Enhanced Target Localization in the Internet of Underwater Things through Quantum-Behaved Metaheuristic Optimization with Multi-Strategy Integration

Abstract

Underwater localization is considered a critical technique in the Internet of Underwater Things (IoUTs). However, acquiring accurate location information is challenging due to the heterogeneous underwater environment and the hostile propagation of acoustic signals, especially when using received signal strength (RSS)-based techniques. Additionally, most current solutions rely on strict mathematical expressions, which limits their effectiveness in certain scenarios. To address these challenges, this study develops a quantum-behaved meta-heuristic algorithm, called quantum enhanced Harris hawks optimization (QEHHO), to solve the localization problem without requiring strict mathematical assumptions. The algorithm builds on the original Harris hawks optimization (HHO) by integrating four strategies into various phases to avoid local minima. The initiation phase incorporates good point set theory and quantum computing to enhance the population quality, while a random nonlinear technique is introduced in the transition phase to expand the exploration region in the early stages. A correction mechanism and exploration enhancement combining the slime mold algorithm (SMA) and quasi-oppositional learning (QOL) are further developed to find an optimal solution. Furthermore, the RSS-based Cramér–Raolower bound (CRLB) is derived to evaluate the effectiveness of QEHHO. Simulation results demonstrate the superior performance of QEHHO under various conditions compared to other state-of-the-art closed-form-expression- and meta-heuristic-based solutions.

1. Introduction

The Internet of Underwater Things (IoUTs) has been widely applied in various civilian and military research areas, including ocean resource exploration, climate monitoring, and marine security [1,2,3,4,5]. In the context of the IoUTs, underwater acoustic wireless sensor networks (UAWSNs) are the key technology for establishing connections among underwater nodes, where location information is crucial for communication, data analysis, and transmission [6,7,8,9]. However, direct location determination using the global positioning system (GPS) is challenging due to severe radio signal attenuation underwater [10,11]. Consequently, various underwater localization techniques have been developed, typically comprising two main procedures: range acquisition and location computation. The range acquisition phase employs various ranging techniques, such as time of arrival (TOA), time difference of arrival (TDOA), angle of arrival (AOA), and received signal strength (RSS)-based schemes [12,13,14,15]. The location computation phase then utilizes algorithms to calculate the precise position. In addition to these techniques, there is renewed interest in combining communication and echo-location, known as integrated sensing (ranging) and communication, for underwater acoustic communication systems [16]. These advancements have significantly enhanced underwater target localization technologies. Compared to other methods, RSS-based schemes can obtain location information without requiring additional facilities or time synchronization [17]. Therefore, there has been a surge of interest in RSS-based underwater target localization [18,19,20,21].

Despite the development of several RSS-based localization techniques, accurately determining location information remains a challenge [22,23,24]. One contributing factor is the highly dynamic underwater environment, which introduces significant measurement noise [14,25]. Another challenge arises from the heterogeneous underwater environment, which causes stratified signal propagation and signal absorption [26,27]. These factors can severely impact localization accuracy [28]. Some research in the literature has been investigated to address these adverse impacts. For instance, Mei et al. developed a coarse-to-fine localization method (CFLM) that formulates the problem within an alternating non-negative constrained least squares (ANCLS) framework and offers an iterative closed-form solution [18] to address the challenges. Similarly, Chang et al. presented another closed-form expression that transforms the underwater localization problem into a generalized trust region sub-problem (GTRS) framework. A square range weighted least squares (SRWLS) approach was then further proposed to derive the solution [29].

However, many of these methods rely on highly strict mathematical assumptions. For example, the SRWLS method in [29], which is based on the Lagrange multiplier method, performs satisfactorily when the Karush–Kuhn–Tucker conditions are met [30]. Nevertheless, these assumptions may not always hold, resulting in inconsistent localization performance across different scenarios [31]. To address these limitations, various meta-heuristic algorithms have been proposed for the localization problem [32]. Sivakumar et al. developed several meta-heuristic algorithms to tackle localization issues using mobile anchor positioning (MAP) [33]. Saha et al. introduced an adaptive virtual anchor node technique based on particle swarm optimization (PSO) that derives the shortest path to minimize multi-hop errors [34].

Notably, according to no-free-lunch theorem [35], not a single method can be suitable for every scenario due to each method suffering from stratified propagation and absorption simultaneously. The more complex the problem is, the higher probability that a meta-heuristic algorithm may fall into a local minimum [36], especially for the Harris hawks optimization (HHO) [37]. The HHO is inspired by the hunting behavior of Harris hawks and has been proved to have relatively good performance in terms of convergence and solution acquirement in the literature [38]. Although HHO performs well in specific scenarios, its effectiveness is limited in the context of underwater localization [39]. Stratification and absorption effects in underwater environments can cause significant measurement divergence, resulting in a highly non-convex optimization problem. The exploration and exploitation phase of the original HHO is not efficient for this type of problem and may converge prematurely, leading to a local optimum.

The strict mathematical assumptions of current localization techniques and the limitations of the HHO algorithm have motivated us to develop a global or near-global search meta-heuristic-based underwater localization method for IoUTs. To the best of our knowledge, a meta-heuristic-based approach that effectively addresses RSS-based localization issues, particularly when dealing with stratification and absorption effects, has not been fully explored. In this context, this study proposes a quantum enhanced HHO (QEHHO) for the RSS-based underwater target localization problem. Four strategies are introduced to enhance HHO, with the aim of achieving a satisfactory localization solution. The main contributions of this work are as follows:

• By considering stratified propagation and absorption effects, this study reformulates the RSS-based underwater measurement model. Subsequently, an optimization framework is constructed using the first-order Taylor series expansion.
• A QEHHO with multiple strategies is proposed to enhance HHO and identify the optimal solution to the problem. These strategies include quantum-computing-inspired initiation, nonlinear transition, exploration enhancement with SMA and QOL, and a correction mechanism: each applied during different phases of HHO.
• The RSS-based Cramér–Rao lower bound (CRLB) with stratification and absorption effects is conducted as a benchmark for the problem. Several simulations are performed under different conditions, and we compare the results with those of other meta-heuristic and closed-form methods.

The remainder of this work is organized as follows: Section 2 provides an overview of related works on underwater localization. Section 3 presents the formulation of the underwater localization problem, considering stratified propagation and absorption effects. Subsequently, Section 4 introduces the proposed QEHHO method. Section 5 discusses the RSS-based CRLB, incorporating stratification and absorption considerations. Section 6 presents simulation results that validate and compare the proposed approach with various methods in different scenarios. Finally, concluding remarks and future directions are discussed in Section 7.

2. Related Works

Underwater localization based on acoustic signals typically involves two main components: (1) measurement acquisition and (2) location determination [22]. Measurements can be derived from time, angle, or signal intensity: known respectively as TOA/TDOA-, AOA-, and RSS-based techniques. Location determination uses various algorithms, including closed-form expressions and meta-heuristic approaches, to solve the localization problem. Research has demonstrated the advantages of RSS-based localization over AOA, TOA, and TDOA, particularly regarding cost and time synchronization [17,40]. Consequently, RSS-based localization techniques have garnered significant attention in the literature [41]. For instance, Aubry et al. explored optimal sensor placement using RSS-based measurements to minimize localization error, employing the trace/determinant of CRLB as the objective function and developing a closed-form expression [42]. Sun et al. tackled the RSS-based localization problem in mixed line-of-sight and non-line-of-sight environments by developing a robust weighted least squares method with an iterative approach [43]. He et al. proposed a unified analytical framework for RSS-based localization and developed a tractable expression for the localizability model [44].

However, obtaining accurate location information using RSS-based techniques becomes infeasible when accounting for stratification and absorption effects simultaneously [18]. These effects, which are prevalent in underwater acoustic signal communications, generally arise from the heterogeneous nature of the underwater environment [45]. To address these challenges, several innovative approaches have been developed. Tao et al. introduced a multiple-RSS-frequency-based underwater localization technique and employed an interior-point algorithm to solve the mapping problem [20]. Poursheikhali et al. proposed an array-RSS localization method that uses an iterative approach to address location determination [21]. Xu et al. developed an efficient underwater localization method based on semi-definite programming (SDP) [40]. Pourkabirian et al. integrated RSS measurements and angle information to create a hybrid localization scheme using SDP efficiently [46]. Gao et al. proposed a similar hybrid scheme that combines RSS-based localization with time-of-flight information, introducing a consensus fusion estimator to mitigate falsification attacks [47]. Furthermore, Ma et al. researched scenarios with unknown transmit power and presented a two-step closed-form localization approach [48].

The aforementioned solutions rely on highly strict mathematical assumptions. For instance, the algorithms proposed in [29,48] are variants of the Lagrange multiplier method, which function only when the Karush–Kuhn–Tucker conditions are satisfied [49]. Similarly, the SDP and its variants require the formulation to be relaxed to a convex expression [50]. To address these limitations, meta-heuristic-based localization approaches have been proposed that waive these assumptions and iteratively find solutions [51,52]. For instance, Draz et al. used a bio-inspired meta-heuristic algorithm to schedule nodes, developing a trust-based hybrid-blockchain-enabled localization method [53]. Kumari et al. considered anchor failures and investigated an NSGA-based meta-heuristic scheme for underwater localization [54]. Liu et al. developed a three-step localization approach using a meta-heuristic algorithm, the gray wolf optimizer, in the final round [55]. A single meta-heuristic algorithm may have limitations in obtaining the global minimum, prompting research into hybrid schemes. Xu et al. investigated a hybrid scheme using particle swarm optimization and genetic algorithms with quantum behavior to address underwater cooperative localization [56].

In addition to the algorithms previously mentioned, numerous meta-heuristic algorithms have been proposed in recent years, with several being applied to localization. These include but are not limited to the whale optimization algorithm (WOA) [57], seagull optimization algorithm (SOA) [58], sparrow search algorithm (SSA) [59], and HHO [38]. A summary of the aforementioned localization methods is shown in

Table 1. Summary and comparison of the aforementioned related localization approaches.

Reference	Stratification Consideration	Absorption Consideration	Time-Synchronization-Free	Strict Mathematical Assumption	Meta-Heuristic-Based Approach
[18]	✓	✓	✓	✓	×
[29]	×	✓	✓	✓	×
[33]	×	×	✓	×	✓
[34]	✓	×	×	×	✓
[42]	×	×	✓	✓	×
[43]	×	×	✓	✓	×
[44]	×	×	✓	✓	×
[20]	×	✓	✓	✓	×
[40]	✓	✓	✓	✓	×
[46]	×	✓	✓	✓	×
[47]	✓	✓	×	✓	×
[48]	×	✓	✓	✓	×
[53]	×	×	×	×	✓
[54]	×	×	×	×	✓
[55]	×	✓	✓	×	✓
[56]	×	×	×	×	✓
[60]	×	×	✓	×	✓

. Notably, HHO has demonstrated satisfactory performance in terms of convergence and efficiency in various scenarios [38], leading to its application in diverse areas such as localization [60], feature selection [61], and economic load dispatch problems [62]. Variants of HHO have also been proposed to address different engineering problems [39].

However, according to the no-free-lunch theorem [35], no single method is universally suitable for every scenario. HHO, in particular, is not well-suited for complex, non-convex underwater localization problems that simultaneously suffer from stratified propagation and absorption effects [37]. The exploration and exploitation phases of the original HHO algorithm are prone to falling into local minima in such complex problems. Furthermore, HHO uses a linear decreasing method to adjust the escape energy factor, which can lead to an imbalance between the exploration and exploitation phases. This method fails to accurately represent the multi-round escape dynamics between the hawk and prey in real scenarios. As the energy inevitably falls below 1 in the later iterations, the algorithm predominantly engages in a local search, thereby lacking global search capabilities. Consequently, if the population converges near a local optimum early on, the algorithm is prone to becoming trapped in that local optimum at a later stage.

The limitations imposed by strict mathematical assumptions and the tendency of the HHO algorithm to fall into local minima have prompted the development of a more suitable variant of HHO for underwater localization. To address these challenges, we propose the integration of four novel strategies designed to enhance the performance of HHO. These strategies are incorporated into each phase of the algorithm and aim to improve the quality of the initial population, diversify transition mechanisms, increase exploration efficiency, and bolster the robustness of the solutions.

3. Problem Formulation

To formulate the localization problem underwater, this section first introduces the RSS-based signal attenuation measurement model. By quoting the ray tracing theorem, the stratification-aware RSS framework is then formulated. According to the first-order Taylor series expansion, the fitness function of underwater localization is finally derived.

3.1. RSS-Based Stratified Propagation Model

We consider a three-dimensional UAWSNs scenario set up with N anchors and a target. Assume the $i^{\mathrm{th}}$ anchor location is ${\mathit{a}}_i={[a_{i1},a_{i2},a_{i3}]}^T$, where $i=1,\cdots ,N$ and T represents the transpose operation. In the network, anchors are assumed to be aware of their location and the target location needs to be determined [28]. Let the target’s location be $\mathit{x}={[x_1,x_2,x_3]}^T$. In addition, the underwater target can transmit an acoustic signal with RSS information to anchors, and the signal intensity may decrease with distance, modeled as [40]

$$P_{\mathrm{r}i}=P_0-10\alpha {\log }_{10}\frac{d_i}{d_0}-{\alpha }_f\left(d_i-d_0\right)+{\eta }_i,$$

(1)

where $P_{\mathrm{r}i}$ is the received signal power of the $i^{\mathrm{th}}$ anchor from the target, $P_0$ denotes the transmit power, $\alpha$ represents the path loss factor, $d_i$ is the distance between the $i^{\mathrm{th}}$ anchor and target, $d_0$ is known as the reference distance and is normally 1 m, ${\eta }_i$ indicates the Gaussian distributed noise with variance ${\sigma }_i^2$ and zero mean, and ${\alpha }_f$ denotes the absorption coefficient that is derived from Thorp’s formula following [63]

$${\alpha }_f=\frac{0.11f^2}{1+f^2}+\frac{44f^2}{4100+f^2}+2.75\times {10}^{-4}{f}^2+3\times {10}^{-3},$$

(2)

with f indicating the frequency.

Following the study by [45], the frequency f in UAWSNs typically ranges from 10 kHz to 1000 kHz. Consequently, the absorption coefficient ${\alpha }_f$ varies from 0.001 dB/m to 0.32 dB/m [23]. Therefore, the absorption effect when the reference distance $d_0=1$ m can be ignored, and (1) can be further converted into $P_{\mathrm{r}i}=P_0-10\alpha {\log }_{10}\frac{d_i}{d_0}-{\alpha }_f{d}_i+{\eta }_i$ [18].

Further, it is also important to note that the distance $d_i$ is typically assumed to be the Euclidean distance, which presupposes straight-line propagation of the acoustic signal. However, this assumption is invalid in an inhomogeneous underwater environment [26], where the acoustic signal propagates in a stratified manner according to Snell’s law [64], such that

$$\frac{\cos \theta }{C\left(z\right)}=\frac{\cos {\theta }^x}{C\left(x_3\right)}=\frac{\cos {\theta }^i}{C\left(a_{i3}\right)}=\kappa ,\mathrm{and}{\theta }^x,{\theta }^i\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right],$$

(3)

where ${\theta }^x$ denotes the transmission angle of the target, ${\theta }^i$ is the received angle of the $i^{\mathrm{th}}$ anchor, $\kappa$ is a constant, and $C(·)$ denotes the isogradient model [65] such that

$$C\left(z\right)=\rho z+b,$$

(4)

with $\rho$ indicating the steepness of the sound speed profile (SSP), z representing the depth, and b denoting the sound speed on the sea surface.

Also, from Figure 1, we have

$$\partial r=\frac{\partial z}{\tan \theta },\partial l=\frac{\partial z}{\sin \theta },$$

(5)

where r is the horizontal Euclidean distance such that $r=r^x-r^i=\sqrt{{(x_1-a_{i1})}^2+{(x_2-a_{i2})}^2}$, and l is the arc length of a ray. Given t is the travel time, then we have $\partial t=\frac{\partial l}{C\left(z\right)}$.

The underwater ray length is then derived with an integral manipulation under an isogradient SSP such that

$$l=-(\rho x_3+b)\frac{{\theta }^x-{\theta }^i}{\rho \cos {\theta }^x},$$

(6)

where ${\theta }^x={\beta }_0+{\alpha }_0,{\theta }^i={\beta }_0-{\alpha }_0$ with ${\beta }_0=\arctan [(a_{i3}-x_3)/(r^x-ri)]$ and ${\alpha }_0=\arctan [\left(0.5\rho (r^x-r^i)\right)/(b+0.5\rho (a_{i3}+x_3))]$.

Subsequently, the RSS-based stratified propagation measurement model can be expressed as

$$P_{\mathrm{r}i}=P_0-10\alpha {\log }_{10}\frac{l_i}{d_0}-{\alpha }_f{l}_i+{\eta }_i,$$

(7)

where $l_i$ is the length traveled between the $i^{\mathrm{th}}$ anchor and the target. Notably, the distance an acoustic signal can travel depends on the acoustic communication equipment and the transmission frequency. For instance, in civilian applications, a 1 kHz sound wave can travel tens or even hundreds of kilometers, a 10 kHz sound wave can travel approximately ten kilometers, a 100 kHz sound wave can travel a few hundred meters, and a 1 MHz sound wave can only travel a few meters [45].

3.2. Fitness Function Formulation

By addressing a simple transformation, (7) is then converted into

$$l_i=d_0{10}^{\frac{P_0-P_{\mathrm{r}i}-c_i+{\eta }_i}{10\alpha }},$$

(8)

where $c_i={\alpha }_f{l}_i$. Note that the maximum of $c_i$ can be known in some specific situations when the area is determined [23].

With the first-order Taylor series expansion theorem, (8) can be further expressed as

$$l_i\approx {10}^{\frac{P_0-P_{\mathrm{r}i}-c_i}{10\alpha }}\left(1+\frac{ln10}{10\alpha }{\eta }_i\right),$$

(9)

where $d_0$ is 1 m.

Let $\parallel \mathit{x}-{\mathit{a}}_i\parallel$ be the true distance between the $i^{\mathrm{th}}$ anchor and the target. Then, the localization problem can be formulated as

$$\underset{\mathit{x}}{\arg \min }\sum _{i=1}^N{\left(l_i-\parallel \mathit{x}-{\mathit{a}}_i\parallel \right)}^2.$$

(10)

The problem in (10) can be solved using various closed-form expressions, such as SDP or second-order cone programming (SOCP) through convex optimization as well as least squares and its variants [22]. However, these closed-form solutions rely on stringent assumptions. To address this limitation, this study explores solving the underwater localization problem using a metaheuristic optimization method. In this approach, (10) serves as the fitness function, and each population member calculates its fitness value in every iteration. The iterative procedure continues until the stopping criteria are met.

4. Proposed Quantum-Behaved Metaheuristic Optimization

To determine the target location, this section proposes a quantum-behaved metaheuristic method: a variant of HHO. Several strategies are incorporated to enhance performance and achieve an optimal solution for (10). This section focuses on fundamental mathematical expressions rather than detailing the specific mechanics of HHO, which can be referenced in [38]. Initially, a brief overview of the original HHO is provided, followed by the introduction of four strategies designed to improve different phases of the HHO algorithm.

4.1. Original HHO Algorithm

HHO comprises three phases in the search for the optimal solution: exploration, transition, and exploitation. The transition phase involves determining whether the process should shift from exploration to exploitation based on an energy criterion. The exploration phase simulates the behavior of searching for prey, while the exploitation phase emulates the act of hunting the prey. In this context, the position of the Harris hawk represents a candidate solution, with the best candidate solution in each iteration identified as the prey.

HHO commences with the exploration phase, which introduces a parameter q to guide the hawk’s strategy selection. Specifically, when $q<0.5$, each hawk’s movement is influenced by the positions of other hawks and the prey. Conversely, when $q\ge 0.5$, a random selection method is employed, as illustrated in

$$X(t+1)=\left\{\begin{array}{cc}{X}_{\mathrm{rand}}\left(t\right)-rand\left|X_{\mathrm{rand}}\left(t\right)-2randX\left(t\right)\right|& q\ge 0.5\\ \left(X_{\mathrm{prey}}\left(t\right)-X_m\left(t\right)\right)-rand\left(lb+rand(ub-lb)\right)& q<0.5\end{array}\right.$$

(11)

where $X(t+1)$ is the position of the hawk in the $t+1$ iteration, $X_{\mathrm{rand}}\left(t\right)$ is a randomly selected individual in the population of iteration t, $X_{\mathrm{prey}}\left(t\right)$ denotes the prey location, $X_m\left(t\right)$ indicates the average location of the current Harris’s hawk population such that $X_m\left(t\right)=\frac{1}{M}{\sum }_{j=1}^M{X}_j\left(t\right)$, with M representing the population size, $ub$ and $lb$ are the upper and lower bounds of the search range, respectively, and $rand$ and q are random numbers following the uniform distribution within the interval (0,1).

Afterwards, HHO employs an energy transition step to figure out whether it enters into the exploitation phase, as shown in

$$E=2E_0\left(1-\frac{t}{{\mathrm{Max}}_t}\right),$$

(12)

where E indicates the energy of the prey, $E_0$ is the initial energy, which varies from −1 to 1 at each iteration, and ${\mathrm{Max}}_t$ denotes the maximum iteration. The hawks enter into the exploration phase when $\left|E\right|\ge 1$, while they get to the exploitation phase when $\left|E\right|<1$.

Four strategies are proposed for the exploitation phase to address various prey statuses: soft besiege, hard besiege, soft besiege with progressive rapid dives, and hard besiege with progressive rapid dives. A tuning parameter $sp$, representing the prey’s escape probability, is introduced to determine the appropriate hunting behavior. The tuning parameter $sp$ is randomly selected from the interval $\left(0,1\right)$. When $sp<0.5$, it indicates that the prey has a probability of escaping.

Both the energy E and $sp$ impact the hunting strategy. For instance, the hawks execute a soft besiege when $0.5\le \left|E\right|<1$ and $sp\ge 0.5$, as shown in

$$X(t+1)=\Delta X\left(t\right)-E\left|\tau X_{\mathrm{prey}}\left(t\right)-X\left(t\right)\right|,$$

(13)

where $\Delta X\left(t\right)$ is the divergence between the position of the prey and the current location at t such that $\Delta X\left(t\right)=X_{\mathrm{prey}}\left(t\right)-X\left(t\right)$, and $\tau =2\left(1-rand\right)$.

If the prey does not have enough energy and a chance to escape, i.e., $\left|E\right|<0.5$ and $sp\ge 0.5$, the hawks then implement a hard besiege for hunting, as shown by

$$X(t+1)=X_{\mathrm{prey}}\left(t\right)-E|\Delta X\left(t\right)|.$$

(14)

The hawks will perform the soft besiege with progressive rapid dives strategy when $\left|E\right|\ge 0.5$ and $sp<0.5$, indicating that the prey has both the chance and energy to escape. This behavior is modeled by

$$X(t+1)=\left\{\begin{array}{cc}\hfill \mathrm{\Omega }\hfill & \hfill \mathrm{if}F\left(\mathrm{\Omega }\right)<F\left(X\right(t\left)\right)\hfill \\ \hfill \mathrm{{\rm Y}}\hfill & \hfill \mathrm{if}F\left(\mathrm{{\rm Y}}\right)<F\left(X\right(t\left)\right)\hfill \end{array}\right.,$$

(15)

where $\mathrm{\Omega }=X_{\mathrm{prey}}\left(t\right)-E\left|\tau X_{\mathrm{prey}}\left(t\right)-X\left(t\right)\right|$, $F(·)$ indicates the fitness function of (10), and $\mathrm{{\rm Y}}=\mathrm{\Omega }+\mathrm{\Psi }\times \mathrm{Levy}\left(D\right)$, with D, $\mathrm{\Psi }$, and $\mathrm{Levy}(·)$ denoting the dimension, random vector with $1\times D$, and levy flight function, respectively. Specifically, the levy flight function can be expressed as [66]

$$\mathrm{Levy}\left(x\right)=0.01\times \frac{\mu \times \varrho }{{\left|v\right|}^{\frac{1}{\beta }}},$$

(16)

where $\mu$ and v are random numbers ranging from $\left(0,1\right)$, and

$$\varrho ={\left(\frac{\mathrm{\Gamma }(1+\beta )\times \sin \left(\frac{\pi \beta }{2}\right)}{\mathrm{\Gamma }\left(\frac{1+\beta }{2}\right)\times \beta \times 2^{\left(\frac{\beta -1}{2}\right)}}\right)}^{\frac{1}{\beta }},$$

(17)

with $\beta$ and $\mathrm{\Gamma }\left(·\right)$ representing a preset parameter and gamma function, respectively.

If the prey is exhausted but still has a chance to escape, i.e., $\left|E\right|<0.5$ and $sp<0.5$, the hawks then execute a hard besiege with progressive rapid dives to reduce the distance between them and the mean position of the prey. The mathematical formulation is similar to the soft besiege but utilizes the average position of the population, as shown in

$$X(t+1)=\left\{\begin{array}{cc}\hfill \tilde{\mathrm{\Omega }}\hfill & \hfill \mathrm{if}F\left(\tilde{\mathrm{\Omega }}\right)<F\left(X\left(t\right)\right)\hfill \\ \hfill \widetilde{\mathrm{{\rm Y}}}\hfill & \hfill \mathrm{if}F\left(\widetilde{\mathrm{{\rm Y}}}\right)<F\left(X\left(t\right)\right)\hfill \end{array}\right.,$$

(18)

where $\tilde{\mathrm{\Omega }}=X_{\mathrm{prey}}\left(t\right)-E\left|\tau X_{\mathrm{prey}}\left(t\right)-X_m\left(t\right)\right|$, and $\widetilde{\mathrm{{\rm Y}}}=\tilde{\mathrm{\Omega }}+\mathrm{\Psi }\times \mathrm{Levy}\left(D\right)$.

The hunting behavior is carried out iteratively according to $\left|E\right|$ and $sp$ until the hawks catch the prey or the stop criteria has been reached.

4.2. Multi-Strategy Integration-Based QEHHO

In this section, four strategies are proposed to enhance the performance of HHO in the underwater localization scenario, including quantum-computing-inspired initiation, a nonlinear transform strategy, exploration enhancement, and a correction mechanism.

4.2.1. Quantum-Computing-Inspired Initiation

To enhance the diversity of the initial population, this section proposes a quantum-behaved approach by incorporating good point set theory [67] and quantum computing principles [68]. Let $G_s$ denote the unit cube in s dimensional Euclidean space, and $r=\left(r_1,r_2,\cdots ,r_s\right)\in G_s$; then, we have

$$P_M\left(k\right)=\left\{\left(\left\{r_1^{\left(M\right)}·k\right\},\left\{r_2^{\left(M\right)}·k\right\},\cdots ,\left\{r_s^{\left(M\right)}·k\right\}\right),1\le k\le M\right\},$$

(19)

where $\left\{r_m^{\left(M\right)}·k\right\}$ is the decimal part of $r_m^{\left(M\right)}·k$.

Assume $r=\left\{2\cos \left(2\pi k/p\right),1\le k\le s\right\}$, where p is the smallest prime that satisfies $s\le \left(p-3\right)/2$. $P_M\left(k\right)$ is the good points set, and r is the good point if the deviation satisfies

$$\varphi \left(M\right)=C(r,\epsilon )M^{-1+\epsilon },$$

(20)

where $\epsilon$ is an arbitrary positive number, and $C(r,\epsilon )$ represents a constant related to r and $\epsilon$.

To further augment the quality of the initial population, quantum computing is introduced to enhance diversity. Each individual m within the population can be expressed as a superposition of two states using quantum bits, thereby yielding:

$$m=\left({\phi }_1\left|{\phi }_2\right|\dots \mid {\phi }_o\right),$$

(21)

where $\left({\phi }_1,\cdots ,{\phi }_o\right)$ represents a quantum bit of length o with

$$\left|{\phi }_o\right.〉={\varpi }_0|0\rangle +{\varpi }_1|1\rangle ,$$

(22)

where ${\varpi }_0$ and ${\varpi }_1$ are the amplitudes of the probability and satisfy ${\varpi }_0^2+{\varpi }_1^2=1$.

When a quantum state in superposition is observed, it collapses to zero with a probability of ${\varpi }_0^2$ or to one with a probability of ${\varpi }_1^2$. The quantum bit can be represented in a matrix form using sine and cosine based on the probability amplitude as $\left|{\phi }_o\right.〉={\left[\cos {\theta }_o,\sin {\theta }_o\right]}^T$, where $\cos {\theta }_o={\varpi }_0$ and $\sin {\theta }_o={\varpi }_1$. Then, (21) can be further converted into

$$m=\left(\begin{array}{cccc}\cos \left({\theta }_1\right)& \cos \left({\theta }_2\right)& \cdots & \cos \left({\theta }_o\right)\\ \sin \left({\theta }_1\right)& \sin \left({\theta }_2\right)& \cdots & \sin \left({\theta }_o\right)\end{array}\right|),$$

(23)

where ${\theta }_1,\cdots ,{\theta }_o$ are randomly picked from $[0,2\pi ]$. Each individual occupies two positions in the search space, with each position representing an optimization solution that can be decomposed into

$$m_x=\left(\cos \left({\theta }_o\right),\cos \left({\theta }_2\right),\cdots ,\cos \left({\theta }_o\right)\right);$$

(24)

$$m_y=\left(\sin \left({\theta }_o\right),\sin \left({\theta }_2\right),\cdots ,\sin \left({\theta }_o\right)\right).$$

(25)

Consequently, a mapping relationship between the quantum space and the solution space can be established through a linear transformation. Given a quantum bit ${\left[\cos {\theta }_o,\sin {\theta }_o\right]}^T$ with a value domain of $[-1,1]$, a transformation is derived as

$$m_x=\frac{1}{2}(\cos \theta +1)(ub-lb)+lb.$$

(26)

$$m_y=\frac{1}{2}(\sin \theta +1)(ub-lb)+lb.$$

(27)

Given that the initial population has been determined using $P_M\left(k\right)$ from the previous stage, this corresponds to collapsing the quantum state to a definite state following a single observation. To revert to the superposition state, an inverse transformation can be applied, assuming that the current state is represented as a sinusoidal position, such that

$${\theta }^{*}=\arcsin \left(\frac{2\left(m_y-lb\right)}{ub-lb}-1\right),$$

(28)

where another state can be derived using ${\theta }^{*}$, as expressed by

$$m_x^{*}=\frac{1}{2}\left(\cos {\theta }^{*}+1\right)(ub-lb)+lb.$$

(29)

Combining the quantum computing and good point set theories based on a set of quantum bits generates a larger initial population. This approach significantly enhances the population diversity and distribution across the search space, thereby improving global search capabilities.

4.2.2. Nonlinear Transition Strategy

As demonstrated in the original HHO, the transition between the exploration and exploitation phases is governed by the energy $\left|E\right|$ as defined in Equation (12). Extensive simulations reveal that $\left|E\right|$ consistently falls below 1 during the latter half of the iterations ($t>{\mathrm{Max}}_t/2$). This results in an interesting phenomenon where only the exploitation phase is executed in the later stages. Consequently, if the global optimum or a near-global optimum is not identified during the early phase, HHO is prone to becoming trapped in a local minimum. Therefore, a nonlinear transition strategy is proposed, as represented by

$$E=\frac{1}{2}+\frac{1}{2}\sin \left(\frac{\pi }{2}+\frac{\pi t}{{\mathrm{Max}}_t}\right),$$

(30)

where the exploration phase is conducted when $E>rand$, and the exploitation phase is executed otherwise.

The nonlinear transition strategy facilitates extensive region exploration in the early stages when E decreases slowly and maintains the capability for global exploration in the later stages when E is small. This approach enhances the algorithm’s ability to escape local optima.

4.2.3. Exploration Enhancement

The slime mold algorithm (SMA) is a powerful optimization algorithm known for its excellent global optimization capabilities [69]. SMA adjusts its search modes based on the fitness value and isolates a subset of organisms for exploration in other domains. Additionally, the oscillating effect of $v_b$ increases the likelihood of global exploration. This multi-exploration mechanism endows SMA with robust global optimization search capabilities. To address the tendency of the HHO to fall into local optima when facing complex problems, this paper incorporates the multi-search mechanism of the SMA into the HHO’s exploration phase to enhance its global search capabilities. Assume $z_0$ is the preset slime parameter. In traditional SMA, when $rand<z_0$, the slime mold population randomly generates individuals from the solution space, simulating the behavior of slime molds isolating some individuals to explore other potential food sources. While this approach prevents premature convergence, it does not fully utilize the information of the current solution.

Therefore, we further integrate quasi-oppositional learning (QOL) into the exploration phase. QOL allows for a broader and more rational range of exploration, increasing the chances of approaching the optimal or sub-optimal points in the search space [70]. Consequently, in this paper, we replace the random individual strategy at $rand<z_0$ in the slime mold algorithm with QOL. This enhancement not only helps to avoid premature convergence but also improves the likelihood of approaching the optimal food source. By combining the SMA and QOL, the improved exploration phase can be expressed as

$$X(t+1)=\left\{\begin{array}{cc}{X}_{\mathrm{prey}}\left(t\right)+v_b\left(W·X_A\left(t\right)-X_B\left(t\right)\right),& rand>z_0,ran{d}_0\ge p\\ \left(X_{\mathrm{prey}}\left(t\right)-X_m\left(t\right)\right)-rand\left(lb+rand(ub-lb)\right),& rand>z_0,ran{d}_0<p\\ \mathrm{rand}\left(\frac{lb+ub}{2},lb+ub-X\left(t\right)\right),& rand<z_0\end{array}\right.,$$

(31)

where $v_b$ is a parameter with a range of $\left[-\left(\mathrm{atanh}\left(-\left(t/{\mathrm{Max}}_t\right)+1\right)\right),\left(\mathrm{atanh}\left(-\left(t/{\mathrm{Max}}_t\right)+1\right)\right)\right]$, with $\mathrm{atanh}\left(·\right)$ representing the atanh function, W indicates the weight of the slime mold, $X_A\left(t\right)$ and $X_B\left(t\right)$ are two individuals that are randomly selected from the slime mold, $ran{d}_0$ is the random number from $(0,1)$, and $p=\tanh |F\left(X\left(t\right)\right)-F\left(X_{\mathrm{prey}}\left(t\right)\right)|$.

4.2.4. Correction Mechanism

As the hawks in the population converge towards the optimal individual during the iteration process, they risk gathering around a local optimum. This convergence, which intensifies with an increasing number of iterations, can lead to a decline in population diversity. To mitigate the risk of the algorithm becoming trapped in local optima and to prevent premature convergence, this paper introduces the mean differential evolution mutation [71] to adjust both individual and global optima of the population. This method employs two distinct strategies tailored to different stages of the iteration process such that

$$X(t+1)=\left\{\begin{array}{cc}X\left(t\right)+w·F_1\left(X_{c1}+X_{c2}-2·X\left(t\right)\right),& rand<pm\mathrm{or}t<0.6{\mathrm{Max}}_t\\ X_{\mathrm{prey}}+w·F_2\left(X_{c1}+X_{c2}-2·X\left(t\right)\right),& rand\ge pm\mathrm{or}t\ge 0.6{\mathrm{Max}}_t\end{array}\right.,$$

(32)

where $pm$ is the mutation control parameter, $F_1$ and $F_2$ are random numbers in the ranges $[0,0.35]$ and $[-0.15,0.15]$, respectively [71], w denotes the inertia coefficient, which decreases with each iteration, $X_{c1}$ indicates the mean value of two random Harris hawks, and $X_{c2}$ represents the mean value of one random Harris hawk and the prey. This mutation strategy initially searches in the vicinity of individual solutions and later shifts to exploring around the global optimum, thereby enhancing the algorithm’s ability to escape local optima.

By integrating the four strategies in their corresponding phases, a quantum-behaved HHO with enhanced global search capability is developed. The flowchart of the proposed algorithm for underwater localization is presented in Figure 2, and the related parameters can be referred to in the simulation section.

5. RSS-Based Stratified CRLB

The CRLB is commonly considered the theoretical limit of localization accuracy [72], which is represented by the trace of the Fisher information matrix. Given the observed vector $\mathit{P}={\left[P_{\mathbf{r}i}\right]}^T$, the RSS-based stratified CRLB is then obtained by

$$CRLB=\mathrm{Trace}{\left[FIM\right]}^{-1},$$

(33)

where $\mathit{\vartheta }=\mathit{\eta }{\mathit{\eta }}^T$, with $\mathit{\eta }={[{\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_N]}^T$, $\mathrm{Trace}\left(·\right)$ is the trace function, and $FIM={\left(\frac{\partial \ln (\mathit{P};\mathit{x})}{\partial \mathit{x}}\right)}^T{\mathit{\vartheta }}^{-1}\left(\frac{\partial \ln (\mathit{P};\mathit{x})}{\partial \mathit{x}}\right)$, with

$${\left(\frac{\partial \ln (\mathit{P};\mathit{x})}{\partial \mathit{x}}\right)}_{N\times 3}=\left[\begin{array}{ccc}\frac{\partial P_{\mathrm{r}1}}{\partial x_1}& \frac{\partial P_{\mathrm{r}1}}{\partial x_2}& \frac{\partial P_{\mathrm{r}1}}{\partial x_3}\\ \frac{\partial P_{\mathrm{r}2}}{\partial x_1}& \frac{\partial P_{\mathrm{r}2}}{\partial x_2}& \frac{\partial P_{\mathrm{r}2}}{\partial x_3}\\ ⋮& \ddots & ⋮\\ \frac{\partial P_{\mathrm{r}N}}{\partial x_1}& \frac{\partial P_{\mathrm{r}N}}{\partial x_2}& \frac{\partial P_{\mathrm{r}N}}{\partial x_3}\end{array}\right].$$

(34)

In (34), the partial derivatives of $P_{\mathrm{r}i}$ with respect to $\mathit{x}$ can be expressed as

$$\frac{\partial P_{\mathrm{r}i}}{\partial x_1}=-10\alpha \frac{1}{l_i\ln 10}\frac{\partial l_i}{\partial x_1}-{\alpha }_f\frac{\partial l_i}{\partial x_1},$$

(35)

$$\frac{\partial P_{\mathrm{r}i}}{\partial x_2}=-10\alpha \frac{1}{l_i\ln 10}\frac{\partial l_i}{\partial x_2}-{\alpha }_f\frac{\partial l_i}{\partial x_2},$$

(36)

$$\frac{\partial P_{\mathrm{r}i}}{\partial x_3}=-10\alpha \frac{1}{l_i\ln 10}\frac{\partial l_i}{\partial x_3}-{\alpha }_f\frac{\partial l_i}{\partial x_3},$$

(37)

where the partial derivatives of $l_i$ with respect to $\mathit{x}$ are $\frac{\partial l_i}{\partial r^x}\frac{\partial r^x}{\partial x_1}$, $\frac{\partial l_i}{\partial r^x}\frac{\partial r^x}{\partial x_2}$, and $\frac{\partial l_i}{\partial x_3}$ are

$$\frac{\partial l_i}{\partial x_1}=-\frac{\rho x_3+b}{\rho \cos {\theta }^x}\left\{\left(1+\left({\theta }^x-{\theta }^i\right)\tan {\theta }^x\right)\frac{\partial {\theta }^x}{\partial r^x}-\frac{\partial {\theta }^i}{\partial r^x}\right\}\frac{x_1-a_{i1}}{r^x-r^i},$$

(38)

$$\frac{\partial l_i}{\partial x_2}=-\frac{\rho x_3+b}{\rho \cos {\theta }^x}\left\{\left(1+\left({\theta }^x-{\theta }^i\right)\tan {\theta }^x\right)\frac{\partial {\theta }^x}{\partial r^x}-\frac{\partial {\theta }^i}{\partial r^x}\right\}\frac{x_2-a_{i2}}{r^x-r^i},$$

(39)

$$\frac{\partial l_i}{\partial x_3}=-\frac{\rho x_3+b}{\rho \cos {\theta }^x}\left\{\left(1+\left({\theta }^x-{\theta }^i\right)\tan {\theta }^x\right)\frac{\partial {\theta }^x}{\partial x_3}-\frac{\partial {\theta }^i}{\partial x_3}\right\}-\frac{{\theta }^x-{\theta }^i}{\cos {\theta }^x},$$

(40)

The partial derivatives of ${\theta }^x$ and ${\theta }^i$ with respect to $r^x$ and $x_3$ are then obtained such that $\frac{\partial {\theta }^x}{\partial r^x}=-\frac{\tilde{\psi }\tilde{\varphi }}{1-\tilde{\varphi }}$, $\frac{\partial {\theta }^i}{\partial r^x}=\frac{\tilde{\psi }}{1-\tilde{\varphi }}$, $\frac{\partial {\theta }^x}{\partial x_3}=\frac{\tilde{\upsilon }-\tilde{\varphi }\tilde{\phi }}{1-\tilde{\varphi }}$, and $\frac{\partial {\theta }^i}{\partial x_3}=\frac{\tilde{\phi }-\tilde{\upsilon }}{1-\tilde{\varphi }}$,

where

$$\tilde{\psi }=-\frac{x_3+a_{i3}}{{\left(r^x-r^i\right)}^2}\frac{{\left(\sin {\theta }^x-\sin {\theta }^i\right)}^2}{1-\cos \left({\theta }^x-{\theta }^i\right)},$$

(41)

$$\tilde{\varphi }=-\frac{b+\rho x_3}{b+\rho a_{i3}}\frac{\sin {\theta }^i}{\sin {\theta }^x},$$

(42)

$$\tilde{\phi }=\frac{1}{r^x-r^i}\frac{{\left(\sin {\theta }^x-\sin {\theta }^i\right)}^2}{1-\cos \left({\theta }^x-{\theta }^i\right)},$$

(43)

$$\tilde{\upsilon }=-\frac{\rho }{b+\rho a_{i3}}\frac{\cos {\theta }^i}{\sin {\theta }^x},$$

(44)

6. Simulation Results and Discussion

To rigorously assess the proposed method, a series of simulations under various conditions were conducted using MATLAB R2022b. The study focuses exclusively on evaluating the performance of QEHHO for the underwater localization problem as outlined in Equation (10) rather than testing it on other functions or problems. The root mean square error (RMSE) is employed as the metric for evaluating localization accuracy, as described in [18], such that

$$RMSE=\sqrt{\frac{{\sum }_{mc=1}^{MC}{\left(\widehat{\mathit{x}}-\mathit{x}\right)}^2}{MC}},$$

(45)

where $\widehat{\mathit{x}}$ is the estimated location, $MC$ denotes the total number of Monte Carlo trials (MCTs), which is set as 1000 in the simulations, and $mc$ indicates the current MCT.

The area for simulation is $100\mathrm{m}\times 100\mathrm{m}\times 100\mathrm{m}$. All underwater nodes are randomly selected in each MCT so as to mimic the highly dynamic underwater environment [23]. Additionally, other fixed parameters are set according to [18,38] as follows: $\rho =0.1$, $b=1500$ m/s, $P_0=-55$ dBm, $\alpha =3$, ${\alpha }_f=0.06$ dB/m, $z_0=0.03$, $lb=-100\times {\mathbf{1}}_{1\times 3}$, with $\mathbf{1}$ representing a matrix with all elements being 1, $ub=100\times {\mathbf{1}}_{1\times 3}$, $w=w_{\min }+\left(w_{\max }-w_{\min }\right)\times t/{\mathrm{Max}}_t$, with $w_{\max }=0.9$ and $w_{\min }=0.6$, and $pm=0.5$. Several methods are introduced for comparison, including the CFLM [18], the original HHO-based localization (HHO) [60], SRWLS [29], WOA-based localization [73], a hybrid parallel HHO (HPHHO) localization [74], NOA-based localization [75], SOA-based localization [76], SSA-based localization [77], and the CRLB in (33).

6.1. Variable Iterations

To evaluate the performance of QEHHO, simulations were conducted under varying ${\mathrm{Max}}_t$ conditions with $N=8$, $M=30$, and ${\sigma }_i^2=3$ dB. It is important to note that CFLM, SRWLS, and CRLB are not meta-heuristic-based algorithms and therefore do not involve population or iteration parameters; thus, these methods are excluded from this subsection of the analysis. Meta-heuristic algorithms typically begin with a random initial guess, which can result in significant localization errors in the early stages. As illustrated in Figure 3, localization accuracy is unsatisfactory when ${\mathrm{Max}}_t$ is small. However, performance improves as the number of iterations increases, with the optimal solution being refined through alternating exploration and exploitation phases. Additionally, the NOA-based scheme exhibits lower localization accuracy compared to other methods. Even when ${\mathrm{Max}}_t$ reaches 50, the disparity between NOA and the other algorithms remains substantial. Similar performance trends are observed when ${\mathrm{Max}}_t=1$ for NOA, WOA, SOA, and HHO, with the key difference being the sensitivity to ${\mathrm{Max}}_t$. The performance of WOA, SOA, and HHO improves significantly compared to NOA as ${\mathrm{Max}}_t$ increases.

The proposed QEHHO experiences a performance decline when ${\mathrm{Max}}_t$ is small, with localization accuracy lower than that of HPHHO and SSA. However, by incorporating quantum computing theory, the initial population quality surpasses that of HHO, leading to improved localization accuracy for QEHHO. Additionally, three strategies—nonlinear transition, exploration enhancement, and correction mechanism—are integrated at each phase, enabling QEHHO to achieve the optimal solution with fewer iterations (${\mathrm{Max}}_t$). As shown in Figure 3, QEHHO reaches the optimal solution at ${\mathrm{Max}}_t=10$, outperforming HPHHO and SSA. Beyond this point, performance oscillates slightly with increasing ${\mathrm{Max}}_t$. Figure 4 further illustrates this superiority through cumulative distribution function (CDF) analysis at ${\mathrm{Max}}_t=10$, where QEHHO achieves $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 3.56$ m at 90 percent probability. In comparison, NOA, HHO, WOA, HPHHO, and SSA reach $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 6.63$ m, $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 4.92$ m, $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 4.49$ m, $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 4.62$ m, $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 3.67$ m, and $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 3.63$ m, respectively, at the same probability.

6.2. Variable Populations

Population size is another crucial factor when evaluating the performance of a meta-heuristic algorithm. In this context, a simulation with variable population sizes M was conducted, with parameters set to $N=8$, ${\mathrm{Max}}_t=10$, and ${\sigma }_i^2=3$ dB, as depicted in Figure 5. Notably, the HPHHO algorithm imposes a strict assumption on population size; thus, the simulation varies the population from 8 to 50, considering only even numbers. A larger population size generally increases the probability of successful optimization. As such, localization accuracy is found to be inversely proportional to population size, with NOA showing a significant deviation from the other methods. Interestingly, it is observed that all algorithms tend to converge to a specific error level and then fluctuate around this error, with the convergent population size differing among algorithms. For example, HPHHO, SSA, and the proposed QEHHO converge at an error around a population size of 30.

The QEHHO algorithm, which incorporates a superposition state in the population initialization, enhances population quality and diversity. As a result, QEHHO can achieve optimal performance even with smaller population sizes compared to HPHHO and SSA. Furthermore, QEHHO demonstrates superior localization accuracy with increasing M, maintaining a convergent error of approximately 2.65 m, outperforming other methods. This superior performance of QEHHO is also evidenced in Figure 6, where it achieves $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 3.60$ m with a 90 percent probability when $M=28$. In contrast, NOA, HHO, WOA, SOA, HPHHO, and SSA reach $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 6.67$ m, $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 5.07$ m, $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 4.59$ m, $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 4.78$ m, $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 3.74$ m, and $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 3.71$ m, respectively, at the same probability level.

6.3. Variable Anchors

Based on the previous experiments, we have determined that QEHHO performs well with low iterations and population sizes compared to other meta-heuristic algorithms. In this subsection, we conduct further simulations to compare both meta-heuristic algorithms and closed-form-expression-based approaches, including CFLM, SRWLS, and CRLB, under variable anchor conditions with $M=30$, ${\mathrm{Max}}_t=10$, and ${\sigma }_i^2=3$ dB. The results are depicted in Figure 7. The available information for localization increases with the number of anchors, thus improving localization accuracy as N rises.

Compared to some meta-heuristic algorithms, CFLM and SRWLS achieve better localization performance, as NOA, SOA, HHO, and WOA may converge to local optima during iterations. In contrast, the performance of HPHHO and QEHHO improves with the implementation of multiple strategies during iterations, outperforming CFLM and SRWLS, although there remains a significant gap between them and the CRLB. Despite no new strategies being introduced in SSA, it achieves satisfactory performance close to that of QEHHO. The superior performance of QEHHO is further illustrated in Figure 8, which depicts the CDFs of different methods at $N=12$. It is observed that QEHHO achieves $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 3.45$ m with a 98 percent probability, whereas the other algorithms achieve the same probability with larger errors. For instance, NOA reaches $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 7.09$ m, while the rest are $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 4.95$ m for SOA, $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 5.84$ m for HHO, $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 5.06$ m for WOA, $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 4.08$ m for SRWLS, $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 4.36$ m for CFLM, $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 3.803$ m for HPHHO, and $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 3.80$ m for SSA at the same probability.

6.4. Variable Noise

Noise can also adversely affect localization performance. In this context, we conduct simulations under varying ${\sigma }_i^2$ values with $M=30$, ${\mathrm{Max}}_t=10$, and $N=8$. The corresponding results are depicted in Figure 9. As expected, localization accuracy decreases as ${\sigma }_i^2$ increases. Integrating the findings from the previous three subsections, it is evident that NOA is not well-suited for underwater scenarios. This conclusion is supported by Figure 9, which shows a significant performance gap between NOA and other algorithms. SOA, HHO, and WOA demonstrate moderate performance, particularly when ${\sigma }_i^2$ is small.

Additionally, similar localization accuracy is observed for CFLM, HPHHO, SSA, and QEHHO at ${\sigma }_i^2=1$ dB. Although the margin is minimal, the superiority of QEHHO is evident when comparing Figure 9 and Figure 10. Figure 10 presents the CDF values of different methods at ${\sigma }_i^2=12$ dB, showing that QEHHO achieves $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 4.71$ m with a 90 percent probability. In contrast, other algorithms reach the same probability at larger errors: NOA at $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 6.80$ m, SOA at $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 5.28$ m, HHO at $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 5.55$ m, WOA at $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 5.17$ m, SRWLS at $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 5.21$ m, CFLM at $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 5.06$ m, HPHHO at $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 4.81$ m, and SSA at $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 4.72$ m.

However, this performance advantage diminishes at higher probabilities. For instance, SSA reaches $\parallel \widehat{\mathit{x}}-\mathit{x}\parallel \le 5.41$ m with a 95 percent probability, whereas QEHHO reaches the same probability at an error of 5.42 m. This indicates that the proposed method’s improvement is not as significant under high noise conditions. Nonetheless, considering all previous experiments, QEHHO remains a strong candidate for underwater localization.

6.5. Computational Time

Computational time, which reflects the complexity of an algorithm, is another crucial factor for evaluating performance. This subsection presents simulations conducted under the same conditions as the previous two subsections, with the corresponding results illustrated in Figure 11. Balancing accuracy and efficiency is particularly challenging for most localization methods, especially meta-heuristic-based approaches [32]. For instance, NOA demonstrated relatively low localization accuracy across different scenarios in the previous subsections; however, it exhibited the best efficiency, as shown in Figure 11. In contrast, the proposed QEHHO incorporates multiple strategies at each step to find the solution, resulting in increased computational time. Consequently, QEHHO has the highest computational time among the compared methods. Additionally, computational time is related to the population size and iteration count—higher values for these factors lead to increased computational time. Fortunately, despite having the highest computational time in the simulations, the average time for QEHHO is less than $8\times {10}^{-3}$ s. Furthermore, QEHHO can achieve satisfactory results with a low population and fewer iterations, as demonstrated in the first two subsections. Therefore, QEHHO is a promising solution for underwater localization, particularly when accuracy is of utmost importance.

7. Conclusions

In this paper, we conduct a comprehensive study on RSS-based underwater localization in IoUTs. A stratification-aware RSS model is constructed to compensate for measurement errors caused by stratification and absorption effects. By applying the first-order Taylor series expansion, the study reformulates underwater target localization as a non-convex problem. Unlike previous works that employ traditional mathematical expressions with strict assumptions, we utilize a meta-heuristic algorithm based on quantum-behaved theory and multiple-strategy integration. This algorithm, inspired by the hunting behavior of Harris hawks, addresses the tendency of the original HHO to fall into local minima, motivating the development of QEHHO. In QEHHO, four strategies are integrated into every phase of HHO to avoid local minima. During the initiation phase, we incorporate good point set theory and quantum computing to enhance population quality. In the transition phase, a random nonlinear technique replaces the linear scheme to facilitate extensive region exploration in the early stages. A correction mechanism and an exploration enhancement combining SMA and QOL are further developed to find an optimum. The proposed QEHHO eliminates the need for strict mathematical problem formulations and yields satisfactory solutions for underwater localization, as demonstrated by simulation results under various conditions.

However, there remains a significant gap between the performance of QEHHO and the CRLB, which we aim to address in future work to enhance localization accuracy. Additionally, the highly dynamic underwater environment and equipment aging may cause changes in the transmit power and path loss exponent, leading to performance loss. Developing satisfactory solutions that account for these uncertainties and evaluating the proposed method in real underwater environments will also be focus areas of our future research.