An Optimization Algorithm for Forward-Scatter Radar Network Node Deployment Based on BFGS and Improved NSGA-II

Abstract

Recently, forward-scatter radars (FSRs) utilizing the Global Navigation Satellite System (GNSS) as a radiation source have gained increasing attention. The radar system enables aerial target surveillance by deploying multiple receiving nodes on the ground. It offers a low-cost and easily deployable solution. Therefore, how to deploy the receiving nodes to achieve efficient utilization of node resources is an urgent problem to be addressed. In this paper, a deployment method was proposed for receiving nodes in a single-transmitter and multiple-receiver configuration. First, the problem was reformulated as an optimal equal-circle covering problem via geometric approximation. A multi-objective optimization model was subsequently established with the objective functions of minimizing node cost and maximizing spatial detection area. Second, a method based on the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm was introduced to obtain the sub-optimal solution of node cost, thereby reducing the computational complexity of the optimization process. Finally, an improved Non-dominated Sorting Genetic Algorithm II (NSGA-II) was proposed to derive the deployment schemes. Then, these schemes were ranked using the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) based on the Entropy Weight Method (EWM). The results indicate that the proposed method can obtain the optimal deployment scheme compared to the existing method and enhance the diversity of the solutions.

1. Introduction

The Global Navigation Satellite System (GNSS) provides all-weather, all-time services with global coverage. When utilized as a radar illuminator, it offers the advantages of low cost and high stealthiness [1,2,3,4]. However, the ground radiation power is relatively low, which significantly limits the target detection range. For instance, the received power level of the Global Positioning System (GPS) is −130 dBm [5,6]. Fortunately, the high-level radar cross section (RCS) achievable by the forward-scatter radar (FSR) can mitigate the issue of low target echo power [7]. It is generally acknowledged that a bistatic angle greater than 135° forms a forward scatter detection structure. Under such conditions, the RCS of the target is predominantly determined by its shadow profile, with minimal influence from the material and three-dimensional shape [8,9,10]. However, limitations in the structural configuration of FSRs result in a narrow detection zone, necessitating the deployment of multiple receiving nodes. Consequently, the optimization of node layout for GNSS-FSR systems warrants significant attention.

FSR represents a particular form of bistatic radar (BR). The optimal deployment of BR has been widely investigated. Barrier coverage is a critical issue in boundary surveillance and intrusion detection for wireless sensor networks. Utilizing BR enables the realization of various barrier coverage types, including circular [11,12], sector [13], linear [14,15], and strip barriers. The optimization method generally minimizes node costs while ensuring 100% area coverage. Maximizing the target detection probability is used as the optimization goal for linear barrier coverage [16]. Furthermore, the issues of fault tolerance and energy consumption within line barrier coverage are considered, and the collaboration of transmitters and receivers along a continuous line is optimized to achieve minimal deployment costs in [17]. However, due to the absence of a minimum width constraint, high-speed intruders could potentially traverse the barrier at specific locations without being detected. To address this problem, the detection region is divided into multiple sub-rectangles of equal width and barrier coverage for each using a uniform pattern [18].

The detection area of BR can be described using a Cassini oval, where the relative positions of the transmitter, receiver, and target significantly influence the shape of the coverage area. In contrast, the detection area of ground-based FSRs has a uniform shape and can be deployed using common network configurations, such as star, rectangular, and triangular structures. Currently, research on the node layout optimization of FSR systems in space-to-ground transmission mode is relatively limited. The deployment optimization of the system necessitates addressing the following issues:

(1)

As illustrated in Figure 1, due to the elevation angle of the satellite relative to the receiver, the detection area at a certain altitude is an ellipse. Therefore, a geometric approximation is required to simplify the node deployment problem.

(2)

When the deployment region is determined, it is necessary to maximize the aerial detection area with minimal node cost to optimize the utilization of node resources.

(3)

The detection areas of single-baseline systems vary in size at different altitudes and resemble a spindle shape. Due to the considerable baseline length, the detection area tends to increase with the height of the aerial target. The spacing of the receiving nodes directly affects the coverage efficiency of the detection area. Missed detections may occur if the target operates at a low altitude. Therefore, node deployment must be implemented based on the typical flight altitudes of the specified targets.

Figure 1. FSR detection area.

Figure 1. FSR detection area.

The node layout optimization can be implemented offline, with no stringent requirements on real-time performance. Moreover, the optimization objectives are often mutually exclusive. Therefore, multi-objective optimization methods are required. Commonly employed multi-objective optimization methods include Non-dominated Sorting Genetic Algorithm II (NSGA-II) [19,20], Non-dominated Sorting Genetic Algorithm III (NSGA-III) [21,22], Multi-objective Evolutionary Algorithm Based on Decomposition (MOEA/D), Strength Pareto Evolutionary Algorithm (SPEA), and SPEA2. In the context of radar deployment optimization, a layout optimization method is proposed for weather radar networks in [23], and the optimization performances of Multi-Objective Particle Swarm Optimization (MOPSO), NSGA-II, Multi-objective Gray Wolf Optimizer (MOGWO), and SPEA2 are compared. The NSGA-II is employed to rapidly accomplish multi-radar collaborative mission planning [24]. The resultant solutions exhibit notable characteristics, including uniform data distribution, robust convergence, and enhanced robustness. Addressing the problem of anti-jamming resource optimization for networked radar systems, an improved NSGA-II based on a hierarchical clustering truncation strategy is proposed, with the optimization objectives of minimizing the probability of radar deception and maximizing the coverage area [25]. Multi-objective optimization algorithms have also been applied to the scheduling of synthetic aperture radar (SAR) observation satellites. Two adaptive multi-objective evolutionary algorithms are designed, ALNS + NSGA-II and ALNS + NSGA-III, with the objectives of minimizing the loss rate of ground targets, invalid observations, and sensor activation time [26].

In this study, NSGA-II is employed for optimizing node deployment. The NSGA-II demonstrates certain advantages in terms of computational efficiency, population diversity, and convergence. Based on the framework of the NSGA-II, a deployment optimization method for receiving nodes is proposed. The main contributions are as follows:

(1)

A node cost optimization method based on the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm was proposed. The initial node cost is obtained using a honeycomb network coverage strategy, and the number of nodes is gradually reduced until the BFGS algorithm can no longer improve the solution. The suboptimal node cost significantly reduces the search space of the NSGA-II.

(2)

An improved NSGA-II was proposed to tackle the deployment optimization problem, which is capable of obtaining the optimal layout under various node cost scenarios, including the minimum node cost solution.

(3)

The effectiveness and adaptability of the proposed method were validated through node deployment optimization in regions of varying sizes and shapes, and the deployment scenario of heterogeneous nodes has been considered. The comparisons were made with other algorithms.

The remainder of this paper is organized as follows: Section 2 establishes the mathematical model for the deployment optimization problem. Section 3 proposes an algorithm based on the BFGS method to determine the suboptimal cost of nodes. Section 4 presents an improved NSGA-II for deriving the optimal deployment scheme. Section 5 provides a simulation analysis, and Section 6 provides the conclusions.

2. Node Deployment Optimization Problem Model

2.1. Geometric Approximation

As shown in Figure 1, when the satellite elevation angle is not equal to 90°, the detection area of a single-baseline system at a given altitude is an ellipse. However, due to the GNSS satellites being at a considerable distance from the ground and the altitude of aerial targets being relatively small compared to the baseline length, the detection area can be approximated as a circle. The detection areas of individual receiving nodes are approximately equal. Consequently, the station layout optimization problem can be transformed into an optimal equal-circle covering problem within the designated deployment region [27]. This problem can be further decomposed into two subproblems: whether the given area can be completely covered by circular regions, and the removal of redundant circles. In the node layout optimization problem, maximizing the area of the circular regions is also essential to achieve broader aerial surveillance coverage.

The optimal equal-circle covering problem is prevalent in various fields, including the construction of sensor networks, forest fire prevention, cellular tower placement, irrigation of crops, and the control of artificial satellite circle orbits [28,29,30]. Sensors are typically initially deployed in designated areas according to a fixed template, followed by suboptimal deployment using Voronoi diagrams and Delaunay triangulation. A mathematical model is developed based on Voronoi polygons, and an enhanced Zoutendijk algorithm for local minimum search is proposed in [31]. The model is expressed as a nonlinear programming problem in [32]. For covering an arbitrary polygon with circles of varying radii, the complexity of the problem dramatically increases [29,33]. Intelligent optimization algorithms are frequently used, capable of achieving optimized results based on random initial positions, such as ant colony algorithms (ACA) [34], genetic algorithms (GA) [35], and simulated annealing (SA) [36,37].

As illustrated in Figure 2, the relationship between the spacing of receiving nodes $d$ and the distance to the center of the circular detection area $d_1$ can be derived. Based on the theory of similar triangles, the spacing between the intersection points of each baseline $\left|P_1{P}_2\right|=d_1$ with the height plane at a given altitude is approximately equal to the spacing of the receiving nodes $d$; that is, it satisfies

$${\displaystyle \frac{d_{\mathrm{T}}}{d_{\mathrm{T}}+d_{\mathrm{R}}}}={\displaystyle \frac{d_1}{d}}\approx 1$$

(1)

Common high-value aerial targets typically fly at altitudes of 6 to 10 km, while the distance between the satellite and the ground is approximately 20,200 km. Under these conditions, the calculation results in Equation (1) range from 0.9995 to 0.9997.

2.2. Calculation of the Detection Radius

By considering the echo characteristics when the target crosses the baseline [38], the detection radius of the single-baseline system at different altitudes was simulated. The results are shown in Figure 3, with the parameters listed in

Table 1. Simulation parameters for the detection range of GNSS-FSR.

Parameter	Value
Equivalent isotropically radiated power EIRP	30 dBw [39]
Receiving antenna gain GR	20 dB
Carrier frequency f	1561.098 MHz
Receiver bandwidth B	4.092 MHz
Equivalent noise temperature Teff	344 K
Noise figure Fn	3 dB
System loss LS	3 dB
Elevation of satellite α	90°
Spread spectrum code chip width Tc	10,230 ms
Coherent integration time T	20 ms
Rectangular target length a	20 m
Rectangular target width b	17 m
Baseline length L	21,528 km
SNR detection threshold SNRcol	0 dB

. Given that the target RCS exhibits a sinc envelope pattern, the detection radius undergoes oscillatory changes with variations in target altitude. The inter-node distance corresponding to this detection radius satisfies the communication distance constraint in practice.

2.3. Approximation Error of Circular Detection Areas

The simulation scenario is shown in Figure 4, where $\beta$ represents the bistatic angle, $\alpha$ is the satellite elevation angle, and $h$ denotes the target altitude. The simulation parameters are shown in

Table 2. Simulation parameter settings.

Parameter	Value
Target height	10 km
Satellite elevation angle	90°, 60°, 45°
Baseline length	21,528 km
Critical value of the bistatic angle	176.56°

. Figure 3 indicates that the detection radius can be approximated as 0.6 km. When the target altitude is 10 km, the critical value of the bistatic angle ${\beta }_{col}$ can be calculated using the following equation:

$${\beta }_{col}=\mathrm{atan}\left({\displaystyle \frac{h}{r}}\right)+\mathrm{atan}\left({\displaystyle \frac{L-h}{r}}\right)$$

(2)

where $L$ is the baseline length, and $r$ is the detection radius.

Due to the satellite motion causing the satellite to deviate from the position directly above the receiver, the horizontal distance between the points corresponding to the same bistatic angle and the baseline is not equal for a given target altitude. Consequently, the aerial detection area is elliptical. Taking point $O$ in Figure 4 as the origin, the region composed of points with bistatic angles greater than the critical value is simulated. When the radius of the circular detection area is set to the absolute value of the minimum y-coordinate of the elliptical region, denoted as $\left|y_{\min }\right|$, the results are shown in Figure 5. At this point, the geometric approximation may result in coverage-blind spots. When the radius of the circular detection area is set to the absolute value $\left|x_{\min }\right|$ of the minimum x-coordinate of the elliptical region, the results are shown in Figure 6. At this point, the geometric approximation leads to partial redundancy.

Define the area ratio $S_{ratio}$ as the ratio of the area of the circular detection region $S_{circle}$ to the area of the elliptical detection region $S_{ellipse}$, that is

$$S_{ratio}={\displaystyle \frac{S_{circle}}{S_{ellipse}}}$$

(3)

Figure 7a presents the variation in the area ratio $S_{ratio}$ with the satellite elevation angle when the radius of the circular detection area is $\left|x_{\min }\right|$. The results indicate that as the satellite elevation angle increases, the redundant area of the detection region gradually decreases. However, when the satellite elevation angle is less than 50°, the redundant area of the detection region resulting from this approximation method accounts for more than 24% of the area of the circular detection region. In practice, the satellite’s motion causes continuous changes in the elevation angle. To improve the utilization rate of the detection area of a single node, the radius of the circular detection region can be appropriately increased within an acceptable level of detection blind spots.

Subsequently, an analysis is conducted on the detection blind spot area caused when the radius of the circular detection region is $\left|y_{\min }\right|$. Figure 7b presents the variation in the ratio of the detection blind spot area to the circular detection region area with satellite elevation angle. When the satellite elevation angle is less than 50°, the ratio of the detection blind spot area to the circular detection region area is greater than 17%. So, the radius of the circular detection region can be selected within the range $\left[\left|x_{\min }\right|,\left|y_{\min }\right|\right]$. Figure 7c presents the calculation results of the detection blind spot area when the detection region radius is $0.5\ast \left(\left|x_{\min }\right|+\left|y_{\min }\right|\right)$. It is evident that this approximation method effectively alleviates the issue of large blind spot areas when the satellite elevation angle is small.

2.4. Mathematical Model

Integrating the above assumptions and conclusions, the node layout optimization problem for GNSS-FSR can be described as follows: Deploy receiving nodes within a designated polygonal area to construct an FSR network based on a single GNSS satellite. The objective is to maximize the aerial detection area while minimizing the node cost.

Let the communication distance between node $i$ and its adjacent node $j$ be denoted as $d_{ij}$. The communication link must satisfy the following condition:

$$d_{ij}<d_{\max }$$

(4)

where $d_{\max }$ is the maximum communication distance.

The deployment scheme must ensure the connectivity of the network; that is, any node should be able to communicate with the central node through either single-hop or multi-hop transmission.

Suppose there is a polygon $P$, defined by a set of vertices $p_k$, $k=1,2,\dots ,n$. There exists a set of circles $\mathsf{\Lambda }\left(u\right)=\left\{C_i,i=1,2,\dots ,n\right\}$, with their centers defined as $u_i=\left(x_i,y_i\right)$.

Definition 1. 

The set  $\gamma =\underset{i=1}{\overset{n}{\mathrm{U}}}{C}_i\left(u_i\right)$ represents the coverage area formed by the union of the circles. When the set  $\gamma$  is capable of achieving coverage over the region  $P$, it should satisfy

$$P\cap \left(\underset{i=1}{\overset{n}{\mathrm{U}}}{C}_i\left(u_i\right)\right)=P.$$

(5)

Definition 2. 

If there exists a subset ${\mathsf{\Lambda }}_k\left(\mathit{v}\right)$ of $\mathsf{\Lambda }\left(\mathit{u}\right)$, with a corresponding vector $\mathit{v}=\left(u_1,u_2,\dots ,u_k\right)$, $s.t.\hspace{0.33em}k\le n$, that fulfills the following condition:

$$P\cap \left(\underset{i=1}{\overset{k}{\mathrm{U}}}{C}_i\left(u_i\right)\right)=P.$$

(6)

Then, ${\mathsf{\Lambda }}_k\left(\mathit{v}\right)$ is referred to as a sub-coverage of $P$.

Definition 3. 

Let  $A\left(S\right)$ be the area of the set  $S$, and the area of the set  $\mathsf{\Lambda }\left(\mathit{u}\right)$  outside the polygonal region  $P$  is denoted as

$$O_k\left(\mathit{v}\right)=A\left(\underset{i=1}{\overset{k}{\mathrm{U}}}{C}_i\left(u_i\right)\backslash P\right).$$

(7)

Therefore, the deployment optimization problem is as follows: (1) to determine the smallest subset of circles ${\mathsf{\Lambda }}_k\left(\mathit{v}\right)$ that constitutes a sub-coverage; (2) to maximize $O_k\left(\mathit{v}\right)$, which can be expressed as

$$\begin{array}{l}\left.\begin{array}{l}\underset{k}{\min } {\mathsf{\Lambda }}_k\left(\mathit{v}\right)\\ \underset{k}{\max } O_k\left(\mathit{v}\right)\end{array}\right\}\\ s.t.\hspace{1em}{\displaystyle {\sum }_{j\in k}\mathbf{1}\left(d_{ij}\le d_{\max }\right)\ge 1\hspace{1em}\forall i}\end{array}$$

(8)

where $\mathbf{1}\left(\cdot \right)$ is an indicator function, defined as $\mathbf{1}\left(x\right)=\left\{\begin{array}{ll}1,& d_{ij}\le d_{\max }\\ 0,& d_{ij}>d_{\max }\end{array}\right.$.

3. Node Cost Optimization Based on the BFGS Algorithm

Determining the initial number of receiving nodes required for a given deployment area is of significant importance, as it can greatly reduce the computational time of subsequent node layout optimization algorithms. Here, the BFGS algorithm is employed to obtain a suboptimal number of nodes [29]. The BFGS algorithm requires only first-order gradient information, making it suitable for large-scale node systems. In contrast, methods based on second-order gradient information necessitate the computation of an explicit Hessian matrix. The quasi-Newton approximation employed by BFGS circumvents the need to store the full Hessian matrix. Its update formula relies solely on the gradient information from the previous iteration, thereby consuming relatively less memory [40,41]. The update formula of the BFGS algorithm ensures that the Hessian matrix remains positive definite after each iteration, thereby avoiding matrix singularity. The update formula for the Hessian matrix is

$$B_{k+1}=B_k-{\displaystyle \frac{B_k{s}_k{s}_k^{\mathrm{T}}{B}_k}{s_k^{\mathrm{T}}{B}_k{s}_k}}+{\displaystyle \frac{y_k{y}_k^{\mathrm{T}}}{y_k^{\mathrm{T}}{s}_k}}$$

(9)

where $s_k=x_{k+1}-x_k$, $y_k=\nabla f\left(x_{k+1}\right)-\nabla f\left(x_k\right)$, $f\left(\cdot \right)$ is the objective function, and $x_k$ is the current solution. If the initial matrix $B_0$ is positive definite and $y_k^{\mathrm{T}}{s}_k>0$, then $B_{k+1}$ remains positive definite.

The specific steps for optimizing node cost are as follows:

(1)

The honeycomb method is used to obtain an initial node count.

The input parameters include the boundary coordinates of the deployment region $P=\left\{p_1\left(x_1,y_1\right),\dots ,p_k\left(x_k,y_k\right)\right\}$ and the detection radius $r$. The bounding rectangle for the region $P$ is first identified. Then, starting from its lower-left vertex, the circle’s center is moved right by the height of the circle’s inscribed regular hexagon, that is $\sqrt{3}r$, until the circle’s edge exceeds the right boundary of the rectangle. Following this, the circle’s center is raised by $1.5r$ and shifted left to right until the circle’s upper boundary meets the rectangle’s top edge. Ultimately, check whether the obtained circles intersect with $P$. If an intersection is found, increment the node count by one. See Algorithm 1 for details.

(2)

Randomly generate initial point positions within the deployment area.

(3)

Maximize the detection area overlap with the deployment area using the BFGS algorithm to optimize node positions for a coverage rate of 100%, as shown in Figure 8.

As the BFGS algorithm may converge to a local optimum, a certain number of repairs must be performed for each node count. In the simulation, the number of repairs is set to 3.

(4)

Incrementally reduce the number of nodes, repeat steps (2) and (3) until the BFGS algorithm can no longer repair, and yield an initial node count of $num\_initial$. It can be inferred that $num\_initial$ is approximately equal to the minimum node cost.

The detailed procedure is outlined in Algorithm 2.

Figure 8. Individual repair based on BFGS: (a) Unrepaired. (b) Repaired.

Figure 8. Individual repair based on BFGS: (a) Unrepaired. (b) Repaired.

Algorithm 1. Node count calculation based on the honeycomb method

Input: P = {p1(x1,y1),…,pk(xk,yk)}; r;

Algorithm 2. BFGS-based node count optimization method

4. Improved NSGA-II for Deployment Optimization

4.1. Framework of NSGA-II

The NSGA-II algorithm is a bio-evolutionary optimization method inspired by natural selection, capable of handling multi-objective optimization problems with conflicting goals. It is well-suited for solving the deployment optimization problem of GNSS-FSR receiving nodes. Unlike the Genetic Algorithm (GA), which yields a single optimal solution, the NSGA-II produces multiple Pareto-optimal solutions, representing the optimal layouts under different node cost scenarios. When a larger detection area is desired, a higher node cost can be appropriately selected to achieve this goal, a flexibility that GA cannot provide.

After obtaining the initial number of nodes using the BFGS algorithm, the search space of the NSGA-II is significantly reduced. Each individual is defined by the detection radius and node positions. The crossover and mutation processes enable individuals to transition from suboptimal solutions to more optimal ones with greater diversity. This characteristic is crucial for solving the optimal coverage problem. The NSGA-II typically comprises five steps: initialization, sorting, selection, crossover, and mutation. For the “sorting” step, the algorithm employs the fast non-dominated sorting method and introduces the concept of crowding distance to prevent the loss of high-quality individuals. In the improved algorithm, we utilize the mean absolute deviation distance to calculate the crowding distance, thereby comprehensively reflecting the density of individual solutions. In the “selection” operation, an elitist strategy is employed, where the parent and offspring populations are combined and subjected to non-dominated sorting to ensure the retention of high-quality individuals. Crossover is the process of combining the characteristics of two individuals to produce a new offspring, which typically exhibits higher fitness compared to the parent individuals. Here, the Voronoi diagram method is employed to implement crossover [35]. “Mutation” refers to the process of introducing slight perturbations to the values of individuals in the population. This mechanism prevents the algorithm from converging to local optima and enables a more comprehensive search of the entire solution space. In the improved algorithm, the “mutation” operation includes both node relocation and node removal. To ensure the detection area is free of holes, individuals require additional repair operations. If an individual cannot be repaired, it is removed from the population. The improved NSGA-II flowchart is shown in Figure 9.

4.2. Feasible Solution Generation

To obtain the optimal deployment strategy across varying node cost scenarios, half of the population will consist of individuals possessing a node count of $num\_initial+1$, while the remaining half will be composed of individuals with a node count of $num\_initial$. Given the deployment region $P$ and target detection radius $r$, node positions are randomly generated, and then the BFGS algorithm is used to repair the individual. Sufficient individuals must be initialized to maintain population diversity. To minimize the repair effect, we employ a method that generates uniformly distributed random points within a polygonal region, allowing the BFGS algorithm to initiate the repair process from a superior initial point. The steps for generating the initial node positions are as follows:

(1)

The polygonal area is subdivided into a series of triangles, with each triangle being assigned a unique identifier.

(2)

The areas of the triangles are determined in numerical order, with the vertices of each triangle specified as $\left(x_1,y_1\right)$, $\left(x_2,y_2\right)$, $\left(x_3,y_3\right)$. The area is given by

$$S_{\Delta }={\displaystyle \frac{1}{2}}\left|x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)\right|$$

(10)

(3)

Construct a sequence representing the distribution of triangle areas, denoted as $area$, initializing the index $are{a}_1$ to 0. $are{a}_i$ is the ratio of the cumulative area of triangles numbered from 1 to $i-1$ to the total area of the polygonal region.

(4)

Generate a random number, $random$, that is uniformly distributed within the interval $\left(0,1\right]$. The $m\mathrm{th}$ triangle is then chosen as the region for generating the random point, according to $are{a}_{m-1}<random\le are{a}_m$.

(5)

Inside the $m\mathrm{th}$ triangle, produce two random numbers, $rando{m}_1$ and $rando{m}_2$, which follow a uniform distribution within the range $\left[0,1\right]$.

①

If $rando{m}_1<rando{m}_2$, then generate two new random numbers;

②

If $rando{m}_1\ge rando{m}_2$, then generate a new random number, $rando{m}_3$, which is uniformly distributed within the interval $\left[0,1\right]$.

(6)

Calculate the coordinates of the random point $\left(x_r,y_r\right)$ within the triangular area as follows:

$$\left\{\begin{array}{l}{x}_r=x_1+rando{m}_2\cdot \left(x_3-x_1\right)+\\ \hspace{1em}\hspace{1em}\left(1-rando{m}_2\right)\cdot rando{m}_3\cdot \left(x_2-x_1\right)\\ y_r=y_1+rando{m}_2\cdot \left(y_3-y_1\right)+\\ \hspace{1em}\hspace{1em}\left(1-rando{m}_2\right)\cdot rando{m}_3\cdot \left(y_2-y_1\right)\end{array}\right.$$

(11)

Since the first step of generating the initial solution requires the division of the polygonal region into multiple triangles, the widely used Delaunay triangulation method encounters some difficulties when dealing with concave polygonal regions, resulting in some nodes being located outside the region. This is because Delaunay triangulation requires that the circumcircle of each generated triangle does not contain any other points, a condition that is generally well satisfied in convex polygons but may not be met in concave polygons.

To generate feasible solutions for concave polygons, the following method is employed: The position of the first node $\left(x_{r1},y_{r1}\right)$ is randomly generated. Subsequently, the position $\left(x_{r1},y_{r1}\right)$ of the $i$-th node must satisfy the condition that the distance to $\left\{\left(x_{r1},y_{r1}\right),\dots ,\left(x_{r\left(i-1\right)},y_{r\left(i-1\right)}\right)\right\}$ is greater than $r$.

4.3. Selection, Crossover, and Mutation

The selection operation considers eliminating 50% of the lower-ranked individuals, a proportion that can be adjusted based on the initial population size and the desired convergence rate. The crossover operation helps retain the characteristics of high-quality individuals. In the optimal equal-circle covering problem, randomly selecting nodes from two parent individuals does not yield more optimal offspring and cannot be repaired. We employ a crossover method based on Voronoi diagrams [35]. To obtain superior offspring, the crossover probability is set to 1.

The specific procedure is as follows:

(1)

Traverse the individuals in the current population. For each individual $A_i$, randomly select another remaining individual from the population to perform the crossover operation. The two individuals $A_i$ and $A_j\left(j\ne i\right)$ are, respectively, regarded as Parent 1 and Parent 2.

(2)

Obtain the Voronoi diagrams of Parent 1 and Parent 2. Given the node positions $\left\{p_1,p_2,\dots ,p_n\right\}$, the plane is segmented into $n$ subregions. Each subregion $S_i$ encompasses points that are closer to a designated node $p_i$ than to any other node.

(3)

Based on the Voronoi diagram of Parent 1, the node positions of Parent 2 are superimposed onto this diagram. For each polygonal region, if it contains nodes from both Parent 1 and Parent 2, a node is randomly selected from the contained nodes for retention. If it contains only nodes from Parent 1, the current node is retained. Subsequently, the offspring individual is subjected to a repair operation. If the repair is successful, the offspring is incorporated into the population.

(4)

Based on the Voronoi diagram of Parent 2, the node positions of Parent 1 are displayed on this diagram. For each polygonal region, if it contains nodes from both Parent 1 and Parent 2, a node is randomly selected from the contained nodes for retention. If it contains only nodes from Parent 2, the current node is retained. After the aforementioned processing, a repair operation is performed on the offspring individual. If the repair is successful, the offspring is incorporated into the population.

(5)

Operations (2), (3), and (4) are executed for each individual in the population to generate a new population.

Each crossover results in two offspring, as depicted in Figure 10. When the detection radii of the receiving nodes are not identical, and the number of available receiving nodes and their detection performance are fixed, the crossover algorithm may face insufficient node resources when randomly selecting a node to retain in a polygonal subregion. To mitigate this, the algorithm first checks whether the detection radii of the nodes from Parent 1 and Parent 2 within the subregion are equal. If they are equal, a node is randomly selected for retention; otherwise, the node from the original parent is retained.

The flowchart of the crossover operation is depicted in Figure 11.

Mutation consists of two types: circle removal and circle movement. The removal of circles is designed to identify deployment schemes that minimize node costs. Consequently, the probability of this mutation operation is set to 1, ensuring that every individual in the population undergoes this operation. The rule for selecting nodes to be removed is based on the intersection area between the node’s detection region and the deployment area. Specifically, nodes with either the maximum or minimum intersection area are chosen. A larger intersection area typically indicates redundant nodes within the deployment area, while a smaller intersection area provides an opportunity to achieve 100% coverage through the repair operation by adjusting the positions of the remaining nodes.

The movement of circles is intended to seek more optimal deployment schemes while maintaining the current node cost. Since individuals in the population have previously undergone repair operations, the probability of obtaining superior individuals through random node relocation is relatively low. Therefore, the probability of this mutation operation is set to 0.2 to decrease the algorithm’s runtime. The index of the node to be relocated is randomly selected, and random offsets are generated in both the x-axis and y-axis.

The flowcharts of the mutation operations are depicted in Figure 12.

4.4. Fitness Function

Considering three fitness functions, which are ① maximizing the area of intersection between circles and the exterior of the deployment region; ② minimizing the area of self-intersection among circles; ③ minimizing the circle count. These functions are normalized to the range of 0 to 1 and can be expressed as follows:

$$\left\{\begin{array}{l}\min \left(f_1=-{\displaystyle \frac{C_{outside}}{R}}\right)\\ \min \left(f_2={\displaystyle \frac{C_{self\_intersection}}{C}}\right)\\ \min \left(f_3={\displaystyle \frac{num}{num\_initial}}\right)\end{array}\right.$$

(12)

where $C_{outside}$ refers to the area of the circle outside the polygonal region, $R$ is the area of the polygonal area, $num$ is the number of nodes in the current individual, $C_{self\_intersection}$ represents the area of self-intersection of the circles, and $C$ indicates the total area of circles.

4.5. Crowding Distance Calculation

In the conventional NSGA-II, the crowding distance calculation only considers the distance between adjacent individuals at the same rank, which may lead to a reduction in population diversity. To mitigate this, we utilize the mean absolute deviation to compute the crowding distance, thereby providing a comprehensive representation of the solution density distribution. The formula for computing the crowding distance is expressed as follows:

$$d_i=\left\{\begin{array}{l}\infty ,\hspace{1em}\hspace{1em}i=1\hspace{0.33em}\hspace{0.33em}or\hspace{0.33em}\hspace{0.33em}i=end\\ {\displaystyle \frac{1}{k}}{\displaystyle \sum _{l=1}^k\frac{1}{M}{\displaystyle \sum _{m=1}^M\frac{\left|f_l\left(i\right)-f_l\left(m\right)\right|}{f_l^{\max }-f_l^{\min }}}},others\end{array}\right.$$

(13)

where $k$ is the number of objective functions, $M$ denotes the number of solutions in the current layer of the Pareto front, and $f_l\left(\cdot \right)$ represents the $l\mathrm{th}$ fitness function. For function $f_3$ in Equation (12), the crowding distance calculation may encounter a zero denominator when individuals of the same rank achieve $f_3^{\max }=f_3^{\min }$. Therefore, when $l=3$, the denominator in Equation (13) is considered to be 1 to prevent this situation.

4.6. TOPSIS Analysis Based on the Entropy Weight Method

The Technique for Order Preference by similarity to an Ideal Solution (TOPSIS) as a comprehensive evaluation method, is suitable for decision-making problems involving multiple criteria. In the evaluation of various schemes, higher scores indicate greater superiority. However, in the original method, equal weights are assigned to all criteria, which does not allow for adaptive weighting. The Entropy Weight Method (EWM) is an objective weighting approach that calculates the information entropy of each criterion. It determines the weight of a criterion based on its impact of relative variability on the overall evaluation.

The steps of the EWM are as follows:

(1)

Normalization of the criterion matrix: According to the definitions of the fitness functions in Equation (12), where smaller values are preferred, all criteria are of the minimization type. Therefore, the following operation is performed to normalize the criteria:

$$y=\max -x$$

(14)

where $\max$ represents the maximum value of the current criterion, and $x$ denotes the value of the current criterion.

(2)

Normalization of the positive matrix: The purpose of this operation is to eliminate the influence of different scales. Assuming the number of schemes is $n$ and the number of evaluation criteria is $m$, the normalized positive matrix can be represented as follows:

$$Y=\left[\begin{array}{cccc}{y}_{11}& y_{12}& \dots & y_{1m}\\ y_{21}& y_{22}& \dots & y_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ y_{n1}& y_{n2}& \cdots & y_{nm}\end{array}\right]$$

(15)

The matrix obtained through the normalization process is denoted as $Z$, where each element can be expressed as

$$z_{ij}=y_{ij}/\sqrt{{\displaystyle {\sum }_{i=1}^n{y}_{ij}^2}}$$

(16)

If the matrix $Z$ contains negative values, normalization is performed using the following method:

$$z_{ij}={\displaystyle \frac{y_{ij}-\min \left\{y_{1j},y_{2j},\dots ,y_{nj}\right\}}{\max \left\{y_{1j},y_{2j},\dots ,y_{nj}\right\}-\min \left\{y_{1j},y_{2j},\dots ,y_{nj}\right\}}}$$

(17)

(3)

Calculate the proportion of each element in the normalized matrix:

$$p_{ij}={\displaystyle \frac{z_{ij}}{\sqrt{{\displaystyle {\sum }_{i=1}^n{z}_{ij}}}}}$$

(18)

(4)

Calculate the information entropy of each criterion and determine the weights: For the j-th criterion, its information entropy can be expressed as

$$e_j=-{\displaystyle \frac{1}{\ln n}}{\displaystyle \sum _{i=1}^n{p}_{ij}\ln p_{ij}}$$

(19)

The entropy weights of the criteria are given by

$$w_j={\displaystyle \frac{1-e_j}{{\displaystyle {\sum }_{j=1}^{m}1-e_j}}}$$

(20)

(5)

Construct the weighted normalized matrix.

$$v_{ij}=w_j\cdot y_{ij}$$

(21)

The specific steps of the Topsis analysis are as follows:

From the normalized matrix $V$, the ideal optimal and worst solutions are obtained. The ideal optimal solution vector is constructed by extracting the maximum values from each column, while the ideal worst solution vector is formed by selecting the minimum values from each column, which are, respectively, denoted as

$$v^{+}=\left[v_1^{+},v_2^{+},\dots ,v_m^{+}\right]$$

(22)

$$v^{-}=\left[v_1^{-},v_2^{-},\dots ,v_m^{-}\right]$$

(23)

The unnormalized score of the i-th evaluation object can be expressed as

$$S_i={\displaystyle \frac{D_i^{-}}{D_i^{+}+D_i^{-}}}$$

(24)

where $D_i^{+}$ represents the distance of the evaluation object from the maximum value, and $D_i^{-}$ represents the distance from the minimum value, which are, respectively, given by

$$D_i^{+}=\sqrt{{\displaystyle \sum _{j=1}^m{\left(v_j^{+}-v_{ij}\right)}^2}}$$

(25)

$$D_i^{-}=\sqrt{{\displaystyle \sum _{j=1}^m{\left(v_j^{-}-v_{ij}\right)}^2}}$$

(26)

By normalizing $S_i$, the final scores of the evaluation objects can be obtained.

5. Simulation Analysis

5.1. Algorithm Validity Verification

The deployment area vertices are (−0.5, 0.8), (0.5, 1.0), (1.5, 0.5), (1.6, 0), (1.0, −1.0), (0, −1.0), and (−0.5, −0.5) km, with a detection radius of 0.6 km. The initial node count is determined to be 6 through the BFGS algorithm. We employ a variety of methods to search for the optimal stationing scheme, including the honeycomb method, the rectangle area coverage [42], the Nelder–Mead, the BFGS, the improved GA [35], and the improved NSGA-II. Among these, the Nelder–Mead and BFGS, operating from the initial node count, yield 10 repaired individuals. The individual with the smallest values for both $f_1$ and $f_2$ is chosen as the optimal solution. The initial population for both the improved GA and the improved NSGA-II is identical.

Due to the extensive repair required for the initial population, a larger population size is not recommended. In simulations, an initial population of 20 and 6 iterations is sufficient to obtain an optimal stationing configuration. The initial population consists of 10 individuals with a node cost of 7 and 10 individuals with a node cost of 6. The node deployment is illustrated in Figure 13, and the optimization results of each method are presented in

Table 3. Receiving-node deployment optimization results 1.

Method	f1	f2	Num_Initial * f3	Remark
Honeycomb method	−0.5882	0.0963	5	Coverage rate < 100%
Rectangle area coverage	−0.2113	0.2376	4	Coverage rate < 100%
Nelder–Mead	−0.5886	0.2792	6	–
BFGS	−0.6239	0.2524	6	–
Improved GA	−0.6288	0.2497	6	–
Improved NSGA-II	−0.6362	0.2430	6	–
	−0.8421	0.2870	7	–

. The symbol * represents multiplication, the same applies in the following text. The self-intersection area within the circles, the area outside the polygonal region covered by the circles, and the area intersected by the circles with the polygonal region are distinctly marked with different colors. The honeycomb strategy and the approximate rectangular coverage method do not yield a stationing configuration that achieves a coverage rate of 100%. The Nelder–Mead and BFGS algorithms yield satisfactory outcomes. However, the Nelder–Mead algorithm exhibits a longer repair time, leading to lower optimization efficiency. The proposed improved NSGA-II demonstrates the smallest values for $f_1$ and $f_2$ at a node cost of 6, and it further offers an optimized deployment strategy for a node cost of 7, thereby augmenting the variety of potential solutions.

5.2. Algorithm Adaptability Verification

5.2.1. Polygonal Deployment Area

Increase the vertex coordinates of the previous section’s deployment area by 25% to define a new polygonal area, with all other parameters unchanged. The initial number of nodes is 9. To obtain stationing solutions with various node costs, the initial population consists of 10 individuals with a node cost of 10 and 10 individuals with a node cost of 9. The deployment illustrations are shown in Figure 14, with the optimization outcomes presented in

Table 4. Receiving-node deployment optimization results 2.

Method	f2	f2	Num_Initial * f3	Remark
Honeycomb method	−0.2872	0.1042	6	Coverage rate < 100%
Rectangle area coverage	−0.2678	0.2891	7	Coverage rate < 100%
Nelder–Mead	−0.4655	0.3324	9	–
BFGS	−0.5238	0.2850	9	–
Improved GA	−0.4880	0.3155	9	–
Improved NSGA-II	−0.3639	0.2768	8	–
	−0.5285	0.2820	9	–
	−0.6272	0.3368	10	–

. The honeycomb method and the approximate rectangular coverage method do not achieve 100% area coverage. The BFGS achieved a superior solution compared to the improved GA, indicating that repeatedly invoking the BFGS might also lead to better stationing solutions. The proposed improved NSGA-II not only exhibits the best performance when the node cost is 9 but also provides stationing solutions for a node cost of 8.

5.2.2. Circular Deployment Area

For a deployment area with a circular radius of 1.3 km, and with all other parameters as previously stated, the initial number of nodes is 9. The initial population consists of 10 individuals with a node cost of 10 and 10 individuals with a node cost of 9. The optimization results for station layout are presented in

Table 5. Receiving-node deployment optimization results 3.

Method	f1	f2	Num_Initial * f3	Remark
Honeycomb method	−0.0588	0.0862	4	Coverage rate < 100%
Rectangle area coverage	−0.4654	0.3366	9	Coverage rate < 100%
Nelder–Mead	−0.4530	0.3299	9	–
BFGS	−0.4868	0.3048	9	–
Improved GA	−0.5226	0.2737	9	–
Improved NSGA-II	−0.5226	0.2737	9	–
	−0.6235	0.3252	10	–

, with a schematic diagram shown in Figure 15. The stationing optimization outcomes for the improved GA and the improved NSGA-II are the same. As both methods employ the same initial population, the optimal solution is already present within the population. Consequently, genetic operations like crossover and mutation do not yield a superior individual, leading to identical optimization results.

5.2.3. Concave Polygonal Deployment Area

The vertices of the concave polygon are set as (−0.5, 0.75), (0.5, 0.5), (0.5, −0.5), (1.5, −0.5), (1.5, 0.5), (2, 0.5), (2, −1), (0, −1), and (−0.5, −0.5) km, with a detection radius of 0.6 km. The initial node cost obtained using the BFGS algorithm is 6. The initial population consisted of 10 individuals with a node cost of 7 and 10 individuals with a node cost of 6, with the number of iterations set to 6. The deployment schemes obtained by each optimization method are shown in Figure 16, and the values of the fitness function are summarized in

Table 6. Receiving-node deployment optimization results 4.

Method	f1	f2	Num_Initial * f3	Remark
Honeycomb method	−1.2107	0.0940	9	Coverage rate < 100%
Rectangle area coverage	−1.3555	0.3916	8	Coverage rate < 100%
Nelder–Mead	−1.1293	0.1546	6	–
BFGS	−1.1008	0.1705	6	–
Improved GA	−1.1347	0.1513	6	–
Improved NSGA-II	−1.1453	0.1459	6	–
	−1.4109	0.1891	7	–

. It can be seen that the improved NSGA-II is capable of providing two deployment schemes, and the fitness function value of the deployment scheme with a node cost of 6 is superior to those of other algorithms.

5.2.4. Heterogeneous Node Deployment Optimization

Based on the initial node cost ${num}_0$ identified using the BFGS algorithm, it is assumed that there are ${num}_0+1$ available radar receivers. Among these receiving nodes, a few nodes exhibit varying detection performance due to hardware differences. Four deployment scenarios for heterogeneous nodes are discussed in different deployment areas. The simulation parameters are listed in

Table 7. Simulation parameters for heterogeneous node deployment scenarios.

Deployment Area	Number of Available Receiving Nodes	Node Detection Radius
Convex polygon	7	[0.7,0.6,0.6,0.6,0.5,0.4] (km)
Larger convex polygons	10	[0.7,0.7,0.6,0.6,0.6,0.6,0.6,0.6,0.5,0.5] (km)
Circle	10	[0.7,0.7,0.6,0.6,0.6,0.6,0.6,0.6,0.5,0.5] (km)
Concave polygon	7	[0.7,0.6,0.6,0.6,0.5,0.4] (km)

. The optimized deployment results are shown in Figure 17, and the fitness function values corresponding to the optimization results are provided in

Table 8. Fitness function values corresponding to the optimization results of heterogeneous node deployment scenarios.

Deployment Area	BFGS			Improved NSGA-II
	f1	f2	Num_Initial * f3	f1	f2	Num_Initial * f3
Convex polygon	−0.6040	0.2683	6	−0.3083	0.2913	5
				−0.6102	0.2640	6
				−0.8717	0.2660	7
Larger convex polygons	−0.4806	0.3246	9	−0.3715	0.2735	8
				−0.5176	0.2895	9
				−0.6091	0.3494	10
Circle	−0.4770	0.3061	9	−0.3962	0.2307	8
				−0.5220	0.2725	9
				−0.5558	0.3792	10
Concave polygon	−1.0485	0.2006	6	−0.7387	0.1804	5
				−1.1954	0.1199	6
				−1.3101	0.2440	7

. Compared with the results of the BFGS algorithm, the improved NSGA-II algorithm successfully achieves the deployment optimization of heterogeneous nodes.

5.3. Scheme Optimization

The optimization results of the four deployment regions were evaluated using the TOPSIS analysis method based on the EWM. The calculation results are presented in

Table 9. The TOPSIS analysis results based on the EWM.

	Region	Polygonal Region 1		Polygonal Region 2			Circular Region		Concave Polygonal
Algorithm
Nelder–Mead		0.1116		0.1062			0.1028		0.1627
BFGS		0.1825		0.2120			0.1626		0.1235
Improved GA		0.1918		0.1411			0.2602		0.1717
Improved NSGA-II		0.2104 (Node cost: 6)	0.3037 (Node cost: 7)	0.1953 (Node cost: 8)	0.2173 (Node cost: 9)	0.1281 (Node cost: 10)	0.2602 (Node cost: 9)	0.2142 (Node cost: 10)	0.1878 (Node cost: 6)	0.3543 (Node cost: 7)

. It can be observed that the improved NSGA-II algorithm is capable of providing a more optimal deployment scheme. The EWM can assign weights based on the distribution of the data, thereby reducing subjectivity. However, in practical decision-making, the preferences of decision-makers may need to be considered. For example, selecting a deployment scheme with the lowest node cost to reduce expenses, or choosing a deployment scheme with the highest node cost to increase the probability of target detection.

5.4. Analysis of the Effects of Satellite Motion

As illustrated in Figure 18, the movement of the satellite radiation source causes the displacement of the aerial detection region. Consequently, continuous surveillance of a specified airspace cannot be achieved using a single satellite. If the receiving nodes can move flexibly, for a specific satellite radiation source, assuming that the satellite is directly above the receiving station at time zero, the displacement of the aerial detection area corresponding to a change in the satellite elevation angle $\alpha /2$ is $\Delta x=h\cdot \tan \left({\displaystyle \raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)$. Based on the principle of similar triangles, the distance $\Delta x^{\prime }$ that the ground receiving station needs to move is

$$\Delta x^{\prime }=\Delta x{\displaystyle \frac{L}{L-\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$\cos \left(\alpha /2\right)$}\right.}}$$

(27)

Assuming a baseline length of 20,200 km and a target altitude ranging from 6 to 10 km (approximately corresponding to $d_{\mathrm{R}}$ in Figure 2); the displacement of the detection region under different satellite elevation angle variations is shown in Figure 19. The results indicate that the displacement of the detection region increases with the increase in target altitude. When $\alpha ={1.5}^{\circ }$ and the target altitude is 10 km, the displacement of the detection region has already reached 260 m. The velocity of the BeiDou Medium Earth Orbit (MEO) satellites is approximately 4730 m/s. Ignoring the variation in baseline length during satellite motion, the corresponding satellite movement times are 37.27 s ($\alpha ={0.5}^{\circ }$), 74.54 s ($\alpha =1^{\circ }$), and 111.80 s ($\alpha ={1.5}^{\circ }$).

In practical applications, under open and unobstructed conditions, ground-based receiving nodes typically can simultaneously receive signals from 8 to 12 GNSS satellites. The elevation angles of these satellites relative to the ground-receiving station vary, and consequently, the positions of the corresponding aerial detection regions differ. The deployment scheme obtained using the method proposed can optimize the detection region of a single satellite, achieving a larger aerial detection area at a lower node cost. The detection regions of multiple satellites can be connected to form continuous coverage, thereby enabling the surveillance of airspace. Full 100% detection coverage over the deployment region can be achieved even with some nodes in an inactive state, which can further reduce the system’s energy consumption. Assuming the detection areas corresponding to three satellites in the air are as shown in Figure 20, it can be determined from the numbering of the nodes that Node 5 can be in an inactive state at this time.

6. Conclusions

This paper addresses the optimization of ground-receiving-node deployment for aerial target surveillance applications in the FSR network based on GNSS. To reduce the computational workload of the optimization algorithm, a node cost optimization method based on the BFGS algorithm is proposed. Subsequently, an improved NSGA-II is introduced for deployment optimization. The effectiveness and adaptability of the algorithm were verified by applying it to deployment areas of convex polygons, circles, and concave polygons, and comparisons are made with other algorithms. The TOPSIS analysis results based on the EWM indicate that the improved NSGA-II can obtain the optimal deployment solution. The algorithm is also applicable to deployment optimization involving a small number of heterogeneous nodes. This algorithm aims to maximize the aerial detection area at the minimum node cost. It can be applied to the node deployment of aerial target surveillance systems over designated maritime areas or islands. Future research may focus on the deployment optimization of receiving nodes using multiple satellites as radiation sources, which could further enhance the system detection capability.