Contents Introduction 介绍
Underwater long baseline (LBL) localization systems [1], [2] are renowned for their wide localization area, high localization precision, and excellent scalability. As such, they have been successfully employed in various underwater engineering applications. Traditional LBL localization methods are typically based on a two-step paradigm consisting of two independent steps: 1) the time of arrival (TOA) of the target signal is estimated independently at each buoy using threshold processing and 2) the distance between the target and the buoy can be calculated from the estimated TOA and sound speed and is used for localization based on relevant geometric relationships [3]. The physical basis that the TOAs estimated from different buoys all correspond to the same target position is ignored in the first step. Therefore, the strategy is suboptimal. In addition, threshold settings present extra challenges. First, threshold settings introduce an undesirable false alarm/detection trade-off [4], [5]. Second, affected by the complex underwater environment and underwater acoustic channel, multiple TOA measurements may exceed the threshold. These excessive measurements may originate from direct sound, reflected sound, or noise. To choose a measurement corresponding to the direct sound for localization, traditional LBL localization schemes must perform direct sound selection (DSS), which is a persistent challenge that severely limits localization performance [6]. Worse still, the direct sound may be mixed with the reflected sounds from neighboring signal periods, further increasing the hardness of DSS.
水下长基线（LBL）定位系统[1]、[2]以其广泛的定位区域、高定位精度和出色的可扩展性而闻名。因此，它们已成功应用于各种水下工程应用。传统的 LBL 定位方法通常基于由两个独立步骤组成的两步范式：1）目标信号的到达时间（TOA）使用阈值处理在每个浮标上独立估计，2）目标与浮标之间的距离可以根据估计的 TOA 和声速计算出来，并用于基于相关几何关系的定位[3]。第一步忽略了从不同浮标估计的 TOA 都对应于同一目标位置的物理基础。因此，该策略是次优的。此外，阈值设置带来了额外的挑战。首先，阈值设置引入了不良的误报/检测权衡[4]、[5]。其次，受复杂的水下环境和水下声道的影响，多次 TOA 测量可能会超过阈值。这些过度测量可能源于直接声音、反射声或噪音。为了选择与直接声音相对应的测量值进行定位，传统的 LBL 定位方案必须进行直接声音选择（DSS），这是一个持续存在的挑战，严重限制了定位性能[6]。更糟糕的是，直接声音可能会与相邻信号周期的反射声音混合，进一步增加 DSS 的硬度。
Existing DSS methods include three main types. The first type utilizes an underwater acoustic multipath channel model to select direct sound [7], [8]. To improve the robustness of multipath TOA estimation, a TOA estimation method based on the delay-and-sum of cross correlation functions is proposed [7]. In [8], a high-resolution technique for jointly estimating the multipath TOAs, Doppler scales, and attenuation amplitudes of a time-varying underwater acoustic channel is proposed, which formulates the estimation of channel parameters as a sparse representation problem. The second type is based on the improved matched filter (MF), such as multidimensional MF (MMF) [9], phase MF (PMF) [10], and adaptive MF (AMF) [11]. Specifically, the MMF separates the multipath to determine each path’s TOAs and amplitudes. PMF allows the recognition and removal of reflected signals. The AMF adaptively adjusts the structure of the reference signal according to the channel. The third type develops the classification network for DSS based on the underwater propagation characteristics of the target signal [12]. In [6], a high-rate direct sound classification network based on a decision tree is established to extract the propagation characteristics of the direct sound accurately.
现有的 DSS 方法包括三种主要类型。第一种类型利用水声多径通道模型来选择直达声 [7]， [8]。为了提高多径 TOA 估计的鲁棒性，提出了一种基于互相关函数的延迟和和 TOA 估计方法 [7]。在[8]中，提出了一种联合估计时变水声信道的多径 TOA、多普勒标度和衰减幅度的高分辨率技术，将信道参数的估计表述为稀疏表示问题。第二种类型基于改进的匹配滤波器 （MF），例如多维 MF （MMF） [9] 、相位 MF （PMF） [10] 和自适应 MF （AMF） [11]。具体来说， MMF 分离多路径以确定每条路径的 TOA 和幅度。PMF 允许识别和消除反射信号。AMF 根据通道自适应地调整参考信号的结构。第三种类型根据目标信号的水下传播特性开发 DSS 分类网络 [12]。在[6]中，建立了基于决策树的高速率直达声分类网络，以准确提取直达声的传播特性。
Track-before-detect (TBD) technology localizes targets utilizing raw sensor data rather than only threshold exceedances. Since TBD technology works without the threshold setting, the persistent and challenging problems mentioned earlier arising from threshold setting can be circumvented. For these reasons, we advocate using the TBD technique to localize underwater targets. Two typical TBD techniques are single-frame recursive TBD (SF-TBD) [13], [14], [15], [16], [17], [18], [19] and multiframe batch style TBD (MF-TBD) [20], [21], [22]. Among these methods, MF-TBD utilizes the correlation of the target position between frames to accumulate the target energy for better performance of weak target detection [20]. However, in the focused LBL localization scenario, the superiority of MF-TBD may be obscured by the relatively high signal-to-noise ratio (SNR) of the target. In addition, since the performance of MF-TBD is strongly related to the accuracy of the motion model, this performance deteriorates significantly when the model is mismatched [21]. Hence, we employ SF-TBD for LBL localization, focusing on its implementation with a Monte Carlo (MC) approximation technique called particle filtering (PF) to adequately model the nonlinear relationship between estimates and measurements. To date, various PF-style TBD (PF-TBD) methods have been proposed. However, most are applied in the radar and optics fields [13], [14], with only a few involved sonar [15], [16], [17], [18], [19]. PF-TBD for multitargets suffers from the curse of dimensionality [23], which is caused by the high-dimensional multitarget particle states, implying that the particle number needs to grow exponentially as the target number increases to guarantee performance. In [15], the posterior independence assumption was proposed to overcome the curse of dimensionality. Based on this assumption, a PF-TBD method for passive sonar bearing tracking was proposed in which the required particle number is proportional to the target number. In [16], the PF-TBD method was employed for passive sonar bearing-only localization, demonstrating that the PF-TBD approach can completely circumvent the challenge of measurement-to-track association. However, the above algorithms ignore the interference from neighboring targets and, thus, perform poorly when targets are close. In [17] and [18], the effect of neighboring targets is considered in the likelihood function of PF to produce better neighboring target tracking performance. Recent investigations of the PF-TBD method also involve active sonar applications [19].
先跟踪后检测 （TBD） 技术利用原始传感器数据定位目标，而不仅仅是阈值超标。由于 TBD 技术无需阈值设置即可工作，因此可以避免前面提到的由阈值设置引起的持续且具有挑战性的问题。由于这些原因，我们提倡使用 TBD 技术来定位水下目标。两种典型的 TBD 技术是单帧递归 TBD （SF-TBD） [13]， [14]， [15]， [16]， [17]， [18]， [19] 和多帧批处理式 TBD （MF-TBD） [20]， [21]， [22]。在这些方法中，MF-TBD 利用帧之间目标位置的相关性来积累目标能量，以获得更好的弱目标检测性能 [20]。然而，在聚焦 LBL 定位场景中，MF-TBD 的优势可能会被目标相对较高的信噪比 （SNR） 所掩盖。此外，由于 MF-TBD 的性能与运动模型的准确性密切相关，因此当模型不匹配时，这种性能会显着恶化 [21]。因此，我们采用 SF-TBD 进行 LBL 定位，专注于使用称为粒子过滤 （PF） 的蒙特卡洛 （MC） 近似技术实现它，以充分模拟估计和测量之间的非线性关系。迄今为止，已经提出了各种 PF 型 TBD （PF-TBD） 方法。然而，大多数应用于雷达和光学领域[13]、[14]，只有少数涉及声纳[15]、[16]、[17]、[18]、[19]。多目标的 PF-TBD 受到维数诅咒 [23] 的影响，这是由高维多目标粒子状态引起的，这意味着粒子数需要随着目标数的增加呈指数级增长，以保证性能。 在 [15] 中，提出了后独立性假设来克服维度的诅咒。基于这一假设，提出了一种用于被动声纳方位跟踪的 PF-TBD 方法，其中所需的粒子数与目标数成正比。在 [16] 中，PF-TBD 方法用于仅被动声呐方位定位，表明 PF-TBD 方法可以完全规避测量与航迹关联的挑战。但是，上述算法忽略了来自相邻目标的干扰，因此当目标靠近时，性能不佳。在 [17] 和 [18] 中，在 PF 的似然函数中考虑了相邻目标的影响，以产生更好的相邻目标跟踪性能。最近对 PF-TBD 方法的研究也涉及主动声呐的应用 [19]。
This work presents a PF-TBD method for LBL localization directly using MF outputs as inputs. The proposed method can overcome some persistent and troublesome issues faced by the traditional LBL localization derived from the two-step paradigm, such as unacceptable false alarm detection trade-offs and DSS. Moreover, this method allows us to consider the physical basis that the TOAs estimated from different buoys all correspond to the same target position to produce better localization performance. Specifically, the following contributions are made.
这项工作提出了一种直接使用 MF 输出作为输入进行 LBL 定位的 PF-TBD 方法。所提方法可以克服传统 LBL 定位从两步法衍生而来的一些持续存在且棘手的问题，例如不可接受的误报检测权衡和 DSS。此外，这种方法允许我们考虑从不同浮标估计的 TOA 都对应于同一目标位置的物理基础，以产生更好的定位性能。具体来说，做出了以下贡献。
To achieve automatic initialization of the target, the proposed method utilizes a discrete grid to determine the target’s approximate location. Once the approximate position of the target is determined, an adaptive grid (particles) is utilized to track the target.
为了实现目标的自动初始化，所提出的方法利用离散网格来确定目标的大致位置。一旦确定了目标的大致位置，就会使用自适应网格（粒子）来跟踪目标。In general, the likelihood function is designed with a specific probability distribution, which allows interference from neighboring targets to be considered. This consideration of neighboring target effects is necessary for many noncooperative target scenarios. However, in the LBL localization scenario of interest, we can design the emission signals of the targets. With coherent signal processing techniques, interference from neighboring targets can be sufficiently avoided. More importantly, the form of the probability distribution used for designing the likelihood function is determined by the statistical properties of the measurement. In this work, the measurements are MF outputs, which exhibit different statistical properties for different signal types. Therefore, using a specific probability distribution to design the likelihood function may be applicable to only one or several signal types. For the above two reasons, we define the likelihood function as the product of the MF outputs of multiple buoys conditioned on the particle’s state. Compared with the likelihood function designed using a specific probability distribution, the proposed likelihood function is applicable to signals of any waveform.
通常，似然函数设计有特定的概率分布，这允许考虑来自相邻目标的干扰。对于许多不合作的目标场景，这种对相邻目标效应的考虑是必要的。然而，在感兴趣的 LBL 定位场景中，我们可以设计目标的发射信号。使用相干信号处理技术，可以充分避免来自相邻目标的干扰。更重要的是，用于设计似然函数的概率分布形式由测量的统计属性决定。在这项工作中，测量值是 MF 输出，对于不同的信号类型，它们表现出不同的统计特性。因此，使用特定概率分布来设计似然函数可能仅适用于一种或多种信号类型。基于上述两个原因，我们将似然函数定义为以粒子状态为条件的多个浮标的 MF 输出的乘积。与使用特定概率分布设计的似然函数相比，所提出的似然函数适用于任何波形的信号。To reduce the performance loss caused by the target maneuver, we sample particles with multiple motion models, and the transfer between models is controlled by a first-order Markov chain. However, the use of multiple models may contain invalid models, meaning that particle sampling with multiple models may be inefficient. To address this issue, we employ an auxiliary variable [24] to optimize particle sampling for the purpose of eliminating particles sampled by invalid models.
为了减少目标机动造成的性能损失，我们使用多个运动模型对粒子进行采样，模型之间的传输由一阶马尔可夫链控制。但是，使用多个模型可能包含无效模型，这意味着使用多个模型进行粒子采样可能效率低下。为了解决这个问题，我们采用了一个辅助变量 [24] 来优化粒子采样，以消除无效模型采样的粒子。
The rest of this article is organized as follows. In Section II, a brief overview of the traditional LBL localization method is provided. In Section III, the proposed algorithm is introduced in detail. In Section IV, a comprehensive performance analysis of the proposed algorithm utilizing simulated and real data, including the effect of different SNRs and severe multipath effects, is presented. Finally, conclusions are given in Section V.
本文的其余部分组织如下。在第 II 节中，简要概述了传统的 LBL 定位方法。在第 III 节中，详细介绍了所提出的算法。在第 IV 节中，利用仿真和真实数据对所提出的算法进行了全面的性能分析，包括不同 SNR 的影响和严重的多径效应。最后，第 V 节给出了结论。
Problem Statement 问题陈述
Here, we utilize a global coordinate system as follows. The
在这里，我们使用全局坐标系，如下所示。-
Diagram of the LBL localization system.
LBL 定位系统示意图。
A. Detection A. 检测
In a Gaussian white noise background, the MF is optimal in terms of the output SNR [25]. Therefore, the MF is a classic detection scheme for estimating TOA. In the classical LBL localization method, the TOA is extracted by setting thresholds. However, multiple TOA measurements may exceed the threshold due to multipath effects (see Fig. 2). Moreover, the study of [4] showed that the performance of traditional localization methods is more sensitive to TOA errors than buoy position errors and sound velocity errors [6]. Consequently, selecting the TOA measurement corresponding to direct sound is essential for the traditional LBL localization method. The commonly used DSS methods select a direct sound by establishing a classification network based on the underwater propagation characteristics of the direct sound signal. For example, direct sound has a smaller TOA and higher intensity than reflected sounds. This class of methods has been deployed in many LBL positioning systems and is often applicable. However, such methods struggle to provide reliable TOA estimations for challenging environments, such as direct sound mixed with reflected sounds from neighboring periods, strong multipath effects, and low SNR.
在高斯白噪声背景下，MF 在输出 SNR 方面是最佳的 [25]。因此，MF 是估计 TOA 的经典检测方案。在经典的 LBL 定位方法中，TOA 是通过设置阈值来提取的。但是，由于多径效应，多个 TOA 测量可能会超过阈值（见图 2）。此外，对 [4] 的研究表明，传统定位方法的性能对 TOA 误差比浮标位置误差和声速误差更敏感 [6]。因此，选择与直达声相对应的 TOA 测量值对于传统的 LBL 定位方法至关重要。常用的 DSS 方法根据直达声信号的水下传播特性，通过建立分类网络来选择直达声。例如，与反射声相比，直达声具有更小的 TOA 和更高的强度。这类方法已部署在许多 LBL 定位系统中，并且通常适用。然而，这些方法难以为具有挑战性的环境提供可靠的 TOA 估计，例如直接声与来自相邻周期的反射声混合、强烈的多径效应和低 SNR。
First buoy’s MF output of Real Data 1 at 400 s.
第一个浮标在 400 s 时对真实数据 1 的 MF 输出。
B. Localization B. 本地化
Two conditions are necessary for applying such LBL localization scenarios: 1) the sound source is equipped with a synchronous device and 2) the signal period is given. In this case, the localization is essentially a circular intersection problem, as shown in Fig. 3, commonly referred to as TOA-based localization [3]. In Fig. 3, the distances
应用此类 LBL 定位场景需要两个条件：1） 声源配备同步设备，以及 2） 给定信号周期。在这种情况下，定位本质上是一个圆形交集问题，如图 3 所示，通常称为基于 TOA 的定位 [3]。在图 3 中，
Schematic of TOA-based localization.
基于 TOA 的定位示意图。
Proposed Algorithms 建议的算法
To overcome these drawbacks of the traditional LBL localization approach, the TOA-based auxiliary variable optimization multiple model TBD (TOA-AMTBD) algorithm is presented in this section. Essentially, the TOA-AMTBD algorithm considers the physical basis that the TOAs estimated from different buoys all correspond to the same target position, which is ignored in the traditional LBL localization approach. A more detailed explanation is given in Section III-D. Since PF is the core element of our algorithm, we first briefly introduce PF, formulate the proposed algorithm based on the PF framework, and then describe the main steps required to implement PF.
为了克服传统 LBL 定位方法的这些缺点，本节介绍了基于 TOA 的辅助变量优化多模型 TBD （TOA-AMTBD） 算法。从本质上讲，TOA-AMTBD 算法考虑了从不同浮标估计的 TOA 都对应于同一目标位置的物理基础，这在传统的 LBL 定位方法中被忽略了。第 III-D 节给出了更详细的解释。由于 PF 是我们算法的核心元素，我们首先简要介绍 PF，基于 PF 框架制定所提出的算法，然后描述实现 PF 所需的主要步骤。
A. Target Tracking Based on PF
A. 基于 PF 的目标跟踪
The LBL localization system aims to estimate multitarget states, and the multitarget states at time
LBL 定位系统旨在估计多目标状态，时间
查看源码 \begin{equation*} {\mathbf{X}}_{k} =\left [{ {\left ({\boldsymbol{ x_{k,1}} }\right)^{T},\left ({\boldsymbol{ x_{k,2}} }\right)^{T},\ldots,\left ({\boldsymbol{ x_{k,\eta }} }\right)^{T}} }\right]^{T}\in {\mathbb {R}}^{\gamma \cdot \eta }. \tag{1}\end{equation*}
In (1), 在 （1） 中，
PF is a sequential processing technique based on the Bayesian criterion, approximating the target posterior probability density,
PF 是一种基于贝叶斯准则的顺序处理技术，通过一组粒子及其相应的权重来近似目标后验概率密度
查看源码 \begin{equation*} {\mathbf{X}}_{k}^{n} =\left [{ {\left ({\boldsymbol{ x_{k,1}^{n}} }\right)^{T},\left ({\boldsymbol{ x_{k,2}^{n}} }\right)^{T},\ldots,\left ({\boldsymbol{ x_{k,\eta }^{n}} }\right)^{T}} }\right]^{T}\in {\mathbb {R}}^{\gamma \cdot \eta }. \tag{2}\end{equation*}
In (2), 在 （2） 中，
查看源码 \begin{equation*} w_{k}^{n} \propto w_{k-1}^{n} p\left ({\boldsymbol{ z_{k} \vert {\mathbf{X}}_{k}^{n}} }\right)p\left ({\boldsymbol{ X_{k}^{n} \vert {\mathbf{X}}_{k-1}^{n}} }\right)/q\left ({\boldsymbol{ X_{k}^{n} \vert {\mathbf{X}}_{k-1}^{n},{\mathbf{z}}_{k}} }\right). \tag{3}\end{equation*}
在（3）中，
查看源码 \begin{align*} p\left ({\boldsymbol{ X_{k}^{n} \vert {\mathbf{X}}_{k-1}^{n}} }\right)&=\prod _{j=1}^{\eta} {p\left ({\boldsymbol{ x_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{n}} }\right)} \tag{4}\\ q\left ({\boldsymbol{ X_{k}^{n} \vert {\mathbf{X}}_{k-1}^{n},{\mathbf{z}}_{k}} }\right)&=\prod _{j=1}^{\eta} {q_{j} \left ({\boldsymbol{ x_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{n},{\mathbf{z}}_{k}} }\right)} \tag{5}\\ q_{j} \left ({\boldsymbol{ x_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{n},{\mathbf{z}}_{k}} }\right)&\propto p\left ({\boldsymbol{ z_{k} \vert {\mathbf{x}}_{k,j}^{n}} }\right)p\left ({\boldsymbol{ x_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{n}} }\right). \tag{6}\end{align*}
将 （4）、（5） 和 （6） 代入 （3），粒子权重更新由下式给出
查看源码 \begin{equation*} w_{k}^{n} \propto w_{k-1}^{n} p\left ({\boldsymbol{ z_{k} \vert {\mathbf{X}}_{k}^{n}} }\right)/\prod _{j=1}^{\eta} {p\left ({\boldsymbol{ z_{k} \vert {\mathbf{x}}_{k,j}^{n}} }\right)} \tag{7}\end{equation*}
其中
In practice, we cannot always request the target to keep a motion state. Thus, a mismatch between the predefined motion model and the practical motion of the target is unavoidable. To reduce the performance loss caused by model mismatch, we use multiple motion models to match the target motion. The motion model describes the transfer and extrapolation of the target states over time and determines the transfer density function. Different motion models correspond to different transition density functions. The
在实践中，我们不能总是要求 target 保持运动状态。因此，预定义的运动模型与目标的实际运动之间存在不匹配是不可避免的。为了减少模型不匹配导致的性能损失，我们使用多个运动模型来匹配目标运动。运动模型描述目标状态随时间的传递和外推，并确定传递密度函数。不同的运动模型对应于不同的过渡密度函数。
查看源码 \begin{align*} p\left ({\boldsymbol{ X_{k}^{n},{\mathbf{r}}_{k}^{n} \vert {\mathbf{X}}_{k-1}^{n},{\mathbf{r}}_{k-1}^{n}} }\right)&=\prod _{j=1}^{\eta} {p\left ({\boldsymbol{ x_{k,j}^{n},r_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{n},r_{k-1,j}^{n}} }\right)} \tag{8}\\ p\left ({\boldsymbol{ x_{k,j}^{n},r_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{n},r_{k-1,j}^{n}} }\right)\!&=\!p\left ({\boldsymbol{ x_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{n},r_{k,j}^{n}} }\right)p\left ({{r_{k,j}^{n} \vert r_{k-1,j}^{n}} }\right). \tag{9}\end{align*}
The transfer of model states is controlled by a first-order Markov chain. Equations (5) and (6), thus, become 模型状态的传递由一阶马尔可夫链控制。因此，方程 （5） 和 （6） 变为
查看源码 \begin{align*} &\hspace {-1.5pc}q\left ({\boldsymbol{ X_{k}^{n},{\mathbf{r}}_{k}^{n} \vert {\mathbf{X}}_{k-1}^{n},{\mathbf{r}}_{k-1}^{n},{\mathbf{z}}_{k}} }\right) \\ &=\prod _{j=1}^{\eta} {q_{j} \left ({\boldsymbol{ x_{k.j}^{n},r_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{n},r_{k-1,j}^{n},{\mathbf{z}}_{k}} }\right)} \tag{10}\\ &\hspace {-1.5pc}q_{j} \left ({\boldsymbol{ x_{k,j}^{n},r_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{n},r_{k-1,j}^{n},{\mathbf{z}}_{k}} }\right) \\ &\propto p\left ({\boldsymbol{ z_{k} \vert {\mathbf{x}}_{k,j}^{n}} }\right)p\left ({\boldsymbol{ x_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{n},r_{k,j}^{n}} }\right)p\left ({{r_{k,j}^{n} \vert r_{k-1,j}^{n}} }\right). \tag{11}\end{align*}
We refer to the TOA-TBD algorithm that samples particles with multiple models as the TOA-based multiple model TBD (TOA-MTBD) algorithm. Although sampling with multiple models can reduce the performance loss caused by model mismatch, it may be inefficient because multiple models may contain invalid models. Thus, we employ an auxiliary variable to eliminate particles sampled by invalid models. Equations (8) and (9) are further written as
我们将具有多个模型的粒子采样的 TOA-TBD 算法称为基于 TOA 的多模型 TBD（TOA-MTBD）算法。虽然使用多个模型进行采样可以减少模型不匹配造成的性能损失，但由于多个模型可能包含无效模型，因此效率可能较低。因此，我们使用辅助变量来消除无效模型采样的粒子。方程 （8） 和 （9） 进一步写为
查看源码 \begin{align*} &\hspace {-2pc}p\left ({\boldsymbol{ X_{k}^{n},{\mathbf{r}}_{k}^{n},{\mathbf{a}}_{k}^{n} \vert {\mathbf{X}}_{k-1}^{n},{\mathbf{r}}_{k-1}^{n}} }\right) \\ &=\prod _{j=1}^{\eta} {p\left ({\boldsymbol{ x_{k,j}^{n},r_{k,j}^{n},a_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{n},r_{k-1,j}^{n}} }\right)} \tag{12}\\ &\hspace {-2pc}p\left ({\boldsymbol{ x_{k,j}^{n},r_{k,j}^{n},a_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{n},r_{k-1,j}^{n}} }\right) \\ &=p\left ({\boldsymbol{ x_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{a_{k,j}^{n}},r_{k,j}^{a_{k,j}^{n}}} }\right)p\left ({{r_{k,j}^{a_{k,j}^{n}} \vert r_{k-1,j}^{a_{k,j}^{n}}} }\right) \tag{13}\end{align*}
where 式中
查看源码 \begin{align*} &\hspace {-0.5pc}q\left ({\boldsymbol{ X_{k}^{n},{\mathbf{r}}_{k}^{n},{\mathbf{a}}_{k}^{n} \vert {\mathbf{X}}_{k-1}^{n},{\mathbf{r}}_{k}^{n},{\mathbf{z}}_{k}} }\right) \\ &=\prod _{j=1}^{\eta} {q_{j} \left ({\boldsymbol{ x_{k,j}^{n},r_{k,j}^{n},a_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{n},r_{k-1,j}^{n},{\mathbf{z}}_{k}} }\right)} \tag{14}\\ &\hspace {-0.5pc}q_{j} \left ({\boldsymbol{ x_{k,j}^{n},r_{k,j}^{n},a_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{n},r_{k-1,j}^{n},{\mathbf{z}}_{k}} }\right) \\ &\propto p\left ({\boldsymbol{ z_{k} \vert {\mathbf{u}}_{k,j}^{a_{k,j}^{n}}} }\right)p\left ({\boldsymbol{ x_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{a_{k,j}^{n}},r_{k,j}^{a_{k,j}^{n}}} }\right)p\left ({{r_{k,j}^{a_{k,j}^{n}} \vert r_{k-1,j}^{a_{k,j}^{n}}} }\right) \tag{15}\end{align*}
其中
查看源码 \begin{align*} &\hspace {-0.1pc}w_{k}^{n} \\ &\propto w_{k-1}^{n} \frac {p\left ({\boldsymbol{ z_{k} \vert {\mathbf{X}}_{k}^{n}} }\right)\prod _{j=1}^{\eta} {p\left ({\boldsymbol{ x_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{a_{k,j}^{n}},r_{k,j}^{a_{k,j}^{n}}} }\right)p\left ({{r_{k,j}^{a_{k,j}^{n}} \vert r_{k-1,j}^{a_{k,j}^{n}}} }\right)}}{\prod _{j=1}^{\eta} {p\left ({\boldsymbol{ z_{k} \vert {\mathbf{u}}_{k,j}^{a_{k,j}^{n}}} }\right)p\left ({\boldsymbol{ x_{k,j}^{n} \vert {\mathbf{x}}_{k-1,j}^{a_{k,j}^{n}},r_{k,j}^{a_{k,j}^{n}}} }\right)p\left ({{r_{k,j}^{a_{k,j}^{n}} \vert r_{k-1,j}^{a_{k,j}^{n}}} }\right)}} \\ &\propto w_{k-1}^{n} p\left ({\boldsymbol{ z_{k} \vert {\mathbf{X}}_{k}^{n}} }\right)/\prod _{j=1}^{\eta} {p\left ({\boldsymbol{ z_{k} \vert {\mathbf{u}}_{k,j}^{a_{k,j}^{n}}} }\right)}. \tag{16}\end{align*}
Using the auxiliary variable enables low-likelihood particles sampled based on the invalid model to be discarded. In other words, the particles at time
使用 auxiliary 变量可以丢弃基于无效模型采样的低似然粒子。换句话说，在 time
B. Initialization
To initialize the particles autonomously, we grid the localization region to determine the initial target location. After meshing, the likelihoods of all meshes can be calculated using the calculations provided in Section III-D, and the coordinates of the mesh with the highest likelihood are expressed as
The range is uniformly distributed between 0 and
Rmax The bearing is uniformly distributed between
0∘ 360∘ The velocity is uniformly distributed between 0 and
vmax The course is uniformly distributed between
0∘ 360∘
The particle motion model state is initialized according to the preset probabilities of motion models, and the preset probability of the
Schematic of the particle position initialization.
C. Dynamic Modeling
In the given LBL localization scenario, the target is characterized by low speed and poor maneuverability. We assume that the target obeys one of the following motion models at any moment: 1) uniform linear motion model; 2) clockwise turning motion model; and 3) counterclockwise turning motion model. The motion equation is presented as
\begin{equation*} {\mathbf{x}}_{k,j}^{n} ={\mathbf{F}}^{r_{k,j}^{n}}{\mathbf{x}}_{k-1,j}^{n} +\boldsymbol{ \Gamma w}_{k,j}^{n} \tag{17}\end{equation*}
where \begin{align*} {\mathbf{F}}^{1}&=\left [{ {{\begin{array}{cccccccccccccccccccc} 1 &\quad T &\quad 0 &\quad 0 \\ 0 &\quad 1 &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 1 &\quad T \\ 0 &\quad 0 &\quad 0 &\quad 1 \\ \end{array}}} }\right],\quad {\rm {\boldsymbol{\Gamma }}}=\left [{ {{\begin{array}{cccccccccccccccccccc} {T^{2} /2} &\quad 0 \\ T &\quad 0 \\ 0 &\quad {T^{2} /2} \\ 0 &\quad T \\ \end{array}}} }\right] \\ {\mathbf{F}}^{2}&=\left [{ {{\begin{array}{cccccccccccccccccccc} 1 &\quad {\sin \left ({{\omega T} }\right)/\omega } &\quad 0 &\quad {-\left [{ {1-\cos \left ({{\omega T} }\right)} }\right]/\omega } \\ 0 &\quad {\cos \left ({{\omega T} }\right)} &\quad 0 &\quad {-\sin \left ({{\omega T} }\right)} \\ 0 &\quad {\left [{ {1-\cos \left ({{\omega T} }\right)} }\right]/\omega } &\quad 1 &\quad {\sin \left ({{\omega T} }\right)/\omega } \\ 0 &\quad {\sin \left ({{\omega T} }\right)} &\quad 0 &\quad {\cos \left ({{\omega T} }\right)} \\ \end{array}}} }\right] \\ {\mathbf{F}}^{3}&=\left [{ {{\begin{array}{cccccccccccccccccccc} 1 &\quad {\sin \left ({{\omega T} }\right)/\omega } &\quad 0 &\quad {\left [{ {1-\cos \left ({{\omega T} }\right)} }\right]/\omega } \\ 0 &\quad {\cos \left ({{\omega T} }\right)} &\quad 0 &\quad {\sin \left ({{\omega T} }\right)} \\ 0 &\quad {-\left [{ {1-\cos \left ({{\omega T} }\right)} }\right]/\omega } &\quad 1 &\quad {\sin \left ({{\omega T} }\right)/\omega } \\ 0 &\quad {-\sin \left ({{\omega T} }\right)} &\quad 0 &\quad {\cos \left ({{\omega T} }\right)} \\ \end{array}}} }\right] \tag{18}\end{align*}
\begin{align*} {\rm {\boldsymbol{\Pi }}}=\left [{ {{\begin{array}{cccccccccccccccccccc} {u_{11}} &\quad {u_{12}} &\quad {u_{13}} \\ {u_{21}} &\quad {u_{22}} &\quad {u_{23}} \\ {u_{31}} &\quad {u_{32}} &\quad {u_{33}} \\ \end{array}}} }\right] \tag{19}\end{align*}
D. Likelihood Function
Typically, the likelihood function is constructed using a specific distribution, which allows effects from neighboring targets to be considered. In noncooperative multitarget tracking scenarios, neighboring target tracking is a challenge due to the lack of a priori information about targets. Therefore, it is necessary to consider the effects of neighboring targets in these scenarios. However, the above constraint is relaxed in the LBL localization scenario of interest. Because we have rich prior target information, effects from neighboring targets can be suppressed by coherent processing techniques, such as the MF technique used here. More importantly, the MF outputs corresponding to different waveform signals have different statistical properties (as shown in Fig. 5), signifying that employing a specific probability distribution to design the likelihood function may apply to only one (or several) signal types. For the above two reasons, we define the likelihood function as the product of the MF outputs of multiple buoys conditioned on the particle’s state. With this design approach, a processing gain is achieved by exploiting the fact that the TOAs estimated from different buoys all correspond to the same target position.
MF outputs for two signal types. (a) Linear frequency modulation (LFM) signal. (b) Continuous wave (CW) signal.
The available information on a target’s TOA is embedded in the MF output, as depicted in Fig. 2. Thus, the particle TOA is employed as a link to establish the relationship between the particle states and the MF outputs. For the
\begin{equation*} \tau _{m,k}^{n,j} =\sqrt {\left ({{x_{k,j}^{n} -x_{m,k}} }\right)^{2}+\left ({{y_{k,j}^{n} -y_{m,k}} }\right)^{2}} /\hat {c} \tag{20}\end{equation*}
where \begin{equation*} i_{m,k}^{n,j} =\text {floor}\left ({{\tau _{m,k}^{n,j} \times f_{s}} }\right)+1 \tag{21}\end{equation*}
\begin{equation*} p\left ({\boldsymbol{ z_{k} \vert {\mathbf{x}}_{k,j}^{n}} }\right)=\prod _{m=1}^{m=M} {z_{k,m}^{n,j} =} \prod _{m=1}^{m=M} \boldsymbol{ z_{k} \left [{ {m,i_{m,k}^{n,j}} }\right]}. \tag{22}\end{equation*}
Since the TOAs estimated from different buoys all correspond to the same target position, the likelihood of the
\begin{equation*} p\left ({\boldsymbol{ z_{k} \vert {\mathbf{x}}_{k,j}} }\right)=\prod _{m=1}^{m=M} {a_{k,m}^{j}} =\prod _{m=1}^{m=M} {\left ({{s_{k,m}^{j} +n_{k,m}} }\right)}. \tag{23}\end{equation*}
In (23), \begin{equation*} p\left ({\boldsymbol{ z_{k} \vert {\mathbf{n}}_{k}} }\right)=\prod _{m=1}^{m=M} {n_{k,m}} \tag{24}\end{equation*}
\begin{align*} \text {SNR}_{l} &=20\lg \left [{ {\frac {p\left ({\boldsymbol{ z_{k} \vert {\mathbf{x}}_{k,j}} }\right)-p\left ({\boldsymbol{ z_{k} \vert {\mathbf{n}}_{k} } }\right)}{p\left ({\boldsymbol{ z_{k} \vert {\mathbf{n}}_{k}} }\right)}} }\right] \\ &=20\lg \left [{ {\prod _{m=1}^{m=M} {\left ({{\frac {s_{k,m}}{n_{k,m}}+1} }\right)} -1} }\right]. \tag{25}\end{align*}
\begin{equation*} \text {SNR}_{l} =20\lg \left ({{\frac {s_{k,1}}{n_{k,1}}+1-1} }\right)=20\lg \left ({{\frac {s_{k,1}}{n_{k,1}}} }\right)=\text {SNR}_{MF}. \tag{26}\end{equation*}
\begin{align*} G_{l} &=\text {SNR}_{l} -\text {SNR}_{MF} \\ &=20\lg \left \{{{\left [{ {\prod _{m=1}^{m=M} {\left ({{\frac {s_{k,m}}{n_{k,m}}+1} }\right)} -1} }\right]/\frac {s_{k,1} }{n_{k,1}}} }\right \}. \tag{27}\end{align*}
We further mention that the above discussions are based on the assumption that the MF outputs corresponding to the interference of different buoys cannot converge in one region, which is valid when the SNR is sufficiently high. However, as the SNR decreases, the interference effects become remarkable, and the above assumption is no longer reasonable. Therefore, when the SNR is poor, the practical processing gain may be lower than the processing gain given by (27).
The designed likelihood function can obtain a certain processing gain, which becomes higher with an increasing number of buoys. To further explain this point, the data at 400 s in Section IV-B Real Data 1 are utilized to exhibit the likelihood distribution under different numbers of buoys. A detailed description of these data is provided in Section IV-B. Since the MF output provides TOA information, the multipath effects are reflected in a series of highlighted concentric circles centered on the buoy, as shown in Fig. 6(a). Since the TOAs estimated from different buoys all correspond to the same target position, as the number of buoys increases, the MF output corresponding to the direct sound consistently converges at the target position, resulting in a persistent increase in the processing gain. In other words, the interference is suppressed. In the case of four buoys, the region with high likelihood only exists near the real
Likelihood distribution under the different numbers of buoys. The partial enlargement near the real location is in the upper left corner. (a) Number of buoys is 1, and the corresponding MF output is shown in Fig. 2. (b) Number of buoys is 2. (c) Number of buoys is 3. (d) Number of buoys is 4. (e) Schematic of the 3-D log-likelihood distribution.
E. Resampling
The PF suffers from particle degeneracy, and the existing solution is resampling [26]. However, if resampling is done too frequently, it will lead to the loss of sample diversity [27]. Furthermore, performing resampling for the entire particles
\begin{equation*} \hat {N}_{k,j}^{\text {eff}} =\frac {1}{\sum _{n=1}^{n_{p}} {\left ({{\varpi _{k,j}^{n}} }\right)^{2}}},\quad \varpi _{k,j}^{n} =\frac {p\left ({\boldsymbol{ z_{k} \vert {\mathbf{x}}_{k,j}^{n}} }\right)}{\sum _{i=1}^{n_{p}} {p\left ({\boldsymbol{ z_{k} \vert {\mathbf{x}}_{k,j}^{i}} }\right)}}. \tag{28}\end{equation*}
If the number of effective target particles is fewer than F. Precision Analysis
This Section utilizes MC experiments with 500 runs to analyze the proposed TOA-AMTBD algorithm’s precision and test its robustness for different input errors. When testing one of the input errors, the other input errors are consistent with those shown in Table I. Other parameters are also given in Table I. We choose the root-mean-square error (RMSE) as the evaluation metric. Precision analysis results indicate that the proposed algorithm achieves high accuracy, with a localization error of less than 1.5 m for most areas inside the array, as shown in Fig. 7(a). The robustness test results reveal that the proposed algorithm is more sensitive to the TOA error relative to the buoy position error and the sound speed error, as shown in Fig. 7(b). However, the proposed TOA-AMTBD algorithm can guarantee a localization error of fewer than 17 m under such severe conditions.
Results of precision analysis. (a) Precision distribution for different target locations. (b) Robustness for different types of input errors.
G. Complexity Analysis
This Section analyzes the computational complexity theoretically by giving the number of multiplications used by the developed PF-TBD algorithms. The numbers of multiplications required to generate Gaussian distributed random numbers, compute the likelihood, resample the particles and estimate the target state are denoted by
\begin{align*} C_{\text {TBD}} \left ({{n_{x},n_{p},\eta } }\right)\!=\!\left ({{n_{x}^{2}\!+\!n_{x} ^{2}/2+c_{1} +c_{2} +n_{x} c_{3} +n_{x} c_{4}} }\right)n_{p} \eta. \tag{29}\end{align*}
The PF-TBD algorithms optimized by auxiliary variables require additional operations on the auxiliary variables, including sampling, likelihood calculation, and resampling. Therefore, the number of multiplications required by the PF-TBD algorithms with auxiliary variable optimization can be expressed as
\begin{align*} C_{\text {ATBD}} \left ({{n_{x},n_{p},\eta } }\right)\!=\!\left ({{2n_{x}^{2}\!+\!n_{x} ^{2}\!+\!2c_{1} \!+\!2c_{2} \!+\!2n_{x} c_{3} \!+\!n_{x} c_{4}} }\right)n_{p} \eta. \tag{30}\end{align*}
The PF-TBD algorithms optimized by auxiliary variables require approximately twice as many multiplications as the PF-TBD algorithms without auxiliary variable optimization. Therefore, the particle number of the TOA-ATBD algorithm and the TOA-AMTBD algorithm is made to be half that of the TOA-TBD algorithm and the TOA-MTBD algorithm in the subsequent experiments.
Performance Evaluation
In this section, the performance of the proposed TOA-AMTBD algorithm is evaluated. The traditional TOA-based localization algorithm, referred to as the TOA algorithm, is chosen as the comparison algorithm. The TOAs (inputs of the TOA algorithm) are selected by the algorithm proposed in [12]. Additionally, the localization results of the TOA algorithm are filtered using the Kalman filtering method, referred to as the TOA + KF algorithm, for simplicity. In Section IV-A, the performance is examined with simulated data, and in Sections IV-B and IV-C, the performance is examined with real data. The results indicate that the proposed algorithms perform well.
A. Simulated Data
Section IV-A consists of utilizing a series of MC simulation experiments to examine the influence of SNR on the performance of the proposed algorithms. Each MC simulation experiment is based on 500 runs of simulated data. To examine the influence degree of different SNRs on the performance of the proposed algorithms, different intensity band-limited white noise is utilized to superimpose a direct sound signal to generate simulated data.
An LFM signal is selected as the transmitted signal in the simulated experiment. The parameters of the transmitted signal are as follows: the center frequency is
\begin{align*} {\rm {\boldsymbol{\Pi }}}=\left [{ {{\begin{array}{cccccccccccccccccccc} {0.8} &\quad {0.1} &\quad {0.1} \\ {0.1} &\quad {0.8} &\quad {0.1} \\ {0.1} &\quad {0.1} &\quad {0.8} \\ \end{array}}} }\right]. \tag{31}\end{align*}
Geometrical configuration of the simulated experiment.
Because ground truths that correspond to the simulated data are known, performance evaluation can be readily implemented by state estimation errors. The results after 15 s are used to quantify the error and ensure convergence of the proposed algorithms. The RMSEs during the entire movement and turning movement are given in Fig. 9(a) and (b), respectively, indicating that the proposed PF-TBD algorithms demonstrate competitive performance with the TOA and TOA + KF algorithms in the case of high SNR. The KF improves localization precision by using historical information and filtering outliers. However, it suffers from the model mismatch, resulting in the TOA + KF algorithm performing inferior to the TOA algorithm in high SNR conditions. For low SNR conditions, the proposed PF-TBD algorithms outperform the TOA and TOA + KF algorithms. This is attributed to the following reasons. The TOA algorithm solves the analytic solution of the target position by mathematical equations. Its performance depends on the accuracy of the TOA. Under a high SNR [see Fig. 10(a)], the TOA estimation is accurate, and the direct sound can be easily selected; together, these features lead to the excellent performance of the TOA algorithm. However, under a low SNR [see Fig. 10(b)], the above two features (especially the second) are not achieved, which leads to many outliers in the localization results. In contrast to the TOA algorithm, the proposed PF-TBD algorithms circumvent DSS and obtain the processing gain from multiple buoys. Therefore, they perform better under low SNR conditions. Among them, the TOA-AMTBD algorithm gathers the best features of the TOA-ATBD algorithm and the TOA-MTBD algorithm and, therefore, can outperform them.
RMSEs at different SNRs. (a) RMSEs during the entire movement. (b) RMSEs during the turning movement. (c) RMSEs during the linear movement. The partial enlargement is in the top-right corner.
MF outputs under two SNRs. (a) SNR is −3 dB. (b) SNR is −11 dB.
Sampling with multiple motion models may be a burden when the target performs linear motion. Therefore, the performance of the TOA-MTBD algorithm during linear movement is inferior to that of the TOA-TBD algorithm, as shown in Fig. 9(c). The TOA-AMTBD algorithm uses auxiliary variables to eliminate particles sampled by invalid models, and its performance is thus inferior to that of the TOA-ATBD algorithm but better than that of the TOA-TBD algorithm. This indicates that although the use of auxiliary variables cannot completely eliminate the performance loss caused by invalid models, it can effectively alleviate their effects. We also mention that the particles sampled by the invalid model may present a high likelihood under the effect of clutter and thus may be retained in resampling. As the SNR increases, the clutter density decreases, and this effect of the invalid model is weakened. We attribute this phenomenon to why the performance difference between the TOA-AMTBD (TOA-MTBD) algorithm and the TOA-ATBD (TOA-TBD) algorithm decreases as the SNR increases, as seen in Fig. 9(c).
Figs. 11(a)–(d) and 12(a)–(d) show the distributions of the particle position and velocity as a function of time under an SNR of −11 dB. As depicted in Figs. 11 and 12, the states of the particles gradually converge to the real state over time. Similarly, the particle TOAs converge, as seen in Fig. 13(a)–(d). The particle TOAs are distributed over a very minimal time range, which indicates that the performance of the proposed algorithms depends only on the MF output within this range. In other words, the proposed algorithms avoid interference outside this range.
Distribution of particle positions as a function of time under the SNR is −11 dB. Distribution of particle positions at (a) 1 s, (b) 5 s, (c) 15 s, and (d) 40 s.
Distribution of particle velocities as a function of time under an SNR of −11 dB. Distribution of particle velocities at (a) 1 s, (b) 5 s, (c) 15 s, and (d) 40 s.
Distribution of particle TOAs in the MF outputs at 40 s. (a) Buoy #1. (b) Buoy #2. (c) Buoy #3. (d) Buoy #4.
B. Real Data 1
In Section IV-B, a set of data collected at Qiandao Lake is utilized to validate the effectiveness of the proposed algorithms. In this experiment, the LBL localization system consists of four buoys and a sound source towed by rope in the stern of a ship traveling at approximately 1 m/s. The position of the ship is provided by GPS. Fig. 14 shows the trajectory of the ship and the position of the buoys. The sound speed was 1478 m/s, measured by the hydrological instrument. The initialization was the same as that of the simulated data.
Geometrical configuration of Real Data 1.
A double-pulse LFM signal is selected as the transmitted signal, in which the first pulse is utilized for localization, and the second pulse is utilized for sounding. The parameters of the transmitted signal are the same as those of the simulated data. There is a certain functional relation between the time interval of the two pulses and the depth measured by the pressure sensor. Therefore, the target depth can be obtained by measuring the time interval. It is worth noting that sound is not considered in this article. If desired, a state denoting the time interval of two pulses can be added to the particle states.
Fig. 15 shows the distributions of particle TOAs in the MF outputs at 908 s. Because of the superposition of the reflected sounds, there is a series of strong correlation peaks in the MF outputs of Buoy #1 and Buoy #3. The correlation peaks of some reflected sounds are higher than those of the direct sound, especially at Buoy #1. Even under this harsh condition, the TOAs of the particles were evenly distributed around the correlation peak of the direct sound. We attribute this result to two reasons, as discussed earlier.
The likelihood function considers the fact that the TOAs estimated from different buoys all correspond to the same target position. Therefore, only the vicinity of the GPS measurement has a high likelihood, as shown in Fig. 16.
The TOAs of the particles gradually converge to a small range over time. Interference outside this range has no significant effect on the performance.
Distributions of particle TOAs in the MF outputs at 908 s. (a) Buoy #1. (b) Buoy #2. (c) Buoy #3. (d) Buoy #4.
Likelihood distribution at 908 s. A partial enlargement near the real location is shown in the top left corner.
The localization results shown in Fig. 17 indicate that the proposed algorithms can track the target continuously and accurately, even under severe multipath effects. By comparison, there are many outliers in the TOA and TOA + KF algorithm results due to the selection errors of direct sound. In addition, the TOA and TOA + KF algorithms utilize the TOA information of the three buoys for localization. For the scenario with four buoys, these four combinations correspond to four sets of results. Choosing different combinations may decrease the continuity of localization results because of the difference in the sound speed of the transmission paths between the target and the different buoys.
Localization results for Real Data 1.
C. Real Data 2
To further verify the tracking accuracy of the proposed algorithms, we changed the installation mode of the sound source from dragging on the stern to a rigid connection to the GPS, as shown in Fig. 18. The experimental conditions are consistent with Real Data 1, except that a double-pulse CW signal is selected as the transmitted signal. The parameters of the transmitted signal are as follows: the frequency is
Installation of the sound source and GPS.
Geometrical configuration of Real Data 2.
Distributions of particle TOAs in the MF outputs at 912 s. (a) Buoy #1. (b) Buoy #2. (c) Buoy #3. (d) Buoy #4.
The localization results are shown in Fig. 21(a). Under such harsh conditions, the performance of the TOA and TOA + KF algorithms deteriorates severely, with many outliers in the results. However, the results of the proposed PF-TBD algorithms are without any outliers. Since the sound source is rigidly connected to the GPS and fixed to the ship, a precision evaluation can be implemented based on the position estimation error, as depicted in Fig. 21(b). The precision of the proposed PF-TBD algorithms is higher than that of the TOA and TOA + KF algorithms, and most errors are less than 1.5 m in magnitude. Along with presenting the statistical results, it is worth emphasizing that the GPS accuracy utilized in this experiment is 0.10 m. Unexpectedly, the mean error of position estimation for the TOA-AMTBD algorithm is 0.69 m, for the TOA-ATBD and TOA-MTBD algorithms is 0.70 m, and for the TOA-TBD algorithm is 0.71 m. In contrast, the mean error of position estimation for the TOA algorithm and the TOA + KF algorithm is 3.1 m. Compared with the TOA-TBD algorithm, the TOA-ATBD and TOA-MTBD algorithms optimize particle sampling to further improve localization precision. The TOA-AMTBD algorithm gathers the best properties of the TOA-ATBD and TOA-MTBD algorithms and thus can surpass both. The results of both the simulated experiment and the real experiment are consistent with the theoretical analysis and prove the effectiveness of the TOA-AMTBD algorithm.
Results for Real Data 2. (a) Localization results. (b) Position estimation error.
Conclusion
In this article, LBL localization with low SNR and severe multipath effects is addressed, and improving the localization precision and trajectory continuity is the focus. The TOA-TBD algorithm is developed to avoid the challenges of DSS. The TOA-TBD algorithm utilizes the previously neglected fact that the TOAs estimated from different buoys all correspond to the same target position to achieve a processing gain. Moreover, the TOA-ATBD algorithm and the TOA-MTBD algorithm are developed by using auxiliary variables and multiple models to optimize particle sampling. To combine the strengths of both algorithms above, we have also presented the TOA-AMTBD algorithm, which exploits auxiliary variables, multiple models, and the neglected fact to achieve favorable performance in LBL localization scenarios with low SNRs and severe multipath effects. Using simulations and real experiments, we have verified the effectiveness of the proposed PF-TBD algorithms. These results show that the proposed PF-TBD algorithms outperform the traditional TOA algorithm under harsh conditions.
Although these algorithms provide an alternative approach for solving the LBL localization problem, they require the sound source to be equipped with a synchronous device, which can be a limitation in certain applications, such as searching for a sunken black box in the ocean. In our future work, we will concentrate on developing the proposed algorithms for such an asynchronous use case.