Sensor Placement Strategies for Target Localization via 3-D TOA Measurements in Underwater Acoustic Sensor Networks

1. System Model and Problem Formulation

In this section, we first introduce the TOA-based 3-D localization measurement model and the involved parameters. Subsequently, we adopt the isogradient sound speed profile and varied noise measurement to quantify the localization error in terms of the CRLB.

1.1. System Scenario

In this paper, we consider a 3-D underwater acoustic localization scenario, where a stationary target is to be located using N cooperative sensors. Let $\mathcal{C}=\{1,2,3,\dots ,N\}$ denote the set of sensors, $\mathbf{q}={[x,y,z]}^{⊺}\in {\mathbb{R}}^3$ represent the position of the target, and ${\mathbf{p}}_i={[x_i,y_i,z_i]}^{⊺}$ denote the location of the ith sensor, where $i\in \mathcal{C}$. The distance between the target and the ith sensor is given by $r_i=∥\mathbf{q}-{\mathbf{p}}_i∥$, where $\parallel ·\parallel$ denotes the Euclidean norm. Assuming that the acoustic signal generated at time $t_0$ arrives at sensor ${\mathbf{p}}_i$ at time is $t_i$, and the signal propagation speed is c. The signal arrival time of the ith sensor in a noisy environment can be expressed as

$$t_i=t_0+{\displaystyle \frac{r_i}{c}}+{\chi }_i=t_0+{\widehat{t}}_i+{\chi }_i$$

(1)

Here ${\chi }_i$ denotes the measurement noise, which is assumed to be mutually independent and follows a zero-mean Gaussian distribution with variance ${\sigma }_i^2$. The measurement noise covariance matrix can be expressed as a diagonal matrix given by

$$\mathbf{Q}=cov\left(\chi {\chi }^{\mathrm{T}}\right)=diag\left({\sigma }_1^2,{\sigma }_2^2,\dots ,{\sigma }_N^2\right)$$

(2)

In order to facilitate the analysis, we assumed that the signal occurrence time $t_0=0$. Based on this assumption, the vector form of the measurement model can be expressed as

$$\mathbf{t}=\widehat{\mathbf{t}}+\chi$$

(3)

with

$$\begin{array}{cc}\hfill \mathbf{t}& ={\left[\begin{array}{cccc}{t}_1\hfill & t_2\hfill & \dots \hfill & t_N\hfill \end{array}\right]}^{\mathrm{T}}\hfill \\ \hfill \widehat{\mathbf{t}}& ={\left[\begin{array}{cccc}{\widehat{t}}_1\hfill & {\widehat{t}}_2\hfill & \dots \hfill & {\widehat{t}}_N\hfill \end{array}\right]}^{\mathrm{T}}\hfill \\ \hfill \chi & ={\left[\begin{array}{cccc}{\chi }_1\hfill & {\chi }_2\hfill & \dots \hfill & {\chi }_N\hfill \end{array}\right]}^{\mathrm{T}}\hfill \end{array}$$

(4)

The underwater sound speed is a hierarchical variable. Generally, it follows an isogradient depth-dependent SSP and can be expressed as

$$c\left(z\right)=b+az$$

(5)

where b is the sound speed at the sea surface, and a denotes the steepness of the SSP. Meanwhile, the target signal fails to satisfy the isotropic conditions, and the actual acoustic path does not propagate along a straight line but follows a curved trajectory [29]. The deflection angle of the actual acoustic path relative to the straight line connecting the target and the ith sensor is derived from Snell’s law and can be expressed as

$${\alpha }_i=arctan{\displaystyle \frac{\sqrt{{\left(x-x_i\right)}^2+{\left(y-y_i\right)}^2}}{\frac{2b}{a}+\left(z+z_i\right)}}$$

(6)

and the angle of this straight line with respect to the horizontal axis can be expressed as

$${\theta }_i=arctan{\displaystyle \frac{z-z_i}{\sqrt{{\left(x-x_i\right)}^2+{\left(y-y_i\right)}^2}}}$$

(7)

Above all, the actual ray traveling time (RTT) from the ith sensor to the target along the bent curve is formulated as

$${\widehat{t}}_i={\displaystyle \frac{1}{a}}\left[ln\left({\displaystyle \frac{1+sin\left({\theta }_i+{\alpha }_i\right)}{cos\left({\theta }_i+{\alpha }_i\right)}}\right)-ln\left({\displaystyle \frac{1+sin\left({\theta }_i-{\alpha }_i\right)}{cos\left({\theta }_i-{\alpha }_i\right)}}\right)\right]$$

(8)

1.2. Problem Formulation

From the above definition, we can deduce the correlation between the observation performance of the sensors with respect to the target and the localization error. The CRLB establishes a fundamental limit for unbiased estimators in parameter estimation problem. Mathematically, the CRLB is defined as the inverse matrix of the FIM. Given an unbiased estimator for $\widehat{\mathbf{u}}$ the parameter vector $\mathbf{u}$, the covariance matrix of $\widehat{\mathbf{u}}$ is bounded from below by the CRLB, then we can obtain

$$E\left[(\widehat{\mathbf{u}}-\mathbf{u}){(\widehat{\mathbf{u}}-\mathbf{u})}^{\mathrm{T}}\right]\ge \mathbf{CRLB}\left(\mathbf{u}\right)={\mathbf{FIM}}^{-1}$$

(9)

where $\mathbf{FIM}$ is defined as the inverse matrix of $\mathbf{CRLB}$. The natural logarithm of the joint probability density function (PDF) of the localization measurement vector $\mathbf{t}$ and the target location estimation is derived as follows

$$lnp(\mathbf{u};\mathbf{t})=-{\displaystyle \frac{1}{2}}{(\mathbf{t}-\widehat{\mathbf{t}})}^T{\mathbf{Q}}^{-1}(\mathbf{t}-\widehat{\mathbf{t}})$$

(10)

Owing to the spatial heterogeneity of the marine environment, the ambient noise variance and underwater sound speed exhibit significant depth-dependent variations. Accordingly, we denote the noise variance and sound speed associated with the ith sensor as ${\sigma }_i$, $c_i$, respectively. The corresponding $\mathbf{FIM}$ is given by

$$\mathbf{FIM}=\sum _{i=1}^N\left[\begin{array}{ccc}{\displaystyle \frac{1}{c_i^2{\sigma }_i^2}}{\displaystyle \frac{{\left(x-x_i\right)}^2}{r_i^2}}& {\displaystyle \frac{1}{c_i^2{\sigma }_i^2}}{\displaystyle \frac{\left(x-x_i\right)\left(y-y_i\right)}{r_i^2}}& {\displaystyle \frac{1}{c_i^2{\sigma }_i^2}}{\displaystyle \frac{\left(x-x_i\right)\left(z-z_i\right)}{r_i^2}}\\ {\displaystyle \frac{1}{c_i^2{\sigma }_i^2}}{\displaystyle \frac{\left(x-x_i\right)\left(y-y_i\right)}{r_i^2}}& {\displaystyle \frac{1}{c_i^2{\sigma }_i^2}}{\displaystyle \frac{{\left(y-y_i\right)}^2}{r_i^2}}& {\displaystyle \frac{1}{c_i^2{\sigma }_i^2}}{\displaystyle \frac{\left(y-y_i\right)\left(z-z_i\right)}{r_i^2}}\\ {\displaystyle \frac{1}{c_i^2{\sigma }_i^2}}{\displaystyle \frac{\left(x-x_i\right)\left(z-z_i\right)}{r_i^2}}& {\displaystyle \frac{1}{c_i^2{\sigma }_i^2}}{\displaystyle \frac{\left(y-y_i\right)\left(z-z_i\right)}{r_i^2}}& {\displaystyle \frac{1}{c_i^2{\sigma }_i^2}}{\displaystyle \frac{{\left(z-z_i\right)}^2}{r_i^2}}\end{array}\right]$$

(11)

Thus, the trace of $\mathbf{CRLB}$ can be expressed as

$$T_{ar}={\displaystyle \frac{1}{det\left(\mathbf{FIM}\right)}}\left[{\left(\sum _{i=1}^N{\displaystyle \frac{\left(z-z_i\right)}{c_i{\sigma }_i^2{r}_i}}\right)}^2-{\left(\sum _{i=1}^N{\displaystyle \frac{\left(x-x_i\right)}{c_i{\sigma }_i^2{r}_i}}\right)}^2-{\left(\sum _{i=1}^N{\displaystyle \frac{\left(y-y_i\right)}{c_i{\sigma }_i^2{r}_i}}\right)}^2\right]$$

(12)

Given a narrowband acoustic signal characterized by a bandwidth of $B\left(\mathrm{Hz}\right)$, a center frequency of $f_c\left(\mathrm{Hz}\right)$, and a duration of $T_s\left(\mathrm{s}\right)$, the theoretical $\mathbf{CRLB}$ for TOA estimation can be obtained [30]

$${\sigma }_i^2⩾\mathrm{CRLB}\left(\mathrm{TOA}\right)={\displaystyle \frac{1}{8{\pi }^{2}B{T}_s{f}_c^2{\mathrm{SNR}}_i}}$$

(13)

where ${\mathrm{SNR}}_i$ represent the received signal-to-noise ratio at the ith sensor. Based on the analysis of underwater acoustic channel characteristics, the received signal-to-noise ratio at each sensor can be calculated once the system operating parameters are determined.

According to the classical passive sonar equation, neglecting the directivity of the acoustic transducer, the signal-to-noise ratio at the ith sensor in UASNs is given by

$${\mathrm{SNR}}_i^s=\mathrm{SL}-{\mathrm{TL}}_i-{\mathrm{NL}}_i$$

(14)

where $\mathrm{SL}$ represents the source level of the acoustic signal radiated by the target, ${\mathrm{TL}}_i$ denotes the propagation loss from the target to the ith receiving sensor, and ${\mathrm{NL}}_i$ corresponds to the ambient noise ocean level at that sensor. Note that (14) is the decibel scale signal-to-noise ratio, which is converted to linear power signal-to-noise ratio as

$${\mathrm{SNR}}_i^P={10}^{{\mathrm{SNR}}_i^s/10}$$

(15)

Substituting (14) and (15) into (13), the TOA estimation variance in the underwater environment is given by

$${\sigma }_i^2\ge \mathrm{CRLB}\left(\mathrm{TOA}\right)={\displaystyle \frac{1}{8{\pi }^{2}B{T}_s{f}_c^2{\mathrm{SNR}}_i^P}}$$

(16)

It can be seen from (16) that the variance of TOA estimation is determined by the propagation loss and ambient noise in the marine environment.

2. Sensor Placement Solutions

In the preceding section, we first established the TOA measurement model for UASNs based on a realistic isogradient sound speed profile and actual ray traveling time. Subsequently, we systematically investigated the relationship between TOA estimation variance, propagation loss, ambient noise. Nevertheless, all the above analyses are predicated on the unrealistic assumption that the exact target location is known in advance, which is invalid in practical localization scenarios. To achieve consistent and superior localization performance over the entire test region, we adopt the optimization criterion of minimizing the trace of the average CRLB over all possible target locations. The corresponding objective function is formulated as

$$T_{AVG}={\displaystyle \frac{1}{\Omega }}\underset{\Omega }{\int \int \int }{T}_{ar}dxdydz$$

(17)

where $\Omega$ represents the test region, and $T_{AVG}$ is defined as the trace of the average CRLB at all possible target positions within $\Omega$. It should be noted that due to the nonlinearity of the underwater acoustic propagation model and the CRLB expression, the integral in the above (17) has no closed-form solution when substituting (12).

To facilitate numerical solution, we discretize the continuous test region into a uniform grid of candidate target locations. Under this discretization scheme, the corresponding optimization objective function for sensor placement can be rewritten as

$$arg\underset{{\mathbf{p}}_1,{\mathbf{p}}_2,\dots ,{\mathbf{p}}_N}{min}{T}_{AVG}=arg\underset{{\mathbf{p}}_1,{\mathbf{p}}_2,\dots ,{\mathbf{p}}_N}{min}{\displaystyle \frac{1}{M}}\underset{M}{\sum ^{m=1}}\sqrt{T_{ar}\left({\mathbf{q}}_m\right)}$$

(18)

where M denotes the total number of discrete target points, ${\mathbf{q}}_m$ is the mth target position in the test region, and the vector $\mathbf{p}={\left[p_{x,1},p_{y,1},p_{z,1},p_{x,2},p_{y,2},p_{z,2},\dots ,p_{x,N},p_{y,N},p_{z,N}\right]}^{\mathrm{T}}$ is formed by concatenating the three-dimensional coordinates of all N underwater acoustic sensors. Given the non-convex nature of the optimization problem (18), we employ the MinMax k-Means optimization algorithm to find a suboptimal solution through iterative computation [31]. The detailed implementation steps of the algorithm are outlined below

1.

Initialization

The initial positions of N sensors are selected using the MinMax k-Means initialization strategy:

• Randomly select a point in the test region $\Omega$ as the initial position of the first sensor ${\mathbf{p}}_1$,
• Select the point in $\Omega$ farthest from ${\mathbf{p}}_1$ as the initial position of the second sensor ${\mathbf{p}}_2$,
• For $i=3\dots N$: selecting the point that maximizes the minimum Euclidean distance to all ready selected sensors.

2.

Iterative Optimization

The iterative framework is similar to the standard k-Means algorithm, but the update rule does not compute the centroid of each cluster. Instead, the position of each sensor is optimized individually to minimize the overall objective function. The detailed steps are as follows:

• Allocation Step: Assign each virtual target point ${\mathbf{q}}_m$ to its nearest sensor to form clusters:

  $${\mathcal{C}}_k^{\left(t\right)}=\left\{{\mathbf{q}}_m:k=arg\underset{j}{min}{\theta }_m-{\mathbf{p}}_j^{\left(t\right)}\right\},k=1,\dots ,K$$

  (19)

  where ${\mathcal{C}}_k^{\left(t\right)}$ denotes the set of target points assigned to the k sensor at iteration t.
• Update Step: For each cluster, $k=1,\dots ,K$, if $\left|{\mathcal{C}}_k^{\left(t\right)}\right|=0$, randomly generate a new position in the test region. Otherwise, fix the positions of all other sensors and update the position of sensor ${\mathbf{p}}_k^{\left(t\right)}$ by solving the following sub-optimization problem numerically

  $${\mathbf{p}}_k^{(t+1)}=arg\underset{\mathbf{p}\in \Omega }{min}\sum _{{\mathbf{q}}_m\in {\mathcal{C}}_k^{\left(t\right)}}\mathrm{Tr}\left(\mathbf{CRLB}\left({\mathbf{q}}_m;{\mathbf{P}}_{-k}^{\left(t\right)}\cup \left\{\mathbf{p}\right\}\right)\right)$$

  (20)

  with ${\mathbf{P}}_{-k}^{\left(t\right)}=\left\{{\mathbf{p}}_1^{\left(t\right)},\dots ,{\mathbf{p}}_{k-1}^{\left(t\right)},{\mathbf{p}}_{k+1}^{\left(t\right)},\dots ,{\mathbf{p}}_K^{\left(t\right)}\right\}$

Mathematically, this update rule implements a block coordinate descent optimization scheme [32]. Specifically, we decompose the original high-dimensional optimization problem into N independent low-dimensional subproblems. For each subproblem, we fix the positions of all sensors except the j-th one, and adjust the position of the j-th sensor to minimize the sum of $\mathrm{tr}\left(\mathbf{CRLB}\right)$ over all virtual target points assigned to that sensor.

3.

Convergence Judgment

Terminate the iteration if the maximum position change of all sensors is less than a predefined threshold $ϵ$

$$\underset{k}{max}∥{\mathbf{p}}_k^{(t+1)}-{\mathbf{p}}_k^{\left(t\right)}∥<ϵ$$

(21)

4.

Output

Return the optimized sensor placement matrix ${\mathbf{P}}^{\left(t+1\right)}$.

3. Simulation Studies

In this section, comprehensive simulations are carried out to evaluate the performance of the proposed sensor placement strategies for UASNs. Specifically, we first configure the simulation parameters, then employ the MinMax k-Means algorithm to optimize sensor placement in UASNs, and finally a systematic comparison of the root mean squared error (RMSE) performance is presented for three representative sensor placement strategies.

The simulation parameters are configured as follows. Bulleted lists look like this:

• Signal Parameters: The center frequency is set to $f_c=1000\mathrm{Hz}$, the sampling interval $T_s=0.001\mathrm{s}$, the bandwidth $B=600\mathrm{Hz}$, and the source level $SL=180\mathrm{dB}$. The ambient noise at each sensor is a random value in the interval of $60-70\mathrm{dB}$.
• Underwater Acoustic Propagation Parameters: The depth range of the acoustic target is set to $0-1000\mathrm{m}$. The distance between the target and each sensor ranges from $0\mathrm{m}$ to $1500\mathrm{m}$, and the acoustic emission angle ranges from $-{90}^{\circ }$ to ${90}^{\circ }$. The surface sound speed is $b=1480\mathrm{m}/\mathrm{s}$, and the underwater sound speed profile is assumed to be isogradient, increasing linearly with depth z as (5), and linear sound speed profile with steepness $a=0.1$.
• MinMaxk-Means Algorithm Parameters: The target location probability is uniformly distributed in the test region. The region is discretized into a regular grid with a spacing of $10\mathrm{m}$, where each grid intersection is treated as a virtual target point. The algorithm is terminated when the maximum number of iterations $t=1000$ is reached or the convergence condition $\epsilon =0.01$ is satisfied.

In a $1000\mathrm{m}\times 1000\mathrm{m}\times 1000\mathrm{m}$ test region, the MinMax k-Means algorithm is employed to optimize the sensor placement under the assumption of uniform target occurrence probability. All algorithmic parameters are set in accordance with the configurations given above. The surveillance region is discretized into a 3D grid with a spacing of $10\mathrm{m}$, where each grid intersection is regarded as a virtual target point. Simulations are performed with the number of sensors set to 5, 10, 20, and 30, respectively. The optimal sensor placement for UASNs obtained by the proposed algorithm is illustrated in Figure 1.

Figure 1 presents the optimal sensor placement results obtained by the MinMax k-Means algorithm with different numbers of sensors, where the distribution of sensors and virtual target points varies with the number of sensors. It can be observed from Figure 1(a) to (d) that the number of sensors increases gradually. Specifically, Figure 1(a) exhibits 5 sensors with a sparse yet approximately uniform distribution within the test region. As the number of sensors increases stepwise in Figure 1(b) and (c), the sensors are distributed more extensively and uniformly across the entire region. In Figure 1(d), the spatial distribution of sensors becomes further homogenized, which ensures more thorough coverage of the surveillance region and effectively covers all virtual target points.

These results demonstrate that the optimized placeyment positions obtained via the MinMax k-Means algorithm present an approximately uniform distribution in the test region for different numbers of sensors, which well satisfies the requirements for accurate localization of virtual target points.

In the follows, we evaluate the localization performance of the proposed placement strategies by comparing its localization accuracy with two typical benchmark methods, namely, random placement and cube placement. To quantitatively assess the localization performance, the RMSE between the actual target positions and their estimated positions is calculated as follows

$$\mathrm{RMSE}=\sqrt{\mathbb{E}\left\{{∥\widehat{\mathbf{q}}-\mathbf{q}∥}^2\right\}}$$

(22)

Figure 2 presents a comparison of the RMSE versus the number of sensors for three different sensor placement stratagies. As observed in Figure 2, while the RMSE curves of all three placement schemes exhibit a decreasing trend with an increasing number of sensors, the proposed MinMax k-Means scheme outperforms the other two methods by consistently achieving the lowest localization error. A key advantage of the proposed method is that it not only achieves an approximately uniform spatial distribution of sensors but also incorporates the inherent characteristics of the marine environment into the placement optimization process. This unique combination of features makes the proposed algorithm more robust and suitable for practical applications in UASNs.

Furthermore, we conduct a systematic investigation into the influence of critical environmental parameters on the performance of the proposed MinMax k-Means algorithm, with the corresponding experimental results depicted in Figure 3 and Figure 4. Specifically, Figure 3 presents a quantitative analysis of the RMSE for the three placement schemes as a function of the ToA estimation variance ${\sigma }_i^2$.

For this set of simulations, the number of placed sensors is fixed at $N=8$, and the noise variance is varied in the range of $0.1m^2$ to $1m^2$. As clearly observed from the figure, the RMSE values of all three placement strategies increase linearly with the growth of ${\sigma }_i^2$, while the proposed method maintains a consistently lower RMSE and outperforms the other two schemes in terms of estimation performace. Importantly, the proposed algorithm retains its superior localization performance even when subjected to large measurement noise, demonstrating its robustness to noise perturbations.

Additionally, Figure 4 evaluates the impact of the SSP steepness on the localization performance of the three placement strategies. It is evident that the RMSE curve of the proposed algorithm exhibits the least fluctuation when the SSP steepness changes, indicating its strong adaptability to variations in the underwater sound speed profile. Furthermore, across the entire range of SSP steepness values, the proposed algorithm consistently achieves superior localization performance compared to the other two placement strategies. Collectively, the simulation results validate that the MinMax k-Means placement strategy offers optimal localization performance and enhanced robustness, making it well-suited for application in dynamically changing underwater acoustic environments.

4. Conclusions

In this paper, the optimal sensor placement problem for 3-D TOA-based localization in UASNs is considered. A localization measurement model with varied noise measurements and a realistic isogradient SSP is proposed, and the trace of the average CRLB is used as the optimization criterion. Additionally, the MinMax k-Means algorithm is employed to solve the optimal sensor placement problem by minimizing the average of the trace of CRLB. Simulation results show that the proposed method achieves better localization accuracy while maintaining satisfactory localization performance in an underwater environment with dynamic changes.

In future research, we will extend our work to the the scenario of multiple uncertain targets with distinct probability distributions, as this scenario is more consistent with the practical requirements of underwater surveillance applications. Furthermore, we will explore the sensor placement problem for uncertain moving targets in UASNs, aiming to develop a more flexible and practical placement strategy that can adapt to the dynamic nature of underwater targets.