Deployment Strategy Analysis for Underwater Geodetic Networks

Abstract

Seafloor geodetic network (SGN) is the foundation for building an underwater positioning, navigation and timing (PNT) system. Traditional network deployment mainly focuses on the deployment of underwater sensor network nodes. However, for SGN, there is no surface buoy node and submarine buoy node, and the number of anchors is limited because it is quite expensive to fully cover large scale areas. To achieve wide coverage and good positioning service of each set of underwater base stations, we focus on the network design of a single set of reference stations in this paper. We propose several deployment plans for a local SGN and then analyze their service quality indicators by considering the stratification effect caused by non-uniformly distributed sound speed. To evaluate the performance of each topology of SGN, we compare their coverage range, horizontal dilution precision (HDOP) and accuracy performance of positioning tests. Based on the overall performance in our simulation, we believe that the star five-node topology is a good topology design under sufficient economic conditions.

1. Introduction

Seafloor geodetic networks (SGNs) contain many sets of acoustic-based reference stations distributed on the seabed, forming a positioning system similar to the global navigation satellite system (GNSS). It can provide time and spatial information for various equipment on the water surface and underwater [1]. The SGN is the foundation for building an underwater positioning, navigation and timing (PNT) system. It is also an important infrastructure for marine safety, marine economic development, and marine environmental monitoring, and an important support for marine geological research and seabed resource exploration.

In 1991, James proposed a buoy-based long baseline positioning system in [2], which extended the concept of global positioning system (GPS) to underwater systems for the first time. Unlike terrestrial radio, underwater localization faces with many challenges due to the special underwater environment, such as the difficulty in clock synchronization caused by long signal propagation delay [3,4], the multipath effect caused by signal reflection at the ocean surface or bottom [5], insufficient reference nodes due to limited communication coverage of nodes [4,5], and the signal propagation path bending, which is called stratification effect, caused by the spatio-temporal variety of sound velocity [6,7].

For accurate target localization and navigation, different methods have been developed using angle of arrival (AOA) [8], time of arrival (TOA) [9], time difference of arrival (TDOA) [10], received signal strength (RSS) [11], among which the TOA and TDOA are the two most popular methods adopted in underwater environment. Beck et al. [9] developed the least squares (LS) approach for positioning a radiating source from circular TOA-based measurements. Carevic [10] proposed a nonlinear LS technique in computing the motion parameters for each target in a TDOA-based multi-target localization in the 3D space.

To solve the clock asynchronization problem, Cheng et al. proposed a time of arrival (TOA)-based silent underwater positioning scheme (UPS) in [12], which transforms the clock asynchronization problem between target node and reference nodes into the clock asynchronization problem among reference nodes. Carroll et al. and Yan et al., respectively, proposed localization algorithms in [13,14,15] that utilize asymmetric signal round-trip processes to reduce clock synchronization requirements.

With the scale expansion of underwater sensor networks, large-scale positioning has received widespread attention. However, due to the expensive underwater equipment and sparse node deployment, underwater positioning faces the problem of insufficient reference nodes. To tackle this problem, Zhou et al. proposed an efficient localization algorithm for large-scale underwater sensor network in [16], in which a multi-hop 3D Euclidean distance estimation method was proposed, and new reference nodes were searched among two hop neighbors. Based on [16], Zhang et al. proposed a top–down positioning scheme in [17] that optimized the selection of new reference nodes.

To tackle the issue of the underwater sound velocity distribution exhibiting spatio-temporal variability, resulting in a significant Snell effect in the signal propagation mode [4,6], Diamant and Lampe proposed a localization algorithm in [18] that regards the sound velocity as an invariant parameter to be solved. Liu et al. proposed a joint synchronization and localization method in [19], Zhang et al. proposed a signal trajectory correction localization method based on Gaussian Newton solution in [20], Zhang et al. proposed a Cramer–Rao lower bound-based optimality localization (CRLB–OL) method in [21], and Zhang et al. combined TDOA and AOA measurements for localization with a constrained weighted LS estimator in [22].

The aforementioned algorithms have proposed good solutions for the problem of high latency, limited node coverage, and stratification compensation (uniform sound speed distribution) for underwater positioning. However, apart from above localization algorithms, target-sensor geometry also plays a significant role in determining target localization performance. Many underwater deployment methods that concentrate on improving coverage, latency, and energy efficiency of underwater sensor networks were proposed ten years ago [23,24,25,26,27].

Recently, Iqbal et al. [28] proposed a deployment strategy that achieves full coverage of a given area with minimum sensor nodes. Wang et al. [29] proposed a node sinking algorithm for 3D coverage and connectivity in underwater sensor networks. Zhang et al. [30] proposed an improved cuckoo optimization algorithm to achieve a fair and efficient surface gateway layout optimization. Liu et al. [31] proposed a dynamic surface gateway placement scheme for mobile underwater networks that maximizes the coverage and minimizes end-to-end latency. Xu et al. [32] proposed an efficient deployment scheme to provide reliable data transmission.

Existing deployment works [28,29,30,31,32] mainly focus on underwater wireless sensor networks. However, the deployment strategy may not suitable for SGNs because of several reasons. First of all, the SGN is oriented towards large-scale ocean regions, even the global ocean, so the network construction cannot be seamlessly connected due to equipment production costs. Secondly, the element nodes in SGN are designed to serve for predetermined areas for a long time, so the buoys and submerged sensor nodes are not suitable for SGN as they move with the water flow. Moreover, for each set of acoustic-based reference stations distributed on the seabed, the stations should form a symmetrical structure to provide fair and consistent positioning and navigation services for all directions. The scenario of SGNs is shown in Figure 1.

In order to achieve wide coverage and good positioning service of each set of underwater base stations, we focus on the network design of a single set of reference stations in this paper. We propose several deployment plans for local SGN, and then analyze their service quality indicators, including the serving coverage and accuracy performance for localization. The contribution of our work can be summarized as follows:

• To offer a suitable design of local SGN, we propose several deployment strategies and compare their coverage and accuracy performance.
• To make the design more realistic, we consider the stratification effect of sound speed and combine it with the active sonar equation to calculate the coverage characteristics, where we derive the sound trajectory as a function of the initial grazing angle.

The organization of this paper is elaborated below. Section 2 provides the deployment strategy of SGN and shows the methods for performance analysis. Comparisons of different structures of local SGNs are given in Section 3, and conclusions are drawn in Section 4.

2. Deployment Strategy of SGN

2.1. Network Structures

Due to the sparsity of discontinuous coverage in the deployment of the SGN, the service range of a set of anchor nodes has become a very important indicator. It has been proven that the optimal geometry for multi-static TOA localization is symmetric [33]; thus, we mainly consider the symmetric deployment strategy of SGNs. Moreover, the symmetric deployment strategy can provide consistency services in all directions.

In this section, we illustrate the detail network model, where the symmetric deployment strategy with anchor Nodes from 3 to 8 is employed. In fact, the underwater positioning is a 3D problem, thus it requires at least four reference nodes for locating a target. However, the depth information can be obtained with depth sensors, so the number of reference anchors can be decreased to three. The deployment strategies are shown in Figure 2. For symmetric networks, we mainly consider polygon networks as shown in Figure 2a; however, there is a problem of poor reference node configuration at the edge of their coverage range. Therefore, we propose a star network for four-node and five-node scenarios as shown in Figure 2b.

2.2. Anchor Coverage

2.2.1. Single Anchor Coverage

When the signal propagation speed is uniformly distributed in the medium, the coverage range of a single anchor is a hemisphere as shown in Figure 3a. However, real underwater sound speed is non-uniformly distributed due to the effect of temperature, salinity, and static pressure, resulting in a significant Snell refraction effect during signal propagation. Meanwhile, under the same scale conditions, the variation in sound speed with depth is more significant compared to that in the horizontal direction, so we assume that the horizontal direction of the medium is isotropic, and the actual coverage range is similar to that of a half-ellipsoid as shown in Figure 3b.

We assume the sound signal source power is $P_a$ in Watt, the detective threshold of any receiver in this paper is $D_t$ in dB; then, the sonar function satisfies

$$D_t=SL-TL-NL,$$

(1)

where $SL=10log\frac{I}{I_0}$, I is the sound intensity in ${\mathrm{W}/\mathrm{m}}^2$, and $I_0$ is the reference sound intensity that satisfies $I_0=0.67\times {10}^{-18}$ [34]. In fact, I describes the sound intensity when signal is transmitted to a distance of 1 m; then, $I=\frac{P_a}{4\pi }$ for spherical broadcasting. Therefore, $SL$ can be calculated as $SL=170.77+10log{P}_a$, the acoustic source level, $TL$ is the transmission loss, and $NL$ is the noise interference level. The transmission loss caused by spreading and absorption can be expressed as

$$TL=n·10log\left(r\right)+\alpha r\times {10}^{-3},$$

(2)

where r is the transmission distance in meters, n is the propagation factor, and $\alpha$ is the absorption coefficient measured in decibels per kilometer. The value of n depends on the type of channel [35]; when considering the spherical propagation mode of the light-of-sight signal, there are $n=2$. Absorption coefficient $\alpha$ is an empirical monotonic increasing function of signal frequency f [36], which can be calculated by

$$\alpha \left(f\right)=\frac{0.109f^2}{1+f^2}+\frac{40.7f^2}{4100+f^2}+2.75\times {10}^{-4}{f}^2+0.003,$$

(3)

where f is measured in kilohertz. By solving Equations (1)–(3), the maximum propagation distance of the light-of-sight signal can be obtained, which is marked as $r_{max}$ in the following parts of this paper.

As mentioned earlier, the variety of sound speed causes signal propagation path bending, and sound speed changes are typically described using a layered ray model according to [7,37]. We let the sound speed profile be $S=[s_0,s_1,\cdots ,s_i],i=0,1,\cdots ,I$. For a single linear sound speed layer unit, as shown in Figure 4, there is a relationship between the signal propagation path and the horizontal propagation distance:

$$dx=\frac{dz}{tan{\beta }_i}.$$

(4)

According to Snell’s law, Equation (4) can be modified as

$$dx=\frac{cos{\beta }_i}{\sqrt{n_i^2-co{s}^2{\beta }_i}}dz,$$

(5)

where $n_i$ is the refractivity at the ith depth layer. Since the depth information is often measured with depth sensors, the target localization can be achieved when knowing the horizontal propagation distance of signals between the target and each reference node. After integrating, the horizontal propagation distance of the signal from target to the jth reference node is [34]

$$\Delta l_{i,j}=\left|\frac{sin{\beta }_{i-1,j}-sin{\beta }_{i,j}}{a_{i}cos{\beta }_{i-1,j}}\right|,$$

(6)

where $a_i$ is the relative gradient of sound speed that satisfies

$$s\left(z_i\right)=s\left(z_{i-1}\right)(1+a_i\Delta z_i),\Delta z_i=z_i-z_{i-1}.$$

(7)

The real signal propagation path is

$$\Delta r_{i,j}=\left|\frac{sin{\beta }_{i-1,j}-sin{\beta }_{i,j}}{a_{i}co{s}^2{\beta }_{i-1,j}}\right|.$$

(8)

According to Snell’s law [7,34],

$$n_0{c}_0=n_i{s}_i=n_k{s}_k,$$

(9)

$$\frac{n_i}{n_k}=\frac{s_k}{s_i}=\frac{cos{\beta }_{k,j}}{cos{\beta }_{i,j}}.$$

(10)

Equation (7) can be rewritten as

$$\Delta l_{i,j}=\left|\frac{\sqrt{1-{\left(cos{\beta }_{i-1,j}\right)}^2}-\sqrt{1-{\left(cos{\beta }_{i,j}\right)}^2}}{a_{i}cos{\beta }_{i-1,j}}\right|.$$

(11)

According to (10), there is

$$cos{\beta }_{i,j}=\frac{cos{\beta }_{0,j}}{s_0}{s}_{.}$$

(12)

Then, the total signal propagation distance in the horizontal direction is

$$l_j=\sum _{i=1}^I\Delta l_{i,j}=\sum _{i=1}^I\left|\frac{\sqrt{1-{\left(\frac{cos{\beta }_{0,j}}{s_0}{s}_{i-1}\right)}^2}-\sqrt{1-{\left(\frac{cos{\beta }_{0,j}}{s_0}{s}_i\right)}^2}}{a_{i-1}\left(\frac{cos{\beta }_{0,j}}{s_0}{s}_{i-1}\right)}\right|,$$

(13)

where horizontal signal propagation distance $l_j$ is a function of the initial grazing angle $cos{\beta }_{0,j}$. Meanwhile, based on (8), the real signal propagation distance can be calculated as

$$r_j=\sum _{i=1}^I\Delta r_{i,j}=\sum _{i=1}^I\left|\frac{\sqrt{1-{\left(\frac{cos{\beta }_{0,j}}{s_0}{s}_{i-1}\right)}^2}-\sqrt{1-{\left(\frac{cos{\beta }_{0,j}}{s_0}{s}_i\right)}^2}}{a_{i-1}{\left(\frac{cos{\beta }_{0,j}}{s_0}{s}_{i-1}\right)}^2}\right|,$$

(14)

where $r_j$ should satisfy $r_j\le r_{max}$.

2.2.2. SGN Coverage

The positioning service area is an irregularly shaped intersection of multiple anchor coverage areas. Letting $P(x,y,z)$ be a random point, and $P_j(x_j,y_j,z_j)$ be the jth anchor node, the positioning service area V can be modeled as

$$\begin{array}{cc}\hfill V& =V(x,y,z)\hfill \\ \hfill s.t.\exists x,y& \in \mathbb{R},0\le z\le h,3\le \dot{J}\le J,{(x-x_j)}^2+{(y-y_j)}^2+{(z-z_j)}^2\le r_j^2,j=1,2,\cdots ,\dot{J},\hfill \end{array}$$

(15)

where h is the maximum ocean depth. When the target depth is known, through projection technique, Equation (15) can be rewritten as

$$\begin{array}{cc}\hfill V& =V(x,y,z)\hfill \\ \hfill s.t.\exists x,y& \in \mathbb{R},0\le z\le h,3\le \dot{J}\le J,{(x-x_j)}^2+{(y-y_j)}^2\le l_j^2,j=1,2,\cdots ,\dot{J}.\hfill \end{array}$$

(16)

Model Equation (16) can be solved by many methods, such as enumeration or probability methods.

2.3. Positioning Service

The SGN is mainly designed to provide PNT services for cooperative targets, so the depth information is usually known. Therefore, when the pseudo distance is known, the horizontal distance between the target and the reference node can be calculated using the trigonometric relationship, which is called projection technique. For positioning based on projection technique, there should be at least three references for successful localization of the target. However, when the depth information of the target is inaccurate due to sensor problem, there are multiple solutions in acoustic ray tracking. In order to obtain accurate positioning results when the depth is unknown, we propose an iterative ray tracking positioning algorithm in [38], which can convert pseudo distance information into horizontal distance multiple times, iteratively completing accurate target positioning.

We assume there are $\dot{J}$ nodes within the communication coverage of the target, and the rough measured distance between node j and the target is $d_j=\overline{s}{t}_j$, where $\overline{s}$ is the average sound speed and $t_j$ is the signal propagation time. If the depth information of the target is known, and the depth difference between the jth node and the target is $\Delta z$, the horizontal distance is $l_j=\sqrt{d_j^2-\Delta z^2}$. If the depth information of the target is unknown, a rough positioning process assuming the signal is propagated straightly is needed first according to [38] to obtain the initial depth estimation of the target; then, the depth is adjusted iteratively, during which the depth is always assumed to be known, and the horizontal distance can also be calculated by $l_j=\sqrt{d_j^2-\Delta z^2}$. After positioning of the target, the multilateration method adopted in this paper is

$$\left\{\begin{array}{c}\hfill {(x_1-x)}^2+{(y_1-y)}^2=l_1^2\\ \hfill {(x_2-x)}^2+{(y_2-y)}^2=l_2^2\\ \hfill \cdots \\ \hfill {(x_j-x)}^2+{(y_j-y)}^2=l_j^2\end{array}\right.,j=1,2,\cdots ,\dot{J},$$

(17)

where $l_j$ denotes the horizontal distances between the target and anchor nodes, respectively. Subtracting the last equation from the other equations, we have

$$\left\{\begin{array}{cc}\hfill 2x(x_1-x_j)+2y(y_1-y_j)& =x_1^2-x_j^2+y_1^2-y_j^2+l_j^2-l_1^2\hfill \\ \hfill 2x(x_2-x_j)+2y(y_2-y_j)& =x_2^2-x_j^2+y_2^2-y_j^2+l_j^2-l_2^2\hfill \\ \hfill \cdots \\ \hfill 2x(x_{j-1}-x_j)+2y(y_{j-1}-y_j)& =x_{j-1}^2-x_j^2+y_{j-1}^2-y_j^2+l_j^2-l_{j-1}^2\hfill \end{array}\right.,j=1,2,\cdots ,\dot{J}.$$

(18)

Equation (18) can be simply rewritten as

$$A{\widehat{P}}_{xy}=B,$$

(19)

$$A=\left[\begin{array}{c}\hfill 2(x_1-x_j)2(y_1-y_j)\\ \hfill 2(x_2-x_j)2(y_2-y_j)\\ \hfill \cdots \\ \hfill 2(x_{j-1}-x_j)2(y_{j-1}-y_j)\end{array}\right],$$

(20)

$${\widehat{P}}_{xy}={\left[xy\right]}^T,$$

(21)

$$B=\left[\begin{array}{c}\hfill x_1^2-x_j^2+y_1^2-y_j^2+l_j^2-l_1^2\\ \hfill x_2^2-x_j^2+y_2^2-y_j^2+l_j^2-l_2^2\\ \hfill \cdots \\ \hfill x_{j-1}^2-x_j^2+y_{j-1}^2-y_j^2+l_j^2-l_{j-1}^2\end{array}\right],$$

(22)

where ${\widehat{P}}_{xy}=(\widehat{x},\widehat{y})$ is the estimated horizontal location of the target. Thus, the position of the target can be calculated through the least square method by

$${\widehat{P}}_{xy}={\left(A^{T}A\right)}^{-1}{A}^{T}B.$$

(23)

The final located position of the target is $\widehat{P}=(\widehat{x},\widehat{y},\widehat{z})$, where $\widehat{z}$ is measured by the depth sensor.

When the target is at the edge of the SGN network coverage, although it can be located, due to the poor configuration of the reference node, it is easy to cause high positioning errors. Geometric dilution precision (GDOP) is a performance metric commonly used to describe the quality of service of reference nodes [39]. Therefore, for SGN networks, it is necessary to calculate their GDOP to evaluate positioning service quality.

According to Equation (23), the real position of target can be expressed as

$$\left\{\begin{array}{c}\hfill x=\widehat{x}+\Delta x\\ \hfill y=\widehat{y}+\Delta y\end{array}\right..$$

(24)

We let $l_j=f_j(x,y)=f_j(\widehat{x}+\Delta x,\widehat{y}+\Delta y)$; the first-order Taylor expansion at point ${\widehat{P}}_{xy}$ related to anchor node j is

$$f_j(x,y)=f_j(\widehat{x},\widehat{y})+\frac{\partial f_j}{\partial x}{{|}_{{\widehat{P}}_{xy}}\Delta x+\frac{\partial f_j}{\partial y}{\displaystyle |}}_{{\widehat{P}}_{xy}}\Delta y,$$

(25)

where $\frac{\partial f_j}{\partial x}{{\displaystyle |}}_{{\widehat{P}}_{xy}}=\frac{-(x_j-\widehat{x})}{\sqrt{{(x_j-\widehat{x})}^2+{(y_j-\widehat{y})}^2}}$, $\frac{\partial f_j}{\partial y}{{\displaystyle |}}_{{\widehat{P}}_{xy}}=\frac{-(y_j-\widehat{y})}{\sqrt{{(x_j-\widehat{x})}^2+{(y_j-\widehat{y})}^2}}$. Let $a_{xj}=\frac{x_j-\widehat{x}}{\sqrt{{(x_j-\widehat{x})}^2+{(y_j-\widehat{y})}^2}}$, $a_{yj}=\frac{y_j-\widehat{y}}{\sqrt{{(x_j-\widehat{x})}^2+{(y_j-\widehat{y})}^2}}$; then, we have

$$\Delta r_j=f_j(\widehat{x},\widehat{y})-f_j(x,y)=a_{xj}\Delta x+a_{yj}\Delta y,$$

(26)

which can be fully written as

$$\left\{\begin{array}{cc}\hfill \Delta r_1& =a_{x1}\Delta x+a_{y1}\Delta y\hfill \\ \hfill \Delta r_2& =a_{x2}\Delta x+a_{y2}\Delta y\hfill \\ \hfill \cdots \\ \hfill \Delta r_j& =a_{xj}\Delta x+a_{yj}\Delta y\hfill \end{array}\right..$$

(27)

Equation (27) can also be expressed in matrix form: $\Delta r=H\Delta p$, where H is

$$H=\left[\begin{array}{c}\hfill a_{x1}{a}_{y1}\\ \hfill a_{x2}{a}_{y2}\\ \hfill \cdots \\ \hfill a_{xj}{a}_{yj}\end{array}\right],$$

(28)

and $\Delta p$ is

$$\Delta p={[\Delta x\Delta y]}^T.$$

(29)

Finally, the matrix related to GDOP is

$$G={\left(H^{T}H\right)}^{-1},$$

(30)

by which the horizontal DOP (HDOP) can be calculated as $HDOP=\sqrt{tr\left(G\right)}$.

3. Comparisons of Different Network Structures

In this section, we offer parameter settings in our simulations and describe the criteria for evaluating the performance of different network structures. The performances of different kinds of anchor deployment are also compared and analyzed in this section.

3.1. Parameter Settings and Performance Evaluation Criteria

Deployment networks are evaluated using MATLAB 2023a. We consider various symmetric SGN structures, including three to eight anchor nodes that form a circular or star-shaped distribution. Detailed parameter settings are given in

Table 1. Simulation parameter settings.

Parameters	Values
Sonar power $P_a$	30 W [40]
Detective threshold $D_t$	20 dB [41]
Sea state level	4
Signal frequency	13.5–16 kHz
Ocean depth	3000 m
Standard deviation of distance measurement error	0.1% × path length

. The level of sonar power and detective threshold can be easily achieved through modern underwater communication modems such as [40,41]. According to Figure 6.2.2 in [34], the Noise level is $NL=50+10log(16000-13500)=83.979$ dB. For a signal with frequency of 16 kHz, the coverage is around 5000 m. For the circular distributed anchors, the distance between each node and the network center is set to be 0.5, 1.0, 1.5 times of ocean depth, respectively. To ensure reliability, simulation results are taken as the average of 10,000 runs.

The performances of SGNs are evaluated by the following three criteria: localization service coverage, GDOP values, and localization errors, which are defined as follows.

• Localization service coverage represents the scope of location services covered by the designed SGN networks. For any spatial coordinate, when the number of receivable anchor signals at its location is not less than three, this coordinate is considered within the positioning coverage range.
• GDOP values represent the distance vector amplification factor between the signal source and the receiver by sonar ranging errors. The GDOP value offers a theoretical description of the impact of network type on positioning accuracy. When adopting projection techniques, the depth information can be measured by the depth sensor, so we mainly focus on HDOP values in this paper.
• Localization error is the average distance bias between the estimated positions of underwater vehicles and their real positions. The localization error can be computed as

  $$P_{er}=\frac{{\sum }_{p=1}^{N_p}\sqrt{{(x_p-{\widehat{x}}_p)}^2+{(y_p-{\widehat{y}}_p)}^2+{(z_p-{\widehat{z}}_p)}^2}}{N_p},$$

  (31)

  where $(x_p,y_p,z_p)$ are real positions of underwater vehicles, $({\widehat{x}}_p,{\widehat{y}}_p,{\widehat{z}}_p)$ are corresponding estimated positions, $N_p$ is total statistical positioning times. The simulation of localization error can numerically provide intuitive statistical characteristics of positioning errors.

3.2. Simulation Results

It is well known that increasing the number of anchors can increase network coverage, but it also means increasing network construction costs. Therefore, it is necessary to utilize a small number of nodes to maximize network coverage. On the other hand, improving network coverage does not necessarily mean that effective service quality can be provided, so it is also necessary to evaluate quality of service, which is reflected in positioning accuracy.

3.2.1. Coverage Performance

Figure 5 shows the coverage range performance of different SGNs with different numbers of anchors, where in Figure 5b–d, “4c” means four anchors forming a square distribution (circular shape), “4s” means four anchors forming a star shape. From Figure 5a, the coverage range of the SGN expands as the number of nodes increases, and the growth trend significantly weakens after exceeding five nodes. With less than six anchors, the result shows that reducing the distance between the anchor and the center can increase the coverage range, but as the number of nodes increases further, reducing the distance between the anchor and the center reduces the coverage range. From Figure 5b–d, the results of “3c” and “4s” indicate that adding an anchor at the network center can increase the coverage range, and the range increase effect is related to the distance from outer anchors to the center. According to the results of “4s” and “4c”, “5s” and “5c”, it can be seen that under the same number of anchors, the star-shaped network has a larger coverage range. Comparing “4s” and “5s”, the coverage improvement brought by increasing nodes is not as significant as that of circular SGNs. For coverage performance, a star network composed of four or five nodes is a better choice for SGN construction.

3.2.2. HDOP Distribution

The purpose of SGN deployment design is to provide PNT service for underwater vehicles, so the accuracy performance is of great importance. To evaluate the positioning accuracy performance of different network shapes, the HDOP is shown in Figure 6 and Figure 7 when the receiver moves around at a depth of 500 m. In Figure 6 and Figure 7, a smaller value indicates better positioning accuracy, because small ranging errors do not cause large positioning errors at these positions. From Figure 6, it can be seen that as the number of nodes increases, the area of high-quality service regions also increases. However, the growth rate slows down when it exceeds five nodes. Though the service areas grow when the number of nodes increases, the service quality in the surrounding areas is not satisfactory that the colors of these areas are yellow and red. This is mainly because the reference node positions are too concentrated or even approximated to be on the same straight line, which makes the positioning results very susceptible to sonar measurement errors.

Comparing Figure 7a and Figure 6b, it can be seen that the service coverage area is roughly the same, but the high-quality service area of the circular configuration is larger, while the service quality of the surrounding areas of the star configuration is seriously deteriorating due to the topological issues of reference anchors. When adding an anchor at the center of Figure 6b to form a five-point star network as shown in Figure 7b, the service coverage becomes larger and has better symmetry. By comparing Figure 7b and Figure 6c, the high-quality service areas (low HDOP values) of the five-node star network and the five-node circular network are roughly equivalent, but the low-quality service areas of the star network are more balanced, which means that the average service quality of the star shape is more suitable for SGN. Considering the coverage performance and economic cost factors, adding an anchor at the center of Figure 6b to form a five-point star network is a good choice for anchor deployment.

3.2.3. Localization Error

To further evaluate the accuracy performance of different kinds of SGNs, we conduct some simulations as shown in Figure 8, where (a), (c), (e) are results with no position constraints of target nodes. (b), (d), (f) are results in which the position of the target node locates around the center of the network topology when it is projected onto the seafloor. The root mean square error (RMSE) of localization is adopted in this paper to evaluate the accuracy performance. From Figure 8a,c,e, localization error increases obviously as anchor nodes increase. This is because when the target node is located at the edge of the coverage range, the network structure of the anchor node is not good, so although the target node can be located, it cannot guarantee the quality of positioning service. If we focus on target nodes near the perpendicular in the network topology with difference depth, the positioning error can be reduced according to Figure 8b,d,f. Comparing “4s” with “4c”, “5s” with “5c” in Figure 8b,d,f, adding an anchor in the middle of SGN can significantly improve the quality of positioning services in the central area as shown in Figure 8b,d,f, which further indicates that the “5s” star topology is a network configuration with good overall performance.

3.2.4. Non-Flat Seafloor Situation

In fact, the seabed terrain is complex and diverse, making it difficult to meet completely flat conditions. In this section, we consider a “stepped” terrain as shown in Figure 9, and positions of five anchor nodes are given in

Table 2. Positions of anchor nodes.

Scenario 1: Circular Topology
Items	Anchor 1	Anchor 2	Anchor 3	Anchor 4	Anchor 5
X	980	309	−793	−793	309
Y	0	951	576	−576	−951
Z	3200	3000	2800	2800	3000
Scenario 1: Star Topology
Items	Anchor 1	Anchor 2	Anchor 3	Anchor 4	Anchor 5
X	1487	464	−−1203	−1203	464
Y	0	1427	874	−874	−1427
Z	3200	3000	2800	2800	3000
Items	Anchor 1	Anchor 2	Anchor 3	Anchor 4	Anchor 5
X	1487	0	−1487	0	0
Y	0	1500	0	−1500	0
Z	3200	3000	2800	2800	3000
Scenario 2: Star Topology
Items	Anchor 1	Anchor 2	Anchor 3	Anchor 4	Anchor 5
X	1470	0	−1470	0	0
Y	0	1500	0	−1500	0
Z	3300	3000	2700	2800	3000

, where Scenario 1 is when $\Delta z=200$ m and Scenario 2 is when $\Delta z=300$ m.

Figure 10 gives the HDOP simulation results of circular and star-shaped SGN with five anchor nodes, and the $\Delta z$ is set to be $0,200,300$, respectively. In Figure 10a,c,e, the maximum value of HDOP increases as $\Delta z$ grows, indicating that a non-flat terrain deteriorates the positioning performance around the service area. This phenomenon also occurs in the star-shaped topology according to Figure 10b,d,f. Comparing Figure 10d,f with Figure 10b, it can be seen that the symmetry in the x-axis direction is disrupted, and the low HDOP value shifts towards the direction of terrain descent. If symmetrical service quality needs to be maintained, it may be necessary to achieve it through methods such as adjusting node spacing (how to adjust the node spacing is beyond the scope of this article, and we will conduct further research in future work). However, by comparing the HDOP value between circular topology and star-shaped topology, the high-quality service areas (low HDOP values) of the five-node star network and the five-node circular network are roughly equivalent, but the low-quality service areas of the star network are more balanced, which also means that the average service quality of the star shape is more suitable for SGN.

Table 3. Accuracy performance of anchor nodes on a non-flat seabed.

Scenario	Average Localization Error without Position Constraints (m)	Standard Deviation of Positioning Error without Position Constraints (m)	Average Localization Error with Position Constraints (m)	Standard Deviation of Positioning Error with Position Constraints (m)
Circular topology when $\Delta \mathbf{z}=\mathbf{0}$ m	7.9798	6.6530	2.8181	2.0711
Star topology when $\Delta \mathbf{z}=\mathbf{0}$ m	7.9601	7.2190	1.9772	1.3244
Circular topology when $\Delta \mathbf{z}=\mathbf{200}$ m	8.0957	6.8508	2.8146	2.0039
Star topology when $\Delta \mathbf{z}=\mathbf{200}$ m	7.5988	7.2533	2.0034	1.3148
Circular topology when $\Delta \mathbf{z}=\mathbf{300}$ m	7.9110	6.4076	2.8114	2.0311
Star topology when $\Delta \mathbf{z}=\mathbf{300}$ m	7.8054	6.9600	1.9971	1.3023

offers accuracy results under 10,000 tests for each situation. The result shows that the accuracy performance is not affected by the position change of anchor nodes, which is mainly because the change in anchor nodes is symmetric. For areas with deteriorating positioning and areas with improved positioning, the volume is equivalent, so the overall error statistics remain unchanged. This situation may also occur where the seabed is inclined and is symmetric in relation to the center of SGN topology.

4. Conclusions

SGN is the foundation for building an underwater PNT system. To achieve wide coverage and good positioning service of SGNs, we focus on the network design of a single set of reference stations in this paper. We propose several deployment plans for local SGN, and then analyze their service quality indicators, including the serving coverage and accuracy performance for localization. To offer a suitable design of a local SGN, we propose several deployment strategies and compare their coverage and accuracy performance. To make the design more realistic, we consider the stratification effect of sound speed and combine it with the active sonar equation to calculate the coverage characteristics where we derive the sound trajectory as a function of the initial grazing angle. To evaluate the performance of each topology of SGN, we compare their coverage range, HDOP and accuracy performance of positioning tests. In terms of overall performance, we believe that the star five-node topology is a better structure under sufficient economic conditions.