# Selection of Optimal Deployment and Routing Configurations in Underwater Acoustic Sensor Networks for Avoiding Energy Holes

Authors: Fatma Bouabdallah
Journal: Mobile Information Systems
 (2019)

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## Abstract

Improving the energy efficiency of underwater acoustic sensor networks (UW-ASNs) is a crucial issue due to the reduced and nonrechargeable energy resource of the underwater sensor nodes. In this work, we address the energy sink hole problem in UW-ASNs while considering the unique and harsh characteristics of the underwater channel. Our goal is to determine the optimal deployment and routing settings that surmount the energy sink hole problem and hence maximize the network lifetime. We prove that sensors can evenly consume their initial battery power provided that first they adjust their transmission power when they transmit the route through traffic and second they are appropriately placed while deployed. Mainly, we propose a deployment scheme and the corresponding balanced routing strategy that lead to uniform energy consumption among all underwater sensors subject to a predefined reliability level at the sink. Specifically, we look for the optimal deployment settings especially in terms of nodes’ separation distances that help achieving uniform energy consumption in the network while satisfying the application requirement especially in terms of desired information reliability. Jointly, at the routing layer, we assume that each sensor is provided with the possibility of dynamically adjusting its transmission power up to a given number of levelsN. To this goal, we mainly deal with two main cases: fixed and variable nodes separation distance. For the fixed case, we suppose that any two successive nodes in the network are equally spaced, and we strive for deriving the optimal distance as well as the optimal number of transmission power levels along with optimal load weight corresponding to every possible transmission power level for every sensor node. For the variable case, we deal with two subcases: first, we suppose that the distance separating successive nodes follows an arithmetic progression, and second, we assume that the distance separating successive nodes is following a geometric sequence. Note that for both cases, namely, fixed and variable, we succeed to determine the optimal distances separating successive nodes and the optimal number N of transmission power levels along with the corresponding optimal load weight that overcome the energy holes problem, and hence the network lifespan is maximized while respecting the desired reliability level.


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## Body

## 1. Introduction

Underwater sensor networks are attracting an increasing interest of the research community since they enable a broad range of applications in various domains such as military, environmental, and scientific [1]. Acoustic communication is the most appropriate technology for these applications.An underwater channel requires exclusive and tough characteristics. Indeed, the channel’s bandwidth is limited, the attenuation is frequency-dependent and significant, the propagation delays are high and variable, and the battery power is limited [2]. For these reasons, proposing network protocols specifically dedicated for the underwater environment confronts serious challenges. Moreover, the initially provided battery power for underwater sensors is limited. Even worse, the battery cannot be recharged, since it is impossible to use solar energy. In addition to that, replacing flat sensors risks to cost too much since the underwater sensors deployment is quite difficult. Even worse, acoustic underwater communications use much more energy than the terrestrial sensor networks using radio communications. Indeed, the deployment is quite sparse in underwater environment. This makes the underwater communication involve transmission over long distances. Moreover, in order to surmount the impairments of the underwater channel, more developed signal-processing solutions are needed at receivers.For the above reasons, UW-ASNs need protocols that pertinently consume the limited initial battery power of the underwater sensor nodes. To this end, load balancing is the most efficient yet delicate technique that prolongs the UW-ASN lifetime. Its ultimate objective is to guarantee to the most possible extent the equitable and soft energy consumption among all the sensors. Note that, wireless sensor networks research community showed that the nearest sensors to the sink suffer from a harsh depletion of their initially provided battery power [3–7], as these sensors are relays of all the other sensors in the network. This may result in energy holes which cause drastic reduction of the network lifespan. Consequently, balancing the energy utilization in the network, by evenly distributing the traffic load, is the most promising energy efficient technique.In this paper, we address the energy sink holes problem in UW-ASNs while considering the unique and characteristics of the underwater channel. Our goal is to jointly determine the optimal deployment and routing settings that surmount the energy hole problem and hence maximize the lifetime of UW-ASNs. To this aim, we strive for determining the optimal distances separating subsequent sensor nodes during deployment that helps achieving uniform energy expenditure among all sensor nodes while satisfying the desired information reliability. Jointly, we propose variable transmission range-based data forwarding task. Accordingly, each underwater sensor is allowed to dynamically adjust its transmission range (or interchangeably transmission power) among several possible levels (N). Indeed, the number of transmission power levels (N) is a key factor influencing the energy consumption. First, increasing N allows the network traffic load distribution to be more balanced giving a more uniform energy utilization of the network’s sensors. Second, adopting a large value for N augments the transmission energy utilization since the farthest nodes may be reached. Therefore, the number N has an optimal value for which the energy consumption is optimized, and hence the network lifetime is maximized. In this study, we propose a joint optimization problem that aims at deriving the optimal deployment configuration along with the optimal value of the number N’s of transmission levels and their corresponding optimal load weights for each sensor in the network that surmounts the sink hole problem, and hence the network energy efficiency is maximized. As a recap, our work is a three-fold optimization study where the optimal distances between subsequent sensors and the optimal value of power levels N along with the optimal load weights are jointly derived.A recap of our contributions can be stated as follows. First, we propose a study case deployment pattern for UW-ASN for which we aim at deriving the optimal settings that surmount the energy hole problem. More precisely, considering this proposed deployment strategy, our goal is to determine the optimal distances separating subsequent sensors that help achieving uniform energy consumption through the network while satisfying the reliability requirement at the sink. Second, at the routing layer, we suppose that each sensor node is capable to dynamically tune its transmission power up to a predefined numberN of levels while sending or forwarding data. Consequently, we determine the optimal value of N, with the corresponding optimal load weights that balance the energy depletion among the network’s sensors. Therefore, the network lifetime is maximized since the energy hole problem is solved. To summarize, we endeavor to determine the optimal deployment configuration along with the optimal value of N and their corresponding optimal load weight distribution that balance the energy depletion among sensor nodes. Our joint optimization solution is specifically tailored for the underwater environment as it considers the intrinsic features of the underwater acoustic propagation. More precisely, in this work, we adopt the underwater channel model proposed by Qarabaqi and Stojanovic in [8] and implemented in [9] which is considered in the literature as one of the most recent and pertinent mathematical analyses that meticulously describes the underwater acoustic time-varying channel model.The rest of the paper is organized as follows. Section2 summarizes work related to the focus of this paper. Section 3 describes the network and channel models under study. Section 4 formally states the general energy-balancing optimization problem. Different study cases are addressed in Section 5. Numerical results are provided in Section 6 for all proposed study cases. Section 7 concludes this paper.


## 2. Related Work

Underwater acoustic networks are gaining a noteworthy interest within the research community. The unique and severe features of the underwater channel usually impose the design of dedicated strategies. Pompili et al. [10] and Climent et al. [11] present a state-of-the-art networking protocols for underwater networks. While extensive routing protocols have been conducted up to date [12, 13], much less work has been dedicated to surmount the energy hole problem, and thus there is still much room for innovation [14]. Note however that the energy hole problem has been recently investigated in industrial WSNs and the Internet of things [15–19] but rather from a coverage point of view. Indeed, authors in [15, 18, 19] focus on how to detect and localize coverage holes while taking into consideration the network energy efficiency. However, studies proposed in [16, 17] strive to overcome the coverage hole problem using mobile sensor nodes. More precisely, they aim at optimally selecting a subset of the randomly deployed mobile sensors to overcome the coverage holes. It is worth pointing out that authors in [17] solve the coverage holes problem while taking into consideration also the network connectivity. Although the aforementioned approaches may be inspiring, some fundamental differences with our objective and context impose the design of a new approach. Indeed, in this work, rather than dealing with coverage holes, we are interested in overcoming the energy sink hole problem that may lead to sink isolation and hence network partition. Moreover, in this paper, instead of using mobile sensors to heal the energy sink hole problem, we opt for evenly distributing the traffic load inside the network while equipping the sensors with multiple transmission power levels in order to alleviate the forwarding task of the one-hop away sensors, and hence the energy sink hole problem is surmounted.In this section, we summarize work mainly related to energy-efficient routing and energy hole problem encountered in UW-ASNs.


### 2.1. Energy Efficient Routing in UW-ASNs

Energy efficiency is a crucial issue in UW-ASNs. Conceiving routing protocols that make wise utilization of the finite and nonrechargeable energy budget of the underwater sensor nodes is a decisive factor for the network lifetime. Many UW-ASNs routing protocols have been proposed to improve the energy efficiency but not from an energy-balancing point of view in order to surmount the energy hole problem which restrains their contributions. In [20], a geographical routing scheme was proposed for underwater acoustic networks and joined with a power-control approach in order to increase the protocol energy efficiency. Indeed, this routing strategy called FBR aims at dynamically establishing routes only on demand while preserving the scarce energy resources of the underwater sensors. In fact, by gradually and carefully increasing the transmission power, FBR tries to find the appropriate next hop toward the sink while reducing the total energy consumption. However, for its well functioning, FBR requires from every source node to know its own location as well as one of the final destinations. Another approach would rather rely on depth information in order to reduce energy consumption throughout the network, since depth information is much easier to acquire than location information in UW-ASNs. For instance, in [21], the authors propose a depth-based routing (DBR), which uses depth information to reduce the invalid broadcasts. Indeed, by only allowing nodes with lower depth to forward a packet destined to a surface sink, the number of forwarder is highly decreased, and hence energy saving is achieved. Another depth-based routing protocol called EUROP was introduced in [22]. Thanks to the use of depth sensor, EUROP will eliminate the requirement of hello messages for control purposes which improves the energy efficiency. According to EUROP, underwater nodes are divided into different layers based on the depth information such that only nodes in the same layer (namely, in the same depth) can communicate with each other. The forwarding process is straightforward and simply dictates that data packets are forwarded from deeper layers to shallower layers. In addition to that, it is assumed that nodes can move to the upper layer and back to their predefined place for successful delivery to the surface sink. Although the use of the depth sensor will completely eliminate the need for control packets which may reduce the energy consumption, the use of these costly sensors may compromise the total energy consumption as these sensors consume a large amount of energy to move from one depth to another.


### 2.2. Sink Hole Solutions in UW-ASNs

One of the early studies on the sink hole problem in UW-ASNs was conducted by Chen et al. [23], who proposed an energy-efficient routing protocol, called REBAR, to surmount the sink hole problem. REBAR protocol supposes that by default, every generated data packet has to undergo a flooding process in order to reach the sink. Based on this, REBAR tries to highly reduce the flooding region for each source node by most importantly involving much less number of the sink one-hop away sensors while guaranteeing a high delivery rate. Therefore, the closest nodes to the sink are much less solicited especially compared to the integral flooding process. Consequently, the network lifetime is augmented. It is completely true that, thanks to REBAR protocol, the closest sensors to the sink are much less involved in the routing process especially compared to the complete flooding process, but they keep acting as relays on behalf of all other sensors, and hence keep suffering from severe energy depletion. Indeed, as it was shown by terrestrial wireless sensor networks research community, the energy sink hole problem is inevitable in static always-on sensor networks where sensors perform continuous monitoring of a given field using a nominal communication range [5, 24–28]. Consequently, using adjustable communication power was the most undertaken research strategy to make the energy consumption more uniform among the network’s sensors. Indeed, by endowing each sensor with the ability of dynamically adjusting its transmission power, the traffic load distribution among sensors is much more balanced, and thus sensors, which are in neighborhood of the sink, are eased from the relying task. In this perspective, authors in [29] have proposed a routing strategy assuming that every sensor node has two transmission ranges: the smallest one used to reach the next hop in a linear topology and the farthest one used to directly reach the sink. According to EBH, each sensor node has to alternate between the two possible transmission ranges based on the residual nodes’ energy such that the network lifetime is optimized. Indeed, accordingly, a sensor node has to keep sending to its upstream neighbor in the linear topology as long as its own residual energy is greater than the one of its neighbor. Otherwise, the node has to proceed sending directly to the sink node.Another possible approach to surmount the sink hole problem is by using a mobile sink node such that the set of the closest node to the sink is constantly changing. In this direction, most of the research studies, especially in WSNs, will rather focus on finding the optimal sink trajectory to maximize the network lifetime. Similarly, authors in [30] consider a UW-ASN context where an autonomous underwater vehicle (AUV) is acting as a data collector and hence try to find the best trajectory by taking into consideration the underwater environment constraints. In the same way, another recent energy-efficient routing protocol proposed in [31] is MobiCast. This protocol also aims at maximizing data collection while overcoming the energy hole problem by the use of mobile AUVs acting as a data collector.Authors in [32, 33] propose to surmount the energy sink hole problem by using multiple transmission power levels rather than the use of mobile sink. More precisely, and as a distinguishing feature, each sensor node is endowed with multiple transmission power levels in order to optimally distribute the traffic load through the network, and hence balanced uniform energy depletion is achieved among all sensors including the closest nodes to the sink. Authors in [33] strive for analytically deriving for each sensor node the optimal number N of transmission power levels as well as the optimal load weight corresponding to every possible transmission power such that uniform energy consumption is achieved. Indeed, in order to derive their optimal configuration settings, authors in [33] adopted the time-varying channel model proposed in [8], which closely reflects most of the underwater channel impairments such as bottom-surface reflections, frequency-dependent attenuation, and Doppler effects which are caused by random local displacements.This paper can be seen as a continuation of [33], where we not only determine the optimal number of transmission levels N along with the corresponding load weights but also we seek the optimal deployment settings in terms of nodes initial positioning. In this study, we adopt the same channel model as in [33] since it is the most realistic one but we rather opt for the numerical resolution of our threefold optimization problem that aims at determining the optimal distance between two subsequent nodes as well as the optimal value of N along with the optimal corresponding load weights.


## 2.1. Energy Efficient Routing in UW-ASNs

Energy efficiency is a crucial issue in UW-ASNs. Conceiving routing protocols that make wise utilization of the finite and nonrechargeable energy budget of the underwater sensor nodes is a decisive factor for the network lifetime. Many UW-ASNs routing protocols have been proposed to improve the energy efficiency but not from an energy-balancing point of view in order to surmount the energy hole problem which restrains their contributions. In [20], a geographical routing scheme was proposed for underwater acoustic networks and joined with a power-control approach in order to increase the protocol energy efficiency. Indeed, this routing strategy called FBR aims at dynamically establishing routes only on demand while preserving the scarce energy resources of the underwater sensors. In fact, by gradually and carefully increasing the transmission power, FBR tries to find the appropriate next hop toward the sink while reducing the total energy consumption. However, for its well functioning, FBR requires from every source node to know its own location as well as one of the final destinations. Another approach would rather rely on depth information in order to reduce energy consumption throughout the network, since depth information is much easier to acquire than location information in UW-ASNs. For instance, in [21], the authors propose a depth-based routing (DBR), which uses depth information to reduce the invalid broadcasts. Indeed, by only allowing nodes with lower depth to forward a packet destined to a surface sink, the number of forwarder is highly decreased, and hence energy saving is achieved. Another depth-based routing protocol called EUROP was introduced in [22]. Thanks to the use of depth sensor, EUROP will eliminate the requirement of hello messages for control purposes which improves the energy efficiency. According to EUROP, underwater nodes are divided into different layers based on the depth information such that only nodes in the same layer (namely, in the same depth) can communicate with each other. The forwarding process is straightforward and simply dictates that data packets are forwarded from deeper layers to shallower layers. In addition to that, it is assumed that nodes can move to the upper layer and back to their predefined place for successful delivery to the surface sink. Although the use of the depth sensor will completely eliminate the need for control packets which may reduce the energy consumption, the use of these costly sensors may compromise the total energy consumption as these sensors consume a large amount of energy to move from one depth to another.


## 2.2. Sink Hole Solutions in UW-ASNs

One of the early studies on the sink hole problem in UW-ASNs was conducted by Chen et al. [23], who proposed an energy-efficient routing protocol, called REBAR, to surmount the sink hole problem. REBAR protocol supposes that by default, every generated data packet has to undergo a flooding process in order to reach the sink. Based on this, REBAR tries to highly reduce the flooding region for each source node by most importantly involving much less number of the sink one-hop away sensors while guaranteeing a high delivery rate. Therefore, the closest nodes to the sink are much less solicited especially compared to the integral flooding process. Consequently, the network lifetime is augmented. It is completely true that, thanks to REBAR protocol, the closest sensors to the sink are much less involved in the routing process especially compared to the complete flooding process, but they keep acting as relays on behalf of all other sensors, and hence keep suffering from severe energy depletion. Indeed, as it was shown by terrestrial wireless sensor networks research community, the energy sink hole problem is inevitable in static always-on sensor networks where sensors perform continuous monitoring of a given field using a nominal communication range [5, 24–28]. Consequently, using adjustable communication power was the most undertaken research strategy to make the energy consumption more uniform among the network’s sensors. Indeed, by endowing each sensor with the ability of dynamically adjusting its transmission power, the traffic load distribution among sensors is much more balanced, and thus sensors, which are in neighborhood of the sink, are eased from the relying task. In this perspective, authors in [29] have proposed a routing strategy assuming that every sensor node has two transmission ranges: the smallest one used to reach the next hop in a linear topology and the farthest one used to directly reach the sink. According to EBH, each sensor node has to alternate between the two possible transmission ranges based on the residual nodes’ energy such that the network lifetime is optimized. Indeed, accordingly, a sensor node has to keep sending to its upstream neighbor in the linear topology as long as its own residual energy is greater than the one of its neighbor. Otherwise, the node has to proceed sending directly to the sink node.Another possible approach to surmount the sink hole problem is by using a mobile sink node such that the set of the closest node to the sink is constantly changing. In this direction, most of the research studies, especially in WSNs, will rather focus on finding the optimal sink trajectory to maximize the network lifetime. Similarly, authors in [30] consider a UW-ASN context where an autonomous underwater vehicle (AUV) is acting as a data collector and hence try to find the best trajectory by taking into consideration the underwater environment constraints. In the same way, another recent energy-efficient routing protocol proposed in [31] is MobiCast. This protocol also aims at maximizing data collection while overcoming the energy hole problem by the use of mobile AUVs acting as a data collector.Authors in [32, 33] propose to surmount the energy sink hole problem by using multiple transmission power levels rather than the use of mobile sink. More precisely, and as a distinguishing feature, each sensor node is endowed with multiple transmission power levels in order to optimally distribute the traffic load through the network, and hence balanced uniform energy depletion is achieved among all sensors including the closest nodes to the sink. Authors in [33] strive for analytically deriving for each sensor node the optimal number N of transmission power levels as well as the optimal load weight corresponding to every possible transmission power such that uniform energy consumption is achieved. Indeed, in order to derive their optimal configuration settings, authors in [33] adopted the time-varying channel model proposed in [8], which closely reflects most of the underwater channel impairments such as bottom-surface reflections, frequency-dependent attenuation, and Doppler effects which are caused by random local displacements.This paper can be seen as a continuation of [33], where we not only determine the optimal number of transmission levels N along with the corresponding load weights but also we seek the optimal deployment settings in terms of nodes initial positioning. In this study, we adopt the same channel model as in [33] since it is the most realistic one but we rather opt for the numerical resolution of our threefold optimization problem that aims at determining the optimal distance between two subsequent nodes as well as the optimal value of N along with the optimal corresponding load weights.


## 3. Network Model and Problem Statement

### 3.1. Recap of the Adopted Time-Varying Underwater Channel

In this study, we use the same underwater channel model adopted by [33] and initially proposed by [8]. Accordingly, the transmitter and the receiver are subject to a uniform displacement around their nominal positions by some heights ΔhT and ΔhR and a distance Δd. Moreover, we similarly suppose that the water surface level can vary by some height Δh due to water waves. In particular, the sensors’ displacements are supposed to vary according to a random uniform distribution, each within the interval ±5m. For instance, assume that two sensor nodes are initially placed at a height 80m and within a distance of 1000m from each other where the water depth equals 100m. We suppose that each sensor will undergo an independent random independent drift from their height by ΔhT=±5m and ΔhR=±5m due to water current and waves. Moreover, we suppose that this pair of node will be subject to random displacement Δd from their nominal locations, where Δd=±5m. Furthermore, we suppose that the water surface that initially equals 100m may randomly vary by Δh=±5m. Figure 1 clearly depicts the different parameters with their corresponding variation ranges.Figure 1
Nominal channel parameters: distance between the transmitter and receiverd=1km; water depth h=100m; transmitter and receiver are at depth 80m, ΔhT=ΔhR=Δh=±5m.As introduced in [33], in this work, we adopt an adaptive power allocation model where the transmit power P^T is calculated according to(1)P^T=KPT0G¯G^,where G¯ is the average of the real channel gain G, G^ is an estimate of the true actual gain G, K is a margin introduced to avoid any gap between the true and the estimated channel gains that may yield interruption, and PT0 denotes the minimum power needed to obtain a desired SNR0 in the absence of fading (G=G¯).In this study, we also adopt the success probability definition as stated in [33]. The success probability Pji over a link i,j denotes the probability that a node j successfully receives a packet transmission started by i using a transmission power P^Tji over a bandwidth B. Pji is computed with the purpose of considering the impact of the underwater acoustic channel time variability on the successful reception of a transmitted packet. In order for this paper to be self-contained, we present next a recap on how to compute Pji. For each link in the network, Pji is numerically determined using the underwater acoustic simulator developed in [33]. Pji simply denotes the ratio of successfully received packets over a link i,j. According to [33], a data packet is considered successfully received if the average reception power during the reception time is greater than a given threshold SNR0. As expected, Pji will affect the total amount of traffic received by each node. Therefore, the total energy expenditure will be impacted. Given that our goal is to closely approach the equitable and smooth energy depletion among sensor nodes in the network while taking into consideration the time variability of the underwater channel, Pji along with P^Tji will inevitably affect the load weights distribution for every sensor in the network as detailed in Sections 4 and 5.


### 3.2. Approach to Overcome the Energy Sink Hole Problem

In this paper, an exhaustive investigation of the energy sink hole problem in UW-ASNs is conducted. Specifically, the closest sensors, which are in the neighborhood of the sink, are excessively solicited in forwarding generated reports to it. Indeed, sensors in the vicinity of a static sink work as relays to it on behalf of all other sensors. Thus, those sensors are traffic hot spots enduring heavy energy depletion. Wadaa et al. [7] revealed that by the time the closest sensors to the sink exhaust their energy budget, the farthest sensors still store up to 93% of their initially provided battery power. Another striking result reveals that the energy hole problem is inevitable in static always-on WSNs where the sensor nodes perform continuous monitoring using their nominal communication range [5, 24–28]. Hence, most of the past works dealing with balancing the energy consumption target adjustable communication range solutions aiming at relieving the sensors, which are in the sink’s neighborhood, from the relaying task.Our approach to surmount the energy sink hole problem encountered in UW-ASNs revolves mainly around two axes: (i) deriving the optimal deployment settings that helps overcoming this problem and (ii) conceiving a multiple transmission power levels data forwarding strategy aiming at balancing the energy consumption among all sensor nodes. By thoroughly investigating the joint feasibility of these two methods, we aim at tightly approaching the perfect equitable energy utilization among all the underwater network’s sensors.The goal of the first method is to pertinently deploy sensor nodes according to a well-studied deployment pattern that helps achieving fair energy utilization through the network while respecting the application reliability. Indeed, we aim at deriving the optimal deployment settings in terms of the optimal number of sensors along with the optimal distance separating subsequent nodes that give uniform energy consumption among all sensors in the network while satisfying the application reliability requirements.The goal of the second method is to allow several subsets of sensors to work as relay runners to the sink such that the set of traffic hot spots is continuously changing over time. Indeed, by adopting a variable transmission powers solution, not only is the total data forwarding load (generated plus received) on each underwater sensor distributed uniformly but also the set of sink forwards will change.To recapitulate, in our work, once the target deployment is obtained and thus the desired information reliability is satisfied, we provide each sensor with the set of possible next hops and their respective load weight that closely approach the perfect equitable energy consumption among underwater sensors.


### 3.3. Network and Energy Model

In this study, we opt for the manual deployment of the underwater acoustic sensor networks for the sake of closely approaching the uniform energy depletion among sensor nodes. Indeed, achieving perfect uniform energy consumption in a completely free-floating underwater sensor network is hard as in such network the objective is rather limited to the successful delivery of data packets to the sink. Seeking to achieve any further objectives especially in terms of optimization rather requires a barely static topology of the underwater sensor network. Thus, we turn to the task of studying the optimal deployment settings in order to closely approach the uniform energy depletion through the network, and hence the problem of the energy sink hole is better solved; thus, the network lifetime is maximized. That being said, conceiving the optimal deployment pattern of the underwater sensors should not be indefinitely sought after since it may break the reliability requirement of the application. For this reason, we aim at designing the optimal deployment settings such that the desired reliability level is satisfied.In this paper, we consider a two-dimensional shallow underwater sensor network. Our deployment strategy states that sensors are anchored to the water bottom and tied with long cables. A floating buoy is used to keep the underwater node at the water surface. Note that in this study, the sensors are assumed to be maintained tied to their ropes, which will lightly restrict their displacement but not completely. Recall that, in this study, we assume that the water surface height, the transmitter, and the receiver heights as well as the distance separating them are slightly moved around their nominal locations by some heightsΔh, ΔhT, ΔhR, and Δd, respectively, each on the interval ±5m. Consequently, assuming that sensor nodes will be attached to their cable, all the time, will only help to have a barely fixed network topology that allows the investigation of the optimal configuration parameters in terms of deployment and routing that surmount the energy hole problem.Our network model suggests that sensors are placed in a circular field of radiusR centered at the sink as shown in Figure 2. For mathematical modeling convenience, the sensor field is virtually partitioned into disjoint concentric coronas of variable widths obtained as follows. Let us consider K concentric circles of radiuses r1<r2<,…,<rK=R centered at the sink. For every i, (1≤i≤K), corona Ci is the subarea surrounded by the circles of radius ri−1 and ri. According to our network model, the width of each corona should not exceed dtx−max, the maximum transmission range of a sensor in order to guarantee communication between adjacent coronas. For example, in Figure 2, K=6. Thus, the sensor field is partitioned into six coronas C1, C2, C3, C4,C5, and C6. Hence, K denotes the number of coronas. The width of corona Ci is wi=ri−ri−1. We assume that a sensor in corona Ci uses a transmission range of wi=ri− ri−1 to reach a sensor in corona Ci−1.Figure 2
Underwater network model.According to the abovementioned network pattern, routing is predictable. Each report is forwarded from the source to the sink by crossing upstream coronas in the same wedgeW. Indeed, the field can be seen as a set of wedges. Each wedge W is virtually partitioned into K sectors, S1,S2,…,SK. The K sectors are the results of the wedge W intersection with the K coronas C1,C2,…,CK, as shown in Figure 3. In our work, we suppose that each sector contains exactly one sensor which is responsible of forwarding the cumulative traffic coming from its predecessors to one of its possible successors. According to our deployment pattern depicted in Figure 3, our circular field will be partitioned into sets of wedges where every wedge W represents a possible path to reach the sink. Thus, our circular field is seen as a set of linear nonintersecting paths. By doing so, we think that, first it is feasible from a practical point of view as underwater sensor networks are sparse by design, and hence the probability of having just one sensor in a given sector is relatively high. Second and most importantly, such deployment pattern will be of great help in order to mathematically and optimally configure our underwater sensor networks in order to surmount the energy hole problem. Indeed, analytically solving this problem in completely random meshed networks is extremely hard. To make it easier, and as a start, we proposed a rather simplified underwater network topology that can be first justified by the sparse deployment of the underwater sensors and second may be intentionally conceived to satisfy given application requirements especially in terms of energy efficiency and hence network lifetime. Indeed, our objective is to take advantage of the sparse deployment of UW-ASNs, in order to propose a target deployment pattern along with the offline-derived optimal load weight distribution which lead to balanced energy use among the network’s sensors, especially for severely energy constrained applications and hence maximize the network lifespan. Figure 3 depicts a possible path where a packet generated by the sensor in the outermost sector is routed to the sink by crossing adjacent sectors. Note that, in this example, each hop involves the immediately adjacent neighbor from the adjacent sector. However, according to our balanced routing strategy, this straightforward routing path can be modified by allowing sensor nodes to have more than one transmission power level, and hence the next hop forwarder can be located in a far away sector and not only the adjacent one. By doing so, we aim at fairly distributing the traffic load such that the energy hole problem is surmounted.Figure 3
Wedge-based network model.In this paper, we suppose that the two main sources of energy consumption are data reception and transmission. More precisely, the energy consumed in transmitting one packet of lengthPl bits over a distance d is given by(2)Etxd=P^Tji×T^txji,where T^txji is the transmission time, which writes(3)T^txji=PlC^ji,where C^ji denotes the channel capacity over the bandwidth B and is derived as follows:(4)C^ji=B×log21+P^Tji×G^jiPNji,where PNji is the noise power over the link. i,j.Similarly, the consumed energy to receive a packet of sizePl bits is given by(5)Erxji=Prx0×T^txji,where Prx0 is the electronics power.


## 3.1. Recap of the Adopted Time-Varying Underwater Channel

In this study, we use the same underwater channel model adopted by [33] and initially proposed by [8]. Accordingly, the transmitter and the receiver are subject to a uniform displacement around their nominal positions by some heights ΔhT and ΔhR and a distance Δd. Moreover, we similarly suppose that the water surface level can vary by some height Δh due to water waves. In particular, the sensors’ displacements are supposed to vary according to a random uniform distribution, each within the interval ±5m. For instance, assume that two sensor nodes are initially placed at a height 80m and within a distance of 1000m from each other where the water depth equals 100m. We suppose that each sensor will undergo an independent random independent drift from their height by ΔhT=±5m and ΔhR=±5m due to water current and waves. Moreover, we suppose that this pair of node will be subject to random displacement Δd from their nominal locations, where Δd=±5m. Furthermore, we suppose that the water surface that initially equals 100m may randomly vary by Δh=±5m. Figure 1 clearly depicts the different parameters with their corresponding variation ranges.Figure 1
Nominal channel parameters: distance between the transmitter and receiverd=1km; water depth h=100m; transmitter and receiver are at depth 80m, ΔhT=ΔhR=Δh=±5m.As introduced in [33], in this work, we adopt an adaptive power allocation model where the transmit power P^T is calculated according to(1)P^T=KPT0G¯G^,where G¯ is the average of the real channel gain G, G^ is an estimate of the true actual gain G, K is a margin introduced to avoid any gap between the true and the estimated channel gains that may yield interruption, and PT0 denotes the minimum power needed to obtain a desired SNR0 in the absence of fading (G=G¯).In this study, we also adopt the success probability definition as stated in [33]. The success probability Pji over a link i,j denotes the probability that a node j successfully receives a packet transmission started by i using a transmission power P^Tji over a bandwidth B. Pji is computed with the purpose of considering the impact of the underwater acoustic channel time variability on the successful reception of a transmitted packet. In order for this paper to be self-contained, we present next a recap on how to compute Pji. For each link in the network, Pji is numerically determined using the underwater acoustic simulator developed in [33]. Pji simply denotes the ratio of successfully received packets over a link i,j. According to [33], a data packet is considered successfully received if the average reception power during the reception time is greater than a given threshold SNR0. As expected, Pji will affect the total amount of traffic received by each node. Therefore, the total energy expenditure will be impacted. Given that our goal is to closely approach the equitable and smooth energy depletion among sensor nodes in the network while taking into consideration the time variability of the underwater channel, Pji along with P^Tji will inevitably affect the load weights distribution for every sensor in the network as detailed in Sections 4 and 5.


## 3.2. Approach to Overcome the Energy Sink Hole Problem

In this paper, an exhaustive investigation of the energy sink hole problem in UW-ASNs is conducted. Specifically, the closest sensors, which are in the neighborhood of the sink, are excessively solicited in forwarding generated reports to it. Indeed, sensors in the vicinity of a static sink work as relays to it on behalf of all other sensors. Thus, those sensors are traffic hot spots enduring heavy energy depletion. Wadaa et al. [7] revealed that by the time the closest sensors to the sink exhaust their energy budget, the farthest sensors still store up to 93% of their initially provided battery power. Another striking result reveals that the energy hole problem is inevitable in static always-on WSNs where the sensor nodes perform continuous monitoring using their nominal communication range [5, 24–28]. Hence, most of the past works dealing with balancing the energy consumption target adjustable communication range solutions aiming at relieving the sensors, which are in the sink’s neighborhood, from the relaying task.Our approach to surmount the energy sink hole problem encountered in UW-ASNs revolves mainly around two axes: (i) deriving the optimal deployment settings that helps overcoming this problem and (ii) conceiving a multiple transmission power levels data forwarding strategy aiming at balancing the energy consumption among all sensor nodes. By thoroughly investigating the joint feasibility of these two methods, we aim at tightly approaching the perfect equitable energy utilization among all the underwater network’s sensors.The goal of the first method is to pertinently deploy sensor nodes according to a well-studied deployment pattern that helps achieving fair energy utilization through the network while respecting the application reliability. Indeed, we aim at deriving the optimal deployment settings in terms of the optimal number of sensors along with the optimal distance separating subsequent nodes that give uniform energy consumption among all sensors in the network while satisfying the application reliability requirements.The goal of the second method is to allow several subsets of sensors to work as relay runners to the sink such that the set of traffic hot spots is continuously changing over time. Indeed, by adopting a variable transmission powers solution, not only is the total data forwarding load (generated plus received) on each underwater sensor distributed uniformly but also the set of sink forwards will change.To recapitulate, in our work, once the target deployment is obtained and thus the desired information reliability is satisfied, we provide each sensor with the set of possible next hops and their respective load weight that closely approach the perfect equitable energy consumption among underwater sensors.


## 3.3. Network and Energy Model

In this study, we opt for the manual deployment of the underwater acoustic sensor networks for the sake of closely approaching the uniform energy depletion among sensor nodes. Indeed, achieving perfect uniform energy consumption in a completely free-floating underwater sensor network is hard as in such network the objective is rather limited to the successful delivery of data packets to the sink. Seeking to achieve any further objectives especially in terms of optimization rather requires a barely static topology of the underwater sensor network. Thus, we turn to the task of studying the optimal deployment settings in order to closely approach the uniform energy depletion through the network, and hence the problem of the energy sink hole is better solved; thus, the network lifetime is maximized. That being said, conceiving the optimal deployment pattern of the underwater sensors should not be indefinitely sought after since it may break the reliability requirement of the application. For this reason, we aim at designing the optimal deployment settings such that the desired reliability level is satisfied.In this paper, we consider a two-dimensional shallow underwater sensor network. Our deployment strategy states that sensors are anchored to the water bottom and tied with long cables. A floating buoy is used to keep the underwater node at the water surface. Note that in this study, the sensors are assumed to be maintained tied to their ropes, which will lightly restrict their displacement but not completely. Recall that, in this study, we assume that the water surface height, the transmitter, and the receiver heights as well as the distance separating them are slightly moved around their nominal locations by some heightsΔh, ΔhT, ΔhR, and Δd, respectively, each on the interval ±5m. Consequently, assuming that sensor nodes will be attached to their cable, all the time, will only help to have a barely fixed network topology that allows the investigation of the optimal configuration parameters in terms of deployment and routing that surmount the energy hole problem.Our network model suggests that sensors are placed in a circular field of radiusR centered at the sink as shown in Figure 2. For mathematical modeling convenience, the sensor field is virtually partitioned into disjoint concentric coronas of variable widths obtained as follows. Let us consider K concentric circles of radiuses r1<r2<,…,<rK=R centered at the sink. For every i, (1≤i≤K), corona Ci is the subarea surrounded by the circles of radius ri−1 and ri. According to our network model, the width of each corona should not exceed dtx−max, the maximum transmission range of a sensor in order to guarantee communication between adjacent coronas. For example, in Figure 2, K=6. Thus, the sensor field is partitioned into six coronas C1, C2, C3, C4,C5, and C6. Hence, K denotes the number of coronas. The width of corona Ci is wi=ri−ri−1. We assume that a sensor in corona Ci uses a transmission range of wi=ri− ri−1 to reach a sensor in corona Ci−1.Figure 2
Underwater network model.According to the abovementioned network pattern, routing is predictable. Each report is forwarded from the source to the sink by crossing upstream coronas in the same wedgeW. Indeed, the field can be seen as a set of wedges. Each wedge W is virtually partitioned into K sectors, S1,S2,…,SK. The K sectors are the results of the wedge W intersection with the K coronas C1,C2,…,CK, as shown in Figure 3. In our work, we suppose that each sector contains exactly one sensor which is responsible of forwarding the cumulative traffic coming from its predecessors to one of its possible successors. According to our deployment pattern depicted in Figure 3, our circular field will be partitioned into sets of wedges where every wedge W represents a possible path to reach the sink. Thus, our circular field is seen as a set of linear nonintersecting paths. By doing so, we think that, first it is feasible from a practical point of view as underwater sensor networks are sparse by design, and hence the probability of having just one sensor in a given sector is relatively high. Second and most importantly, such deployment pattern will be of great help in order to mathematically and optimally configure our underwater sensor networks in order to surmount the energy hole problem. Indeed, analytically solving this problem in completely random meshed networks is extremely hard. To make it easier, and as a start, we proposed a rather simplified underwater network topology that can be first justified by the sparse deployment of the underwater sensors and second may be intentionally conceived to satisfy given application requirements especially in terms of energy efficiency and hence network lifetime. Indeed, our objective is to take advantage of the sparse deployment of UW-ASNs, in order to propose a target deployment pattern along with the offline-derived optimal load weight distribution which lead to balanced energy use among the network’s sensors, especially for severely energy constrained applications and hence maximize the network lifespan. Figure 3 depicts a possible path where a packet generated by the sensor in the outermost sector is routed to the sink by crossing adjacent sectors. Note that, in this example, each hop involves the immediately adjacent neighbor from the adjacent sector. However, according to our balanced routing strategy, this straightforward routing path can be modified by allowing sensor nodes to have more than one transmission power level, and hence the next hop forwarder can be located in a far away sector and not only the adjacent one. By doing so, we aim at fairly distributing the traffic load such that the energy hole problem is surmounted.Figure 3
Wedge-based network model.In this paper, we suppose that the two main sources of energy consumption are data reception and transmission. More precisely, the energy consumed in transmitting one packet of lengthPl bits over a distance d is given by(2)Etxd=P^Tji×T^txji,where T^txji is the transmission time, which writes(3)T^txji=PlC^ji,where C^ji denotes the channel capacity over the bandwidth B and is derived as follows:(4)C^ji=B×log21+P^Tji×G^jiPNji,where PNji is the noise power over the link. i,j.Similarly, the consumed energy to receive a packet of sizePl bits is given by(5)Erxji=Prx0×T^txji,where Prx0 is the electronics power.


## 4. Balancing Energy Consumption

In this paper, we strive for deriving the optimal deployment configuration that satisfies the application reliability requirement along with the optimal traffic load distribution that balances the energy depletion throughout the network. Recall that, in this work, we target continuous-monitoring applications where every sensor periodically generatesA reports per unit of time. In this section, we strive for determining the energy expenditure per sensor in an arbitrary corona Ci of width wi where i≥1. According to our routing strategy, every node in a given wedge W and a generic corona Ci, (1≤i≤K), is supposed to forward two kinds of traffic:(i)
accumulated traffic emanating from downstream sensors belonging to the same wedgeW but located in a different sectors Sj with i<j≤K(ii)
traffic generated by the sensor itself which is located in sectorSiFor each sensor node positioned at sectorSi in a specific wedge W, the possible next hop to forward-accumulated traffic to sink S can be the sensor located in Si−1 or Si−2,…,orSi−ni, in the same wedge W where ni denotes the maximum number of coronas that can be reached by sensor i subject to dtx−max. Let N be(6)N∈1,maxini,N is a system parameter denoting the number of different transmission range levels that can be dynamically adjusted by each underwater sensor. In other words, in our study, we suppose that each underwater sensor can choose among N possible values of transmission range. We aim at deriving the optimal load weight for each possible transmission range that balances the energy utilization among all sensors in the network.Let us consider a wedgeW, and we associate with each possible next hop located in Si−1 or Si−2,…, or Si−N a respective weight β1i,β2i,…,βNi and a respective probability of a successful reception P1i,P2i,…,Pni such that ∑p=1Nβpi=1, ∀i,1≤i≤K. Therefore, the total number of packets per unit of time, Ai, handled by a sensor in sector Si and wedge W, is given by(7)Ai=A+P1i+1β1i+1Ai+1+,…,+Pji+jβji+jAi+j+,…,+PNi+NβNi+NAi+N,ifi+j>Korni+j<jthenβji+j=0.Accordingly, the average transmission energy,Etxi, consumed by a sensor in corona Ci of width wi can be derived as shown in equation (9):(8)Etxi=β1iAiEtxri−ri−1+,…,+βjiAiEtxri−rj+,…,+βNiAiEtxri−ri−N,ifi−j<0thenβji=0.After simplifications,(9)Etxi=β1iAiEtxwi+,…,+βjiAiEtx∑k=0j−1wi−k+,…,+βNiAiEtx∑k=0N−1wi−k,ifi−j<0thenβji=0.Note that the average transmission energy,Etxi, depends on the widths of all the possible upstream coronas wk, 0≤k≤i. Consequently, as a key distinguishing feature from [33], in this study, the corona width is no longer fixed but it can be of any length. Most importantly, in this work, the corona width is a network parameter that will be integrated in our optimization problem and for which we seek to derive the optimal value that helps achieving smooth and uniform energy depletion through the network.Similarly, the average reception energy,Erxi, consumed by a sensor in corona Ci of width wi can be expressed as derived in equation (11). Expressly,(10)Erxi=P1i+1β1i+1Ai+1Erxri+1−ri+,…,+Pji+jβji+jAi+jErxri+j−ri+,…,+PNi+NβNi+NAi+NErxri+N−ri,ifi+j>Kthenβji+j=0.After simplifications,(11)Erxi=P1i+1β1i+1Ai+1Erxwi+1+,…,+Pji+jβji+jAi+jErx∑k=1jwi+k+,…,+PNi+NβNi+NAi+NErx∑k=1Nwi+k,ifi+j>Kthenβji+j=0.It is worth pointing out that here again the reception energy,Erxi, depends on the width of all the possible downstream coronas wk, i+1≤k≤K. Finally, the total energy consumed by a sensor in corona Ci of width wi is(12)Ei=Etxi+Erxi.The objective of our study is to design the coronas widths such that the energy depletion is balanced across all the sensors while achieving the desired reliability levelRL. Note that in our work, the reliability level RL is expressed in terms of the minimum required number of received reports at the sink per unit of time. RL is expressed in our study as RL≥A×K just in order to impose a minimum number of coronas and hence a minimum number of deployed sensor nodes. Therefore, our optimization problem is stated as follows:(13)givenR,dtx−max,RL,A,findN,wi,βi=βli1≤l≤n,∀i,1≤i≤K,such  that,minwi,N,βi1≤i≤KmaxEi,subject  to∑j=1nβji=1,∀i,1≤i≤k,A×K≥RL.As opposed to the result in [33], our optimization problem is now threefold. Accordingly, we are seeking three optimal parameters: (i) the optimal distance separating any two subsequent nodes wi 1≤i≤K, (ii) the optimal number of transmission levels N, and (iii) the optimal load utilization ratio, βi1≤i≤K, for each possible transmission power level for every node in the network. It is worth pointing out that in this study, we rather opt for the numerical resolution as our optimization problem is now threefold. In what follows, we denote βNi=βli1≤l≤N, βN=βNi1≤i≤K, and. wK=wi1≤i≤K.


## 5. Optimizing the Energy Consumption

In this section, we try to solve the above-derived threefold optimization problem of equation (13). For this purpose, we mainly deal with two main cases: fixed and variable corona width. For the fixed corona width case, we suppose that all coronas in the network have fixed width and we aim at deriving the optimal corona width w as well as the optimal number N of transmission levels along with optimal load weight corresponding to every possible transmission power level for every sensor node. For the variable corona width case, we deal with two subcases: first, we suppose that the corona width follows an arithmetic progression, and second, we assume that the corona width is following a geometric sequence. Note that, for both main cases, namely, fixed and variable corona width, the optimal value of N will be derived by browsing the interval 1,maxini. In other words, for each wK configuration, we derive βN that minimizes maxEi1≤i≤K, for each value of N. Once done, we select the optimal N that further minimizes maxEi1≤i≤K for the chosen configuration wK.


### 5.1. Numerical Solution for the Fixed Corona Width Case

Let us start by considering that our network deployment follows a fixed corona width model where the corona width is set equal tow. Consequently,(14)K=Rw,and hence(15)∀i,1≤i≤K,ni=n=dtx−maxw.As such, Etxi can be expressed as follows:(16)Etxi=β1iAiEtxw+,…,+βjiAiEtxjw+,…,+βNiAiEtxNwifi−j<0thenβji=0,and similarly(17)Erxi=P1i+1β1i+1Ai+1Erxw+,…,+Pji+jβji+jAi+jErxjw+,…,+Pni+nβni+nAi+nErxnwifi+j>Kthenβji+j=0.Thence, our problem can be stated as follows:(18)givenR,dtx−max,RL,A,findN,w,βi=βli1≤l≤N,∀i,1≤i≤K,suchthat,minw,N,βimaxEi1≤i≤K,subject  to∑j=1nβji=1,∀i,1≤i≤k,A×K≥RL.Accordingly, given the field radiusR, the desired reliability level RL, and the maximum transmission range dtx−max, we aim at deriving the optimal values for N, w, and βN that balance the energy expenditure among all sensors in the deployed network while respecting the desired reliability level.


### 5.2. Numerical Solution for Variable Corona Width

Corona Width following an Arithmetic Progression. In this section, we suppose that the corona width is following an arithmetic progression with an initial term w1, which is the width of the first corona C1, and a common difference m. In other words,(19)wi=w1+i−1m,∀i,1≤i≤K.It is worth noting that ifwi>dtx−max, then wi will be limited to only dtx−max. In other words, once wi exceeds dtx−max, the remaining field radius will be portioned into coronas of fixed size dtx−max.We aim at finding out the optimal configuration in terms ofw1, m, N, and βN that minimizes maxEi1≤i≤K. Hence, our optimization problem can be expressed as follows:(20)givenR,dtx−max,RL,A,findN,w1,m,βi=βli1≤l≤n,∀i,1≤i≤K,such  that,minw1,m,N,βi1≤i≤KmaxEi,subject  to∑j=1nβji=1,∀i,1≤i≤K,A×K≥RL.Note that our optimization problem is now fourfold as it seeks to determine four optimal parameter, namely, (i)w1, the initial term of the arithmetic progression which represents the first corona width, (ii) m, which is the common difference of the arithmetic progression, (iii) N, the number of the transmission power levels to be adopted by each sensor, and (iv) βi, the utilization ratio of each possible transmission power level for every node in the network.Corona Width following a Geometric Progression. Similar to the previous section, we now suppose that the corona width is following a geometric sequence with an initial term w1, which is the width of the first corona C1, and a common ratio q. In other words,(21)wi=w1×qi−1,∀i,1≤i≤K.Note that, here again, ifwi exceeds dtx−max, then wi will be limited to only dtx−max. In other words, once wi exceeds dtx−max, the remaining field radius will be partitioned into coronas of fixed size dtx−max. Our ultimate goal is to determine the best configuration in terms of w1, q, N, and βN that minimizes maxEi1≤i≤K. Hence, our optimization problem is now fourfold and can be expressed as follows: (22)givenR,dtx−max,RL,A,findN,w1,q,βi=βli1≤l≤n,∀i,1≤i≤K,such thatminw1,q,N,βi1≤i≤KmaxEisubjectto∑j=1nβji=1,∀i,1≤i≤K,A×K≥RL.A straightforward way to solve the above-stated constrained nonlinear optimization problems of equations (18), (20), and (22) is by using the MATLAB optimization toolbox. The MATLAB optimization toolbox is one of the useful toolboxes provided by MATLAB software [34] that provides numerical solutions. This toolbox minimizes a problem which is subjected to some specific constraints. One of the most used solvers of this toolbox is fmincon. Fmincon tries to find the minimum of constrained nonlinear multivariable function. The goal is finding a vector x that is a local minimum to a scalar function fx. More precisely, the optimal values of w,w1,m,q,andN will be derived in a parametric way by browsing an appropriate interval, while we jointly use the fmincon solver to derive the optimal βN for each possible values of w,w1,m,q,andN. It is worth pointing out that fmincon has been applied with several initial values in order to make sure that the obtained optimal values of βi are global ones.


## 5.1. Numerical Solution for the Fixed Corona Width Case

Let us start by considering that our network deployment follows a fixed corona width model where the corona width is set equal tow. Consequently,(14)K=Rw,and hence(15)∀i,1≤i≤K,ni=n=dtx−maxw.As such, Etxi can be expressed as follows:(16)Etxi=β1iAiEtxw+,…,+βjiAiEtxjw+,…,+βNiAiEtxNwifi−j<0thenβji=0,and similarly(17)Erxi=P1i+1β1i+1Ai+1Erxw+,…,+Pji+jβji+jAi+jErxjw+,…,+Pni+nβni+nAi+nErxnwifi+j>Kthenβji+j=0.Thence, our problem can be stated as follows:(18)givenR,dtx−max,RL,A,findN,w,βi=βli1≤l≤N,∀i,1≤i≤K,suchthat,minw,N,βimaxEi1≤i≤K,subject  to∑j=1nβji=1,∀i,1≤i≤k,A×K≥RL.Accordingly, given the field radiusR, the desired reliability level RL, and the maximum transmission range dtx−max, we aim at deriving the optimal values for N, w, and βN that balance the energy expenditure among all sensors in the deployed network while respecting the desired reliability level.


## 5.2. Numerical Solution for Variable Corona Width

Corona Width following an Arithmetic Progression. In this section, we suppose that the corona width is following an arithmetic progression with an initial term w1, which is the width of the first corona C1, and a common difference m. In other words,(19)wi=w1+i−1m,∀i,1≤i≤K.It is worth noting that ifwi>dtx−max, then wi will be limited to only dtx−max. In other words, once wi exceeds dtx−max, the remaining field radius will be portioned into coronas of fixed size dtx−max.We aim at finding out the optimal configuration in terms ofw1, m, N, and βN that minimizes maxEi1≤i≤K. Hence, our optimization problem can be expressed as follows:(20)givenR,dtx−max,RL,A,findN,w1,m,βi=βli1≤l≤n,∀i,1≤i≤K,such  that,minw1,m,N,βi1≤i≤KmaxEi,subject  to∑j=1nβji=1,∀i,1≤i≤K,A×K≥RL.Note that our optimization problem is now fourfold as it seeks to determine four optimal parameter, namely, (i)w1, the initial term of the arithmetic progression which represents the first corona width, (ii) m, which is the common difference of the arithmetic progression, (iii) N, the number of the transmission power levels to be adopted by each sensor, and (iv) βi, the utilization ratio of each possible transmission power level for every node in the network.Corona Width following a Geometric Progression. Similar to the previous section, we now suppose that the corona width is following a geometric sequence with an initial term w1, which is the width of the first corona C1, and a common ratio q. In other words,(21)wi=w1×qi−1,∀i,1≤i≤K.Note that, here again, ifwi exceeds dtx−max, then wi will be limited to only dtx−max. In other words, once wi exceeds dtx−max, the remaining field radius will be partitioned into coronas of fixed size dtx−max. Our ultimate goal is to determine the best configuration in terms of w1, q, N, and βN that minimizes maxEi1≤i≤K. Hence, our optimization problem is now fourfold and can be expressed as follows: (22)givenR,dtx−max,RL,A,findN,w1,q,βi=βli1≤l≤n,∀i,1≤i≤K,such thatminw1,q,N,βi1≤i≤KmaxEisubjectto∑j=1nβji=1,∀i,1≤i≤K,A×K≥RL.A straightforward way to solve the above-stated constrained nonlinear optimization problems of equations (18), (20), and (22) is by using the MATLAB optimization toolbox. The MATLAB optimization toolbox is one of the useful toolboxes provided by MATLAB software [34] that provides numerical solutions. This toolbox minimizes a problem which is subjected to some specific constraints. One of the most used solvers of this toolbox is fmincon. Fmincon tries to find the minimum of constrained nonlinear multivariable function. The goal is finding a vector x that is a local minimum to a scalar function fx. More precisely, the optimal values of w,w1,m,q,andN will be derived in a parametric way by browsing an appropriate interval, while we jointly use the fmincon solver to derive the optimal βN for each possible values of w,w1,m,q,andN. It is worth pointing out that fmincon has been applied with several initial values in order to make sure that the obtained optimal values of βi are global ones.


## 6. Results and Discussion

In order to assess the performance of our proposal, we developed a simulator using MATLAB. This simulator implements the physical layer specifications including attenuation, delays, bandwidth, and power and energy consumption using the model presented in Sections3–5. Note that this MATLAB simulator mainly uses the powerful MATLAB optimization toolbox in order to find the optimal deployment and routing configurations that balances the energy consumption among all underwater sensors in the network while satisfying the application desired reliability level.We want to point out that in our work and for each case study, namely, fixed corona width and corona width following either arithmetic progression or geometric one, we have a number of nested optimization problems. More precisely, we have three nested optimization problems for the fixed corona width case and four nested ones for the corona width following either arithmetic progression or geometric progression cases. Fmincon is only used to determineβi which is a continuous function in 0,1. The other parameters like the optimal N and the optimal wi are derived in a parametric way as explained in the paper by browsing a given interval with a given step.According to our network model, our circular sensor field, centered at the sink and of radiusR, is virtually partitioned into disjoint concentric coronas of variable widths. The width of each corona is at most dtx−max=1000m the maximum transmission range of an underwater acoustic sensor. Each underwater sensor periodically reports with rate A=0.08 packets/s. At each hop, the locally generated traffic along with the route-through traffic is transmitted from the source node by crossing upstream sensor nodes located in the same wedge till reaching the sink. Table 1 provides the parameters setting in our study.Table 1
Parameters setting.
Packet lengthPl
1024 bits
S
N
R
0
25 dB
P
rx
0
0.75 W
BandwidthB
10 Khz
Noise power
20 dB
R
L
5
×
A
=
0.4
packet
/
s


### 6.1. Fixed Corona Width

Let us start by studying the fixed corona width case. It is worth pointing out that in order to derive the optimal configuration parameters, the corona widthw is varied in 50:dtx_max with a step of 10m, where for each possible value of w, N is varied between 1:1:R/w in order to find βi using fmincon. Then the optimal parameters wopt, Nopt, and βN_opt are the ones that jointly achieve the minimum energy consumption among all the possible configurations.By varying the corona widthw in 50:10:dtx−max, N in 1:1:R/w, and the field radius R in 1000,2000,3000,5000,6000, the optimization results show that the optimal corona width wopt that minimizes the total energy consumption of the heavy loaded node is mainly achieved between 100,450 as depicted in Table 2. Table 3 shows βN_opt matrix when the field radius is set equal to 1000m. Note that, in this illustrative case, wopt=190m and Nopt=5 as depicted in Figure 4. Accordingly, our sensor field is partitioned into 6 coronas where βN_opt dictates that each sensor should send its accumulated traffic in equal proportions to all possible upstream coronas.Table 2
Optimal corona width values for various field radii.
Field radius (m)
Optimal corona width (m)
1000
190
2000
460
3000
280
5000
100
6000
370Table 3
β
N
_
opt matrix when R=1000m.
Corona
Corona
Corona
Corona
Corona
Corona
i
i
−
1
i
−
2
i
−
3
i
−
4
i
−
5
Corona 1
1
0
0
0
0
Corona 2
0.5
0.5
0
0
0
Corona 3
0.3332
0.3334
0.3336
0
0
Corona 4
0.25
0.25
0.25
0.25
0
Corona 5
0.2
0.2
0.2
0.2
0.2
Corona 6
0.2
0.2
0.2
0.2
0.2Figure 4
N
opt as function of the field radius R for the three approaches.


### 6.2. Corona Width following an Arithmetic Progression

In this section, we suppose that the corona width is following an arithmetic progression. More precisely,w1∈50:12.5:dtx−max and m∈50:5:dtx−max/2 while the field radius R varies in 1000,2000,3000,5000,6000. Note that, in order to derive the optimal parameters for the arithmetic progression of corona width, w1, m, and N are varied each in a given interval with a given step in a nested parametric optimization problem where βi are obtained using fmincon for each possible configuration. Optimal values of w1opt, mopt, Nopt, and βN_opt are the ones that achieve the minimum energy consumption among all the possible configurations.Table4 shows βN_opt matrix when the field radius is set equal to 1000m. Note that, in this illustrative case, w1opt=50m, mopt=80, and Nopt=5 as depicted in Figure 5. Accordingly, our sensor field is partitioned into 5 coronas as shown in Table 5. According to Tables 4 and 5, optimal balanced energy consumption is achieved when the first corona has a width of 50m, while the second has a width of 130m, and it sends 50% of its total traffic directly to the sink and the remaining 50% to corona 1. Corona 4, for example, should have a width of 290m and should transmit 26% of its traffic to corona 3 and 37% of traffic to corona 2, while the remaining traffic load should be transmitted directly to the sink, leading to no traffic from corona 4 to the first corona in order to alleviate the heavy forwarding task of the one-hop away sensors so that the sink hole problem is avoided.Table 4
ArithmeticβN_opt matrix when R=1000m.
Corona
Corona
Corona
Corona
Corona
Corona
i
i
−
1
i
−
2
i
−
3
i
−
4
i
−
5
Corona 1
1
0
0
0
0
Corona 2
0.5
0.5
0
0
0
Corona 3
0.5
0
0.5
0
0
Corona 4
0.26
0.37
0
0.37
0
Corona 5
0.3
0.3
0
0
0.4Table 5
Arithmetic coronas widths whenR=1000m.
Coronai
Width (m)
Corona 1
50
Corona 2
130
Corona 3
210
Corona 4
290
Corona 5
370The optimization results as shown in Table6 reveal that the optimal w1opt that minimizes maxEi1≤i≤K for various field radii has small values . Recall that w1 is actually the width of the first corona which is the closest one to the sink. Consequently, by reducing the width of the coronas in the close vicinity of the sink, their energy consumption is reduced since their transmission power is reduced. Hence, even if these coronas are handling more traffic compared to the far ones, they will not consume too much greater amount of energy since they will be transmitting over short distances. On the contrary, the farther the corona is from the sink, the higher is its width wi. Hence, these coronas will be in charge of sending over longer distance since they are handling a small amount of packet forwarding, and thus the energy consumption is balanced through the network. Consequently, having a field of radius R, the sensor nodes should be placed such that the closest ones should send at short distances to reach their upstream coronas while the farthest ones should transmit over much higher distances to reach their subsequent coronas. By doing so, the difference in the amount of forwarded traffic between the closest nodes and the farthest ones is balanced by the transmission power expenditure, and hence uniform energy consumption is achieved through the network. Jointly, the energy consumption can be further optimized by deriving the optimal Nopt and βN_opt values.Table 6
Optimalw1 and m values for various field radii.
Field radius (m)
Optimalw1 (m)
Optimal differencem (m)
1000
50
80
2000
62.5
100
3000
50
95
5000
50
100
6000
50
100


### 6.3. Corona Width following a Geometric Progression

In this section, we assume that the corona width is following a geometric progression. Precisely, we varyw1 in 50:12.5:dtx−max and the ration q in 1.5:0.1:2 while the field radius R varies in 1000,2000,3000,5000,6000. Note that, in order to derive the optimal parameters for the geometric progression of corona width, w1, q, and N are varied each in a given interval with a given step in a nested parametric optimization problem where βi are obtained using fmincon for each possible configuration. Optimal values of w1opt, qopt, Nopt, and βN_opt are the ones that achieve the minimum energy consumption among all the possible configurations.The optimization results shown in Table7 reveal that w1opt and qopt that minimize the energy consumption are decreasing function of the field radius R. Indeed, for high field radius, selecting small w1 with small q guarantees to create more coronas with reasonable widths, hence avoiding width explosion of the farthest coronas which may highly increase the transmission power. Recall that, according to Figure 5, the transmission power exponentially increases with distance. A further energy optimization can be achieved by jointly deriving the optimal Nopt and βN_opt values. Table 8 shows βN_opt matrix when the field radius is set equal to 1000m. Note that, in this illustrative case, w1opt=87.5m, qopt=1.5, and Nopt=5 as depicted in Figure 4. Accordingly, our sensor field is partitioned into 5 coronas as shown in Table 9. According to Tables 8 and 9, balanced energy draining through the network is accomplished when the first corona width is set to 87.5m while the second corona width is equal to 131.25m, and the third corona width equals 196.87m. For instance, a sensor in the fifth corona that has a width of 443m should forward 30% of its traffic to the fourth corona and another 30% should be transmitted to the third corona, while the remaining 40% should be transmitted directly to the sink, skipping corona 2 and corona 1 in order to alleviate their forwarding task as they are close to the sink so that the sink hole problem is overcome. Note that, for R=1000m, the optimal parameters of the arithmetic and geometric progressions lead to almost the same coronas widths (Tables 5 and 9) which explains the similarities of Nopt and βN_opt values.Table 7
Optimalw1 and q values for various field radii.
Field radius (m)
Optimalw1 (m)
Optimal ratioq
1000
87.5
1.5
2000
62.5
1.5
3000
62.5
1.4
5000
62.5
1.4
6000
62.5
1.4Figure 5
Transmission power for various values of distance [33].Table 8
GeometricβN_opt matrix when R=1000m.
Corona
Corona
Corona
Corona
Corona
Corona
i
i
−
1
i
−
2
i
−
3
i
−
4
i
−
5
Corona 1
1
0
0
0
0
Corona 2
0.5
0.5
0
0
0
Corona 3
0.5
0
0.5
0
0
Corona 4
0.26
0.37
0
0.37
0
Corona 5
0.3
0.3
0
0
0.4Table 9
Geometric coronas widths whenR=1000m.
Coronai
Width (m)
Corona 1
87.5
Corona 2
131.25
Corona 3
196.875
Corona 4
295.31
Corona 5
442.97Most importantly, according to the results shown in Tables4, 5, 8, and 9, overcoming the energy hole problem in a given sensor field simply comes down to design the corona widths such that the closer the corona to the sink, the smaller its width, whereas the farther the corona from the sink, the larger its width. By doing so, the coronas that are close to the sink will compensate their higher traffic load by transmitting over shorter distances and hence consuming less transmission power. However, the far coronas will be in charge of transmitting over larger distances since their traffic load is reduced. Consequently, balanced energy consumption is achieved through the network, thanks not only to the load weight distribution βN_opt (reported in Tables 4 and 8) but also to the optimal design of coronas widths (reported in Tables 5 and 9).Now, and most importantly, let us compare the three approaches. First, observe that according to Figure4, Nopt for the three approaches is decreasing as a function of the field radius. Indeed, the higher the field radius, the more the coronas created; using high N will inevitably lead to transmission over higher distances, drastically increasing the energy consumption. However, having relatively small field radius will ensure creating coronas of reasonable widths, and hence using large Nopt will balance the energy consumption through the network.Figure6 shows the energy consumption as function of the field radius R for the three approaches. Note that, every spot in the three curves represents the maximum consumed energy (maxEi1≤i≤K) among all the formed coronas for a given field radius. Knowing that our simulation results, for every spot, provide minwi,N,βi1≤i≤KmaxEi, we aim at determining which of the three approaches achieves the most balanced energy consumption by reducing at most maxEi1≤i≤K. First, Figure 6 shows that the energy consumption is increasing as a function of the field radius R, for the three approaches. Indeed, as the field radius increases, the number of coronas increases, and hence the traffic load grows leading to an increase in the energy consumption. Most importantly, note that the case of coronas widths following geometric progression is achieving the lowest energy consumption since the geometric progression is using higher Nopt as shown in Figure 4, and hence it ensures more balanced energy consumption. It is worth noting that for small values of the field radius R, the three approaches are using high Nopt values. Indeed, reducing the size of the field radius will reduce the number of the widths of the coronas, and hence increasing N is sought and justified. Observe also that, for small field radius, geometric and arithmetic progressions are almost achieving the same energy consumption; since as explained above (Tables 5 and 9), both approaches are achieving the same coronas width distribution, and hence they have similar Nopt and βN_opt. Finally, notice that the arithmetic corona width progression is converging to the fixed width case since for large field radius Nopt=1 for both approaches, and hence every node is restricted to send to its upstream node in the subsequent corona.Figure 6
The energy consumption as function of the field radiusR for the three approaches.


## 6.1. Fixed Corona Width

Let us start by studying the fixed corona width case. It is worth pointing out that in order to derive the optimal configuration parameters, the corona widthw is varied in 50:dtx_max with a step of 10m, where for each possible value of w, N is varied between 1:1:R/w in order to find βi using fmincon. Then the optimal parameters wopt, Nopt, and βN_opt are the ones that jointly achieve the minimum energy consumption among all the possible configurations.By varying the corona widthw in 50:10:dtx−max, N in 1:1:R/w, and the field radius R in 1000,2000,3000,5000,6000, the optimization results show that the optimal corona width wopt that minimizes the total energy consumption of the heavy loaded node is mainly achieved between 100,450 as depicted in Table 2. Table 3 shows βN_opt matrix when the field radius is set equal to 1000m. Note that, in this illustrative case, wopt=190m and Nopt=5 as depicted in Figure 4. Accordingly, our sensor field is partitioned into 6 coronas where βN_opt dictates that each sensor should send its accumulated traffic in equal proportions to all possible upstream coronas.Table 2
Optimal corona width values for various field radii.
Field radius (m)
Optimal corona width (m)
1000
190
2000
460
3000
280
5000
100
6000
370Table 3
β
N
_
opt matrix when R=1000m.
Corona
Corona
Corona
Corona
Corona
Corona
i
i
−
1
i
−
2
i
−
3
i
−
4
i
−
5
Corona 1
1
0
0
0
0
Corona 2
0.5
0.5
0
0
0
Corona 3
0.3332
0.3334
0.3336
0
0
Corona 4
0.25
0.25
0.25
0.25
0
Corona 5
0.2
0.2
0.2
0.2
0.2
Corona 6
0.2
0.2
0.2
0.2
0.2Figure 4
N
opt as function of the field radius R for the three approaches.


## 6.2. Corona Width following an Arithmetic Progression

In this section, we suppose that the corona width is following an arithmetic progression. More precisely,w1∈50:12.5:dtx−max and m∈50:5:dtx−max/2 while the field radius R varies in 1000,2000,3000,5000,6000. Note that, in order to derive the optimal parameters for the arithmetic progression of corona width, w1, m, and N are varied each in a given interval with a given step in a nested parametric optimization problem where βi are obtained using fmincon for each possible configuration. Optimal values of w1opt, mopt, Nopt, and βN_opt are the ones that achieve the minimum energy consumption among all the possible configurations.Table4 shows βN_opt matrix when the field radius is set equal to 1000m. Note that, in this illustrative case, w1opt=50m, mopt=80, and Nopt=5 as depicted in Figure 5. Accordingly, our sensor field is partitioned into 5 coronas as shown in Table 5. According to Tables 4 and 5, optimal balanced energy consumption is achieved when the first corona has a width of 50m, while the second has a width of 130m, and it sends 50% of its total traffic directly to the sink and the remaining 50% to corona 1. Corona 4, for example, should have a width of 290m and should transmit 26% of its traffic to corona 3 and 37% of traffic to corona 2, while the remaining traffic load should be transmitted directly to the sink, leading to no traffic from corona 4 to the first corona in order to alleviate the heavy forwarding task of the one-hop away sensors so that the sink hole problem is avoided.Table 4
ArithmeticβN_opt matrix when R=1000m.
Corona
Corona
Corona
Corona
Corona
Corona
i
i
−
1
i
−
2
i
−
3
i
−
4
i
−
5
Corona 1
1
0
0
0
0
Corona 2
0.5
0.5
0
0
0
Corona 3
0.5
0
0.5
0
0
Corona 4
0.26
0.37
0
0.37
0
Corona 5
0.3
0.3
0
0
0.4Table 5
Arithmetic coronas widths whenR=1000m.
Coronai
Width (m)
Corona 1
50
Corona 2
130
Corona 3
210
Corona 4
290
Corona 5
370The optimization results as shown in Table6 reveal that the optimal w1opt that minimizes maxEi1≤i≤K for various field radii has small values . Recall that w1 is actually the width of the first corona which is the closest one to the sink. Consequently, by reducing the width of the coronas in the close vicinity of the sink, their energy consumption is reduced since their transmission power is reduced. Hence, even if these coronas are handling more traffic compared to the far ones, they will not consume too much greater amount of energy since they will be transmitting over short distances. On the contrary, the farther the corona is from the sink, the higher is its width wi. Hence, these coronas will be in charge of sending over longer distance since they are handling a small amount of packet forwarding, and thus the energy consumption is balanced through the network. Consequently, having a field of radius R, the sensor nodes should be placed such that the closest ones should send at short distances to reach their upstream coronas while the farthest ones should transmit over much higher distances to reach their subsequent coronas. By doing so, the difference in the amount of forwarded traffic between the closest nodes and the farthest ones is balanced by the transmission power expenditure, and hence uniform energy consumption is achieved through the network. Jointly, the energy consumption can be further optimized by deriving the optimal Nopt and βN_opt values.Table 6
Optimalw1 and m values for various field radii.
Field radius (m)
Optimalw1 (m)
Optimal differencem (m)
1000
50
80
2000
62.5
100
3000
50
95
5000
50
100
6000
50
100


## 6.3. Corona Width following a Geometric Progression

In this section, we assume that the corona width is following a geometric progression. Precisely, we varyw1 in 50:12.5:dtx−max and the ration q in 1.5:0.1:2 while the field radius R varies in 1000,2000,3000,5000,6000. Note that, in order to derive the optimal parameters for the geometric progression of corona width, w1, q, and N are varied each in a given interval with a given step in a nested parametric optimization problem where βi are obtained using fmincon for each possible configuration. Optimal values of w1opt, qopt, Nopt, and βN_opt are the ones that achieve the minimum energy consumption among all the possible configurations.The optimization results shown in Table7 reveal that w1opt and qopt that minimize the energy consumption are decreasing function of the field radius R. Indeed, for high field radius, selecting small w1 with small q guarantees to create more coronas with reasonable widths, hence avoiding width explosion of the farthest coronas which may highly increase the transmission power. Recall that, according to Figure 5, the transmission power exponentially increases with distance. A further energy optimization can be achieved by jointly deriving the optimal Nopt and βN_opt values. Table 8 shows βN_opt matrix when the field radius is set equal to 1000m. Note that, in this illustrative case, w1opt=87.5m, qopt=1.5, and Nopt=5 as depicted in Figure 4. Accordingly, our sensor field is partitioned into 5 coronas as shown in Table 9. According to Tables 8 and 9, balanced energy draining through the network is accomplished when the first corona width is set to 87.5m while the second corona width is equal to 131.25m, and the third corona width equals 196.87m. For instance, a sensor in the fifth corona that has a width of 443m should forward 30% of its traffic to the fourth corona and another 30% should be transmitted to the third corona, while the remaining 40% should be transmitted directly to the sink, skipping corona 2 and corona 1 in order to alleviate their forwarding task as they are close to the sink so that the sink hole problem is overcome. Note that, for R=1000m, the optimal parameters of the arithmetic and geometric progressions lead to almost the same coronas widths (Tables 5 and 9) which explains the similarities of Nopt and βN_opt values.Table 7
Optimalw1 and q values for various field radii.
Field radius (m)
Optimalw1 (m)
Optimal ratioq
1000
87.5
1.5
2000
62.5
1.5
3000
62.5
1.4
5000
62.5
1.4
6000
62.5
1.4Figure 5
Transmission power for various values of distance [33].Table 8
GeometricβN_opt matrix when R=1000m.
Corona
Corona
Corona
Corona
Corona
Corona
i
i
−
1
i
−
2
i
−
3
i
−
4
i
−
5
Corona 1
1
0
0
0
0
Corona 2
0.5
0.5
0
0
0
Corona 3
0.5
0
0.5
0
0
Corona 4
0.26
0.37
0
0.37
0
Corona 5
0.3
0.3
0
0
0.4Table 9
Geometric coronas widths whenR=1000m.
Coronai
Width (m)
Corona 1
87.5
Corona 2
131.25
Corona 3
196.875
Corona 4
295.31
Corona 5
442.97Most importantly, according to the results shown in Tables4, 5, 8, and 9, overcoming the energy hole problem in a given sensor field simply comes down to design the corona widths such that the closer the corona to the sink, the smaller its width, whereas the farther the corona from the sink, the larger its width. By doing so, the coronas that are close to the sink will compensate their higher traffic load by transmitting over shorter distances and hence consuming less transmission power. However, the far coronas will be in charge of transmitting over larger distances since their traffic load is reduced. Consequently, balanced energy consumption is achieved through the network, thanks not only to the load weight distribution βN_opt (reported in Tables 4 and 8) but also to the optimal design of coronas widths (reported in Tables 5 and 9).Now, and most importantly, let us compare the three approaches. First, observe that according to Figure4, Nopt for the three approaches is decreasing as a function of the field radius. Indeed, the higher the field radius, the more the coronas created; using high N will inevitably lead to transmission over higher distances, drastically increasing the energy consumption. However, having relatively small field radius will ensure creating coronas of reasonable widths, and hence using large Nopt will balance the energy consumption through the network.Figure6 shows the energy consumption as function of the field radius R for the three approaches. Note that, every spot in the three curves represents the maximum consumed energy (maxEi1≤i≤K) among all the formed coronas for a given field radius. Knowing that our simulation results, for every spot, provide minwi,N,βi1≤i≤KmaxEi, we aim at determining which of the three approaches achieves the most balanced energy consumption by reducing at most maxEi1≤i≤K. First, Figure 6 shows that the energy consumption is increasing as a function of the field radius R, for the three approaches. Indeed, as the field radius increases, the number of coronas increases, and hence the traffic load grows leading to an increase in the energy consumption. Most importantly, note that the case of coronas widths following geometric progression is achieving the lowest energy consumption since the geometric progression is using higher Nopt as shown in Figure 4, and hence it ensures more balanced energy consumption. It is worth noting that for small values of the field radius R, the three approaches are using high Nopt values. Indeed, reducing the size of the field radius will reduce the number of the widths of the coronas, and hence increasing N is sought and justified. Observe also that, for small field radius, geometric and arithmetic progressions are almost achieving the same energy consumption; since as explained above (Tables 5 and 9), both approaches are achieving the same coronas width distribution, and hence they have similar Nopt and βN_opt. Finally, notice that the arithmetic corona width progression is converging to the fixed width case since for large field radius Nopt=1 for both approaches, and hence every node is restricted to send to its upstream node in the subsequent corona.Figure 6
The energy consumption as function of the field radiusR for the three approaches.


## 7. Conclusions

In this paper, we address the energy sink hole problem in UW-ASNs while taking into account the severe features of the underwater channel. We strive for determining the optimal distances separating subsequent sensor nodes during deployment that helps achieving fair energy expenditure among all sensor nodes while satisfying the desired information reliability. Jointly, we propose a variable transmission range-based data forwarding task. Accordingly, each underwater sensor is allowed to dynamically modify its transmission power among multiple possible levelsN. Consequently, we derived for each sensor in the network the optimal transmission power levels and their corresponding optimal load weights that surmount the sink hole problem. To achieve this, we numerically solved well-stated optimization problems that jointly derive for each source sensor, the appropriate deployment configuration, and the optimal number of transmission power levels along with their associated optimal load weights that evenly consume the energy budget for each underwater sensor in the network, and hence, the network energy efficiency is maximized.



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*Source: 3597304-2019-02-03.xml*