A Localization Method Based on Nonlinear Batch Processing for Non-Cooperative Underwater Acoustic Pulse Source

Abstract

The position of a non-cooperative underwater pulse signal source can be estimated by applying target motion analysis techniques to the direction of arrival (DOA) and frequency of arrival (FOA) measurements obtained from a hydrophone array. However, the harsh underwater acoustic environment, with its pronounced multipath propagation, high signal attenuation, and sparse detectable pulses, introduces considerable errors into the estimation of DOA and FOA. These errors can degrade the performance of conventional estimators such as the pseudolinear estimation (PLE) method, leading to significant bias and divergence issues. To address these issues, this paper proposes a method based on nonlinear batch processing for underwater non-cooperative target localization. A cost function is constructed based on a nonlinear observation model and the weighted least squares principle to ensure high modeling fidelity. Subsequently, a multi-start grid search combined with a trust region dogleg algorithm is employed for global iterative optimization of the cost function, enhancing the accuracy and stability of the final position estimate. Numerical simulation results demonstrate that the proposed method achieves high convergence speed and localization accuracy under adverse noise conditions and with a limited number of received pulses. Moreover, the sea trial results confirm that the algorithm attained a convergence rate of 93% with only 25 received pulses, and outperformed the conventional PLE method by approximately 80% in terms of positioning accuracy.

1. Introduction

Owing to the complexity of the underwater environment, the localization of non-cooperative underwater signal sources remains a challenging and active research area in underwater acoustic engineering [1,2,3,4]. This process typically involves localizing a source by analyzing its intercepted pulse signal or radiated noise. At this time, the prior characteristics of source signals are unclear or insufficiently understood, which easily causes divergence of the target motion estimation results [5]. Even in radio target localization, the non-cooperative estimation error of target distance remains above 10% of the ground truth [6]. Furthermore, underwater acoustic signal processing faces profoundly greater challenges stemming from the dynamic and complex nature of the propagation medium as well as significantly restricted conditions for acquiring the ground truth of targets, making precise localization inherently more difficult.

In contrast to localizing sources based on continuous radiated noise such as that from ships, localizing targets that emit pulsed signals presents a more challenging problem. Specifically, the intermittent nature of pulsed signals limits the amount of available data, while their finite duration restricts the processing gain that can be achieved through temporal integration. Consequently, these characteristics lead to larger estimation errors of the DOA, time of arrival (TOA), and FOA [7,8]. Moreover, the limited observations are often subject to anomalous interference from ambient ocean noise and sonar equipment disturbances [9,10]. Therefore, for non-cooperative underwater targets, the absence of prior cooperative information in the pulse signals further complicates the localization problem. In contrast to other underwater localization methods, target motion analysis (TMA) utilizes target motion and observation models to construct a state estimation framework, subsequently estimating the target’s kinematic state from a sequence of observation parameters. Offering greater practical utility, the TMA method exhibits lower sensitivity to variations in observation parameters and environmental conditions [11].

TMA methods include two types of algorithms: recursive estimation and batch processing. For real-time tracking of a target’s motion state, recursive Bayesian approaches, typified by the Kalman filter, are commonly employed [12]. These methods require an initial estimate of the target state that is sequentially updated by real-time observations, and their accuracy is highly dependent on the chosen initial values and prior statistical information. Consequently, poor initialization or erroneous statistical priors can cause the subsequent recursive estimates to be biased or even to diverge [3,13]. In contrast, batch processing methods operate on a time window of observations to yield an estimate of the target’s state over its trajectory [14]. A key advantage of this approach is that it avoids the propagation of localization bias and obviates the need for initialization. Common batch processing estimation methods include PLE method, instrumental variable (IV) method, and maximum likelihood estimation (MLE) method.

Passive target localization and motion analysis have been extensively studied. Lindgren and Gong first introduced the PLE method by linearizing the bearing angle equations and embedding the nonlinear components into the noise term [15]. However, this approach induces a correlation between the noise and the measurement matrix, leading to biased estimates [16]. To mitigate this, Chan and Rudnicki constructed an IV method based on historical observations, converting the original problem into a constant parameter estimation task and introducing unbiasedness constraints to effectively resolve the estimation bias [17]. Building on this, Badriasl and Doğançay obtained better performance by constructing auxiliary variables and considering the statistical characteristics of the noise [18]. In recent years, Doğançay proposed a novel three-dimensional pseudolinear estimation algorithm for single-platform target motion analysis using angle observations [19]. This method uses a small-noise approximation to improve the MLE and combines it with the weighted IV (WIV) method to reduce estimation bias. Badriasl further proposed a new batch Bayesian WIV algorithm that leverages prior information, striking a good balance between computational efficiency and estimation accuracy [20].

In parallel, the MLE method treats the TMA problem as a nonlinear least squares (LS) estimation problem capable of providing asymptotically unbiased and efficient estimates. Nardone et al. applied MLE to passive localization problems and analyzed the relationship between its estimation accuracy and the Cramér–Rao lower bound (CRLB) [21]. Ho and Chan used a Lagrange multiplier-constrained LS method to obtain asymptotically unbiased performance without requiring initial value guessing [22]. Furthermore, Ahmed et al. studied target localization methods based on multiple frequency observations and proposed a constrained optimization MLE method that improves noise resistance through semi-definite relaxation techniques [23].

Despite the significant progress achieved by these approaches, critical shortcomings remain in the literature. Both the PLE and IV methods inherently rely on linearizing nonlinear system equations prior to solving for target positions, which inevitably introduces linearization errors that can severely degrade performance. Conversely, although the nonlinear MLE and LS methods avoid these linearization errors, their reliance on iterative search algorithms, including Newton’s method, the Levenberg–Marquardt (LM) algorithm, and the trust region (TR) algorithm [24,25,26], presents several notable drawbacks. First, they often require highly accurate a priori information, such as the statistical properties of the observation noise. Second, the lack of a closed form solution makes these iterative frameworks extremely sensitive to initialization; poor initial guesses frequently lead to local optima or algorithm divergence [27]. Finally, these methods typically demand a substantial amount of observation data to converge reliably, making them largely ineffective for localizing noncooperative sources that emit only a sparse number of pulses.

Therefore, to address the aforementioned issues, the core contributions of this study are formulated to answer the following key research questions:

(1) How can a high-fidelity nonlinear observation model and a weighted LS (WLS) cost function be constructed using DOA and FOA measurements to effectively localize noncooperative underwater acoustic pulsed-signal targets?

(2) How can a robust optimization framework integrating the TR algorithm with a multi-starting grid search strategy be designed to overcome initialization sensitivity, avoid local optima, and ensure global convergence across the entire search space?

By systematically addressing these questions, this paper proposes a robust localization method. Rather than introducing new mathematical theory, the core contribution of this work is an underwater pulse-specific design that strategically combines WLS weighting, a multi-start grid search, and dogleg optimization. This specific integration effectively avoids the biased estimation problems inherent in pseudolinear methods while simultaneously overcoming the severe initialization sensitivity of standard iterative optimization techniques. Extensive numerical simulations and sea trial data demonstrate that the proposed algorithm significantly surpasses traditional methods in both accuracy and stability. To provide an intuitive overview of the proposed approach, its overall flowchart is illustrated in Figure 1.

The remainder of this paper is organized as follows: Section 2 presents the target motion state model and the PLE method; Section 3 details the multistart TR localization algorithm; Section 4 provides the numerical simulation results and analysis; Section 5 validates the performance of the proposed algorithm using sea trial data; and finally, Section 6 concludes the paper.

2. System Model and PLE Algorithm

2.1. State Model and Measurement Model

Assume that the initial position of the observation station is the origin, the due east direction is the X coordinate axis, and the due north direction is the Y coordinate axis, constructing a rectangular coordinate system. The observation station and target move at constant velocities, with specific geometric relationships as shown in Figure 2.

For the target state model, let $t_0$ be the initial observation time; at any time $t_k$, the state variable at this time is recorded as ${\mathbf{x}}_k$, where $k=0,1,2\dots$ is the pulse sequence number. The target state is ${\mathbf{x}}_k^t=\left[x_k^t,y_k^t,{\dot{x}}_k^t,{\dot{y}}_k^t,f_c\right]\in {\mathbb{R}}^m$, where the components $x_k^t$, $y_k^t$, and ${\dot{x}}_k^t$, ${\dot{y}}_k^t$ represent the target’s position and velocity, respectively, and $f_c$ is the center frequency of the emitted pulse signal. The dimension of the target state vector is m, which is set to 5 in this paper. Then, the state equation for the next time step can be expressed as

$${\mathbf{x}}_{k+1}^t={\mathbf{Fx}}_k^t+\mathbf{G}{\mathbf{w}}_k,$$

(1)

where the state transition matrix is given by Equation (2) and T is the pulse repetition interval.

$${\mathbf{F}}_{k+1,k}=\left[\begin{array}{ccccc}1& 0& T& 0& 0\\ 0& 1& 0& T& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$$

(2)

Here, $\mathbf{G}$ is the noise driving matrix, as given in Equation (3). The vector ${\mathbf{w}}_k$ comprises perturbations in the X-axis and Y-axis acceleration and frequency, and is modeled as a zero-mean Gaussian random vector with covariance matrix $\mathbf{Q}=\mathbf{G}{\mathbf{w}}_k{\mathbf{w}}_k^{\mathrm{T}}{\mathbf{G}}^{\mathrm{T}}$. The superscript ${\left(·\right)}^{\mathrm{T}}$ indicates matrix transposition.

$$\mathbf{G}=\left[\begin{array}{ccc}0.5T^2& 0& 0\\ 0& 0.5T^2& 0\\ T& 0& 0\\ 0& T& 0\\ 0& 0& 1\end{array}\right]$$

(3)

In non-cooperative underwater pulse target localization, targets do not provide collaborative localization information; consequently, an observation station must passively intercept the acoustic pulse signals emitted by the target in order to estimate its position. The observation station utilizes a sonar array to receive these pulse signals and extract key parameters, primarily the DOA and FOA. Based on the array’s received signals, beamforming techniques are employed to extract the target’s DOA. Concurrently, time–frequency analysis methods such as the short-time Fourier transform are used to process the received pulses and determine their FOA. Although practical underwater acoustic environments introduce non-Gaussian disturbances and multipath-induced biases to DOA and FOA measurements, these overall errors are commonly approximated by a Gaussian distribution to adequately represent most typical scenarios [21,22]. This assumption allows us to focus on the core performance of the proposed multi-start trust region algorithm. When the observation station conducts the k-th observation, the observation is recorded as ${\mathbf{z}}_k\in {\mathbb{R}}^n$, where n is the dimension of measurement. The observation model is expressed as Equation (4):

$${\mathbf{z}}_k=\mathbf{h}\left({\mathbf{x}}_k,{\mathbf{x}}_k^o\right)+{\mathbf{n}}_k=\left[\begin{array}{c}{\beta }_k\\ f_k\end{array}\right]+{\mathbf{n}}_k,$$

(4)

where $\mathbf{h}\left(·\right)$ is the measurement function and ${\mathbf{n}}_k$ is the measurement noise, also satisfying a Gaussian distribution, with the corresponding covariance matrix denoted as $\mathbf{R}$. Let ${\mathbf{x}}_k^o=\left[x_k^o,y_k^o,{\dot{x}}_k^o,{\dot{y}}_k^o\right]$ denote the state vector of the observer (position and velocity) and let the initial target state ${\mathbf{x}}_0^t=\left[x_0^t,y_0^t,{\dot{x}}_0^t,{\dot{y}}_0^t,f_c\right]\in {\mathbb{R}}^m$ be initialized based on Equations (1)–(3). Hence, the DOA ${\beta }_k$ and FOA $f_k$ are mathematically formulated in Equation (5), as follows:

$$\left\{\begin{array}{c}{\beta }_k=arctan{\displaystyle \frac{x_0^t+kT{\dot{x}}_0^t-x_k^o}{y_0^t+kT{\dot{y}}_0^t-y_k^o}}\hfill \\ f_k=f_c\times \left[1-{\displaystyle \frac{\left({\dot{x}}_0^t-{\dot{x}}_k^o\right)\times \left(x_0^t+kT{\dot{x}}_0^t-x_k^o\right)+\left({\dot{y}}_0^t-{\dot{y}}_k^o\right)\times \left(y_0^t+kT{\dot{y}}_0^t-y_k^o\right)}{\mathrm{c}\sqrt{{\left(x_0^t+kT{\dot{x}}_0^t-x_k^o\right)}^2+{\left(y_0^t+kT{\dot{y}}_0^t-y_k^o\right)}^2}}}\right]\hfill \end{array}\right. ,$$

(5)

where $\mathrm{c}$ is the speed of sound in seawater, with a nominal value of 1500 m/s.

It should be noted that a key modeling constraint in this study is the constant velocity assumption. Approximating the velocity as a constant aligns well with the majority of practical navigation scenarios, where vessels maintain a steady cruising state; however, if the target vessel engages in maneuvering behaviors such as turning or altering its speed, a severe mathematical model mismatch occurs. Under these complex dynamic conditions, the performance of the proposed method will significantly degrade or even completely fail.

2.2. PLE Algorithm for Target Localization

Under small error assumptions, PLE methods linearize observation equations to enable linear LS solution. We establish ideal observation parameter equations under ideal conditions, expressing $f_k$ using ${\beta }_k$ to enable approximate linearization using the trigonometric function

$$f_k=f_{\mathrm{c}}\times \left[1-{\displaystyle \frac{\left({\dot{x}}_0^t-{\dot{x}}_k^o\right)sin{\beta }_k+\left({\dot{y}}_0^t-{\dot{y}}_k^o\right)cos{\beta }_k}{c}}\right].$$

(6)

In actual practice, observations are disturbed by measurement noise. Let the observed values of DOA and FOA be ${\beta }_{nk}$ and $f_{nk}$, respectively. Actual observations are expressed as

$$\begin{array}{c}{\beta }_{nk}={\beta }_k+n_{\beta },\end{array}$$

(7)

$$\begin{array}{c}{f}_{nk}=f_{nk}+n_f,\end{array}$$

(8)

where $n_{\beta }$ and $n_f$ are the noise for the DOA and FOA measurements, respectively, in both cases assumed to follow Gaussian distribution. Substituting actual observation values into the ideal observation equations and taking tangent function values on both sides of the DOA equation yields

$$\begin{array}{c}tan\left({\beta }_{nk}\right)=tan\left(arctan{\displaystyle \frac{x_0^t+kT{\dot{x}}_0^t-x_k^o}{y_0^t+kT{\dot{y}}_0^t-y_k^o}}+n_{\beta }\right),\end{array}$$

(9)

$$\begin{array}{c}{f}_{\mathrm{n}k}-n_f=f_{\mathrm{c}}\times \left[1-{\displaystyle \frac{\left({\dot{x}}_0^t-{\dot{x}}_k^o\right)sin\left({\beta }_{nk}-n_{\beta }\right)+\left({\dot{y}}_0^t-{\dot{y}}_k^o\right)cos\left({\beta }_{nk}-n_{\beta }\right)}{c}}\right].\end{array}$$

(10)

By applying the trigonometric sum and difference formulas, Equations (9) and (10) can be transformed. Furthermore, under the assumption of small observation noise, the following approximation can be made:

$$cos\left(n_{\beta }\right)\approx 1,sin\left(n_{\beta }\right)\approx n_{\beta },$$

(11)

allowing Equations (9) and (10) to be simplified as follows:

$${\displaystyle \frac{cos{\beta }_{nk}+sin{\beta }_{nk}{n}_{\beta }}{sin{\beta }_{nk}-cos{\beta }_{nk}{n}_{\beta }}}={\displaystyle \frac{x_0^t+kT{\dot{x}}_0^t-x_k^o}{y_0^t+kT{\dot{y}}_0^t-y_k^o}},$$

(12)

$$\begin{array}{cc}\hfill f_{\mathrm{n}k}-n_f=& f_c-f_c{\displaystyle \frac{\left({\dot{x}}_0^t-{\dot{x}}_k^o\right)\left(sin{\beta }_{nk}-cos{\beta }_{nk}{n}_{\beta }\right)}{c}}\hfill \\ \hfill & -f_c{\displaystyle \frac{\left({\dot{y}}_0^t-{\dot{y}}_k^o\right)\left(cos{\beta }_{nk}+sin{\beta }_{nk}{n}_{\beta }\right)}{c}}\hfill \end{array}.$$

(13)

Further rearrangement yields the pseudolinear equation given by Equation (14), in which the state variable corresponding to frequency is the reciprocal of the source frequency:

$${\mathbf{\Xi }=\mathbf{Ax}}_0^t+\mathbf{N},$$

(14)

where

$$\begin{array}{c}\mathbf{\Xi }=\left[\begin{array}{c}cos\left({\beta }_{nk}\right)x_k^o-sin\left({\beta }_{nk}\right)y_k^o\\ 1+{\displaystyle \frac{{\dot{x}}_k^{o}sin{\beta }_{nk}}{c}}+{\displaystyle \frac{{\dot{y}}_k^{o}cos{\beta }_{nk}}{c}}\end{array}\right],\end{array}$$

(15)

$$\begin{array}{c}\mathbf{A}=\left[\begin{array}{ccccc}cos\left({\beta }_{nk}\right)& -sin\left({\beta }_{nk}\right)& kTcos\left({\beta }_{nk}\right)& -kTsin\left({\beta }_{nk}\right)& 0\\ 0& 0& {\displaystyle \frac{sin{\beta }_{nk}}{c}}& {\displaystyle \frac{cos{\beta }_{nk}}{c}}& f_{nk}\end{array}\right],\end{array}$$

(16)

$$\begin{array}{c}\mathbf{N}=\left[\begin{array}{c}-tan{n}_{\beta }\left[cos{\beta }_{nk}\left(y_0^t+kT{\dot{y}}_0^t-y_k^o\right)+sin{\beta }_{nk}\left(x_0^t+kT{\dot{x}}_0^t-x_k^o\right)\right]\\ -{\displaystyle \frac{n_f}{f_c}}+{\displaystyle \frac{1}{c}}\left[-\left({\dot{x}}_0^t-{\dot{x}}_k^o\right)cos{\beta }_{nk}{n}_{\beta }+\left({\dot{y}}_0^t-{\dot{y}}_k^o\right)sin{\beta }_{nk}{n}_{\beta }\right]\end{array}\right].\end{array}$$

(17)

Using multiple historical observation data to construct pseudolinear equation sets according to Equation (14), linear LS methods can then be used for solving. However, the pseudolinear process results in the noise $\mathbf{N}$ being correlated with the estimation parameters, which is one of the reasons for the poor performance of the PLE method.

3. Multi-Start Trust Region-Based Localization Algorithm for Non-Cooperative Underwater Pulse Source

In the proposed algorithms, the entire solution process is divided into two stages: model establishment and optimization solving. According to the DOA and FOA observation models, corresponding objective functions are constructed that convert target localization problems into nonlinear LS problems. A multi-starting grid search strategy is adopted to reduce sensitivity to initial iteration points, and trust region algorithms are used for solving to enhance global convergence capabilities.

3.1. Nonlinear Weighted Localization Model for Non-Cooperative Underwater Pulse Source

This localization problem is described as a nonlinear LS problem. Let $\mathbf{s}\left(\mathbf{x}\right):{\mathbb{R}}^m\to {\mathbb{R}}^N$ be a vector function, where its components $s_k\left(\mathbf{x}\right):{\mathbb{R}}^m\to \mathbb{R}$ are scalar functions. Here, N represents the number of received pulses and corresponds to the dimension of the vector function $\mathbf{s}\left(\mathbf{x}\right)$. In order for the state vector to be estimable, the condition $N>m$ must be satisfied. The LS objective function $S\left(\mathbf{x}\right)$ is

$$\underset{\mathbf{x}}{min}S\left(\mathbf{x}\right)={\displaystyle \frac{1}{2}}{∥\mathbf{s}\left(\mathbf{x}\right)∥}^2={\displaystyle \frac{1}{2}}\sum _{k=1}^N{\left[s_k\left(\mathbf{x}\right)\right]}^2,$$

(18)

where the vector function $\mathbf{s}\left(\mathbf{x}\right)$ and scalar function $s_k\left(\mathbf{x}\right)$ are expressed as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathbf{s}\left(\mathbf{x}\right)=\left[s_1\left(\mathbf{x}\right),s_2\left(\mathbf{x}\right),\dots ,s_N\left(\mathbf{x}\right)\right]\hfill \\ s_k\left(\mathbf{x}\right)=q_k\left(\mathbf{x}\right)-q_k\hfill \end{array}\right.,\end{array}$$

(19)

where $q_k\left(\mathbf{x}\right)$ and $q_k$ denote the predicted measurement given the state estimate $\mathbf{x}$ and the actual noisy measurement at time step k, respectively.

The principle of the LS method is to minimize the sum of squared residuals by optimizing an objective function. However, an unweighted approach that treats all observation errors equally gives undue weight to larger residual terms. In pulse signal localization, directly using the raw DOA and FOA data in the optimization process can lead to unbalanced weighting due to the discrepancy in their scales. This in turn causes the optimization results to be dominated by the parameter with the larger magnitude, undermining the effectiveness of the joint estimation.

When jointly using DOA and FOA for state estimation, it is important to consider the discrepancies in their physical dimensions, value ranges, and scales of variation. A numerical example is provided to illustrate the observational characteristics of the pulse signals. In the ideal noise-free case, the target bearing angle and the frequency of received signal at the k-th observation are given by Equation (5). Setting the target initial distance from origin as 10 km, initial bearing angle as $45°$, and traveling westward at 4 m/s, the observation station moves eastward from the origin at constant speed of 4 m/s. The pulse signal’s center frequency is 4000 Hz, while the transmission period is 15 s. The standard deviations of the angle observation noise ${\sigma }_{\beta }$ and frequency observation noise ${\sigma }_f$ are set to $0.5°$ and 0.5 Hz, respectively.

Figure 3a shows the DOA measurements for the first 25 pulse groups, which are centered at approximately $45°$ and span a range of only about $17°$. In contrast, Figure 3b presents the corresponding FOA measurements, which have a magnitude of around 4000 but a variation range of only 5. Thus, the DOA and FOA measurements exhibit significant discrepancies in both magnitude and scale of variation.

To address the aforementioned issues, a weighting method is employed for the joint processing of DOA and FOA data. In the optimization process, this weighting serves the dual purpose of mitigating the impact of scale discrepancies between the parameters and enhancing the numerical stability of the algorithm, which leads to a more effective solution. The final WLS optimization model is expressed as

$$\underset{\mathbf{x}}{min}S\left(\mathbf{x}\right)={\displaystyle \frac{1}{2}}\sum _{k=1}^N\left[w_{\beta }{\left({\beta }_k\left(\mathbf{x}\right)-{\beta }_{nk}\right)}^2+w_f{\left(f_k\left(\mathbf{x}\right)-f_{nk}\right)}^2\right]$$

(20)

where $w_{\beta }$ and $w_f$ are the weighting coefficients used to balance the contributions of the DOA and FOA terms in the optimization model, respectively. The LS problem is fundamentally an optimization problem that estimates unknown parameters by finding the minimum of an objective function. In this context, the objective function not only measures the quality of the current estimate but also determines the direction and step size of the optimization process. As such, its structural properties directly impact the problem’s solvability and the efficiency of the solution. Different weighting schemes influence the accuracy and convergence of the solution by altering the properties of this objective function.

In statistical inference, if the weighting coefficients are selected as the inverse of the noise variance for each observation term, the WLS method becomes equivalent to MLE under the assumption of Gaussian noise. Variance, which reflects the degree of dispersion among observed data samples, is a measure of data uncertainty. A larger variance indicates greater data volatility and stronger observation noise; conversely, a smaller variance signifies that the data are more concentrated, implying less uncertainty and higher reliability in the observation. Therefore, measurement variances are used as the basis for weighting to mitigate the influence of observations with high uncertainty on the final estimate.

Since the DOA and FOA measurements constitute non-stationary sequences, directly applying traditional variance calculation methods can introduce significant bias. Therefore, this section employs the smoothed residual method to calculate the noise variance. It is worth comparing this approach with conventional robust weighting strategies such as Huber M-estimation. Conventional robust weights typically rely on point-wise residual evaluation, which treats each measurement independently. While mathematically straightforward, this point-wise suppression ignores the physical continuity of the target’s motion. In contrast, by exploiting the temporal correlation of the trajectory, the smoothed residual acts analogously to a low-pass filter. It comprehensively evaluates the current measurement within the context of adjacent trajectory points, providing a more reliable distinction between actual target maneuvers and sudden noise spikes. Compared to fully adaptive weighting schemes, the smoothed residual approach also avoids complex hyperparameter tuning, maintaining high computational efficiency suitable for batch processing.

The fundamental principle is to first smooth the observation sequence, then obtain a residual sequence by taking the difference between the smoothed and original observations. The mean of the squared residuals is then used as the basis for the weighting. The specific steps are detailed in Equations (21) and (22). Here, $\mathrm{smooth}(·)$ is a smoothing function that calculates the mean of several adjacent observations to produce the smoothed value. In this study, a simple moving average (SMA) filter is employed as the smoothing function, given its low computational complexity and robust baseline performance. The smoothing window length is empirically set to $L=5$. To appropriately handle the sequence boundaries, an adaptive symmetric shrinking window strategy is utilized, which symmetrically reduces the local window size at the extreme edges to prevent artificial signal transients. This process serves to filter out high-frequency noise and extract the main trend of the sequence.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\beta }_{sk}=\mathrm{smooth}\left({\beta }_{nk}\right)\hfill \\ f_{sk}=\mathrm{smooth}\left(f_{nk}\right)\hfill \end{array}\right.,\end{array}$$

(21)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{w}_{\beta }=1/\phantom{1{\displaystyle \frac{1}{N}}{\displaystyle \sum _{k=1}^N}{\left({\beta }_{sk}-{\beta }_{nk}\right)}^2}{\displaystyle \frac{1}{N}}{\displaystyle \sum _{k=1}^N}{\left({\beta }_{sk}-{\beta }_{nk}\right)}^2\hfill \\ w_f=1/\phantom{1{\displaystyle \frac{1}{N}}{\displaystyle \sum _{k=1}^N}{\left(f_{sk}-f_{nk}\right)}^2}{\displaystyle \frac{1}{N}}{\displaystyle \sum _{k=1}^N}{\left(f_{sk}-f_{nk}\right)}^2\hfill \end{array}\right..\end{array}$$

(22)

Sensitivity Analysis of Smoothing Window Length. From a theoretical perspective, an excessively small window such as $L<3$ fails to achieve adequate smoothing; conversely, an overly large window such as $L>9$ introduces severe measurement lag inherent to the moving average operation, distorting the true trajectory dynamics and degrading overall tracking performance. Consequently, considering the total number of detectable pulses within a typical observation batch and based on extensive empirical tuning, setting $L\in [4,6]$ is deemed appropriate for ensuring the algorithm’s robustness across varied underwater scenarios.

3.2. Multi-Start Trust-Region Solution to the Nonlinear Weighted Localization Model

The essence of model solving is finding the minimum value of objective functions. To improve global search capabilities and avoid local optimal solutions, multi-starting grid search combined with trust region optimization methods are adopted for solving.

The multi-starting grid search consists of two steps:

1.

First, a predefined search space is partitioned into a grid to generate a set of starting points, denoted as ${\mathbf{x}}_{j,i}$. Here, $j=1,2,\dots ,J$ is the index for the starting points, with being their total number. The subscript i in ${\mathbf{x}}_{j,i}$ (where $i=0,1,\dots$) represents the iteration count, so ${\mathbf{x}}_{j,0}$ is the initial position for the j-th starting point.

2.

Second, starting from each of these points, a TR optimization is performed iteratively. The final solution is then selected as the one that yields the minimum objective function value among all starting points.

The search space is partitioned into a grid over the intervals $\left[X_{\min },X_{max}\right]$ and $\left[Y_{\min },Y_{max}\right]$ with step sizes of $X_{\mathrm{step}}$ and $Y_{\mathrm{step}}$, respectively. The coordinates of the resulting grid points are ${\mathbf{x}}_{j,0}$, calculated as shown in Equation (23):

$$\begin{array}{c}\begin{array}{c}{\mathbf{x}}_{j,0}=\left[X_{\min }+{\tau }_x\times X_{\mathrm{step}},Y_{\min }+{\tau }_y\times Y_{\mathrm{step}}\right],\hfill \\ {\tau }_x=0,1,\dots \mathrm{floor}\left({\displaystyle \frac{X_{max}-X_{\min }}{X_{\mathrm{step}}}}\right),{\tau }_y=0,1,\dots \mathrm{floor}\left({\displaystyle \frac{Y_{max}-Y_{\min }}{Y_{\mathrm{step}}}}\right),\hfill \end{array}\end{array}$$

(23)

where $\mathrm{floor}\left(·\right)$ is the floor function.

For notational simplicity within each TR iterative process, ${\mathbf{x}}_{j,i}$ is abbreviated as ${\mathbf{x}}_i$. A trust region is then defined as the area centered at ${\mathbf{x}}_i$ with a radius of ${\Delta }_i$, which serves as the maximum step size for the current search. The initial radius is set to ${\Delta }_0=0.1$ [28]. The algorithm’s final convergence accuracy is inherently robust to the selection of the initial radius due to its adaptive updating mechanism. Specifically, the sensitivity of the algorithm to ${\Delta }_0$ is primarily reflected in the early convergence speed rather than the final localization performance. Empirical evaluations indicate that ${\Delta }_0=0.1$ provides a good balance of search accuracy and convergence efficiency, successfully avoiding the excessive step rejections caused by an overly large initial radius while preventing the sluggish early convergence associated with a strictly conservative small radius.

Within this trust region, we construct a quadratic model $m\left({\mathbf{p}}_i\right)$ of the objective function. Based on this model, a TR subproblem is defined with the goal of finding a step ${\mathbf{p}}_i$, known as the trial step, that minimizes $r\left({\mathbf{p}}_i\right)$. The mathematical formulation of this subproblem is shown in Equation (24):

$$\underset{{\mathbf{p}}_i\in {\mathbb{R}}^m}{min}r\left({\mathbf{p}}_i\right)={\mathbf{g}}_i^{\mathrm{T}}{\mathbf{p}}_i+{\displaystyle \frac{1}{2}}{\mathbf{p}}_i^{\mathrm{T}}{\mathbf{B}}_i{\mathbf{p}}_i,∥{\mathbf{p}}_i∥\le {\Delta }_i,$$

(24)

where ${\mathbf{g}}_i$ is the gradient and ${\mathbf{B}}_i$ is an approximation to the Hessian matrix of the objective function $S\left(\mathbf{x}\right)$, both evaluated at ${\mathbf{x}}_i$:

$$\begin{array}{c}{\mathbf{g}}_i={\mathbf{J}}^{\mathrm{T}}\left({\mathbf{x}}_i\right)\mathbf{s}\left({\mathbf{x}}_i\right),\\ {\mathbf{B}}_i={\mathbf{J}}^{\mathrm{T}}\left({\mathbf{x}}_i\right)\mathbf{J}\left({\mathbf{x}}_i\right),\end{array}$$

(25)

letting $\mathbf{J}\left(\mathbf{x}\right)\in {\mathbb{R}}^{m\times N}$ be the Jacobian matrix

$$\mathbf{J}\left(\mathbf{x}\right)=\left[\nabla s_k\left(\mathbf{x}\right)\right]=\left[\begin{array}{c}\nabla s_1{\left(\mathbf{x}\right)}^{\mathrm{T}}\\ \nabla s_2{\left(\mathbf{x}\right)}^{\mathrm{T}}\\ ⋮\\ \nabla s_N{\left(\mathbf{x}\right)}^{\mathrm{T}}\end{array}\right]=\left[\begin{array}{c}\nabla q_1{\left(\mathbf{x}\right)}^{\mathrm{T}}\\ \nabla q_2{\left(\mathbf{x}\right)}^{\mathrm{T}}\\ ⋮\\ \nabla q_N{\left(\mathbf{x}\right)}^{\mathrm{T}}\end{array}\right],$$

(26)

where ∇ denotes the gradient operator.

The TR subproblem is solved at each current iteration. This subproblem is a constrained quadratic optimization problem, for which finding an exact solution is often computationally expensive; therefore, the dogleg method is employed to solve this subproblem, as it is computationally efficient while exhibiting both stability and convergence. This method constructs a piece-wise linear path based on the steepest descent step ${\mathbf{p}}^{\mathrm{u}}$ and the Gauss–Newton (GN) step ${\mathbf{p}}^{\mathrm{b}}$. The final trial step ${\mathbf{p}}_i$ is the point on this path that intersects the TR boundary of the radius. The specific formulations for ${\mathbf{p}}^{\mathrm{u}}$ and ${\mathbf{p}}^{\mathrm{b}}$ are given in Equations (27) and (28), respectively [29].

$$\begin{array}{c}{\mathbf{p}}^{\mathrm{u}}=-{\displaystyle \frac{{\mathbf{g}}_i^T{\mathbf{g}}_i}{{\mathbf{g}}_i^{\mathrm{T}}{\mathbf{J}}^{\mathrm{T}}\left({\mathbf{x}}_i\right)\mathbf{J}\left({\mathbf{x}}_i\right){\mathbf{g}}_i}}{\mathbf{g}}_i\end{array}$$

(27)

$$\begin{array}{c}{\mathbf{p}}^{\mathrm{b}}=-{\mathbf{B}}_i^{-1}{\mathbf{g}}_i\end{array}$$

(28)

When ${\mathbf{B}}_i$ is positive definite and symmetric, the GN step ${\mathbf{p}}^{\mathrm{b}}$ is the global minimizer of the unconstrained quadratic model according to the first-order optimality conditions. If the length of this step is less than the TR radius (i.e., $∥{\mathbf{p}}^{\mathrm{b}}∥<{\Delta }_i$), then the trial step is simply set to ${\mathbf{p}}^{\mathrm{b}}$. However, this condition is not always satisfied. Conversely, for a sufficiently small trust region, a first-order Taylor expansion provides a good approximation and the steepest descent step ${\mathbf{p}}^{\mathrm{u}}$ is considered optimal. Therefore, the dogleg method finds the solution to Equation (24) by searching along the piece-wise linear path constructed from ${\mathbf{p}}^{\mathrm{u}}$ and ${\mathbf{p}}^{\mathrm{b}}$ to obtain the final trial step ${\mathbf{p}}_i$, determined as follows:

• Case 1: If the GN step ${\mathbf{p}}^{\mathrm{b}}$ is within the trust region (i.e., $∥{\mathbf{p}}^{\mathrm{b}}∥\le {\Delta }_i$), then the trial step is the GN step itself: ${\mathbf{p}}_i={\mathbf{p}}^{\mathrm{b}}$.
• Case 2: If the path along the steepest descent direction intersects the TR boundary before the full step ${\mathbf{p}}^{\mathrm{u}}$ is reached (i.e., $∥{\mathbf{p}}^{\mathrm{u}}∥\ge {\Delta }_i$), then the trial step is a scaled version of the steepest descent step: ${\mathbf{p}}_i={\Delta }_i{\mathbf{p}}^{\mathrm{u}}/∥{\mathbf{p}}^{\mathrm{u}}∥$.
• Case 3: If the steepest descent step is within the trust region but the GN step is outside, then the trial step ${\mathbf{p}}_i$ is the intersection of the dogleg path (the line segment from ${\mathbf{p}}^{\mathrm{u}}$ to ${\mathbf{p}}^{\mathrm{b}}$) and the TR boundary.

Figure 4 illustrates the selection of the trial step (indicated by a red star) for each of these cases.

The gain ratio function ${\rho }_i$ is used to evaluate the acceptability of the trial step, and is defined as

$${\rho }_i={\displaystyle \frac{S\left({\mathbf{x}}_i\right)-S\left({\mathbf{x}}_i+{\mathbf{p}}_i\right)}{r\left(0\right)-r\left({\mathbf{p}}_i\right)}}.$$

(29)

The gain ratio ${\rho }_i$ measures the agreement between the quadratic model $m\left({\mathbf{p}}_i\right)$ and the actual objective function $S\left({\mathbf{x}}_i\right)$ at the new iteration. A value of ${\rho }_i$ close to 1 indicates a good approximation in which the model accurately reflects the behavior of the objective function. Conversely, a small or negative value of ${\rho }_i$ signifies a poor fit, indicating that the optimization strategy must be adjusted.

The decision to accept or reject the trial step ${\mathbf{p}}_i$ and the subsequent update to the TR radius ${\Delta }_i$ are based on the value of ${\rho }_i$. If ${\rho }_i<0$, which signifies that the objective function has increased, the step is rejected, and the trust region has shrunk (e.g., ${\Delta }_{i+1}=0.25{\Delta }_i$). Conversely, if ${\rho }_i>0$, then the step is accepted. The radius is then adjusted based on the quality of the approximation; it is expanded if the approximation is excellent (e.g., ${\rho }_i>0.75$) but shrunk if the agreement is only fair (e.g., $0<{\rho }_i\le 0.75$). Specific trust region radius scaling conditions can be flexibly set according to actual application conditions. The trust region radius update method adopted in this paper is shown in Equation (30):

$${\Delta }_{i+1}=\left\{\begin{array}{cc}min(2{\Delta }_i,{\Delta }_{max}),\hfill & {\rho }_i>0.75\\ {\Delta }_i,\hfill & 0.25\le {\rho }_i\le 0.75\\ 0.5{\Delta }_i,\hfill & {\rho }_i<0.25\end{array}\right..$$

(30)

After each trial step is computed and the trust region is updated, a stopping criterion is checked to determine whether the iteration should terminate. If the criterion is not met, the iterative process continues. If the process has converged, it yields a feasible solution for the target position estimate ${\mathbf{x}}_j^{*}$ along with its corresponding objective function value $S\left({\mathbf{x}}_j^{*}\right)$.

In order to be mathematically rigorous, while the WLS principle defines the local optimization landscape, the global cost function of the proposed multi-start system evaluates the converged solutions from all grid points. Specifically, after convergence from each starting point, the individual objective function values $S\left({\mathbf{x}}_j^{*}\right)$ are compared. The global optimal solution that minimizes this overall system cost function is chosen as the final target position estimate ${\mathbf{x}}^{*}$. This global selection cost function of the proposed system is explicitly described in Equation (31), and the complete optimization procedure is summarized in Algorithm 1.

$$\begin{array}{c}{\mathbf{x}}^{*}=\underset{{\mathbf{x}}_j^{*}}{arg\; min}S\left({\mathbf{x}}_j^{*}\right),\\ S\left({\mathbf{x}}_j^{*}\right)=\sum _{k=1}^N{\left[s_k\left({\mathbf{x}}_j^{*}\right)\right]}^2=\sum _{k=1}^N{\left[q_k\left({\mathbf{x}}_j^{*}\right)-q_k\right]}^2\end{array}$$

(31)

Algorithm 1 Multi-start trust-region-based localization algorithm.

Input: Grid division intervals $[X_{min},X_{max}]$, $[Y_{min},Y_{max}]$, step sizes $X_{\mathrm{step}}$, $Y_{\mathrm{step}}$, pulse observations ${\beta }_{nk}$, $f_{nk}$.

Output: Optimal target position estimate $\widehat{\mathbf{x}}$.

Start

1: Initialize maximum iterations $k_{max}=100$, gradient tolerance ${ϵ}_g={10}^{-6}$, step tolerance ${ϵ}_x={10}^{-8}$, and trust-region radius ${\Delta }_{max}=1.0$.

2: Construct the global objective function $S\left(\mathbf{x}\right)$ using Equation (20).

3: Generate J initial starting points ${\left\{{\mathbf{x}}_{j,0}\right\}}_{j=1}^J$ by grid division using Equation (23).

4: for$j=1,2,\dots ,J$do

5:      Initialize trust-region radius ${\Delta }_0=0.1$, and iteration counter $i=0$.

6:      repeat

7:            Compute gradient ${\mathbf{g}}_i$ and approximate Hessian ${\mathbf{B}}_i$ at ${\mathbf{x}}_{j,i}$.

8:            Solve trust-region subproblem in Equation (24) for trial step ${\mathbf{p}}_i$.

9:            Evaluate gain ratio ${\rho }_i$ using Equation (29).

10:            Update trust-region radius ${\Delta }_{i+1}=min(\mathrm{updated}\mathrm{value},{\Delta }_{max})$ in Equation (30).

11:            Update iteration point ${\mathbf{x}}_{j,i+1}={\mathbf{x}}_{j,i}+{\mathbf{p}}_i$.

12:            $i=i+1$.

13:      until $\parallel {\mathbf{g}}_i\parallel < {ϵ}_g$ or $\parallel {\mathbf{p}}_i\parallel < {ϵ}_x$ or $i \ge k_{max}$

14:      Store the converged solution for the j-th starting point as ${\mathbf{x}}_j^{*}$.

15: end for

16: Select the final target position estimate: $\widehat{\mathbf{x}}=arg{min}_{{\mathbf{x}}_j^{*}}S\left({\mathbf{x}}_j^{*}\right)$.

End

4. Numerical Simulation

This study adopts the northeast coordinate system, setting the target initial distance at 10 km and initial bearing angle at $45°$ while traveling westward at 4 m/s constant speed. The observation station moves eastward from the origin at 4 m/s constant speed. The pulse signal frequency is set at 4000 Hz with a transmission period of 15 s. During simulation, the search interval is 0 to 30 km with a step size of 5 km. The Monte Carlo method is used for statistical analysis with 500 Monte Carlo trials.

The primary performance metrics used are the convergence rate and the root mean square error (RMSE). The RMSE of the position coordinates, calculated over the number of Monte Carlo runs, is given by Equation (32). The convergence rate is defined as the percentage of successful trials out of the total number of Monte Carlo runs. A trial is considered successful (i.e., the estimate has converged) if its relative distance error (RDE) is less than 25%. It is important to clarify that this 25% margin is strategically established as a criterion for escaping local minima rather than as a measure of terminal localization accuracy. Given the role of this batch processing method as an initialization stage, an initial estimate within a 25% error margin is highly valuable, as it securely falls within the pull-in range of subsequent recursive trackers such as Kalman filters in practical underwater systems.

The RDE is defined as the ratio of the Euclidean distance between the estimated and true target positions to the true target-to-observer range, as formulated in Equation (33). The CRLB provides the theoretical minimum mean square error for any unbiased estimator of the state variables. As such, it reflects the optimal estimation performance achievable under a given set of parameter conditions. Specifically, this theoretical bound is derived directly from the exact true noise statistics without incorporating empirical smoothing and weighting mechanisms, thereby serving as a strict mathematical benchmark to evaluate the practical estimator under realistic unknown noise conditions. In performance analysis, CRLB is compared with algorithm performance, with CRLB curves representing square root results of CRLB.

$${\mathrm{RMSE}}_{\mathrm{pos}}=\sqrt{{\displaystyle \frac{1}{Mon}}\sum _{l=1}^{Mon}\left[{\left(x_l^{*}-x^t\right)}^2+{\left(y_l^{*}-y^t\right)}^2\right]}$$

(32)

$${\mathrm{RDE}}_k=\sqrt{{\displaystyle \frac{{\left(x_k^{*}-x_k^o\right)}^2+{\left(y_k^{*}-y_k^o\right)}^2}{{\left(x_k^t-x_k^o\right)}^2+{\left(y_k^t-y_k^o\right)}^2}}}\times 100\%$$

(33)

4.1. Multi-Starting Grid Search Effectiveness Verification

Error distribution histograms are used to compare convergence rate performance differences between fixed single starting point search and multi-starting grid search [30]. The simulation conditions are set as follows: observed pulse quantity = 25, angle observation noise standard deviation = $0.5°$, and frequency observation noise standard deviation = 0.5 Hz. The statistics for the “single-start search” represent the collective results obtained by treating each grid point as an independent starting point across all Monte Carlo runs; in contrast, the statistics for the “multi-start search” are based solely on the final solution selected after the optimization and comparison process.

As shown in Figure 5, the error distribution histogram comparison displays estimation result error distributions for fixed single starting grid search and multi-starting grid search. The horizontal axis represents relative error intervals; the vertical axis represents sample proportions in each interval. Using multi-starting grid search, estimation result proportions in 0% to 20% error intervals are 70.20% and 25.20%, respectively, significantly higher than 59.27% and 21.26% for the single starting point search. This result indicates that the multi-starting grid search strategy can achieve lower localization errors with higher probability, demonstrating stronger stability and global search capabilities in optimization processes.

In high error intervals from 40% to 100%, the single starting point search distribution proportions consistently exceed multi-starting grid search. When using only single fixed starting points, about 8.33% of results have relative errors exceeding 100%, belonging to the category of extreme errors. This situation indicates high sensitivity of single starting point search to initial point selection. Relying on fixed iterative starting points easily causes algorithms to fall into local optimal solutions or even non-convergent situations, seriously affecting algorithm performance. In contrast, multi-starting grid optimization effectively avoids extreme errors, indicating stronger global optimization capabilities of multi-starting grid search strategy.

To explicitly justify the selected grid search design, it is imperative to quantitatively analyze the tradeoff between global convergence, localization accuracy, and computational runtime.

Table 1. Quantitative analysis of grid density sensitivity, convergence rate, and computational runtime.

Initialization Strategy	Start Points	Convergence	Mean Rel. Error	Mean Runtime
Random Single Start	1	31.50%	99.36%	8.88 ms
Fixed Single Start	1	81.00%	30.49%	5.51 ms
Coarse Grid ($5\times 5$)	25	99.50%	7.76%	195.37 ms
Medium Grid ($7\times 7$)	49	99.50%	7.79%	371.80 ms
Fine Grid ($11\times 11$)	121	99.50%	7.78%	974.04 ms

presents a statistical evaluation based on 200 Monte Carlo runs across various initialization strategies. While it is true that increasing the grid density inflates the absolute computational time (for instance, increasing the grid points from 25 to 121 raises the average runtime from approximately 195 ms to 974 ms), this additional computational burden is marginal from a practical systems perspective. Specifically, the maximum runtime remains substantially shorter than the typical pulse period of underwater acoustic sources, which often spans several seconds. Consequently, the adoption of a multi-start grid search ensures highly robust localization without imposing any prohibitive computational bottleneck, easily satisfying the temporal requirements for sequential batch processing.

The multi-starting grid search strategy utilizes results from multiple iterative starting points, reducing the risk of algorithms falling into local optimal points. Therefore, it can effectively reduce the impact of algorithm starting point settings, leading to improved algorithm stability and robustness.

4.2. Effects of Observation Noise and Received Pulse Number on Algorithm Performance

To evaluate the effects of the frequency observation noise, angle observation noise, and number of pulse groups on the performance of different algorithms, a series of simulations were conducted.

The simulation is configured with a fixed set of parameters: 25 pulses were observed, the frequency observation noise standard deviation was 0.5 Hz, the initial target range was 10 km, and the closest point of approach (CPA) was 7 km. Under these conditions, the performance of the proposed TR based algorithm was compared against the GN algorithm, LM algorithm, and PLE algorithm across varying levels of angle observation noise, with the results shown in Figure 6.

As shown in Figure 6a, when the angle noise ranges from $0.1°$ to $0.5°$, all algorithms maintain a convergence rate above 90% and their performance differences are minimal. However, as the noise level exceeds $0.5°$, a significant divergence in performance becomes apparent. Specifically, the LM and GN algorithms exhibit similar performance, which is slightly inferior to that of the proposed TR algorithm but superior to that of PLE algorithm. The convergence rate of the PLE algorithm deteriorates sharply as the noise exceeds $0.5°$, dropping from over 90% to approximately 20%. In comparison, the proposed TR algorithm maintains a higher convergence rate than the PLE algorithm under the same noise conditions. Nevertheless, the convergence rate of the TR algorithm also decreases with increasing noise levels. For noise levels ranging from $0.1°$ to $0.9°$, the convergence rate of the TR algorithm exhibits a gradual decline. Beyond $0.9°$, the rate decreases more rapidly with increasing noise. Despite this trend, the algorithm maintains a convergence rate of over 80% across a wide range of noise levels, and still achieves a rate of over 70% even at a high noise level of $1.9°$.

Figure 6b further reveals that the RMSE of all algorithms increases with the level of angle noise. Among the compared methods, thePLE algorithm exhibits the most significant increase in RMSE, while the proposed TR algorithm shows the smallest increase. The performance of the LM and GN algorithms lies between that of the TR and PLE algorithms. Across the entire tested noise range, the RMSE of the TR algorithm increases by only approximately 1300 m, whereas that of the PLE algorithm increases by about 2700 m. At a noise level of $1.9°$, the RMSE of the PLE algorithm reaches as high as 3800 m. The TR algorithm consistently yields a smaller error and more closely approaches the CRLB.

Figure 7 illustrates the effect of frequency noise on the convergence rate of the algorithms under a fixed set of parameters: 25 observed pulse groups, an angle noise standard deviation of $0.5°$, an initial target range of 10 km, and a CPA of 7 km.

As shown in Figure 7a, the proposed TR algorithm consistently maintains a higher convergence rate than the other methods across various frequency noise levels. Although all algorithms achieve a 100% convergence rate under low-noise conditions, the TR algorithm demonstrates superior robustness as the noise increases. The LM and GN algorithms, which are based on the nonlinear model, both outperform the PLE algorithm. The performance of the PLE algorithm degrades rapidly with increasing noise, with its convergence rate dropping below 50% at a noise level of 0.9 Hz.

As depicted in Figure 7b, while the RMSE for both algorithms increases with frequency noise, the proposed TR algorithm consistently exhibits a lower error that grows at a slower rate across the entire range. At a frequency noise level of 2.0 Hz, the RMSE of the TR algorithm is approximately half that of the PLE algorithm. Furthermore, its performance more closely tracks the CRLB, indicating its superior stability and accuracy in the presence of frequency perturbations.

Figure 8 compares the performance of the various algorithms as a function of the number of received pulses. For this comparison, the fixed parameters include an angle noise standard deviation of $0.5°$, a frequency noise standard deviation of 0.5 Hz, an initial target range of 10 km, and a CPA of 7 km. The results indicate that the TR algorithm makes more effective use of the available information, achieving a higher convergence rate with fewer pulses. As observed in Figure 8a, when the number of pulses increases from 10 to 19, the convergence rate of the TR algorithm improves sharply from 40% to 90%. This improvement is significantly faster than that of the PLE algorithm, for which the convergence rate only increases from 10% to 60% over the same interval. At 25 pulses, the TR algorithm already achieves a convergence rate approaching 100%, whereas the PLE algorithm requires more than 28 pulses to reach a comparable level of performance. Figure 8b illustrates the trend of the RMSE as a function of the number of received pulses. The TR algorithm achieves rapid convergence even with a small number of pulses, as evidenced by its RMSE decreasing more rapidly to a lower final error and approaching the CRLB sooner than the other algorithms.

4.3. Effects of Motion Scenarios on Algorithm Performance

The performance of the algorithm is also highly dependent on the specific target–observer geometry. Therefore, in order to better understand the algorithm’s applicability and provide insights for its practical use, this section investigates the effects of key geometric parameters on performance, including initial target range, speed, course, and initial bearing angle. In the experiments for this subsection, the number of received pulses is fixed at 25, with the standard deviations for angle and frequency observation noise set to $0.5°$ and 0.5 Hz, respectively. A nominal scenario is established with the following standard parameter settings: an initial range of 10 km, a target speed of 4 m/s, a course of due west, and an initial bearing angle of $45°$. To investigate the impact of the target–observer geometry on performance, only the parameter under investigation is varied in each experiment, while all other parameters are held at their nominal values.

Figure 9a shows the effects of initial target distance on algorithm convergence rates. TR demonstrates certain performance advantages under various initial distance conditions. When initial target distance increases to 20 km and above, the TR algorithm still maintains about 60% convergence rate, while the convergence rate for the PLE algorithm drops sharply to near 0%. At 15 km, the TR algorithm maintains about 90% high convergence rate, while the PLE algorithm can only achieve about 30%. Figure 9b presents RMSE comparisons between both algorithms. At the closer distance of 5 km, the RMSEs of the various algorithm are nearly identical. As the initial target distance increases, the RMSE of the PLE algorithm shows approximately exponential growth trends, increasing rapidly. At 30 km, it reaches an error level of about 22 km. In comparison, growth rate for the RMSE of the TR algorithm is significantly lower, reaching only about 10 km at 30 km, approximately 45% that of the PLE algorithm. Overall, the farther the initial distance, the smaller the DOA and FOA changes caused by motion within the same time, making target motion state estimation more difficult for algorithms.

Both the convergence rate and the RMSE metrics indicate that the proposed TR algorithm possesses greater robustness and adaptability, excelling particularly in long-range target tracking scenarios. Furthermore, the RMSE curve of the TR algorithm more closely approaches the theoretical CRLB, signifying that the algorithm can better approximate the theoretical performance limit under various range conditions.

The comparative analysis in Figure 10 indicates that at higher target speeds, the algorithm achieves a higher convergence rate and a lower RMSE. This is because a higher speed results in more significant changes in the observation geometry over time, providing more information for the localization algorithm and thereby improving the estimation accuracy. Furthermore, as observed in Figure 10a, the convergence rate of the TR algorithm is significantly superior to that of the PLE algorithm at low target speeds. Notably, at a speed of 1 m/s, the TR algorithm already achieves a convergence rate of approximately 70%, demonstrating its rapid response capability in low-speed tracking scenarios. As illustrated in Figure 10b, the RMSE of the TR algorithm is consistently lower than that of the PLE algorithm across the entire range of tested speeds, with the performance gap being particularly pronounced at lower speeds. As the target speed exceeds 7 m/s, the performance of the two algorithms begins to converge, although the TR algorithm maintains a slight advantage. Moreover, the RMSE curve of the TR algorithm more closely tracks the theoretical CRLB. While higher speeds are generally more conducive to localization because they produce more significant changes in observation geometry, the TR algorithm demonstrates its ability to extract more information even from subtle changes in relative motion, thereby achieving a higher convergence rate and lower localization error in such challenging scenarios.

Figure 11 compares the effects of the target’s course on algorithm performance, where the course angle is defined as $0°$ for due east and increases counterclockwise to $180°$ for due west. The results show that both algorithms exhibit poor performance at small course angles. This is because, with the observer also traveling eastward, a small target course angle reduces the relative motion between the target and observer, leading to decreased observability. At course angles below $60°$, the PLE algorithm fails to converge, demonstrating its significant limitations under conditions of low observability. Between $60°$ and $120°$, the PLE algorithm’s performance improves rapidly, eventually reaching a level comparable to that of the TR algorithm, although its peak performance remains slightly lower. Notably, the performance of the PLE algorithm exhibits a slight decline at course angles beyond $140°$, which may indicate its potential computational instability in certain geometric configurations. In contrast, the TR algorithm remains capable of localization even at small course angles, achieving a convergence rate of approximately 10% for angles below $40°$. However, the prominent surge in estimation error observed at a course angle of $20°$ occurs because the target engages in nearly pure radial motion relative to the observation array. Under this specific geometric configuration, the lack of sufficient bearing variation leads to weak system observability, which causes the algorithm to occasionally become trapped in poor local optima despite achieving mathematical convergence. Conversely, target courses that introduce larger bearing variations, such as $0°$ or $40°$, provide much stronger observability and yield significantly more accurate localization results. Following this initial fluctuation, the performance of the TR algorithm rapidly improves, reaching a near 100% convergence rate at approximately $120°$ and maintaining this high performance up to $180°$.

Counterintuitively, at small course angles the TR algorithm exhibits a higher RMSE than the PLE algorithm, reaching a maximum of nearly 19 km even though its convergence rate is superior. This suggests that although the TR algorithm converges more frequently, its non-convergent cases produce very large errors, increasing the overall variance of the results. Such behavior is a known characteristic of nonlinear models, which can yield large deviations when the optimization process becomes trapped in a local optimum. In contrast, the PLE algorithm may converge to a biased solution more consistently, resulting in a lower RMSE but an inaccurate estimate. Beyond a course angle of $20°$, the error of the TR algorithm decreases sharply. It begins to outperform the PLE algorithm at approximately $40°$ and continues to improve until it stabilizes at its minimum error level for angles greater than $100°$. The PLE algorithm exhibits a larger overall error but smaller fluctuations, a characteristic consistent with its known properties of being biased yet having an optimistic covariance. It is also worth noting that the CRLB does not have a linear relationship with the target course. In general, the TR algorithm demonstrates superior convergence performance, and with the exception of the small-angle course scenarios its RMSE is also consistently lower than that of the PLE algorithm.

Figure 12a shows the convergence rate as a function of the target’s initial bearing angle. The initial bearing is defined as the angle measured counterclockwise from due north to the line connecting the observer’s origin and the target’s initial position. Both algorithms exhibit a very high convergence rate approaching 100% at small initial bearing angles. This is because such angles correspond to a scenario where the target moves near the observer’s broadside, which maximizes the rates of change of the DOA and FOA. However, the performance of the algorithms begins to diverge significantly as the bearing angle increases. The TR algorithm maintains a convergence rate close to 100% up to a bearing of $50°$, after which its performance begins to decline. However, this decline is relatively gradual, as the algorithm still sustains a convergence rate of approximately 20% even at an angle of $80°$. In contrast, the PLE algorithm exhibits a much steeper decline in the convergence rate beyond $40°$. It maintains a rate of only about 40% at $60°$ before dropping sharply to nearly 0% at $70°$, after which it becomes completely ineffective. This failure occurs because large initial bearing angles correspond to a scenario in which the target and observer move along nearly the same line of sight. According to the principles of observability, non-radial motion between the target and observer is required to solve the TMA problem; therefore, performance is poor in such geometries.

Figure 12b depicts the RMSE for different initial target bearing angles. At smaller bearing angles, both algorithms perform with relative stability, with the TR algorithm yielding a slightly lower error than the PLE algorithm. However, beyond an angle of $50°$, the errors for both methods begin to increase significantly. The error of the PLE algorithm grows at a markedly faster rate than that of the TR algorithm, reaching a peak of nearly 10,000 m at an angle of $80°$. It is also noteworthy that the CRLB exhibits a nonlinear relationship with the initial bearing angle. Specifically, it first increases for angles between $10°$ and $30°$, then decreases after $30°$, before rising again around $60°$. This demonstrates the complex influence that the relative geometry between the target and observer has on the ultimate localization performance.

In summary, the proposed TR algorithm demonstrates superior performance across various target–observer geometries, showcasing its accuracy and robustness. The simulation-based analysis of how these geometries affect algorithm performance also offers valuable guidance on practical application scenarios for the proposed method.

4.4. Discussion on Algorithmic Synergy and Component Contributions

To fully understand the performance gains of the proposed method, it is essential to clarify the individual contributions of its core components: the multi-start grid, the smoothed residual weighting, and the TR optimizer. The superiority of the algorithm is not driven by any single dominant module but rather by their complementary interplay in addressing distinct underwater acoustic challenges.

First, the multi-start grid isolates the contribution of initialization robustness. As evidenced by the initialization strategy analysis in Table 1, removing this component causes the algorithm to frequently become trapped in local minima, causing the convergence rate to plummet to roughly 31.5%. Second, the smoothed residual weighting drives estimation stability. Removing this weighting design would leave the standard TR method highly vulnerable to transient acoustic outliers, as unweighted least-squares objectives are easily skewed by isolated noise spikes. Finally, the TR optimizer guarantees convergence reliability. If replaced by a standard GN approach, the algorithm would struggle with ill-conditioned Jacobian matrices commonly encountered in the flat regions of the highly non-convex underwater observation model, leading to frequent divergence.

Consequently, the proposed method achieves its high localization precision and robustness exclusively through the synergistic combination of these three components.

5. Sea Trial Data

A set of sea trial data from 2018 was processed using an East–North–Up coordinate system, with the origin defined as the observer’s position at the time the first signal was received. The observer starts at the origin and travels due east at approximately 3 m/s, with the target located at (23,000 m, 15,500 m) and drifting without power, representing a special case of the assumed constant-velocity motion model. The target transmits continuous-wave (CW) pulses with a pulse repetition interval of 20 s and a center frequency of 3.8 kHz. Figure 13 provides a schematic of the observer and target trajectories for the sea trial. The observer received signals using a sonar array and obtained DOA measurements via beamforming techniques. FOA measurements were acquired by measuring the frequency of the CW signal. Simultaneously, GPS data from both the observer and the target were recorded to provide the ground truth for the target’s position.

Figure 14 compares the actual measured DOA and FOA data from the sea trial with their corresponding theoretical values. The overall trend of the measurements is largely consistent with the theoretical data; however, some discrepancies are evident. A bias of approximately $1.15°$ is observed in the DOA measurements, which is likely attributable to heading measurement errors of the receiving array during its maneuvers. Similarly, the FOA measurements exhibit both bias (approximately 1 Hz) and fluctuations, deviating from the ideal model due to factors such as platform motion and measurement noise.

Figure 15 presents the localization results for each algorithm as a function of the number of pulse groups. Overall, the results demonstrate a clear downward trend in the relative range error for all algorithms as the number of received pulses increases. At low numbers of pulse groups, all algorithms exhibit a relative range error exceeding 20%, although the TR algorithm’s error is slightly lower. In the range of 15 to 35 pulses, the TR algorithm demonstrates a distinct advantage, with its error decreasing more rapidly. Beyond 45 pulses it maintains a low error level of approximately 5%, outperforming the other methods and exhibiting a flatter curve. As expected, the PLE algorithm performs poorly with fewer pulse groups, and its error gradually decreases as more pulses are incorporated. Figure 15b reveals that the localization results of the TR algorithm are more tightly clustered and closer to the true target position, indicating higher localization accuracy. Importantly, the algorithm demonstrates its ability to converge with a small number of received pulses, yielding a reliable target estimate with fewer measurements.

Figure 16 shows the coordinate errors in the east and north directions for the different algorithms as a function of the number of pulse groups. Two main trends are observed: first, the average error in the east direction is consistently greater than that in the north direction; and second, the overall tracking error for all algorithms decreases as the number of received pulses increases.

Figure 17 and Figure 18 show the localization error evaluated using a sliding window method with a fixed window size of 25 pulses. This approach assesses algorithm performance as a function of the window’s starting position. The results for all algorithms exhibit significant fluctuations, which is expected as each window contains a different target–observer geometry. With the exception of windows starting between pulses 18 and 24, the relative range error of the TR algorithm is consistently smaller than that of the PLE algorithm. The maximum relative error for the TR algorithm is approximately 30%, which is significantly lower than PLE, for which the maximum error exceeds 50%.

The number of trials that achieved convergence for each method is tallied in

Table 2. Algorithm convergence rate for different starting pulses.

Method	TR	GN	LM	PLE
Convergence	93.18%	90.90%	90.90%	50.00%

. The proposed TR algorithm achieves a maximum convergence rate of 93.18%, reflecting its stable estimation performance. In summary, the sea trial data validate the superior localization performance of the TR algorithm. The proposed algorithm can produce more accurate localization results using fewer pulses while simultaneously maintaining a higher convergence rate and more stable error characteristics.

To further evaluate the robustness of the proposed method against the aforementioned systematic measurement errors, a bias sensitivity test was conducted using the sea trial data. Building upon the previously noted discrepancies, it is crucial to recognize that systematic biases are often unavoidable in practical underwater applications. For instance, the heading of a towed receiving array can be significantly affected by ocean currents, which inevitably leads to the constant bias observed in the direction of arrival measurements. To analyze the specific impact of this angle bias, we compared the localization performance using the original biased observations against a manually corrected unbiased observation set. The comparative results are clearly illustrated in Figure 19.

As shown in the scatter plot, the systematic bias primarily causes a spatial offset in the estimated east and north coordinates. However, it does not induce a catastrophic divergence in the overall localization geometry. The relative distance error curve further demonstrates this robust stability. During the initial phase of the first 15 pulses, the difference between the biased and unbiased estimations is completely negligible. Between 15 and 35 pulses, the biased estimation exhibits a slightly lower performance, with an average error increase of only about 3%. Interestingly, between 35 and 53 pulses, the biased estimation actually yields a smaller relative distance error compared to the unbiased case. Furthermore, from a theoretical standpoint, the frequency of arrival bias is exceptionally small compared to the signal center frequency and as such exerts a minimal impact on the overall estimation. This comprehensive analysis confirms that the proposed localization algorithm maintains strong robustness and reliability even in the presence of practical systematic measurement biases.

6. Conclusions

To address the batch processing localization problem for non-cooperative underwater acoustic pulse signals, this paper proposes a localization algorithm based on a multi-starting grid search and TR optimization. Fundamentally, this methodological contribution is an underwater pulse-specific design that strategically combines weighted least squares weighting, multi-start grid searching, and dogleg optimization. This specific combination successfully circumvents both the biased estimation of pseudolinear approaches and the initialization vulnerabilities of basic iterative methods. The proposed method first constructs a WLS localization model by accumulating a batch of pulse data and using the mean squared residuals of the smoothed DOA and FOA for weighting. This model is then solved using a TR algorithm, with the search optimized via the multi-starting method. The multi-starting grid search strategy avoids the need for manual selection of an initial iterate, enhancing robustness against poor initialization. This reduces the probability of converging to outlier solutions, improving the overall stability of the algorithm. This in turn reduces the likelihood of converging to outlier solutions, significantly improving the overall stability of the algorithm. Notably, the current method assumes Gaussian measurement noise; evaluating and enhancing the robustness of the algorithm under strictly non-Gaussian disturbances remains a critical subject for our future research.

The performance of the proposed algorithm has been validated through both simulations and sea trial data. Simulation results demonstrate that the algorithm is both robust against noise and data-efficient, yielding reliable results even with a small number of pulses; for instance, a convergence rate of 80% can be achieved with only 20 pulses. As the number of received pulses increases, the convergence rate improves and the range RMSE decreases, with the relative range error converging to below 20% at 30 pulses. However, it is important to objectively acknowledge the inherent limitations of the proposed system. As observed in the numerical simulations, the algorithm experiences a reduction in observability under extreme geometric or environmental conditions, such as a very small closest point of approach distance or exceptionally high measurement noise. This can lead to noticeable increases in estimation errors. Despite these boundary limitations, the proposed algorithm successfully overcomes the key shortcomings of traditional algorithms such as low modeling accuracy and reliance on a large number of observations by demonstrating superior convergence performance and accuracy under standard operational conditions, thereby offering greater practicality and reliability.