Underwater Sparse Acoustic Sensor Array Design under Spacing Constraints Based on a Global Enhancement Whale Optimization Algorithm

Abstract

Sparse arrays with low cost and engineering complexity are widely applied in many fields. However, the high peak sidelobe level (PSLL) of a sparse array causes the degradation of weak target detection performance. Particularly for the large size of underwater low-frequency sensors, the design problem requires a minimum spacing constraint, which further increases the difficulty of PSLL suppression. In this paper, a novel swarm-intelligence-based approach for sparse sensor array design is proposed to reduce PSLL under spacing constrains. First, a global enhancement whale optimization algorithm (GEWOA) is introduced to improve the global search capability for optimal arrays. A three-step enhanced strategy is used to enhance the ergodicity of element positions over the aperture. In order to solve the adaptation problem for discrete array design, a position decomposition method and a V-shaped transfer function are introduced into off-grid and on-grid arrays, respectively. The effectiveness and superiority of the proposed approach is validated using experiments for designing large-scale low-frequency arrays in the marine environment. The PSLL of the off-grid array obtained by GEWOA was nearly 3.8 dB lower than that of WOA. In addition, compared with other intelligent algorithms, the on-grid array designed using GEWOA had the lowest PSLL.

1. Introduction

Sparse sensor arrays are widely used in observation and communication systems, such as radar, sonar, satellite, etc. [1,2,3]. With fewer array elements, such arrays obtain better array performance, including the narrow main lobe width and the PSLL compared with equally spaced arrays. In addition, such arrays have clear advantages in terms of the installation cost and system complexity.

Generally, the design problems of sparse sensor arrays are mainly divided into two types [4]: (1) The element positions are optimized to minimize the PSLL with a fixed number of array elements and aperture [5]. (2) The array beampattern is designed to match the desired beampattern by optimizing the array element positions and weights while reducing the number of array elements [6,7]. Some arrays, such as large towed arrays and the seabed array, are prone to shift in array position and change in array configuration during operation, which causes a mismatch between the real positions and the design ones.

For example, the towed array is flexible with oil-filled polyurethane tubing. When the array is towed, it is difficult to keep the array straight; therefore, the real positions of the elements deviate from the designed positions. For the seabed array mounted at the bottom of sea, the uncertainties of the element positions and the entire array configuration are induced by ship drift and subsurface currents when the array deploys to the seafloor from the layout vessel. We focus on the first type problem under the marine environment to design a random sparse array with low PSLL to enhance the detection of weak targets [8].

The existing methods to solve the first problem are mainly divided into two categories: deterministic optimization methods and stochastic optimization methods. Deterministic optimization methods, including minimum redundancy array [9], coprime array [10], and difference coarray [11], are easy to implement; however, the designed arrays are generally suboptimal. Stochastic optimization methods use intelligent algorithms to find the element positions that satisfy the design requirements. Genetic algorithm (GA) [12], particle swarm optimization (PSO) algorithm [13], and simulated annealing algorithm [14] have been successfully applied to sparse sensor array design.

However, these intelligent algorithms converge slowly and easily fall into local optima. Some improved algorithms [15] have been proposed to solve the problem for array design—for example, a series of swarm intelligence optimization algorithms mimicking swarm behavior, including the wolf pack algorithm (WPA) [16], bat algorithm (BA) [17], and cuckoo search (CS) algorithm [18], which show good search accuracy and quick convergence speed. Although the stochastic optimization method suffers from global performance degradation during the design process of large-scale arrays, this method outperforms the deterministic optimization method in terms of the array performance as the calculation capacity increases.

The whale optimization algorithm (WOA) [19]—a swarm intelligence algorithm—has been applied successfully to sparse sensor array design for both fixed aperture and non-fixed aperture and has shown its advantages in sidelobe suppression and null steering for small- and medium-scale arrays [20,21]. Combining the salp swarm algorithm (SSA) and WOA, a novel swarm algorithm was proposed to improve its convergence accuracy for conformal antenna array design [22] and for dual-band aperture-coupled antenna design [23].

Nevertheless, the global search capability of WOA decreases with the expansion of the search range in large-scale array design. Researchers have attempted to improve the algorithm from several aspects in successive steps. The initialization strategy played an important role [24]. Chaos initialization improves population diversity due to its ergodic and random nature, and the chaotic particle algorithm was demonstrated to successfully reduce the PSLL of sparse sensor arrays [25].

With the development of intelligent algorithms, more chaotic initialization algorithms, including the chaotic invasive weed optimization algorithm [26], chaotic cuckoo search algorithm [27], and chaotic sparrow search algorithm [28], have been applied to sparse array design for improving global search capability. These success examples lead us to believe that WOA embedded with chaotic initialization will improve the performance of large-scale array design.

In the next step, search strategy optimization can be achieved by introducing adaptive weights [29] and search path planning [30]. WOA with adaptive weights had a good global search capability [31,32]. Search path planning is another effective method. Levy flight strategy, as a random search path strategy with global search capability, was introduced to WOA [33]. Although the method has not been widely applied to array design within WOA, it has been successful within other swarm algorithms for linear and circular array design, including the differential evolutionary algorithm [34], CS algorithm [27], InvasiveWeed Optimization [35], and biogeography-based optimization approach [36]. Unfortunately, the Levy flight strategy destroys the original whale spiral search, which causes the convergence speed to slow down; hence, a new search strategy should be introduced to maintain the speed.

In the third step to produce a new population, it was found that opposition-based learning (OBL) can also improve the search accuracy of WOA [37]. The combined strategy of adaptive weights with OBL was applied to high-dimensional global optimization problems [38]. However, it is easy to fall into the inverse local optimum, since the search range of the inverse solution is limited. A new learning algorithm should be developed to jump out of local minima. In short, the existing methods still have some shortcomings in the large-scale array design, particularly for the insufficient global search capability. To tackle this issue, GEWOA with a three-step enhanced global search strategy is introduced in this paper.

Additionally, the sparse sensor array design is a discrete problem, while the WOA is suitable for the solution of continuous variables. Therefore, WOA needs to be customized for discrete problems. Continuous variables discretization is generally achieved by transfer functions. If the discrete array element is on an integral multiple of the half wavelength, then this is an on-grid problem, otherwise it is an off-grid problem. The former is a discrete optimization problem with popular S-shaped and V-shaped transfer functions [39,40]. The latter uses binary discrete continuous variables with the ability to achieve lower PSLL [41,42]. Considering the physical limit of the array element size, the array design needs to constrain the minimum spacing between adjacent array elements while minimizing the PSLL.

A novel approach based on GEWOA for sparse sensor array design is proposed to reduce PSLL. A three-step enhanced global search strategy is introduced in GEWOA to improve the global search capability. In the initial stage, chaotic initialization is embedded in GEWOA, which enhances the ergodicity of the algorithm. In the search stage, the conventional spiral strategy is replaced by the Archimedean spiral strategy (ASS) to avoid premature algorithm results. In the offspring selection stage, refraction learning (RL) is used to obtain the inverse solution of the offspring, which avoids falling into local optima.

Moreover, to solve the adaptation problem for the discrete array design based on GEWOA, a position decomposition method is applied to make the element positions of off-grid arrays continuous, and a V-shaped transfer function is introduced into the on-grid array design to map the search position into the discrete grid points to obtain the desired array. The effectiveness of the proposed method is validated on the design tasks for sparse sensor arrays. To recap, the main contributions of this paper can be summarized as follows:

(1)

A novel GEWOA with a three-step enhanced global search strategy is proposed for sparse sensor array design. Due to the strong global search capability, the arrays obtained by GEWOA show good performance in PSLL.

(2)

A position decomposition method and a V-shaped transfer function are introduced into off-grid arrays and on-grid arrays, respectively, to make GEWOA suitable for discrete array design.

(3)

The effectiveness of the proposed method is validated on design tasks for large-scale low-frequency sensor arrays in the marine environment. Additionally, comparisons with other representative methods, such as GA, PSO, GWO, and WOA, also demonstrate the superiority of our method.

The rest of this paper is organized as follows: Section 2 presents the problem description and the background. Section 3 introduces the proposed GEWOA. In Section 4, the effectiveness and superiority of the proposed method are demonstrated. Finally, Section 5 concludes this paper.

2. Problem Description

Let us consider that a sparse linear array has N isotropic radiative elements, which are distributed along the x-axis at $\mathbf{D}=[d_0,d_1,\dots ,d_i,\dots ,d_{N-1}],i\in [0,N-1]$. The array aperture is L. The beampattern of the array is given by:

$$P=\sum _{i=1}^N{w}_i{e}^{\mathrm{j}\frac{2\pi }{\lambda }{d}_{i}sin(\theta -{\theta }_0)}$$

(1)

where $\mathrm{j}=\sqrt{-1},\lambda$ denotes the wavelength of the operating frequency, $w_i$ denotes the complex excitation of the ith element in the array, $\theta$ is the steering angle, and ${\theta }_0$ represents the desired beam direction. The schematic of the array elements distribution is shown in Figure 1. Sparse arrays can be divided into on-grid arrays and off-grid arrays according to the positions of the array elements. The grid points are set at positions that are integer multiples of half the wavelength.

The black solid circles in Figure 1 are the actual positions of the array elements. The red hollow circles are grid points. If all array elements fall on the grid points, it is an on-grid array. Otherwise, it is an off-grid array. The PSLL of the off-grid array could be lower than that of on-grid array. In other words, an off-grid array can use fewer elements than an on-grid array to meet the same PSLL. However, the on-grid array is easier to process with less computation charge. Therefore, different design methods can be selected with a trade-off between the PSLL and computation charge.

In this paper, we focus on the layout problem of large-scale sparse sensor arrays, and all weights are set to 1 ($w_n=1$ for all elements). The beampattern of a uniform array is a Dirichlet function with a PSLL of about −13 dB. However, the positions of the sparse array can be distributed nonuniformly. Assuming that the weight vector of a uniform array is $\tilde{\mathbf{w}}$, then the weight vector $\mathbf{w}$ of the sparse array can be expressed as $\mathbf{w}=\tilde{\mathbf{w}}\odot \mathbf{t}$, where $\mathbf{t}=[1,0,1,\dots ,0,1]$ can be considered a tapering vector, which consists of 1 and 0, with 1 for element and 0 for no element, and ⊙ denotes the Hadamard product.

The tapering operation enables the side lobes to be lower than those of the uniform array. The optimal positions by minimizing PSLL are filled with 1 in vector $\mathbf{t}$. The beampattern of a sparse array obtained through Equation (1) is the convolution of the Dirichlet function with the spatial spectrum of weight vector $\mathbf{w}$, which is based on the array positions. Hence, a sparse sensor array with a low PSLL can be obtained by optimizing the array element positions. Considering the physical limit of the sensors’ size, the minimum spacing of the elements must be restricted. Therefore, the array design problem can be formulated as follows:

$$\underset{\mathbf{X}}{min}\mathbf{PSLL}\left(P\right)\mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}{x}_i-x_j\ge d_c>0\\ i,j\in \mathbf{Z},0\le j\le i\le N-1\\ x_0=0,x_{N-1}=L\end{array}\right.$$

(2)

where $x_i$ and $x_j$ denote the position of the ith and jth element, respectively. $d_c$ is the minimum distance between the adjacent elements. $\mathbf{Z}$ denotes the set of positive integers.

3. Methodology

3.1. Overview

The flowchart of the proposed method is shown in Figure 2. For off-grid array design, the position decomposition is used to make the element positions continuous. This strategy scales the array aperture L to the new aperture $L^{*}$, and then the proposed GEWOA is performed to obtain the optimized design result. For on-grid array design, GEWOA is directly performed to update the positions of the population, and then a V-shaped transfer function is used to map the search positions into the discrete grid points to obtain the desired array.

In GEWOA, a three-step enhanced global search strategy is introduced to improve the global search capability of the array element positions as shown in the red dashed box. Chaotic initialization is used to initialize the population. Next, the position updating process is executed, where the three strategies, including the shrinking encircling mechanism, ASS, and random search, are selected according to preset rules.

After the position updating, RL is performed to seek the inverse solution to optimize the offspring. Next, the best offspring is selected by calculating the PSLL according to the objective function. Finally, the iterative optimization process is continued until the convergence condition is satisfied or the maximum number of iterations is reached.

3.2. Global Enhancement Whale Optimization Algorithm

GEWOA is a swarm intelligence optimization algorithm inspired by the foraging process of humpback whales. It is inspired by WOA and continues to follow its strategy execution process. The execution process simulates the behavior of whales attacking prey through a shrinking encircling mechanism and spiral updating position. In addition, a random search strategy is performed to avoid falling into a local solution [19]. However, the global search capability of the original strategy decreases as the search range increases. A three-step enhanced global search strategy, including chaotic initialization, ASS, and RL, is proposed to improve the search performance in GEWOA.

3.2.1. Chaotic Initialization

Chaos is a phenomenon that can traverse all states without repetition in a certain range [24]. We embed the chaos model to calculate the initial value aiming at increases the diversity of the population. Conventional chaotic mapping functions include Logistic mapping, Tent mapping, Kent mapping, Henon mapping, and Sin mapping.

After setting the initial value ${\tilde{s}}_0$ and the parameters, the chaotic sequence $\tilde{\mathbf{S}}({\tilde{s}}_1,{\tilde{s}}_2,\dots ,{\tilde{s}}_{N-2})$ can be obtained by using the equations in

Table 1. Chaotic mapping equations.

Name	Chaotic Map Equation	Parameters
Logistic	${\tilde{s}}_{n+1}=\alpha {\tilde{s}}_n\left(1-{\tilde{s}}_n\right)$	$\alpha$
Henon	${\tilde{s}}_{n+1}=1-\beta {\tilde{s}}_n^2+\zeta {\tilde{s}}_n$	$\beta ,\zeta$
Kent	${\tilde{s}}_{n+1}=\left\{\begin{array}{cc}{\tilde{s}}_n/\mu \hfill & 0<{\tilde{s}}_n\le 0.5\hfill \\ \left(1-{\tilde{s}}_n\right)/\left(1-\mu \right)\hfill & 0.5<{\tilde{s}}_n\le 1\hfill \end{array}\right.$	$\mu$
Tent	${\tilde{s}}_{n+1}=\left\{\begin{array}{cc}\eta {\tilde{s}}_n\hfill & {\tilde{s}}_n<0.5\hfill \\ \eta \left(1-{\tilde{s}}_n\right)\hfill & {\tilde{s}}_n\ge 0.5\hfill \end{array}\right.$	$\eta$
Sin	${\tilde{s}}_{n+1}=\gamma sin\left(\pi {\tilde{s}}_n\right)$	$\gamma$

, where $\alpha \in (0,4]$ is the control parameter of Logistic mapping. $\beta =1.4,\zeta =0.3$ are the control parameters of Henon mapping, which allows the equation to enter a chaotic state and generates a chaotic sequence with strong randomness.

$\mu \in (0,1)$ is the the control parameter of Kent mapping, $\eta \in (0,2]$ is the control parameter of Tent mapping, and $\gamma \in [0,1]$ is the control parameter of Sin mapping. Then, the positions of the initial population $\tilde{\mathbf{X}}({\tilde{x}}_1,{\tilde{x}}_2,\dots ,{\tilde{x}}_{N-2})$ can be obtained by:

$${\tilde{x}}_n={\tilde{s}}_n(ub-lb)+lb,n=0,1,2,\dots ,N-1$$

(3)

In this paper, different chaotic initialization strategies are simulated to determine the best one as the optimal initialization strategy that has the lowest PSLL. Compared with the randomly distributed population in WOA, chaotic initialization overcomes the shortcoming of falling into a local solution to a certain extent.

For a swarm of whales with M individuals, the search space of each whale position is N-dimensional. The position of the jth whale in the population after the qth iteration is $\mathbf{X}\left(q\right)=(x_{0,j},x_{1,j},\dots ,x_{i,j},\dots ,x_{N-1,j}),i\in [0,N-1],j\in [0,M-1]$. Here, $\mathbf{X}\in \left[lb,ub\right]$, $ub$ is the upper boundary of the search range, and $lb$ is the lower boundary of the search range.

3.2.2. Shrinking Encircling Mechanism

The optimal position is regarded as the position of a target prey, and other individuals surround the optimal position. Such behavior is called the shrink encircling mechanism, and its expressions are as follows:

$$\tilde{\mathbf{D}}=\mid \mathbf{C}{\mathbf{X}}^{*}\left(q\right)-\mathbf{X}\left(q\right)\mid$$

(4)

$$\mathbf{X}(q+1)={\mathbf{X}}^{*}\left(q\right)-\mathbf{A}·\tilde{\mathbf{D}}$$

(5)

where q is the number of iterations, and ${\mathbf{X}}^{*}\left(q\right)$ is the optimal position. $\mathbf{X}\left(q\right)$ denotes the individual position in the population after qth iterations. $\mathbf{X}(q+1)$ is the updated position. · denotes the multiplication of the corresponding elements. $\mathbf{A}$ and $\mathbf{C}$ are coefficient vectors:

$$\mathbf{A}=2\mathbf{a}·\mathbf{r}-\mathbf{a}$$

(6)

$$\mathbf{C}=2\mathbf{r}$$

(7)

where $\mathbf{a}$ decreases linearly from 2 to 0, and $\mathbf{r}$ is a random vector in $\left[0,1\right]$. It is worth noting that the individuals move to the optimal position by adjusting $\mathbf{A}$ and $\mathbf{C}$.

3.2.3. Random Search

To avoid falling a local optimum, an individual is randomly selected as a target prey, and the positions of the population are updated by:

$$\tilde{\mathbf{D}}=\mid \mathbf{C}{\mathbf{X}}_{rand}\left(q\right)-\mathbf{X}\left(q\right)\mid$$

(8)

$$\mathbf{X}(q+1)={\mathbf{X}}_{rand}\left(q\right)-\mathbf{A}·\tilde{\mathbf{D}}$$

(9)

where ${\mathbf{X}}_{rand}$ is a randomly selected whale position.

3.2.4. Archimedean Spiral Strategy

Whales swim in a spiral shape while shrinking towards their prey with the spiral position update model:

$$\mathbf{X}(q+1)={\mathbf{X}}^{*}\left(q\right)+{\mathbf{D}}_p{e}^{bl}cos\left(2\pi l\right)$$

(10)

where ${\mathbf{D}}_p$ denotes the distance between candidate whales and their prey, b is a constant used to control the logarithmic spiral shape, and l is a random number between $\left[-1,1\right]$.

The spiral position update method of WOA follows the logarithmic spiral model, which is characterized by equal spiral angles but unequal pitches. Such a method can find the optimal value quickly; however, it likely misses the intermediate optimal solution. Archimedean spiral curve provides a better idea to improve the original spiral search strategy, which avoids the reduced accuracy caused by long search steps. Each position dimension of a whale is generated by (11) instead of (10).

$$\mathbf{X}(q+1)={\mathbf{X}}^{*}\left(q\right)+({\mathbf{D}}_p+b·l)e^{bl}cos\left(2\pi l\right)$$

(11)

Figure 3 shows the Archimedean spiral curve and the logarithmic spiral curve, where the blue solid line is the Archimedean spiral curve, and the red dashed line is the logarithmic spiral curve. It can be seen that the Archimedean spiral curve is characterized by equal pitch, and the search step can be set by setting the pitch.

In the process of WOA, the strategy selection probability is set to 50% to choose encircling prey or the spiral updating position:

$$\mathbf{X}(q+1)=\left\{\begin{array}{c}{\mathbf{X}}^{*}\left(q\right)-\mathbf{A}·\tilde{\mathbf{D}},p<0.5\hfill \\ {\mathbf{X}}^{*}\left(q\right)+({\mathbf{D}}_p+b·l)e^{bl}cos\left(2\pi l\right),p\ge 0.5\hfill \end{array}\right.$$

(12)

where p is a random value between (0, 1).

If p is less than 50%, encircling prey is chosen. Otherwise, the Archimedean spiral strategy is chosen, and the whale position is updated by (11). In encircling prey, the prey is selected to pass by $\mathbf{A}$. If $\left|\mathbf{A}\right|<1$, the optimal individual in the population is selected as the prey to execute encircling prey by (4) and (5). If $\left|\mathbf{A}\right|\ge 1$, an individual is randomly selected as the prey, and the positions of population are updated by (8) and (9). By following the above three mechanisms, the whale swarm can gradually approach the optimal position.

3.2.5. Refraction Learning

RL is an inverse solution solving method that introduces the refraction principle of light into opposition-based learning (OBL). The superior inverse solution is screened out by OBL, and it is used to replace the existing solution for the next iteration. This strategy has a higher probability of reaching the global optimal solution [37]. However, the conventional OBL is not adjustable in the search range and cannot prevent the inverse solution from falling into the local optimal solution. To fix the defect, a scaling factor is introduced to control the search range according to the refraction principle.

Figure 4 shows the process of RL, where the center point of the search interval $[x_{\min },x_{\max }]$ is denoted as O. The light propagates from $P(x,y)$ to the origin O in one medium and enters another medium. Refraction occurs after it enters a new medium, and the refraction point is ${\widehat{P}}^{*}$. The distance $|\overrightarrow{PO}|$ from the incident point to the origin O is h, and the distance $|\overrightarrow{O{\widehat{P}}^{*}}|$ between the origin O and the refraction point is $h^{*}$. According to Shell’s Law, the relationship between the angle of incidence $\alpha$ and that of refraction $\beta$ satisfies the shell formula: $n=sin\alpha /sin\beta$, where n is the refractive index.

 Definition 1. 

The projection ${\widehat{x}}^{*}$ of the refraction point ${\widehat{P}}^{*}$ on the x-axis is defined as the inverse solution of x based on the RL.

According to the shell formula, we find

$$\begin{array}{c}sin\alpha =((x_{min}+x_{max})/2-x)/h\\ sin\beta =\left({\widehat{x}}^{*}-(x_{min}+x_{max})/2\right)/h^{*}\end{array}$$

(13)

Assuming $k=h/h^{*}$, (13) can be rewritten as

$${\widehat{x}}^{*}=(x_{min}+x_{max})/2+(x_{min}+x_{max})/\left(2kn\right)-x/kn$$

(14)

The inverse solution ${\widehat{x}}^{*}$ can be solved using (14), where k is the scaling factor, which is used to adjust the positions of the inverse solutions on the x-axis. n is a fine-tuning factor. It can be seen that, assuming $n=1,k=1$, (14) can be simplified as ${\widehat{x}}^{*}=-x$, which is the expression for OBL. It can be seen that OBL is a special case of RL. The refractive index of light from air to water is about 0.75. Assuming that $n=0.75$ is fixed, the change in incident angle causes the change in the projection point $x^{*}$.

The refraction point can be moved from point ${\widehat{P}}^{*}$ to point ${\widehat{P}}_1^{*}$ by adjusting k, which is closer to the optimal point. If the ${\widehat{P}}_1^{*}$ does not reach the optimal value, the optimal solution can be reached by adjusting the incidence angle and moving the refraction point to the ${\widehat{P}}_2^{*}$. In order to improve the projection range of the inverse solution, we set k in a linearly decreasing trend:

$$k=k_{min}+(k_{max}-k_{min})t/t_{max}$$

(15)

The N-dimensional positions of a whale after position updating are denoted as:

$$\tilde{\mathbf{X}}=\left({\tilde{x}}_0,{\tilde{x}}_1,\dots ,{\tilde{x}}_i,\dots ,{\tilde{x}}_{N-1}\right)$$

(16)

where $x_{i,j}\in \left[x_{min},x_{max}\right],i=0,1,\dots ,N-1$. $x_{min}$ and $x_{max}$ are the minimum and maximum values in the search range, respectively. The inverse solution ${\widehat{\mathbf{X}}}^{*}$ obtained by RL is

$${\widehat{\mathbf{X}}}^{*}=\left({\widehat{x}}_0^{*},{\widehat{x}}_1^{*},\dots ,{\widehat{x}}_i^{*},\dots ,{\widehat{x}}_{N-1}^{*}\right)$$

(17)

If $\mathbf{PSLL}\left({\widehat{\mathbf{X}}}^{*}\right)<\mathbf{PSLL}\left(\tilde{\mathbf{X}}\right)$, ${\widehat{\mathbf{X}}}^{*}$ is chosen as the best candidate solution. Otherwise, $\tilde{\mathbf{X}}$ is chosen as the best candidate solution.

3.3. Off-Grid Arrays Design Based on GEWOA

For off-grid arrays, considering the limit of the array element size, the minimum spacing $d_c$ is constrained. This means that the search range is not the whole aperture L. To solve this problem, the positions $\mathbf{D}(d_0,d_1,\dots ,d_{N-1})$ are decomposed. Since the positions of the two array elements $x_0$ and $x_{N-1}$ are fixed, only the positions ${\mathbf{D}}^{*}(d_1,d_2,\dots ,d_{N-2})$ of the $N-2$ array elements need to be designed. Thus, a position decomposition method is applied to make ${\mathbf{D}}^{*}$ continuous:

$${\mathbf{D}}^{*}=\tilde{\mathbf{X}}+\left[\begin{array}{c}{d}_c\\ 2d_c\\ ⋮\\ (N-2)d_c\end{array}\right]=\left[\begin{array}{c}{\tilde{x}}_1+d_c\hfill \\ {\tilde{x}}_2+2d_c\hfill \\ ⋮\hfill \\ {\tilde{x}}_{N-2}+(N-2)d_c\hfill \end{array}\right]$$

(18)

In (19), ${\tilde{x}}_1\le {\tilde{x}}_2\le ,\dots ,\le {\tilde{x}}_{N-2}$,$d_1>d_c,d_{N-1}<L-d_c$. The solution for the array position ${\mathbf{D}}^{*}$ is converted to the solution for $\tilde{\mathbf{X}}$ by the the position decomposition method. Here, the search range of $N-2$ array elements becomes $L-2d_c$. In addition, $(N-3)d_c$ needs to be subtracted because the spacing between adjacent array elements is not less than $d_c$. Therefore, the search aperture becomes $L^{*}$:

$$L^{*}=L-2d_c-(N-3)d_c=L-(N-1)d_c$$

(19)

Through the above position decomposition, the discontinuous problem can be converted into a continuous problem. Then, GEWOA can be performed to achieve off-grid array design (Algorithm 1).

The algorithmic details of the proposed method for off-grid array design based on GEWOA are summarized in Algorithm 2. First, position decomposition is performed to obtain the search range $L^{*}$. Secondly, GEWOA is performed to search the optimal position. In the initial stage of GEWOA, the population initialization is performed by using chaos sequences to obtain $\tilde{\mathbf{X}}$, and the maximum number of iterations $t_{max}$ is set. The optimal whale in the initialized population is selected by (2), and then the iterative optimization is performed.

The parameters $\mathbf{a},\mathbf{r},l,p$ of each whale are set, and $\mathbf{A},\mathbf{C}$ are calculated for choosing different strategies of position updating. If $p>0.5$, ASS for position updating is performed. Else, if $\left|\mathbf{A}\right|<1$, a shrinking encircling mechanism is executed. Otherwise, a random search is used. After the position updating, the inverse solution ${\widehat{\mathbf{X}}}^{*}$ is generated by using RL. The best solution is selected from the new offspring and its inverse solution. The iterative optimization process is continued until the convergence condition is satisfied or the maximum number of iterations is reached. Finally, the optimal array positions are output by (19).

3.4. On-Grid Arrays Design Based on GEWOA

For on-grid arrays, the positions are discrete on half wavelength grid points. Therefore, the V-shaped transfer function is used to discretize the output of GEWOA. The V-shaped transfer function maps the whale position to the [0,1] interval. When the mapped values fall in different intervals, discrete values are selected. The expression of the V-shaped transfer function is:

$$\begin{array}{c}V\left(x_{ij}\left(t\right)\right)=\mid \frac{e^{x_{ij}\left(t\right)}-1}{e^{x_{ij}\left(t\right)}+1}\mid \\ x_{ij}^{\prime }=\left\{\begin{array}{cc}G\hfill & 0\le V\left(x_{ij}\left(t\right)\right)<r_1\hfill \\ G-1\hfill & r_1\le V\left(x_{ij}\left(t\right)\right)<r_2\hfill \\ \dots \hfill & \\ 0\hfill & r_M\le V\left(x_{ij}\left(t\right)\right)<1\hfill \end{array}\right.\end{array}$$

(20)

where $x^{\prime }$ is the converted discrete value. $r_1,r_2,\dots ,r_{k-1}$ are random numbers between $(0,1)$, which satisfy $0<r_1<r_2<\dots <r_{k-1}<1$. When $0\le V\left(x_{ij}\left(t\right)\right)<r_1$, the whale position is mapped to G. $G=\mathbf{floor}(2\ast L/\lambda )$ is the largest grid point (integer multiple of half wavelength), where $\mathbf{floor}\left(\right)$ is a function to find the nearest integer less than or equal to the parameter. The algorithmic details of the proposed method for on-grid array design based on GEWOA are summarized in Algorithm 2.

4. Experiment

In this section, first, two kinds of numerical experiments on sparse-sensor-array-design tasks are performed to validate the effectiveness and superiority of the proposed method. (1) For off-grid array design, the proposed three-step enhanced global search strategy in GEWOA is simulated step by step to test the effectiveness of each strategy. Furthermore, the designed results are compared with those of WOA.

(2) For on-grid array design, the proposed GEWOA is compared with some other intelligence algorithm-based design methods, including GA, PSO, and WOA, in terms of PSLL. Then, the array designed using the proposed method is verified in the marine environment, and the beampattern obtained by the array based on GEWOA is analyzed.

The basic design requirements in numerical experiments are as follows. The array aperture is 200 m, the number of array elements is 40, and the array operating frequency is 300 Hz with a wavelength of 5 m. The objective function is shown in (2), where PSLL is chosen as the evaluation criterion. The number of whales in the population $M=30$, the dimension of search $N=40$, the number of iterations $t_{max}=1000$.

The minimum spacing of the array $d_c$ is set to 2 m for off-grid arrays, and the array elements are set at half-wavelength grid points for on-grid arrays. In addition, we also conduct numerical experiments for large-scale low-frequency sensor array design, and the design conditions are as follows: the array operating frequency is 300 Hz, the number of array elements is 256, and the array aperture is 1280 m.

4.1. Results of Off-Grid Arrays

4.1.1. Performance of Chaotic Initialization

In this part, different chaotic initialization methods are used to design the off-grid sensor array, and the convergence speed and beampattern performance are analyzed. Five chaotic mapping functions, including Logistic, Henon, Kent, Sin, and Tent, are tested under two termination conditions.

Under the first termination condition, which depends on the maximum number of iterations $t_{max}$, the search accuracy of five chaotic initialization methods is compared. The parameter settings and the obtained PSLLs are shown in

Table 2. PSLLs and the numbers of iterations for five chaotic initialization methods.

Chaotic Initialization	Parameter Settings	PSLL (dB)	Iterations
Logistic	$\alpha =3.9,{\tilde{s}}_0=0.5$	−19.96	659
Henon	$\beta =1.4,\zeta =0.3,{\tilde{s}}_0=0$	−19.63	451
Kent	$\mu =0.625,{\tilde{s}}_0=0.45$	−19.98	800
Sin	$\eta =0.867,{\tilde{s}}_0=0.5$	−19.30	733
Tent	$\gamma =1.41,{\tilde{s}}_0=0.1$	−19.77	488
Origin	-	−18.65	897

, where the PSLL is the minimum value among 10 experiments. The beampatterns of the obtained array are shown in Figure 5. The iterative process of five chaotic initialization methods are shown in Figure 6. It can be seen that the search accuracy is improved after chaotic initialization compared to the original WOA (denoted as Origin in Figure 5 and Figure 6). In addition, the Kent initialization method yields the array with the lowest PSLL.

Under the second termination condition that the algorithm stops if the PSLL remains unchanged beyond 100 iterations, the convergence speed of five chaotic initialization methods is compared. The numbers of iterations are shown in Table 2. It can be seen that Origin<Kent<Sin<Tent<Logistic<Henon in terms of the convergence speed.

Here, the Kent initialization method is adopted in GEWOA because it has the best search accuracy and faster convergence speed compared with the original method. Considering that the search accuracy is the most important criterion for array design, the selection is reasonable.

4.1.2. Performance of Archimedean Spiral Strategy

In this part, we compare three search path planning strategies, including the original spiral position update, ASS, and Levy flight strategy. The Archimedean spiral spacing is set to 10 m. The beampatterns and the iterative process are shown in Figure 7 and Figure 8, respectively. It can be seen that the PSLL of ASS is −21.02 dB, while the PSLL of Levy flight strategy is −20.76 dB. After adding ASS, the PSLL is 2.37dB lower than the original strategy. ASS improves the original strategy, and has a lower PSLL than the Levy flight strategy.

4.1.3. Performance of Refraction Learning

On the basis of Kent chaotic initialization and ASS, RL is used to further optimize the offspring to obtain an array with the lower PSLL. The setting parameters of RL are $k_{max}=1.2,k_{min}=0.8,t_{max}=10$. The beampatterns of 40-element off-grid arrays and the iterative process before and after RL optimization are shown in Figure 9 and Figure 10. It can be seen that the PSLL of array without RL optimization is −21.02 dB, and the PSLL of array with RL optimization is −22.47 dB. Compared with WOA, the PSLL obtained by GEWOA is reduced by 3.81 dB.

In order to test the influence of RL parameter on the search accuracy, experiments with different RL parameters are performed. The three sets of RL parameters are (1) $k=1$; (2) $k_{max}=1.2,k_{min}=0.8,t_{max}=5$; and (3) $k_{max}=1.2,k_{min}=0.8,t_{max}=10$. The array beampatterns and iterative process are shown in Figure 11 and Figure 12, where the PSLLs of three sets of RL parameters are −21.47, −22.30, and −22.47 dB. The PSLLs of the designed arrays improve as the search range of the inverse solution is expanded.

4.2. Results of On-Grid Arrays

For on-grid arrays, the grids are set according to the half wavelength $\lambda /2=2.5$ m, and the number of grid points is 81. The proposed GEWOA discretized by a V-shaped transfer function is used to design on-grid arrays. In order to validate the superiority of the proposed GEWOA, three intelligence algorithms, including GA, PSO, and WOA, are compared with GEWOA.

The beampatterns of 40-element on-grid arrays and iterative process are shown in Figure 13 and Figure 14, respectively. The PSLLs of GA, PSO, WOA, and GEWOA are −14.90, −15.10, −15.66, and −17.14 dB, respectively. It can be seen that the on-grid array designed using GEWOA obtains the lowest PSLL, which proves the advantage of the GEWOA in terms of the global search capability.

The proposed array design method, including the off-grid array design method and on-grid array design method, are successfully applied to the large-scale low-frequency arrays (256-element) design, and the obtained arrays have low PSLL. The beampatterns of the designed arrays and iterative process are shown in Figure 15 and Figure 16, respectively. The results show that the PSLLs of the off-grid array and the on-grid array obtained by the proposed methods are −27.71 and −21.47 dB, which verifies the feasibility of the algorithm in large-scale low-frequency array design.

In order to compare the performance of the proposed method with typical configuration arrays, such as coprime arrays and nested arrays, three different arrays were designed at the same aperture and the same number of elements. The 40-element on-grid array was designed using the proposed method. The nested array has three levels of nesting. The first level is a 20-element uniform array with half-wavelength element spacing. The second level of nesting is a 10-element uniform array with a wavelength element spacing. The third level of nesting is a 10-element uniform array with two times wavelength element spacing.

To ensure the same aperture, the 40-element coprime array was designed, and its elements were arranged on three and four times half-wavelength grids. Figure 17 shows the beampatterns of the three arrays. It can be seen that the PSLL of the array designed using GEWOA are −14.79 dB, and the PSLLs of the coprime array and the nested array are −5.82 and −6.02 dB, respectively. The main lobe widths of three arrays are approximately the same. However, the sparse array designed using GEWOA has a lower PSLL, which proves the advantage of the proposed method compared with the determined methods.

To summarize, the above experimental results show that the proposed GEWOA with a three-step enhanced global search strategy had a good global search capability, and the array designed using GEWOA had a lower PSLL when compared with the other algorithms and deterministic methods. In addition, the proposed array design method was successfully applied to the large-scale low-frequency array design, and the obtained array had a low PSLL.

4.3. Discussion and Analysis

To demonstrate the superiority of the proposed method, we compared the performance of the sparse array designed using GEWOA with those of arrays obtained by traditional deterministic and hybrid methods, including the GWO array [23] and the different coarray (DC) designed using the method in [11]. Array design via these methods was performed at the same aperture.

The number of array elements is 40, and the array aperture is 200 m. The beampatterns of the designed arrays are shown in Figure 18, and their PSLLs are shown in

Table 3. The PSLLs of arrays designed using different algorithms.

Arrays	DC Array	GWO Array	GEWOA
PSLL(dB)	−15.34 dB	−15.82 dB	−17.58 dB
Threshold Time(s)	122.41 s	20.54 s	14.39 s

, which indicates the lowest PSLL of our method. To further illustrate the repeatability of our method, we conducted 50 Monte Carlo simulations of the algorithms under the same conditions.

The results show that our method had a 96% probability of making PSLL below −17 dB. We compared the time efficiency of these methods on a CPU Intel E3-1220 v6 @ 3.00 GHz, and −15 dB was chosen as the convergence threshold. The computation times of GEWOA, GWO, and DC were 14.39, 20.54, and 122.41 s, and it can be seen that our method achieved the best computational efficiency.

4.3.1. Mutual Coupling Analysis

Due to the limitations of the natural frequencies, the mode shapes of frequencies, and inertial effects, the low-frequency acoustic sensors, such as transducers are usually large [43]. For engineering implementation, the minimum spacing of the array elements is constrained. The distance between adjacent element of the off-grid array designed using GEWOA can be less than half the wavelength, which results in the array element falling in the radiated sound field of that adjacent element.

The mutual coupling phenomenon occurs between adjacent array elements. This increases with the decrease of array element spacing, and directly affects the setting of the minimum spacing of the array element. According to the mutual radiation impedance model [44], the mutual radiation impedance $X_{AB}$ of an array element in the radiation field of another array element is

$$X_{AB}=\rho cSk{a}^2/d_{AB}.$$

(21)

where $\rho$ is the medium density. c is the speed of sound. S is the surface area of the array element. a is the radius of the sensor element. $d_{AB}$ is the distance of two elements. k is the wave number assuming the resonant frequencies of the array element is 3.5 kHz. The radius of the array element is 0.4 m. The curve of the radiation impedance changing with the array element spacing is shown in Figure 19.

It can be seen that $d/\lambda \ge 0.4$, mutual-impedance is less than its self-impedance. In the off grid array design, the operating frequency is 300 Hz, and the minimum spacing of array elements is set to 2 m due to engineering requirements. Since the minimum spacing is greater than 0.4 times the wavelength of the resonant frequency, the mutual radiation effect is less than the self-radiation, and it can be disregarded.

4.3.2. Array Gain Analysis

Array gain is an important indicator of array performance. Assuming that the sparse arrays receive a 300 Hz sinusoidal signal with a duration of 1 s. The array gains of the 40-element off-grid array and on-grid array designed using GEWOA are calculated to verify the feasibility of the proposed method under the Gaussian white noise background. The theoretical value of the array gain for the 40-element array is 16.02 dB.

The change trend of the array gain with the input signal-to-noise ratio ($SN{R}_{in}$) is shown in Figure 20. It can be seen that the array gain decreases at a lower input SNR. When the $SN{R}_{in}$ is −30 dB, the array gain of both arrays decreases by about 1.5 dB compared with the theoretical value. When the $SN{R}_{in}$ is higher than −20 dB, the array gain of both arrays decreases by, at most, 0.3 dB compared with the theoretical value, which proves the feasibility of the designed array based on the proposed method from the perspective of array gain.

4.3.3. Position Uncertainty Analysis

To verify the effect of array performance when the array element position is varied, Gaussian perturbation is added to the element position of the on-grid array designed using GEWOA in Section 4.2 to compare the effect of the PSLL performance. The outer layer of most fiber optic towing arrays is the polyurethane tubing, which is filled with oil. This connection limits the offset of the array element position. Assume that the position perturbation with Gaussian distribution $\mathcal{N}(0,{\left(\frac{\lambda }{12}\right)}^2)$, which can guarantee 99.73% probability within a quarter-wavelength error range.

The beampattern of the sparse array designed using GEWOA before and after adding the position perturbation is shown in Figure 21. It can be seen that the PSLL of the designed array is −17.58 dB, while the PSLL of the array is −15.87 dB in the case of perturbation of all array elements. We performed 100 experiments using Monte Carlo simulations. The average value of PSLL for the position perturbation case is 16.23 dB. A quarter-wavelength Gaussian position perturbation causes the PSLL of the array to degrade by about 1.35 dB.

4.4. Experiment in the Marine Environment

To verify the effectiveness of the proposed method, the performance of the designed arrays in the marine environment was analyzed. Due to the long production cycle and high investment of large-scale low-frequency sensor arrays, 40-element on-grid arrays were designed for the experiments. We extracted 40 elements from an 81-element uniform array to obtain the required arrays. The positions of the arrays designed using GA, PSO, WOA, and GEWOA are shown in Figure 22.

All array elements are set on the grid points. The beampatterns of designed arrays are shown in Figure 23. The PSLLs and main lobe widths of the beampatterns are shown in

Table 4. The PSLLs of arrays designed using different algorithms.

Algorithms	GA	PSO	WOA	GEWOA
PSLL (dB)	−15.03	−15.12	−16.03	−17.85
Main lobe width (${}^{\circ }$)	1.32	1.32	1.36	1.38
PSLL in marine environment(dB)	−13.20	−13.95	−14.60	−17.10
Main lobe width in marine environment (${}^{\circ }$)	1.33	1.34	1.38	1.39

. By analyzing the performance of the sparse-sensor array designed based on GEWOA in the marine environment and comparing with the performance of the arrays obtained by GA, PSO, and WOA, the effectiveness of the algorithm was verified. The experimental conditions were as follows: the experimental array was a uniform hydrophone array with a half wavelength of 0.4167 m as shown in Figure 24. The design frequency of the array was 1800 Hz, and the number of array elements was 81.

The experimental data were from the 2020 South China Sea Experiment. The designed arrays were used to receive the transmit signal. The received data of 40-element on-grid arrays obtained by GA, PSO, WOA, and GEWOA were extracted from the 81-element uniform array. The relative azimuth angle between the sound source and the array was ${15}^{\circ }$. The frequency of the transmitted signal was 1800 Hz. Its duration was 1 s, and the transmission period was 40 s.

The beampatterns were obtained by conventional beamforming processing of the direct wave signals. The PSLLs of the beampatterns were analyzed to evaluate the array performance. The beampatterns obtained by the 40-element on-grid array designed using GA, PSO, WOA, and GEWOA are shown in Figure 25. The PSLLs and main lobe widths of the beampatterns in marine environment are shown in Table 4. It can be seen that the PSLLs of the beampatterns in the marine environment are degraded, due to the deviation of the array position from the designed ideal position.

The main lobe widths vary relatively little. The PSLL of the array designed using GEWOA is −17.10 dB in the marine environment. According to Table 4, it can be seen that compared to GA, the main lobe width expanded by nearly 0.06 degrees, which was only increased by 4.5% in the marine experiments. However, the PSLL decreased by 3.9 dB, which is a decrease of nearly 29.5%.

Compared with WOA, the main lobe width expanded by 0.01 degree, which is an increase of only 0.7%, while the PSLL was reduced by 2.5 dB, which is a decrease of 17.1%. The PSLL of GEWOA was 3.9 dB lower than that of the traditional GA algorithm with a small change in the performance of the main lobe width. In comparison with PSLLs obtained using other arrays, the array designed using GEWOA had the lowest PSLL.

5. Conclusions

In this paper, we proposed a novel approach based on GEWOA for sparse sensor array design to suppress PSLL under spacing constrains. A three-step enhanced global search strategy was introduced into GEWOA to improve the global search capability. In the initial stage, chaotic initialization was embedded in GEWOA to enhance the ergodicity of the algorithm.

In the search stage, the conventional spiral strategy was replaced by the ASS strategy to avoid premature algorithm results. In the offspring selection stage, RL was used to obtain the inverse solution of the offspring, which prevents falling into local optima. Moreover, in order to solve the adaptation problem for discrete array design based on GEWOA, a position decomposition method and a V-shaped transfer function were introduced into the off-grid arrays and on-grid arrays, respectively.

The effectiveness and superiority of the proposed method were validated by experiments on large-scale low-frequency sparse sensor array design tasks. The experimental results show that the proposed GEWOA with a three-step enhanced global search strategy had a good global search capability, and the array designed using GEWOA had the lowest PSLL compared with other intelligent algorithms. Additionally, the array designed using the proposed method was further verified in the marine environment, where the proposed GEWOA still achieved the lowest PSLL. In the future, we will attempt to expand our application of the proposed method on 2-D or 3-D arrays.