Capacity Analysis of NOMA-Enabled Underwater VLC Networks

Visible light communication (VLC) has recently emerged as an enabling technology for high capacity underwater wireless sensor networks. Non-orthogonal multiple access (NOMA) has been also proven capable of handling a massive number of sensor nodes while increasing the sum capacity. In this paper, we consider a VLC-based underwater sensor network where a clusterhead communicates with several underwater sensor nodes based on NOMA. We derive a closed-form expression for the NOMA system capacity over underwater turbulence channels modeled by lognormal distribution. NOMA sum capacity in the absence of underwater optical turbulence is also considered as a benchmark. Our results reveal that the overall capacity of NOMA-enabled Underwater VLC networks is significantly affected by the propagation distance in underwater environments. As a result, effective wireless transmission at high and moderate spectral efficiency levels can be practically achieved in underwater environments only in the context of local area networks. Moreover, we compare the achievable capacity of NOMA system with its counterpart, i.e., orthogonal frequency division multiple access (OFDMA). Our results reveal that NOMA system is not only characterized by achieving higher sum capacity than the sum capacity of its counterpart, OFDMA system. It is also shown that the distances between sensor nodes and the clusterhead for achieving the highest sum capacity in these two multiple access systems are different.


I. INTRODUCTION
The increased use of underwater sensor networks (USNs) for various applications such as environmental monitoring, oil exploration, port security, data collection, and tactical surveillance has prompted researchers to investigate underwater wireless connectivity solutions [1]. Acoustic communications have been a common choice for USNs and they can support transmission distances of up to tens of kilometers, albeit at low data rates on the order of kbps. Underwater visible light communication (UVLC) has been proposed as a complementary connectivity solution with data rates in the order of tens of Mbps [2]. As light propagates through water, it suffers from significant attenuation, especially for ultra-The associate editor coordinating the review of this manuscript and approving it for publication was Joewono Widjaja .
violet and infrared wavelengths [3]. Blue-green part of the visible light spectrum is the best wavelength for underwater transmission. While the green part of the spectrum has less attenuation in coastal water, the blue part of the spectrum is more favorable in the open ocean [4].
There has been a growing literature on UVLC where blue or green colored lasers or LEDs are used as wireless transmitters [5]- [29]. Most of these analyses are, however, limited to single user and point-to-point links. Yet, practical implementation of USNs requires the design of multiple access systems for supporting several sensor nodes. Motivated by this, some multiple access schemes for UVLC systems have been further proposed [30]- [39]. For example, in [30], an orthogonal frequency division multiplexing (OFDM)-based multiuser multiple-input multiple-output (MU-MIMO) system was investigated in the context of UVLC. They considered VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ linear pre-coding to serve multiple users at a time. In [32], a shot noise limited interleaver iterative non-orthogonal multiple access (NOMA) UVLC system based on a photon counting receiver is proposed. They have applied repetition coding in order to improve the performance. An orthogonal frequency division multiple access (OFDMA) UVLC system was proposed in [33] where the overall data rate is maximized subject to two constraints, namely the bit error rate (BER) for each user does not exceed the predetermined maximum BER as well as per user data rate is identical for all users.
In [36], experimental validation of NOMA UVLC with blue Laser source has been demonstrated, where the sum rate of 4.686 Gbps has been achieved for two users. Another experimental validation of NOMA Underwater VLC has been demonstrated in [37]. They have compared experimentally the error rate performance of the NOMA system in air and underwater environments and demonstrated that the NOMA system works better in underwater environment. 1 Unlike OFDMA where nodes are allocated orthogonal resources in frequency domain, in NOMA schemes, multiple users simultaneously share the entire available frequency and time resources, with a controlled interference threshold, leading to low latency and significant gains in spectral efficiency [42]. NOMA controlled interference can be realized either in power or code domains. In power-domain NOMA, users are assigned different power levels while, in the code-domain NOMA, multiplexing can be carried out using spreading sequences, which is similar to code division multiple access (CDMA) technology. Generally, power allocation in the commonly adopted NOMA systems is determined based on the channels condition between the transmitter and the receiving nodes. In specific, nodes with strong channel gains are allocated lower power coefficients, while high power coefficients are assigned to nodes with weak channel gain. For reliable data detection, each node in the USN performs successive interference cancellation (SIC) to reduce the multi-user interference. In particular, nodes successively cancel out the interference from those signals with higher power, and then decode their own signals.
In [31], the capacity of multiuser power domain NOMA was investigated numerically over lognormal fading channel, which is typically valid for weak turbulence conditions as experimentally demonstrated in [10], [43]. In [34], the authors provided a numerical evaluation for the capacity and outage probability of power domain NOMA-enabled UVLC system over lognormal fading channels and showed that NOMA outperforms OFDMA in terms of achievable capacity. The authors in [35] investigated the error rate performance and the achievable capacity of power domain NOMA-based UVLC system over Exponential-Generalized Gamma (EGG) distribution, which is valid for turbulence in the presence of air bubbles.
To the best of our knowledge, power domain NOMA for UVLC over lognormal turbulence channels was only addressed in [31], [34]. In [31], the effect of turbulence strength on the channel capacity in addition to the effect of targeted rate and number of nodes on coverage probability have been investigated. In [34], the effect of power allocation coefficient on the achievable capacity and coverage probability has been studied. Note that the above discussed contributions are limited to numerical evaluation, and lack the solid theoretical foundation and derivations.
Motivated by the earlier discussion, the main contributions of this paper are summarized as follows: 1) We derive a novel closed-form expression for the capacity of VLC NOMA system over a lognormal turbulence channel. 2) In an effort to gain more insight into the capacity performance of the underlying system model, we derive asymptotic closed-form capacity expressions, and demonstrate that the asymptotic capacity of the closest node to the clusterhead is the only node that is affected by underwater optical turbulence, i.e., the turbulence is encountered only in the asymptotic capacity of the first node. On the other hand, the asymptotic capacity of other nodes is controlled by the considered power allocation coefficient scheme. 3) To corroborate the efficiency of NOMA in underwater VLC systems, we present the achievable capacity of OFDMA scheme. 4) To further demonstrate the efficiency of NOMA system, we compare the achievable capacity in the presence and absence of underwater optical turbulence. 5) We present numerical and simulation results, with the aim to verify the derived mathematical framework and quantify the system performance under different scenarios, including different power allocation schemes, distances, and water types. The remainder of the paper is organized as follows: In Section II, we present the considered system and channel models. In Section III, we present closed-form and asymptotic expressions for the system capacity over lognormal fading channels. In Section IV, we present the numerical results and finally conclude in Section V.

II. SYSTEM AND CHANNEL MODELS
We consider a NOMA-enabled UVLC where a clusterhead communicates with K underwater sensor nodes, as depicted in Fig. 1. Each node is assumed to be at a distance d k , k = 1, 2, . . . , K , from the clusterhead. Let P T , x k and α k denote the available power budget shared by all nodes, the transmitted unipolar signal to the k th node, and power allocation coefficient of the k th node, respectively. Therefore, the allocated power to the k th node is given by P k = α k P T . Power allocation coefficients are subject to the constraint of K k=1 α k = 1. It should be noted that splitting the transmit power with proper coefficients among nodes allows the practical realization of successive interference cancellation (SIC), i.e., it increases the ability of the receiver to eliminate the effect of high power signals and perform reliable signal detection. In other words, the receiver can successively decode the higher power signals and consider all other signals as interference, subtract the decoded signal from the superimposed signal and finally detect the intended signal.
Mathematically, the transmitted signal from the clusterhead is given by X = K k=1 √ P k x k . The received signal at the k th node is given by where η is the laser diode's electro-optical conversion efficiency and r is the photodetector's opto-electrical responsivity. w k is the additive white Gaussian noise (AWGN) term at the k th receiver, k = 1, 2, . . . , K . It has zero mean and a variance of σ 2 n = N 0 W where N 0 is the noise power spectral density and W is the system bandwidth. In (1), I k , k = 1, 2, . . . , K are random variables characterizing the fading induced by the underwater turbulence, whereas h k , k = 1, 2, . . . , K are deterministic terms that characterize the attenuation loss between the clusterhead and the k th node, k = 1, 2, . . . , K .
Under the assumption of semi-collimated laser sources with Gaussian beam shape, h k is expressed as [11] where θ F and D R denote full-width transmitter beam divergence angle and receiver aperture diameter, respectively. In (2), ρ and c represent correction and extinction coefficients, which are both dependent on water type [11]. Under the assumption of weak turbulence, the probability density function (PDF) of the fading coefficient is given by [6] f I k (I k ) = 1 where µ x k and σ 2 x k denote the mean and variance of the logamplitude coefficient, respectively. To ensure that the fading coefficient does not change the value of average power, the fading amplitude is normalized such that E [I k ] = 1, which implies µ x k = −σ 2 x k [44]. The variance can be written in terms of the scintillation index (σ 2 I k ) as σ 2 x k = 0.25 ln σ 2 I k + 1 . Scintillation index for laser sources with Gaussian beam shape can be calculated by [45,Eq. (7)] and [46,Eq. (16)].
Power allocation in NOMA systems is determined based on the channels condition between the transmitter and the receiving nodes. Specifically, nodes with strong channel gains (i.e., closest nodes to the clusterhead) are allocated lower power coefficients, while high power coefficients are assigned to nodes with weak channel gain. Without loss of generality, we assume that channel gains are ordered in a descending form, i.e., h 1 > h 2 > · · · > h K , hence the power allocation coefficients are then given as α K > · · · > α 2 > α 1 . Since the highest transmit power is assigned to the K th node, this node does not perform SIC. On the other hand, since the least transmit power is assigned to the first node, this node will decode the data of K − 1 nodes before decoding its own signal. The operation at the k th SIC receiver can be described as follows. The receiver first detects/decodes the signal sent for the furthest point (i.e., K th node). After detection/decoding it, k th SIC receiver cancels the interference contributed by the K th node. k th SIC receiver repeats this processes, respectively, for (K-1) th node, (K-2) th node, . . ., (k+1) th node. k th SIC receiver will finally decode/detect its own signal after canceling the effect of k+1, . . ., K nodes.

III. SUM CAPACITY ANALYSIS
The sum capacity of NOMA can be written as R T = K k=1 R k . Due to the fact that the classical Shannon's equation does not work for optical systems, the exact capacity is still unknown for optical channel. Consequently, different bounds on capacity of optical channels were VOLUME 9, 2021 derived [47]- [50]. Based on [50], the gap between the exact and lower bound on channel capacity can be efficiently neglected for high SNR and then the tight lower bound on capacity and assuming perfect SIC, the conditional capacity of the k th node (conditioned on the fading coefficients I k , k = 1, 2, . . . , K ) can be written as 2 where γ k I 2 k is given as The average capacity can be calculated by averaging (4) over the PDF of the turbulence in (3) as (6), as shown at the bottom of the page. To solve the integral in (6), we apply a variable change x = ln (I k ). It can be noticed that x follows the normal distribution with mean 2µ x k and variance 4σ 2 x k . Therefore, (6) can be expressed as where E [·] denotes the expectation operator. Here, g (x) and f x (x) are defined, respectively, as 2 Note: In the absence of underwater optical turbulence, the SNR for the k th node is deterministic and found as Therefore, the capacity of the k th node, in the absence of turbulence, can be calculated by (4) after replacing γ k I 2 k by γ k .
Utilizing Holtzmann's Gaussian approximation [51], (7) can be approximated as weighted sum of g (x) by replacing (10), as shown at the bottom of the page.
In an effort to have further insight into the capacity, we pursue asymptotic analysis in the following. Applying the approximation E log 2 [52,Eq. (35)], we can write (7) as It can be readily verified that E exp (2x) = exp 8σ 2 x k + 4µ x k . Replacing this within (11), we have Noting that the inner argument of the exponential function in (12) is extremely small (i.e., 8σ 2 x k + 4µ x k 1 ). Therefore, exp 8σ 2 x k + 4µ x k can be efficiently replaced by 1. This implies that the capacity of NOMA system in the presence of underwater optical turbulence is almost close to the capacity of NOMA system in the absence of underwater optical turbulence, i.e., R k ≈ W log 2 In the following, we consider asymptotically high transmit power, i.e., P T → ∞. It can be readily found that (12) reduces to (13), as shown at the bottom of the page.
The asymptotic capacity expression in (13) suggests that as P T approaches ∞, the effect of turbulence is encountered only in the first node, which was demonstrated to have a low impact on the NOMA capacity. The asymptotic capacity of other nodes is determined by the power allocation coefficients. Therefore, it can be concluded that the system's asymptotic capacity in the presence of underwater optical turbulence is close to the asymptotic NOMA capacity in the absence of turbulence. In other words, despite the continuous changes in turbulence strength, the NOMA capacity remains almost constant. This further motivates the employment of NOMA as an efficient scheme for multi-access in underwater environments.

IV. NUMERICAL RESULTS
In this section, we present the capacity of wireless nodes in NOMA UVLC system. We also validate our derived closedform expression in (10), closed-form approximate capacity expression in (12) and asymptotic capacity expression in (13). Unless mentioned otherwise, we consider a two sensor nodes scenario, K = 2, which is commonly adopted in the literature in order to ensure limited inter-user interference, e.g., [53]- [56]. We further consider receiver aperture diameter of D R = 5 cm, full-width transmitter beam divergence angle of θ F = 6 • , noise power spectral density of N 0 = 10 −22 W/Hz, a bandwidth of W = 200 MHz and total transmit power of P T = 1 W. Assuming clear ocean and coastal water, the extinction and correction coefficients are given, respectively, as (c , ρ) = (0.15 , 0.05) and (0.305 , 0.13) [11]. Electro-optical efficiency of η = 0.5 W/A and opto-electrical responsivity of r = 0.28 A/W are considered. We further calculate the scintillation index (σ 2 I ) 3 based on [45, Eq. (7)] in conjunction with [46,Eq. (16)] assuming salinity of 35 PPT and temperature of 20 • C.
In Fig. 2, we present the capacity of NOMA and discuss the effect of power allocation coefficient considering Since d 1 is assumed to be shorter than d 2 , we consider the range of d 1 < d T /2. Since the received power at k th node is proportional to h 2 k and given the fact that power allocation coefficient is performed such that low transmits powers are assigned to the nodes with less severe channels and vice versa, we consider α k ∝ 1/h 2 k , k = 1, 2, . . . , K . Utilizing this and K k=1 α k = 1, we consider in simulation α k = 1/h 2 k / K i=1 1/h 2 i , i.e., for two-node case α 1 = h 2 2 / h 2 1 + h 2 2 and α 2 = h 2 1 / h 2 1 + h 2 2 . In order to demonstrate the efficiency of NOMA in underwater VLC networks, the sum capacity of NOMA in the absence of underwater optical turbulence is further included as a benchmark.
It can be observed from Fig. 2 that the derived expression in (10) provides excellent match to the analytical calculation based on (6). It can be noted that when α 1 = 0, i.e., no power is allocated to the first node, its achievable capacity is zero. In this case, the second node achieves capacity of R 2 = 9.57 bps/Hz and R 2 = 5.95 bps/Hz for clear ocean and coastal water, respectively. This also indicates that the highest capacity for the second node in clear ocean is higher than its highest capacity in coastal water. This is due to the fact that the extinction coefficient of coastal water is higher than that of clear ocean, which results in weaker channel coefficient. It can be further observed that, when α 1 3 For computing σ 2 I , we assume a dissipation rate of mean-squared temperature of 1 × 10 −3 K 2 s -3 , a dissipation rate of turbulent kinetic energy per unit mass of fluid of 1 × 10 −2 m 2 s -3 , relative strength of temperature, salinity fluctuation of ω = −3 and wavelength of λ = 530 nm in [46]. increases (i.e., α 2 = 1−α 1 decreases), the capacity of the first node increases and the capacity of the second node decreases. It is also noted that the capacity of the first node (i.e., R 1 ) approaches the NOMA overall capacity while the capacity of the second node (i.e., R 2 ) approaches zero as α 1 increases.
On the other hand, the highest sum capacity is achieved for clear ocean when It can be further observed that the capacity of the NOMA system in the absence of underwater optical turbulence is slightly higher than the NOMA capacity in the presence of turbulence. This gives great importance to the deployment of NOMA system in an underwater environment. For example: While the highest sum capacity is achieved for clear ocean as R T = 23.28 bps/Hz in the presence of under water optical turbulence, it is obtained as R T = 23.49 bps/Hz in the absence of turbulence. Moreover, the distribution of sensor nodes for achieving the best performance (i.e., highest capacity) of the NOMA system in the presence of under water optical turbulence and in the absence of it is almost the same if we consider the same water type. For example, if we consider clear ocean, the highest capacity that can be obtained in the presence of turbulence is achieved when the first node is placed at a distance of d 1 = 3.43 m and the second node at a distance of d 2 = 16.57 m meters. The distances for achieving the highest performance in the same water type and in the absence of turbulence are found as d 1 = 3.37 m and d 2 = 16.63 m. This in turn suggests that it is possible to use the NOMA system in an underwater environment by choosing fixed locations for user nodes, as long as the system is used in the same water type.
In order to demonstrate the superiority of NOMA underwater VLC communication, we consider two sensor nodes in a clear ocean and compare the capacity of NOMA system with OFDMA counterpart considering d T = d 1 + d 2 = 20 m. It can be seen from Fig. 3 that, in most cases, the capacity of the NOMA system exceeds that of the OFDMA system. The capacity of the OFDMA system slightly exceeds the capacity of the NOMA system in cases where the user nodes are very close to each other. This is due to the high interference experienced in NOMA systems as users have almost the same channel coefficient and accordingly, close power coefficients are assigned to all nodes. It can be further observed that, while the highest capacity of NOMA system is obtained as R T = 23.28 bps/Hz when d 1 = 3.43 m and d 2 = 16.57 m, the highest capacity of OFDMA system is obtained as R T = 17.31 bps/Hz when d 1 = d 2 = 10 m.
In the following, we study the effect of distance difference on NOMA capacity considering coastal water. We assume that d T = d 1 +d 2 is fixed and consider two cases: d T = 15 m and d T = 20 m in Figs. 4.a and 4.b, respectively. It can be observed from Fig. 4 that, generally, the overall capacity increases as distance difference increases (i.e., d 2 − d 1 increases). This is because the power allocation coefficient for the farthest point is much larger than the power allocation coefficient for the nearest point (i.e., α 2 α 1 ). Therefore, node 1 has a good capacity due to small propagation distance and node 2 has a good capacity due to very small interference from node 1. On the contrary, we observe a decrease in the overall capacity as d 1 increases, d 2 decreases and hence d 2 − d 1 decreases. This is due to the fact that the closer the nodes to each other the higher the interference.
The general observation of that the overall capacity increases as distance difference increases does not always hold as it can be observed from Figs. 4.a and 4.b that the maximum capacity is obtained when d 1 = 2.59 m and 5.07 m, respectively, for d 1 + d 2 = 15 m and 20 m. This is due to the fact that while assigning most of the power to the farthest point and very low amount of available power to the nearest point, the received power levels at both nodes are low. The nearest node to clusterhead receives a low power signal due to assigning low power to it. At the same time, the furthest point receives a low signal power due to very weak channel coefficient.
It can also be observed in Fig. 4 that, although the transmit power is just 1 W, the asymptotic capacity expression in (12) which is derived assuming high transmit power (i.e., high SNR) provides good match with the exact capacity expression for some ranges of distances. These expressions deviate from the actual capacity for smaller d 1 . because both nodes will have a low SNR. The first node will have low SNR since most of the power is assigned to the farthest node and the second node will have a low SNR due to very low channel gain that is strongly decaying with propagation distance. It can also be observed that the capacity of the first node increases with increasing d 1 , This is due to the considered power allocation strategy. In other words, P 1 η 2 r 2 h 2 1 increases with distance. In the following, we investigate the effect of transceiver parameters on NOMA capacity considering coastal water and d T = 20 m. Particularly, the effect of receiver aperture diameters and transmitter beam divergence angle on the achievable NOMA capacity have been investigated, respectively, in Figs. 5.a and 5.b.
For investigating the effect of receiver aperture diameters, D R = 5 cm, 10 cm and 20 cm are assumed. The corresponding extinction coefficients are ρ = 0.13, 0.16 and 0.21, respectively [11, Table 1]. It is observed from Fig. 5.a that as receiver aperture size increases, the achievable NOMA capacity increases since larger receiver aperture sizes collect more energy. Furthermore, the larger aperture size, the less turbulence strength, yielding higher SNR. For example, at distance of d 1 = 5 m, aperture diameter of 10 cm and 20 cm provides, respectively, 25.33% and 45.22% improvement over aperture diameters of 5 cm. The important point is how aperture diameters affect on the optimal distance where maximum sum capacity is achieved. It can be observed from Fig. 5.a that the maximum sum capacity is obtained for D R = 5 cm, 10 cm and 20 cm, respectively, at distance of d 1 = 5.07 m, 3.80 m, and = 2.63 m. Indicating that, the smaller the aperture diameter, the closer the nodes to each other which are found as d 2 − d 1 = 9.86 m, 12.4 m and 14.74 m, respectively. This is due to the need to reduce the distance of the second node from the clusterhead (i.e., d 2 ) when the receiving area is small in order to be able to receive a suitable power, given that the power received on the small area in the large distance is very little, and therefore the second point will have a very small SNR.
For investigating the effect of transmitter's beam divergence angle, θ F = 6 • , 12 • and 18 • are assumed. The corresponding extinction coefficients are ρ = 0.21, 0.24 and 0.25, respectively [11, Table 1]. It can be observed from Fig. 5.b that as divergence angle increases, the achievable NOMA capacity decreases. This is due to the fact that the focused beam (i.e., more collimated angle) experiences less attenuation and less geometric loss through the propagation medium. For example, the achievable NOMA capacity in coastal water at distance of d 1 = 5 m and beam divergence angle of θ F = 6 • provides 8.83% and 23.57% higher capacity over beam divergence angle of θ F = 12 • and θ F = 18 • , respectively. The important point is how transmitter's beam divergence angles affect on the optimal distance where maximum sum capacity is achieved. It can be observed from Fig. 5. b that the maximum sum capacity is obtained for θ F = 6 • , 12 • and 18 • , respectively, at distance of d 1 = 2.63 m, 2.69 m, and = 2.87 m. Indicating that, the larger the transmitter's beam divergence angles, the closer the nodes to each other which are found as d 2 −d 1 = 14.74 m, 14.62 m and 14.26 m, respectively. Unlike the effect of aperture diameter on the optimal locations of nodes, the effect of transmitter's beam divergence angles is quite negligible.
In all of the above results, we considered the scenario of two sensor nodes. In the following, we study the presence of a number of sensor nodes on the NOMA sum capacity assuming coastal water and d T = K k=1 d k = 20 m. Particularly, we have tabulated the maximum sum capacity that can be achieved in each case, as well as the distances of sensor nodes from the clusterhead and power allocation coefficients  in Table 1. It can be seen that the sum capacity increases as the number of sensor nodes increases. It may be intuitive that increasing the number of sensor nodes increases the capacity. But the capacity does not increase if a new sensor node is added randomly. Rather, all sensor nodes must be redistributed with new distances from the clusterhead and with new power allocation coefficients. Otherwise, adding a new sensor node without re-allocating the old sensor nodes may increase the interference and hence decrease the sum capacity. In other words, adding additional sensor node requires changing the location of other sensor nodes and accordingly the power allocation coefficients. For example, while the maximum sum capacity for the case of two sensor nodes (i.e., K = 2) is obtained for d 1 = 5.048 m and d 2 = 14.916 m, the maximum sum capacity for four sensor nodes is achieved when d 1 = 0.0226 m, d 2 = 0.9575 m, d 3 = 9.1513 m and d 4 = 9.8686 m.
In the following, we assume coastal water and d T = K k=1 d k = 20 m and compare the highest sum capacity of NOMA system with two hypothetical cases, namely when the sensor nodes are separated from each other by the same distance (i.e., d k+1 − d k = Constant, k = 1, . . . , K ) and when all sensor nodes are equidistant from the clusterhead (i.e., d i = d k , k = 1, . . . , K and, i = 1, . . . , K ).
It is noted that, regardless of the number of sensor nodes, the achieved sum capacity in the optimal placement of sensor nodes, given in Table 1, is the highest, while the lowest sum capacity is experienced when all nodes are equidistant from the clusterhead. This is due to the fact that, when sensor nodes are separated from each other by the same distance, the distances of the sensor nodes from the clusterhead are not the same; therefore it can achieve higher sum capacity than second hypothetical case, in which sensor nodes have the same distance from the clusterhead. For example, assume the presence of 5 sensor nodes, the sum capacity of the optimal case is evaluated to 55.02 bps/Hz. This value drops to 27.76 bps/Hz and 22.51 bps/Hz, respectively, for case I in which the sensor nodes are separated from each other by the same distance and case II in which all sensor nodes are equidistant from the clusterhead.

V. CONCLUSION
In this paper, we considered NOMA in the context of underwater sensor networks. To this end, we derived a simple closed-form expression for the NOMA capacity as well as a simple and accurate asymptotic representation. The derived closed-form expressions are in agreement with the corresponding analytic results, while they are insightful and easy to compute.
Analyzing the performance of the considered underwater network set up, we considered various scenarios where we studied the effect of different link distances on the overall NOMA capacity. We observed that the overall capacity is severely worsening with propagation distance which is due to the fact that the VLC channel gain severely drops with propagation distance. Therefore, the considered configuration can exhibit particularly high rates in the context of a local area network. We also compared the performance of the NOMA system with and without the presence of underwater optical disturbance. The results, as well as the theoretical study, proved that the NOMA system exhibits robust capacity performance in the presence and absence of an optical turbulence. This is because the capacity of most nodes depends mainly on the power allocation coefficient.
We further investigated the effect of system parameters on the optimal N nodes' distances, where maximum sum capacity is achieved. It is observed that while aperture diameters hugely influence the optimal distances where maximum sum capacity is achieved, the transmitter's beam divergence angle has almost negligible effect on the optimal nodes' distances. Additionally, one of the valuable observations that must be taken into account is that, for asymptotic analysis, the turbulence is encountered only in the first node and does not affect other nodes. The asymptotic capacity of other nodes is controlled by the power allocation coefficients.

APPENDIX
In order to demonstrate the superiority of NOMA system for achieving the highest capacity as a multiple access technique in an underwater environment, as benchmark, we need to compare it with the well-known multiple access technique of OFDMA. Since in OFDMA rather than sharing the whole bandwidth with all users/sensor nodes similar to the NOMA system, part of the bandwidth is assigned to each user (i.e., users are assigned to a group of subcarriers), we assume that bandwidth is shared among all users equally. For fair comparison with NOMA system counterpart, the same power allocation coefficient is assumed and hence the conditional capacity of the k th , k = 1, 2, . . . , K node in OFDMA system can be written as where W k is the assigned bandwidth for the k th user and given as W k = W /k. In (A1), γ k OFDMA I 2 k is given as Similar to the average capacity of NOMA system over lognormal turbulence channel, Holtzmann's Gaussian approximation in [51] is used. The average capacity per user is then found as (A3), as shown at the top of the page. He has several practical experiences serving as a Designing and Supervising Electrical Engineer with the Engineering Office and Development Department, Ministry of Health, Gaza, Palestine, for four years. His current research interests include visible light communications, optical turbulence, underwater acoustic communication, diversity techniques for fading channels, performance analysis over fading channels and time-varying channels, channel estimation and equalization, multi-input multi-output (MIMO) communications. On these topics, he has authored more than 28 publications (journals and conferences) and received more than 300 Google Scholar citations with an H-index of 10  He received the Best Paper Award at ICUFN 2013. He serves as an Editor for the IEEE COMMUNICATIONS LETTERS. He serves as a regular reviewer for several international journals and has been a member of the technical program committee of numerous IEEE conferences. VOLUME 9, 2021