate quantity. The signal received at the UVs can be expressed as where z k ∈ {M k , E k } and k ∈ {1, 2},ĥ z k is the imprecise channel gain, α 1 and α 2 are the power allocation coefficients. From the security perspective, the confidential signal of user M 1 should be hidden below the signal of user M 2 . Also, more power is allocated to user M 2 under the condition that α 2 > α 1 when α 1 + α 2 = 1, with transmit power P s at T x . η denotes the responsivity and I is denoted as the real valued irradiance fluctuations of the corresponding link. The value of r is related to the used detection technique at the user terminal, which means that r = 1 when Heterodyne Detection (HD) technique is used whereas r = 2 when Intensity Modulation/Direct Detection (IM/DD) technique is used by the receiving UV terminal [3]. Finally, W z is defined as a complex Additive White Gaussian Noise (AWGN) parameter with zero-mean and variance σ 2 z k .

A. LIGHT PROPAGATION MODEL
The geometry of light propagation VLC model is depicted in Fig. 2, which illustrates that the optical signal at z th k UV can be received via two separate paths, namely Line-of-Sight (LoS) (or direct) path and scattering (indirect or NLoS) path following FVT-to-Scatterer and Scatterer-to-Z k links. Let minimum mean square error (MMSE) estimation technique is used to estimate the CSI of each transmitting links under imprecise channel conditions. According to light propagation geometry, the channel-gain at the z th k UV can be given as [24] where β z k is denoted as correlation coefficient between h g/s z k andĥ z k , here, h g z k is the geometric underwater VLC link channel gain in fresh water when no turbulence is present VOLUME 10, 2022 whereas h s z k is the scattered link channel gain obtained under error-free channel conditions and w z k is the Gaussian random error independent toĥ z k with zero-mean and unit variance. Moreover, the β z k can be given as [25] where T t is the length of pilot symbol transmission and α is the normalized pilot power. When T t = 1 and hence, the effect of channel estimation solely characterized by α.

1) CHANNEL GAIN UNDER FRESH WATER CONDITIONS
The direct underwater VLC link gain h g z k can be given by [25] h g where U(ψ z k ) is denoted as optical filter gain and g(ψ z k ) is the gain of the optical concentrator given by where n denotes the refractive index, z k ,FoV denotes the FOV of the z th k UV, A is the detector area, m is the Lambertian radiation pattern which can be given by m = − 1 log(cos(φ 1/2 )) , where φ 1/2 is the semi-angle of each transmitting LED. Therefore, the directivity of the transmitting LED is related to semi-angle φ 1/2 and at φ 1/2 = 60 • , m becomes unity, which corresponds to ideal LED. d z k ,L i is the Euclidian depth from the FVT to z th k .

2) CHANNEL GAIN OVER TURBULENT CONDITIONS
The scattering effect in underwater medium causes two major effects: i) the light beam deviates from its original path which means that it may or may not incident on the legitimate user's FOV and causes probability of eavesdropping occurrence, and ii) it reduces the mean irradiance of light beam at the receiver terminal that indeed reduces the received signal strength at the UV and might deteriorate the system performance. As a consequence, two paths between transceivers can be constructed i.e. LED-to-Scatterer and Scatterer-to-Z th k receiver path. Therefore, an approximate expression for scattering channel gain h s z k can be derived as where d j,L i and d z k ,j are the Euclidian depth from the FVT to the j th scattering molecule and scattering molecule to the z th k user. Moreover, the overall attenuation effect in transmitting light beam can be defined by extinction coefficient c denoted as c = a + b, here, a is defined as an absorption coefficient, and b is defined as a scattering coefficient and their values can be chosen as constant for a particular wavelength of the signal during performance analysis of the proposed underwater VLC network. In order to extract the relevant information from the overlapped signal X k , the UV utilizes the SIC technique. In this technique, when demodulating the X 2 signal, the X 1 signal is viewed as an interference signal. However, when demodulating the signal X 1 , the X 2 signal is first demodulated and then re-modulated after subtracting the interfering signal from the overlapped signal. Assuming that both E k and M k UVs are having the same decoding capabilities, then, by using (1a), Signal-to-Interference plus Noise Ratio (SINR) obtained at M 1 and E 1 , can be defined as where ρ = ηI r P s 2σ 2 and level is residual interference is denoted by ℘ =∈ (0, 1]. Let both M 2 and E 2 can access the second transmission phase, which is an interference-free signal then, by using (1b), the SNR received at M 2 and E 2 , can be defined as Any LED available at the transmitter can transmit the information. During pilot symbol transmission process, some known pilot symbols are been transmitted by the FVT T x over each transmitting link. Then, the MMSE technique is used to receive the CSI information from each receiving legitimate and/or eavesdropping UV terminal. Based on availability and/or non-availability of CSI information at T x , two know and unknown CSI scenarios are defined. In particular, known CSI scenario is defined against active eavesdropper, which shares its CSI with the transmitter and unknown CSI scenario is defined against a passive eavesdropper, which never shares its CSI with the transmitter. The above assumptions on both scenarios are valid because in wireless communication, during pilot symbol transmission process, the receiver first receives the instantaneous CSI of its receiving link and then, when the transmitter demands, the receiver sends its estimated CSI to the transmitter. The CSI estimation is performed by using each transmitting LED. However, from a security perspective, the transmitter T x selects an LED that provides the optimum guarantee of security against both active and passive eavesdroppers. Therefore, the transmit LED selection mechanism is further divided into the following two scenarios;

1) SCENARIO-I: AGAINST ACTIVE EAVESDROPPER
Let E k be an active user. Hence, T x knows the instantaneous CSI of both legitimate and eavesdropping VLC links. Then similar to the Wyner wiretapper channel model [20], the positive secrecy rate can be formulated as where [.] + is used to denoted a non-negative quantity. Moreover, γ X k M k and γ X k E k are the instantaneous SINRs of the legitimate and eavesdropping links given by (9) and (10), respectively. Then, the secrecy rate under scenario-I, can be defined as where i ∈ {1, 2, . . . N }. A pseudo-code of the scenario-I can be given by Algorithm-1.  Let E k is now a passive user and T x does not know the CSI of eavesdropping link. Then, it transmits the signal with a predetermined transmission rate R b,k . The selection of R b,k is constrained with the channel capacity of the legitimate link i.e. C M k = log 2 (1 + γ X k M k ). Furthermore, T x can transmit the information under the certain condition that C M k ≥ R b,k because it cannot transmit the information beyond the channel capacity of the respected link. Otherwise, when C M k < R b,k , connection outage occurs and T x remains silent. In order to transmit the signal to M k , T x selects an LED which can achieve the maximum channel capacity C M k ; that is C i M k = log 2 (1 + γ i,X k M k ), where γ i,X k M k is the received SINR at M th k user from the i th antenna at T x . Then, the secrecy rate for scenario-II, can be formulated as

Algorithm 1 LED Selection in
A pseudo-code of scenario-II can be given by Algorithm-2.
Algorithm 2 LED Selection in Scenario-II 1: Initialization: Instantaneous CSIs of M k link and define R b,k ; 2: Decision:  (13), it can be defined that in order to achieve the highest security against a passive eavesdropper, the value of R b,k should be high or very close to γ X k M k . However, when R b,k increases, the secrecy rate C iSLS s,k decreases. As a consequence, using (12) and (13), it can be concluded that secrecy rate C iSLS s,k can never be greater than C iOLS s,k , which means that knowledge of instantaneous CSI of both legitimate and eavesdropping links at the transmitter can provide better or higher security against the eavesdropping attacks.
In order to model the irrandiance light beam intensity fluctuations I r z k given in (1), a mixed exponential generalized Gamma (EGG) distribution is distribution is used. Then, utilizing (9), (10) and [5, eqn. (21)], the cumulative distribution function (CDF) of γ X k z k can be given by VOLUME 10, 2022 where z k ∈ (M k , E k ). Using (14) and [28, eqn. (8.2.2.31)], the probability density function (PDF) of γ X k z k can be given by where where γ X k z k = E(γ X k z k ) denotes the average electrical SINR, ω denotes the mixture coefficient and λ is the scale parameter of exponential distribution. Moreover, a, b and c quantities are representing the EGG distribution parameters for different underwater scenarios and (.) represents the Gamma function. More specifically, the CDF and PDF expressions given in (14) and (15) are valid for (9), when ϕ 1 z = α 1 ρ z 1 . However, for (9) and (10), both expressions are valid when ϕ 2 z = (α 2 − α 1 γ )ρ z 2 under the condition that γ ≤ α 2 α 1 . Otherwise, when γ > α 2 α 1 , then the CDF expression becomes unity i.e. F γ z k (γ ) = 1 and the PDF expression becomes zero i.e. f γ z k (γ ) = 0.

III. PERFORMANCE ANALYSIS
In order to compare the performance of the proposed underwater multi-LED-VLC system with the conventional underwater single-LED-VLC system, in this section, we derive closed-form secrecy outage probability (SOP) expressions for both single-LED and multi-LED transmission strategies. The SOP is the probability that the secrecy capacity C s,k is less than a target secrecy rate R s,k . Therefore, the security is now constrained with the R s,k parameter which means that the proposed system must provide guaranteed secure communication against eavesdropping attacks up to the desired target secrecy rate R s,k . Otherwise, the system outage condition occurs and the transmitter stops information transmission.

A. SOP OF UNDERWATER SINGLE-LED VLC SYSTEM
In this strategy, we consider that only the LED is transmitting the information and hence, in the first transmission phase for k = 1, the SOP at M 1 can be expressed as where τ 1 = 2 R s, 1 (15), (14) and (17), the derived SOP expression for M 1 UV is given in (18), as shown at the bottom of the next page, where G(.) is the Meijer G function [28].
In practice, the FV transmitter T x does not know the exact location of E k . Therefore, we assume that E k can eavesdrop the information of either M 1 or M 2 . Therefore, the eavesdropping occurrence in both the transmission phases is equal and then, the SOP of the single-LED VLC system can be given as where Pr E eavesdrops M 1 and Pr E eavesdrops M 2 can be obtained according to the practical scenarios. We assume that Pr E eavesdrops M 1 = Pr E eavesdrops M 2 = 1/2. Then, the SOP can be expressed as

B. SOP OF UNDERWATER MULTI-LED VLC SYSTEM
As stated above, the proposed system model is equipped with multiple LEDs and eavesdropping is possible via either an active and/or passive eavesdropper. Hence, in this section, the closed-form SOP expressions are derived for both scenario-I and scenario-II, respectively.

1) SOP ANALYSIS FOR SCENARIO-I
We first consider the case where security of M 1 is more important than M 2 . Then, the SOP at M 1 of the Multi-LED underwater NOMA-VLC transmission against active eavesdropping, can be defined as It is clear here that once an optimal LED is selected for transmitting the signal to M 1 , the SOP at M 1 can be given as P iOLS out,M 1 = P out,M 1 . Then, a random LED can be selected for transmitting the signal to M 2 . Moreover, the SOP at M 2 can be given by (19) and the overall SOP for the VLC link can be given by Similarly, when the optimal LED is selected for M 2 , the overall SOP can be given by

2) SOP ANALYSIS FOR SCENARIO-II
The SOP at M 1 against passive eavesdropping attacks, can be defined as where P iSLS out,M 1 can be given by where γ th,1 = 2 R b,1 − 1. Using (15), the SOP can be derived and given in (28), as shown at the bottom of the next page. As an LED is already selected to transmit the signal to M 1 and hence, T x can select any random LED to transmit the signal to M 2 . Then, the SOP at M 2 can be defined as  Similarly, the SOP that maximizes the channel capacity of the M 2 link can be defined as

IV. NUMERICAL RESULTS
As stated above, the underwater OWC networking systems are mainly affected by air bubbles, which cause attenuation and scattering effects. In previous section, performance metrics of system were defined by considering EGG distribution.   In this section, we will now observe the effect of different air bubble levels, on the security performance of the proposed underwater VLC system. Indeed, we consider the two types of water conditions namely i) fresh water when no (or moderate) turbulence is present and ii) salty water when strong turbulence is present. Turbulence effect variation is estimated by varying the air bubble levels (BL) using different values of ω, λ, a, b, and c, as listed in [5, Tables 1 and 2]. Other parameters such as refractive index of water n = 1.5, gain of optical filter U(ψ k ) = 1, effective area A = 1.0 cm 2 , FOV angle of both M 1 and M 2 UV terminals i.e. ψ z k = 70 • and φ 1/2 = 60 • are kept fixed. For Monte-Carlo simulation analysis, 10 5 random samples are generated and their values are correlated with the numerically obtained values. Moreover, we consider γ M 1 = γ M 2 = γ M and R s,1 = R s,2 = 1 bit/sec/Hz. Using (12) and (13), we first quantify two optimal values of normalized pilot power α that can provide the highest secrecy-rate for transmitting the information to M 1 and/or M 2 UVs for both scenario-I and scenario-II. Fig. 3 demonstrates the achievable secrecy-rate against the normalized pilot signal power α, at γ E = 0 dB and R b = 0.6781 bits/s/Hz. In scenario-I, it can be observed from the curves of γ M = 5 dB and γ M = 10 dB that the secrecy-rate increases fast to a peak value at α = 29 for γ M = 5 dB, and at α = 19 for γ M = 10 as α increases. Similarly, in scenario-II, the secrecy-rate increases to its peak value α = 28 for γ M = 5 dB, and at α = 18 for γ M = 10 dB. Moreover, it is to noticeable from the figure that it is not always good to impose a high pilot power to achieve a maximum secrecy rate because secrecy rate saturates after a certain value of α. This is happening due to the fact that we cannot transmit the information over a VLC link beyond its channel capacity. Hence, for further improvement in the security performance of the system, we need to focus on the other parameters i.e. attenuation, scattering effects, noise, and CSI estimations, etc. With this observation, we have calculated two optimum values of α under which T x achieves the highest secrecy rates. It can also be seen from the plot that the achieved secrecy-rates of both M 1 and M 2 UVs for scenario-I is greater than the achieved rates for scenario-II. More specifically, the secrecy rate of M 2 is higher than the M 1 UV because the allocated power coefficient α 2 = 0.6 at M 2 is higher than the coefficient α 1 = 0.4 at M 1 . Next, we analyze the SOP performance of a conventional single-LED VLC system with the variation in the transmitting signal power P s for two air bubble levels. Also, we assume that the parameters r = 2, γ M = 10 dB and γ E = 5 dB as fixed quantities. One can notice from (9) and (10) that the received SINRs at M k is directly proportional to P s , which means that when we increase the value of P s , the received SINRs at M k will also increase and their consequence effects will also improve secrecy rates C s,k under both scenarios, see eqns. (12) and (13). Therefore, plotted results in Fig. 4 are quite intuitive and validate the equation derived in (17) i.e. increasing the value of P s should deteriorate the secrecy outage effect. Moreover, as power coefficients α 1 and α 2 are different for both UVs and in fact, more power is allocated to M 2 UV than M 1 UV. Hence, the plotted results are also representing that before the floor point, the value of SOP for M 2 UV remains higher than the SOP value for the M 1 UV and then, after the floor point, it represents the opposite effects i.e. higher SOP value for M 1 than M 2 UV. Here, the floor point is occurring when both M 1 and M 2 UVs are having equal power allocation coefficients i.e., α 1 = α 2 . On the other hand, it can also be observed from the same figure that the SOP value is high at air bubble level BL = 16.5 L/min (i.e., for salty water) than the SOP value at BL = 4.7 L/min (i.e., for freshwater) for each individual value of P s . This occurs due to the fact that as the turbulence effect increases, the scattering effect increases, and its consequence effect decreases the value of SOPs for both UVs. Using (22), it can be seen that the total SOP of the system is an average value of the SOPs obtained for M 1 and M 2 UVs, respectively. Therefore, the curve for total SOP approaches near the SOP of M 2 UV before the floor point and then approaches near the SOP value of M 1 UV after the floor point. This complete observation is also indicating that the overall security performance of the system deteriorates when the scattering effect becomes severe. Based on the above-illustrated explanation, one can easily find that Fig. 3 concludes that we cannot increase the normalized power value α after a certain limit whereas Fig. 4 concludes that an increase in value of P s can improve the system performance. Therefore, it is now important to obtain the SOP against the average instantaneous SINR of the legitimate link i.e., γ M because it can quantify the effect the power of a signal received at M k UV on the security performance of the system. We again take r = 2 and γ E = 5 dB parameters as fixed quantities. Fig. 5 illustrates the results of the equations derived in (18) and (20), respectively, which represents that when we increase γ M value, the SOP for M 2 UV remains lower than the SOP value of M 1 UV until it reaches to a floor point and then, as expected, after the floor point, the SOP value for M 2 UV becomes higher than the SOP value obtained for M 1 UV. Here, the floor point exists at around γ M = 33 dB SINR value because we now keep the P s value fixed at 2 dB and it can neglect the turbulence effects and increase the security over a noticeable region. It can also be observed from the same plot that the SOP value is high for air bubble level BL = 16.5 L/min than the SOP value of BL = 4.7 L/min, which is again illustrating the similar effect as described above. Moreover, the total SOP value given by (22) and (24) are also imposing the desired changes, and accordingly, the total SOP curve is approaching the SOP curves of M 1 and M 2 UVs, respectively. Interestingly, it can further be noticed from Figs. 4 and 5 that when γ M increases, it decreases the SOP of M 1 (or M 2 ) in a much faster rate than when P s value increases. The reason is that the secrecy rate becomes a constant at a large value of P s and does not produce any significant contribution in improving the security performance analysis of the system beyond a certain limit. Fig. 6 represents and compares the advantage of imprecise channel estimations over the SOP analysis against when no estimations are performed with correlation coefficient β z k = 0.31, r = 2 and γ E = 5 dB. It can be observed from the plot that CSI estimations represent lower SOP than the SOP with no CSI estimations for both the M 1 and M 2 UVs at each individual value of γ M in dB. The results are pretty obvious because the estimated channel gain defined in (2) is directly proportional to the product of the channel gain parameters of direct and scattered paths that enhances the overall received signal strength at the receiver terminal. The another advantage of using the MMSE technique over LSE is that it avoids the error that occurred during channel estimations. Therefore, the overall SOP performance of the proposed underwater VLC system with imprecise estimations is better than the SOP performance evaluated without CSI estimations. Fig. 7 compares the secrecy outage performance of the proposed underwater VLC system with the conventional single-LED VLC system. Here, we define how the number of LEDs can play an important role in improving the security performance of the system and also present the effect of choosing the detection techniques illustrated before in Section IIA i.e. heterodyne (r = 1) and IM/DD (r = 2) detection techniques. The SOP values are obtained by considering the equations given by (24) and (25) at different values of γ M received at M k UV while considering P s = 2 dB and γ E = 5 dB as constant quantities. At N = 1, we plot the results of a single LED system. It can be seen from the plot that the SOP values of M 1 (or M 2 ) decrease when we increase the value of N = 1 to N = 3 and/or N = 5. In particular, the SOP value of M 2 UV remains less than the SOP value of M 1 UV until it reaches a floor point, but after floor point, the SOP value M 2 UV becomes greater than the SOP value of M 1 UV. Again, the results are quite intuitive and represent that when the value of N increases from N = 3 or N = 5, it can significantly improve the secrecy performance of the proposed system. An important point here is to notice with the same plot that the SOP values for r = 2 are always higher than the values for r = 1, which means that in contrast to the IM/DD detection technique, the heterodyne detection technique can not only improve the decoding ability of the respective UV but it can provide higher security against eavesdropping attacks too.
In Fig. 8, we presents how imprecise channel estimations can affect the overall security performance of the proposed system. The SOP values are obtained by using (28) and (30) for different γ M values while varying the predefined transmission rate R b,k (i.e. illustrated in Section IIB for scenario-II) and P s . We set r = 2, α 1 = 0.4, N = 5 and γ E = 5 dB as fixed quantities. It can be observed from the figure that as data transmission rate R b,k (here, we choose equal data transmission rate in both transmission phases i.e. R b,1 = R b,2 ) increases from 2 bits/s/Hz to 5 bits/s/Hz, the SOP value of M 1 (or M 2 ) decreases. On the other hand, when we increase P s from 2 dB to 7 dB, the SOP value of M 1 (or M 2 ) further decreases. It defines the fact that a large quantity of either R b,k or P s can improve the secrecy performance of the proposed system in the presence of a passive eavesdropper. However, the overall SOP performance of M 2 UV  is lower than the M 1 user as the figure illustrates that the secrecy outage for M 2 UV occurs earlier than M 1 UV. The reason is that R b,k is defined under the constraints of channel capacity of the legitimate M k UV only and its value can never exceed the value of C M k value, as stated in the Section-IIB. Indeed, this reason was discussed earlier in scenario-II, T x has to compromise with the security performance of the system. This observation defines that the CSI estimation process plays an important role in order to identify the eavesdropping UV node presence as well as in defining the strategies to prevent eavesdropping attacks.
Finally, Fig. 9 illustrates a comparison between the results plotted for secrecy outage performance of the proposed NOMA-assisted VLC system in both scenario-I and scenario-II. The results are plotted for the SOP values obtained for different values of γ M by using equations given by (18), (20), (28) and (30), respectively. We take r = 2, α 1 = 0.4, N = 5, γ E = 5 dB, and R b,k i.e. R b,1 = R b,2 = 2 bits/sec/Hz, P s = 2 dB as constant parameters. It can be observed from the plots of both the scenarios that the SOP value of M 2 UV remains lower than the SOP value of M 1 UV at each identical value of γ M . The figure is also representing that under strong turbulence condition (i.e. BL = 16.5 L/min) or moderate (i.e. BL = 4.7 L/min) turbulence condition, the scenario-I is always better than the scenario-II, as we cannot track a scattered photon location when it reaches at the FOV of a passive eavesdropper. Also, it is very clear from the figure that with the use of the heterodyne technique (at r = 1), we can improve the security performance of the proposed underwater VLC system.

V. CONCLUSION
In this paper, according to the geometry of the light propagation mechanism, we have derived the direct and scattered paths' channel gain expressions for an underwater VLC environment. Using the MMSE technique under imprecise underwater channel conditions, we have formulated the secrecy rate for both known CSI scenario-I and unknown CSI scenario-II. Then, the closed-form SOP expressions are derived for both single-LED and proposed multi-LED VLC links. Furthermore, the performance of both systems is evaluated through a variety of different parameters such as underwater turbulence effects, transmit signal power, desired secrecy rate, and predefined transmission rate. The obtained results are validating the fact that the performance of the underwater VLC system in scenario-I is superior to the performance of scenario-II, which means that the CSI knowledge of all transmitting links at the transmitter can improve the security performance of the VLC network. Finally, the validity of the plotted numerical results is verified through a computer-based Monte-Carlo simulation analysis. In the future, investigating optimization algorithms to estimate an optimal pre-determined transmission rate and/or transmitted power values in a underwater VLC scenario with multiple users is an interesting extension to this work. Also, the simulated results can be verified through the experimental setups designed to perform in realistic underwater channel conditions.