**Contents**


#### **Shenghua Zhou, Linhai Wang, Ran Liu, Yidi Chen, Xiaojun Peng, Xiaoyang Xie, et al.** Signal Source Localization with Long-Term Observations in Distributed Angle-Only Sensors Reprinted from: *Sensors* **2022**, *22*, 9655, doi:10.3390/s22249655 .................... **233**

## **About the Editors**

## **Mahendra Mallick**

Mahendra Mallick received an M.S. degree in computer science from Johns Hopkins University, Baltimore, MD, USA, in 1987, and a Ph.D. degree in quantum solid-state theory from the State University of New York, Albany, NY, USA, in 1981. He is a co-editor and a co-author of the book entitled Integrated Tracking, Classification, and Sensor Management: Theory and Applications (New York, NY, Wiley/IEEE, 2012). His research interests include nonlinear filtering, multisensor multitarget tracking, multiple hypothesis tracking, random-finite-set-based multitarget tracking, satellite orbit and attitude determination, over the horizon radar tracking, and distributed fusion.

Dr. Mallick was the Associate Editor-in-Chief of the online journal of the International Society of Information Fusion (ISIF) in 2008–2009. He was the Lead Guest Editor of the special issue on "Multitarget Tracking" in the IEEE Journal of Selected Topics in Signal Processing in June 2013. He is currently an Associate Editor for target tracking and multisensor systems of the IEEE Transactions on Aerospace and Electronic Systems. He is the Lead Guest Editor of the Sensors 2021-2022 Special Issue on Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions. He is a Senior Life Member of IEEE.

## **Ratnasingham Tharmarasa**

Ratnasingham Tharmarasa received the B.Sc. Eng. degree in electronic and telecommunication engineering from University of Moratuwa, Sri Lanka in 2001, and the M.A.Sc and Ph.D. degrees in electrical engineering from McMaster University, Canada in 2003 and 2007, respectively.

Currently he is an assistant professor in the Electrical and Computer Engineering Department at McMaster University, Canada. From 2001 to 2002 he was an instructor in electronic and telecommunication engineering at the University of Moratuwa, Sri Lanka. During 2002-2007 he was a graduate student/research assistant in ECE department at the McMaster University, Canada. In 2008, he has worked as a researcher at DRS Technologies Canada Limited, Ottawa, Canada. From 2008 to 2019 he was a research associate in ECE department at the McMaster University, Canada. He is currently an Associate Editor for Elsevier Signal Processing. He is the Guest Editor of the Sensors 2021-2022 Special Issue on Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions.

His main areas of expertise are target tracking, information fusion, sensor resource managemen<sup>t</sup> and performance prediction with application to surveillance systems, autonomous vehicles and intelligent transportation. He has published more than 60 peer-reviewed journal articles and 65 conference papers in the above areas, in addition to three book chapters. He has also worked on the development of many real-world target tracking, sensor fusion and resource managemen<sup>t</sup> systems.

## **Preface to "Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions"**

Two-dimensional bearing-only filtering (BOF) arises in many real-world tracking problems, including underwater tracking using a passive sonar, aircraft surveillance using a passive radar, robot navigation using a passive sonar, and undersea exploration of natural resources using sonar. Single-sensor BOF is also a challenging nonlinear filtering problem due to poor observability and the nonlinear measurement model. This filtering problem and the associated tracking problem have been studied extensively.

Angle-only filtering (AOF) in 3D is the counterpart of BOF in 2D. Real-world AOF problems include passive ranging using an infrared search and track (IRST) sensor, passive sonar, passive radar in the presence of jamming, ballistic missile and satellite tracking using a telescope, satellite-to-satellite passive tracking, and missile guidance using bearing-only seekers. The number of publications in the AOF and angle-only tracking in 3D is rather limited compared with the corresponding problems in 2D. This Special Issue contains twelve papers, which are grouped into three sections. The first section deals with nonlinear filters, estimators, and localization, and has six papers.

The first paper by Lebon et al. discusses target motion analysis (TMA) using cosine of conical angles acquired by a towed array. It estimates the trajectory of a target moving with constant velocity (CV) at an unknown constant depth. The sound emitted by the target can reach the antenna through the direct path as well as through bounces off the sea bottom and/or off the sea surface. The authors analyze the observability by first identifying the presence of ghost targets and then estimating the target trajectory efficiently.

The second paper by Bu et al. considers the pseudolinear Kalman filter (PLKF) for the bearings-only tracking problem. Current PLKF algorithms use bias compensation to improve the accuracy of the state estimator. The authors present a stable PLKF without bias compensation in the minimum mean square error (MMSE) framework and compare their results with the posterior Cramer–Rao Lower Bound (PCRLB).

The next paper by Li et al. studies the hybrid source localization problem using differential received signal strength (DRSS) and angle of arrival (AOA) measurements. The authors develop a closed-form pseudolinear estimator by incorporating the AOA measurements into a linearized form of DRSS equations. Finally, they propose a selected-hybrid-measurement-weighted instrumental variable (SHM-WIV) estimator that has superior bias reduction and mean-squared error performance.

In the fourth paper, Dogancay studies the optimal sensor placement for target localization using AOA measurements in 2D with a Gaussian prior. Optimal sensor locations are determined analytically for a single AOA sensor using the D- and A-optimality criteria and an approximation of the Bayesian Fisher information matrix (BFIM). Then, these results are extended to a target moving with nearly constant velocity (NCV). For the NCV trajectory of the target, it is observed that the two optimality criteria generate significantly different optimal sensor trajectories.

Luo et al. consider the robust bearings-only source localization in impulsive noise with symmetric distribution based on the Lp-norm minimization criterion. To reduce the bias due to the correlation between system matrices and noise vectors, the authors propose a Total Lp-norm Optimization (TLPO) algorithm by minimizing the errors in all elements of the system matrix and data vector based on the minimum dispersion criterion. They obtain an equivalent form of TLPO and propose two algorithms to solve the TLPO problem by using Iterative Generalized Eigenvalue

Decomposition (IGED) and Generalized Lagrange Multiplier (GLM), respectively.

Oliva et al. investigates the TMA with bearings-only measurements in 2D using an intelligence-aware batch processing algorithm. A constrained maximum likelihood estimation (MLE) is developed by extending the Cramer–Rao lower bound (CRLB) for the MLE problems with ´ inequality constraints. The ownship motion is selected based on the Artificial Potential Fields technique that is typically used by mobile robots to reach a goal while avoiding obstacles. The MLE is performed by evolutionary ant colony optimization software.

Three papers in the next section investigate 3D AOF . The paper by Mitra et al. considers the 3D tracking of a ground target with terrain uncertainty and bias in angle measurements of an airborne sensor. They derive equations for the PCRLB by considering terrain uncertainty and sensor measurement bias in addition to the process and measurement noises. It is shown that the biased PCRLB provides a tighter lower bound when compared with the PCRLB while evaluating the position error. The authors propose an algorithm to select optimal targets of opportunity and optimal platform trajectory to estimate the bias using the biased PCRLB.

The second paper by Mallick et al. studies AOF of a maneuvering target in 3D using bearing and elevation measurements from a passive IRST sensor. The target moves with a nearly constant turn (NCT) in the XY-plane and nearly constant velocity (NCV) along the Z-axis. The NCT motion in the XY-plane cannot be discretized exactly. The continuous-time NCT model is discretized using the first and second-order Taylor approximations and the polar velocity and Cartesian-velocity-based states for the NCT model in a cubature Kalman filter (CKF). Numerical results for realistic scenarios show that the second-order Taylor approximation provides the best accuracy using the polar velocity or Cartesian-velocity-based models.

The third paper by Urooj et al. considers 2D and 3D AOF using the maximum correntropy (MC) Kalman filters, where the measurement noise is non-Gaussian. The authors formulate the UKF and new sigma point Kalman filter (NSKF) using the Gaussian kernel (GK) and Cauchy kernel (CK) in the MC framework. Consequently, they develop the MC-UKF-CK, MC-NSKF-GK and MC-NSKF-CK for the non-Gaussian AOF filtering problems.

Tracking and localization with multiple angle-only sensors are addressed in the third section. Wei et al. consider 2D bearings-only multisensor-multitarget tracking using multidimensional assignment (MDA). They propose a new coarse gating strategy to reduce the computational cost of the MDA algorithm. The two-stage multiple hypothesis tracking framework uses the proposed coarse gating strategy for multisensor-multitarget tracking.

Chen et al. investigate the multitarget localization problem using a sensor network with 3D angle-only sensors. The authors propose a data association technique based on minimum distance and analyze the target localization accuracy as a function of the angle measurement accuracy and platform self-positioning accuracy. Their numerical result shows that the proposed algorithm can achieve a required data association rate and a high positioning accuracy with a low computation cost.

The paper by Zhou et al. considers the position and velocity estimation of a target moving with constant velocity using a passive sensor network with communication links. The sensors move with constant velocity and measure the azimuth and elevation angles. The authors present centralized and distributed fusion algorithms based on least squares (LS) and analyze the communication and computation requirements of each algorithm.

> **Mahendra Mallick and Ratnasingham Tharmarasa** *Editors*

## *Article* **TMA from Cosines of Conical Angles Acquired by a Towed Array**

**Antoine Lebon 1, Annie-Claude Perez 2, Claude Jauffret 2,\* and Dann Laneuville 3**


**Abstract:** This paper deals with the estimation of the trajectory of a target in constant velocity motion at an unknown constant depth, from measurements of conical angles supplied by a linear array. Sound emitted by the target does not necessarily navigate along a direct path toward the antenna, but can bounce off the sea bottom and/or off the surface. Observability is thoroughly analyzed to identify the ghost targets before proposing an efficient way to estimate the trajectory of the target of interest and of the ghost targets when they exist.

**Keywords:** target motion analysis; observability; fisher information matrix; Cramér–Rao lower bound; conical angles; nonlinear estimation

## **1. Introduction**

Bearings-only target motion analysis (BOTMA) is a problem that has been widely studied and various solutions have been proposed in the literature: batch [1–5] or recursive filter (such as extended Kalman filter [6–8], unscented Kalman filter [9], particle filter [10], modified instrumental variable [11–13]), or a mix of recursive and batch methods [14]. Citing all the papers dealing with this topic is now a hard task. Among the abundant literature, most papers share the same assumption: the target is moving in a straight line with a constant speed, while the passive observer is maneuvering adequately in order to ensuretheobservabilityofthetarget[15–17].Thebearingsarethemeasurements.

In this paper, we are concerned with the same problem, except that the available measurements are the cosine of the relative bearings, also called conical angles because the target belongs to the cone of ambiguity whose revolution axis is the line along which the towed array is moving (see [18] p. 39). Implicitly, we consider a target moving in 3D at a constant and unknown depth in near field; in this case, the two more energetic rays are the direct and the reflected paths (bottom or surface). In most cases, the sound bounces off the sea bottom. Therefore, we extend our analysis to surface- and sea bottom-bounced rays.

Indeed, the array detects the cosine of the relative angle of the direction of arrival by means of a suitable spatial filtering method such as beamforming, or more sophisticated techniques (see [19]). In the near field, sound can propagate to the sensor array along the direct path and/or the bottom-reflected path, and/or the surface-reflected path. Most of the time, at most, two rays coming from the same target are detected [18,20].

Unlike Gong [21] and Blanc-Benon [22], who addressed the three-dimensional target motion analysis (TMA) from a sequence of time differences of arrival (TDOA) of a signal traveling by two different paths coupled with a sequence of azimuths, we assume in this paper that the available measurements are the cosines of the conical angles only. In [23], a similar problem was addressed, but observability was not studied. We will consider two situations: the first case is devoted to TMA when sound propagates along a non-direct path at each sampling time. This will be the topic of Section 3: we will conduct observability

**Citation:** Lebon, A.; Perez, A.-C.; Jauffret, C.; Laneuville, D. TMA from Cosines of Conical Angles Acquired by a Towed Array. *Sensors* **2021**, *21*, 4797. https://doi.org/10.3390/ s21144797

Academic Editors: Mahendra Mallick and Ratnasingham Tharmarasa

Received: 18 June 2021 Accepted: 12 July 2021 Published: 14 July 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

analysis and identify all the ghost targets, given a set of noise-free measurements. We will prove that an assumption of the target's depth makes the target's trajectory observable, but not estimable (in the sense that the asymptotic performance given by the Cramér–Rao lower bound—CRLB—of the estimator of the depth is out of the physical constraints, that is, the source is navigating between the surface and the sea bottom).

In the fourth section, we will consider scenarios in which the antenna changes its own route. We will prove that the trajectory of the target is almost certainly observable.

In the fifth section, we will assume that sound will propagate along the direct path and the bottom-reflected path. The two rays will be assumed as being detected. Observability analysis will reveal that only three ghost targets at most exist without maneuvering the antenna. We will check that, in this case, the depth is not "estimable". We will give a palliative, allowing us to propose an estimator which is operationally acceptable, the price being a small bias. Convincing simulations will be given at the end of this section, proving that, even when the duration of the scenario is short, the estimated trajectory is very close to the true one. A conclusion ends the paper.

#### **2. Notation and Problem Formulation**

We consider two underwater vehicles moving at their own constant depth. The first mobile is a surface vessel or a submarine towing a horizontal sensor array, and the second one is the target of interest. Given a Cartesian coordinate system, the acoustic center of the array is located at time *t* at - *xO*(*t*) *yO*(*t*) *zO T*. At the same time, the target is at - *xT*(*t*) *yT*(*t*) *zT T*. The respective horizontal positions of the target of interest and of the center of the array at time *t* are denoted by *PT*(*t*) = - *xT*(*t*) *yT*(*t*) *T* and *PO*(*t*) = - *xO*(*t*) *yO*(*t*) *T*. The sea bottom depth (assumed to be a constant) is denoted as *D*. The source is said to be endfire to the line array if its trajectory is in the same line as the array (which implies that the array and the source are at the same depth, and share the same route). It is broadside to the antenna if it navigates in the vertical plane orthogonal to the line array and passing by the acoustic center of the array. The sensor array detects the line of sight of the target; more precisely, ad hoc array processing (or spatial filtering) delivers at time *t* the cosine of the conical angle *ca*(*t*) given by cos(*ca*(*t*)) = cos(*θ*(*t*) − *h*(*t*)) cos(*φ*(*t*)) - *<sup>m</sup>*(*t*), where *θ*(*t*) and *φ*(*t*) are, respectively, the azimuth (or bearing) and the elevation of the path along which the sound emitted by the source propagates. The angle *h*(*t*) is the heading of the sensor array. Denoting the relative position coordinates of the source with reference to the acoustic center of the array by *xOT*(*t*) = *xT*(*t*) − *xO*(*t*) and *yOT*(*t*) = *yT*(*t*) − *yO*(*t*), we have *θ*(*t*) = *arctan*(*xOT*(*t*), *yOT*(*t*)). Figure 1 displays the different angles and the two actors (the observer reduced to the linear array, and the target).

The ray of the sound (or signal) emitted by the source can be reflected by the bottom and/or the surface or travels in the surface or deep channel. The sound–speed profile makes the paths curve. In this paper, we will consider that the target is in the near field (the distance between the source and the array is less than 20 km), and the bottom depth is in the range 2000–5000 m. Due to the large curvature of the ray (about 80 km), we will approximate the path of the sound as a set of zigzags defined by the reflections on the bottom or on the surface. So, we implicitly use the Snell law widely employed in geometrical optics. An image-source is created whose depth *ζT* will be called "imagedepth". A path is then defined by the triplet (*δ*, *nB*, *nS*), where


**Figure 1.** A typical scenario, viewed from the sky.

Figure 2 illustrates three different paths

**Figure 2.** Three examples of ray paths: the solid line represents the direct path (*δ*, *nB*, *nS*) = (+1, 0, <sup>0</sup>), the dashed-dotted line represents the bottom reflected path (*δ*, *nB*, *nS*) = (+1, 1, <sup>0</sup>), and the dashed line represents the bottom-surface-bottom reflected path (*δ*, *nB*, *nS*) = (+1, 2, <sup>1</sup>).

We have to consider the depth difference between the array and the image-source defined by *ζOT* - *ζT* − *zO* if the ray has been reflected (by the sea bottom or by the surface), or *ζOT* - *zT* − *zO* if the sound wave uses the direct path.

A general expression of *ζOT* based on the triplet (*δ*, *nB*, *nS*) is given by *ζOT*(*<sup>δ</sup>*, *nB*, *nS*) = −2*δnB*(−<sup>1</sup>)*nS*<sup>+</sup>*nBD* − *zO* + (−<sup>1</sup>)*nS*+*nB zT*. Note that, given thepath, thelink between *ζOT*(*<sup>δ</sup>*, *nB*, *nS*) and *zT* is linear: *ζOT*(*<sup>δ</sup>*, *nB*, *nS*) = *azT* + *b*, the constants being a function of the triplet (*δ*, *nB*, *nS*), *D*, and *zO*. Moreover, *ζOT*(*<sup>δ</sup>*, *nB*, *nS*) is null if and only if the antenna and the target are navigating at the same depth (*zT* = *zO*), and sound is traveling in the direct path. In this case, cos(*φ*(*t*)) = 1. For the sakeofsimplicityofthenotations,wewillsimplysubsequentlydenote*ζOT*insteadof*ζOT*(*<sup>δ</sup>*,*nB*,*nS*).

 For the above examples, we have *ζOT*(1, 0, 0) = *zT* − *zO* (direct path), *ζOT*(+1, 1, 0) = 2*D* − (*zT* + *zO*) (bottom-reflected path), and *ζOT*(+1, 2, 1) = 4*D* − (*zT* + *zO*) (bottomsurface-bottom reflected path). Note that *ζOT*(*<sup>δ</sup>*, *nB*, *nS*) can be negative (the imagesource is above the surface). Consequently, the cosine of the elevation is cos(*φ*(*t*)) = √*x*2*OT*(*t*)+*y*2*OT*(*t*)

√*x*2*OT*(*t*)+*y*2*OT*(*t*)+*ζ*<sup>2</sup>*OT*(*<sup>δ</sup>*,*nB*,*nS*).

Figure 3a displays the cone of ambiguity, defined by the set of sources sharing the same cos(*φ*(*t*)). In Figure 3b, we plot a direct ray and a bottom-bounced ray, which allows us to figure out the various angles with which we will work.

**Figure 3.** Cones of ambiguity. (**a**) The cones that the target belongs to, and the one that the image-target belongs to. (**b**) Example of conical angles of the target and of the image-target, for a bottom-reflected ray: *φ<sup>D</sup>* and *φB* are the elevations of the direct path and of the bottom-reflected path, respectively.

We assume that the source is moving in constant velocity (CV) motion during the scenario. Our challenge is to estimate its trajectory, i.e., the state vector defining it, *X* - - *xT*(*t*<sup>∗</sup>) *yT*(*t*<sup>∗</sup>) *zT* .*xT* .*yT T*, for a chosen *t*<sup>∗</sup>, from noisy measurements. We consider two situations:


After the spatial filtering, the antenna supplies a noisy measurement of *m*(*t*) or a noisy measurement of (*<sup>m</sup>*1(*t*), *<sup>m</sup>*2(*t*)). The noisy measurements are regularly acquired at *tk* = (*k* − <sup>1</sup>)<sup>Δ</sup>*t*, *k* ∈ {1, . . . , *<sup>N</sup>*}, for a fixed sampling time Δ*t*.

Before attempting to estimate *X*, we must answer several questions:


For the cases where *X* is observable, we have then to compute the asymptotical performance of an unbiased estimator (given by the CRLB [24]), and the performance of our estimators in terms of bias and the covariance matrix. It is worth noting that using the FIM to prove observability can lead to a wrong conclusion [25]. This why we use an analytic approach.

#### **3. TMA from One Ray**

In this section, we consider the case where the array collects the cosine of a conical angle, the path of the ray being known by the operator. We start by analyzing the observability of the trajectory of the source of interest.

## *3.1. Observability Analysis*

**Theorem 1.** *Let a linear antenna measure the cosine of a conical angle in the direction of a source, both in CV motion. The path of the sound emitted by the source is known, as is also the sea bottom depth.*


Preamble: In the following proof, we choose *t*∗ = 0. Instead of working with the state vector *X* = - *xT*(0) *yT*(0) *zT* .*xT* .*yT T*, we will use the relative state vector of the image source, which is *Y* - - *<sup>x</sup>*0*T*(0) *yOT*(0) *ζOT* .*xOT* .*yOT T*. The reason for this is that we are able to recover *X* from *Y* without ambiguity.

We will prove this theorem in the special case where the heading of the antenna is equal to 0◦, and the value *yOT*(*t*) is positive. This can be easily obtained with an ad hoc rotation of the whole scenario. This will simplify the expression of the measurement, without loss of generality.

**Proof of Theorem 1.** We are seeking the ghost target whose horizontal position at time *t* is - *xG*(*t*) *yG*(*t*) *T*, detected in the same cosine of the conical angle, that is

$$\frac{y\_{\rm OT}(t)}{\sqrt{x\_{\rm OT}^2(t) + y\_{\rm OT}^2(t) + \tilde{\zeta}\_{\rm OT}^2}} = \frac{y\_{\rm OG}(t)}{\sqrt{x\_{\rm OG}^2(t) + y\_{\rm OG}^2(t) + \tilde{\zeta}\_{\rm OG}^2}}, \text{ with } x\_{\rm OG}(t) = x\_{\rm G}(t) - x\_{\rm O}(t), y\_{\rm OG}(t) = \frac{y\_{\rm G}(t)}{\sqrt{x\_{\rm OT}^2(t) + y\_{\rm OF}^2(t) + \tilde{\zeta}\_{\rm OF}^2}}$$

*yG*(*t*) − *yO*(*t*), and *ζOG* is the image-depth of the ghost target. This equality is equivalent to

$$\frac{y\_{OT}^2(t)}{x\_{OT}^2(t) + y\_{OT}^2(t) + \zeta\_{OT}^2} = \frac{y\_{OG}^2(t)}{x\_{OG}^2(t) + y\_{OG}^2(t) + \zeta\_{OG}^2} \tag{1}$$

Note that because the target is moving (as is the ghost target also), the denominators of the left term and of the right term of (1) are two polynomial functions of degree 2.

Case 1: *yOT*(*t*) is a zero function, i.e., ∀*t yOT*(*t*) = 0.

This means the source is broadside to the antenna: *YT* = - *<sup>x</sup>*0*T*(0) 0 *ζOT* .*xOT* 0 *T*. In this case, *yOT*(*t*) = 0, ∀*t* ∈ [0, *<sup>T</sup>*]. Hence, the set of ghost targets is composed of the virtualtargetsbroadsidetotheantenna:*YG*= - *<sup>x</sup>*0*G*(0)0*ζOG*.*xOG*0*T*.

 Case2:*yOT*(*t*)is not azerofunction.

If .*yOT* = 0, then *yOT*(*t*) is a constant. To respect the degrees of the terms of (1), *yOG*(*t*) is a constant too.

If .*yOT* = 0, then there is a root, say *t*, such as *yOT*-*t* = 0, since *yOT*(*t*) is a polynomial function of degree 1. Consequently, *yOG*-*t* = 0, and ∀*t* = *t*, *yOG*(*t*) = 0.

We deduce that, in both cases ( .*yOT* = 0, and .*yOT* = 0), there exists a positive value *β* such that *yOG*(*t*) = *βyOT*(*t*).

$$\begin{array}{l} (1) \Leftrightarrow \left\{ \left[ \mathbf{x}\_{OG}^{2}(t) + \mathbf{y}\_{OG}^{2}(t) + \boldsymbol{\zeta}\_{OG}^{2} \right] - \boldsymbol{\beta}^{2} \left[ \mathbf{x}\_{OT}^{2}(t) + \mathbf{y}\_{OT}^{2}(t) + \boldsymbol{\zeta}\_{OT}^{2} \right] \right\} \boldsymbol{y}\_{OT}^{2}(t) = 0\\ \Leftrightarrow \mathbf{x}\_{OG}^{2}(t) + \boldsymbol{\zeta}\_{OG}^{2} = \boldsymbol{\beta}^{2} \left[ \mathbf{x}\_{OT}^{2}(t) + \boldsymbol{\zeta}\_{OT}^{2} \right] \end{array} \tag{2}$$

The quantity *<sup>x</sup>*2*OT*(*t*) + *ζ*<sup>2</sup>*OT* can be equal to zero at any time, or at one time or never. Subcase 1: ∀*t*, *<sup>x</sup>*2*OT*(*t*) + *ζ*<sup>2</sup>*OT* = 0.

Then, *xOT*(*t*) = 0, ∀*t* and *ζOT* = 0. Note that this case is the one when the target is traveling in the endfire to the array and at the same depth as the antenna and the path is the direct one. For the same reason, *xOG*(*t*) = 0, ∀*t* and *ζOG* = 0. The set of ghost targets is hence composed of virtual targets traveling in the endfire to the array and at the same depth as the antenna.

Subcase 2: ∃˘*t* such that *<sup>x</sup>*2*OT*-˘*t* + *ζ*<sup>2</sup>*OT* = 0. Wededucefrom(2) that

$$\mathbf{x}\_{\rm OG}^2(0) = \boldsymbol{\beta}^2 \mathbf{x}\_{\rm OT}^2(0) + \boldsymbol{\beta}^2 \boldsymbol{\zeta}\_{\rm OT}^2 - \boldsymbol{\zeta}\_{\rm OG}^2 \tag{3}$$

$$\mathbf{x}\_{\rm OG}(0)\dot{\mathbf{x}}\_{\rm OG} = \boldsymbol{\beta}^2 \mathbf{x}\_{\rm OT}(0)\dot{\mathbf{x}}\_{\rm OT} \tag{4}$$

$$
\dot{\mathbf{x}}\_{OG}^2 = \beta^2 \dot{\mathbf{x}}\_{OT}^2 \tag{5}
$$

If .*xOT* = 0, then

tive

$$\begin{aligned} \dot{\mathbf{Y}}\_{G} &= \left( \begin{array}{cc} \pm \sqrt{\beta^{2} \mathbf{x}\_{OT}^{2}(0) + \beta^{2} \zeta\_{OT}^{2} - \zeta\_{OG}^{2}} & \beta y\_{OT}(0) & \zeta\_{OG} & 0 & \beta \dot{y}\_{OT} \end{array} \right)^{T}, \text{for any positive } \pm 1 \text{ constant } \beta \text{ and any positive constant } \zeta\_{OG} \text{ less than } \sqrt{\beta^{2} \mathbf{x}\_{OT}^{2}(0) + \beta^{2} \zeta\_{OT}^{2}}. \text{ Note that,} \\ \dot{\mathbf{y}}\_{\text{new}} &= 0, \text{ the target is nothing less relative to the center of the array (both have the same mean and the other).} \end{aligned}$$

when *yOT* = 0, the target is motionless relative to the center of the array (both have the same velocity); and when .*yOT* = 0, the target has the same heading as the array. If.thentheelementsofanddrawfromthat

 *xOT* = 0, squaring (4), using (5), we (3) *β*<sup>2</sup>*ζ*<sup>2</sup>*OT* = *ζ*<sup>2</sup>*OG*. If *ζOT* = 0, then *ζOG* = 0, and the scalar *β* can take any positive value; else *β* = |*ζOG*| |*ζOT*| . In both cases, the trajectory of a ghost target is defined by the state vector *YG* = - <sup>±</sup>*βxOT*(0) *βyOT*(0) *βζOT* ±*β* .*xOT β* .*yOT T*. 

## **Remark 1.**


0 ≤ *zAs* < *zO, and* [*βMin*, *βMax*] = 0, *zO zT*−*zO . In both cases, the range* [*βMin*, *βMax*] *is too wide to be useful. If the target and the antenna are at the same depth, β can take any positive value.*

#### *3.2. Estimation of the Trajectory*

We run 500 Monte Carlo simulations for a typical scenario described as follows:

The observer starts from -0 0 *T* at the depth *zO* = 200 m. Its speed and heading are, respectively, 5 m/s and 0◦. The initial position of the target is - 5000 7000 *T* and its depth is *zT* = 100 m. Its route is 45◦ and its speed is 4 m/s.


• First, the measurements have been corrupted with an additive Gaussian noise whose standard deviation is *σ* = 1.7 × 10−2.

Then, we choose the least squares estimator, which is identical to the maximum likelihood estimator with these assumptions. Note that, in open literature about TMA, the confidence regions are given by the confidence ellipsoid obtained with the covariance matrix of the estimate. Since the maximum likelihood estimate is asymptotically efficient under nonrestrictive conditions, we use here the Cramér–Rao lower bound to compute such confidence regions.

The result of the simulation is presented in Table 1 and illustrated in Figure 4. Obviously, even if the assumption made on the target's depth makes the state vector observable, it remains inestimable: the hugeness of the diagonal elements of the CRLB does not allow this kind of TMA to be employed. We note in Figure 4 that the cloud of horizontal estimates is hyperbola-shaped. This is because the state vector is "weakly" estimable. The parametric equation of this hyperbola is

$$\begin{cases} \begin{array}{c} x(\omega) = \zeta\_{As} \sinh(\omega) \\ y(\omega) = \frac{\zeta\_{As} \mathrm{w}}{\sqrt{1-\mathrm{w}^{2}}} \cosh(\omega) \end{array}, \text{ with } \zeta\_{As} = 2D - (\mathbbm{z}\_{As} + \mathbbm{z}\_{\bullet}), \text{ and } m = \frac{y\_{\mathrm{OT}}(0)}{\sqrt{\mathfrak{x}\_{\mathrm{OT}}^{2}(0) + \mathfrak{y}\_{\mathrm{OT}}^{2}(0) + \mathfrak{z}\_{\mathrm{OT}}^{2}(0)}} \end{cases}$$

**Table 1.** Performance of the estimator of the reduced state vector when *σ* = 1.7 × <sup>10</sup>−2, in terms of bias, sample standard deviation and the one given by the square root of the diagonal of the CRLB.


**Figure 4.** The cloud of estimated position (in green), a piece of the hyperbola (intersection of the cone of ambiguity and the plane *z* = *zAs*(= 200 <sup>m</sup>), for *σ* = 1.7 × 10−2.

We further reduced the standard deviation to *σ* = 1.7 × 10−<sup>4</sup> in order to appreciate the behavior of the MLE. With this (unrealistic) value, the MLE is efficient, as shown in Table 2 and in Figure 5 (which validates our observability analysis).

**Table 2.** Performance of the estimator of the reduced state vector with *σ* = 1.7 × 10−4.


**Figure 5.** The cloud of estimated position (in green) for *σ* = 1.7 × 10−4. The cloud is no longer hyperbola-shaped. The small black segmen<sup>t</sup> is the 90%-confidence ellipsoid.

Our conclusion is that the state vector is not estimable, even though it is observable with an assumption on the target's depth.

This is why we propose to maneuver the antenna in order to render the state vector observable with no assumption on the target's depth, and to augmen<sup>t</sup> the information about it.

#### **4. TMA with One Ray When the Array Maneuvers**

In this section, the antenna maneuvers, i.e., it changes its own heading. We start by proving that the state vector is observable (without any assumption on the target's depth). Then, we have recourse to perform Monte Carlo simulations to evaluate the performance of the MLE.

## *4.1. Observability Analysis*

**Theorem 2.** *Suppose the antenna's trajectory is composed of two successive legs at constant velocity (however with the same speed). Let the target be in CV motion. The linear array acquires the conical angles of the wave emitted from the target, the path of the ray being known as well as the sea bottom depth. If the target is broadside or endfire to the antenna during a leg, then there is at most a ghost target. Otherwise, there is no ghost target.*

Due to its length, the proof of this theorem is given in the Appendix A.

## *4.2. Estimation*

In this subsection, we present the result of 500 Monte Carlo simulations that are run to illustrate the behavior of the proposed estimators. First, we give the scenario used here.

The center of the array and the initial position of the source are, respectively, at - 0 0 200 *T* and - 5000 7000 100 *<sup>T</sup>*at the very beginning of the scenario. The speed of the array is a constant along the scenario and is equal to 5 m/s. The trajectory of the array is composed of two legs linked by an arc of a circle. The first leg lasts 1 min 40 s, during which the array's heading is 135◦. Then, the array turns to the right with a turn rate equal to 20◦/min to adopt a new heading equal to 270◦. The duration of the maneuver is hence equal to 6 min 44 s. The second leg lasts 5 min, so the total duration of the scenario is 13 min and 20 s. Meanwhile, the target is navigating with a heading equal to 45◦ and a speed of 4 m/s. The bottom depth is *D* = 4000 m.

The state vector we have to estimate is hence *X* = - 5000 7000 100 2.83 2.83 *T* 

.

The array is assumed to measure the cosines of the conical angles of the bottomreflected path given by

$$m(t\_k) = \frac{y\_{OT}(t\_k)}{\sqrt{\mathbf{x}\_{OT}^2(t\_k) + y\_{OT}^2(t\_k) + \left[2D - (z\_T + z\_O)\right]^2}} + \varepsilon\_k$$

Measurements are acquired every Δ*t* = 4 s, with *tk* = (*k* − <sup>1</sup>)<sup>Δ</sup>*t*.

The noise vector *εk* is assumed to be Gaussian, 0-mean and its standard deviation equal to *σ* = 1.7 × 10−2. The vectors *εk* are also assumed to be temporally independent.

Again, we choose the least squares estimator.

#### 4.2.1. Estimation of *X*

The 500 obtained estimates of the initial horizontal position are plotted in Figure 6, together with the trajectory of the target, the 90%-confidence ellipse and the trajectory of the array. Again, the view is from the sky.

**Figure 6.** The cloud of the 500 initial positions estimates and the 90%-confidence ellipse.

The performance of the estimator (bias and standard deviation of each component) is presented in Table 3.


**Table 3.** Performance of the estimator of the plain state vector.

A convenient way to evaluate the behavior of an estimator is to compute the so-called normalized estimation error squared (NEES) [27], defined as *Nl* = -*X*ˆ*l* − *XTF*-*X*ˆ*l* − *<sup>X</sup>*, where *F* is the FIM, and *X* ˆ *l* is the estimate computed at the *l*-th simulation. If *X*ˆ*l* is Gaussiandistributed with *X* as the mathematical expectation and the CRLB as the covariance matrix, then *Nl* is chi-square distributed with d degrees of freedom (*χ*2*d*), where *d* is the dimension of *X* (here 5). From the central limit theorem, the averaged NEES *NS* - 1 *NSim* ∑*NSim l*=1 *Nl* is approximately Gaussian; its mathematical expectation is *d*, and its standard deviation is equal to 2*d NSim* .

From our simulations, we obtain *NS* = 5.34.

In conclusion, the estimator can be declared efficient. However, the minimum standard deviation of the target's depth is not compatible with the physical constraints: with the standard deviation given in Table 3, the target could be up above the sea surface! Therefore, a palliative of this is to impose a depth on the target. Indeed, we saw in Section 3.1 that a supposed depth creates a small bias in estimation of the horizontal position of the target.

#### 4.2.2. Estimation of *X* Reduced When the Depth of the Target Is Fixed

Now, the third component of *X* does not have to be estimated. The new state vector is the denoted as *Xr* = - *xT*(0) *yT*(0) .*xT* .*yT T*. We impose that *zAs* = 200 m (whereas the true depth is still 100 m). Hence, we introduce a bias.

Figure 7 displays the position's estimates in the same manner as Figure 6. The bias is not visible to the naked eye. However, Table 4 reveals this bias, which may be acceptable in a real situation. Even though the averaged NEES (=7.31) is out of its 90% confidence interval, its value remains acceptable.

**Figure 7.** The cloud of the 500 initial positions estimates with the reduced state vector and the 90%-confidence ellipse.


**Table 4.** Performance of the estimator of the reduced state vector.

The main interest of assuming the depth to be known is to economize on the CPU time, and reduce the standard deviation of the remaining components to estimate. We are in the presence of the well-known bias–variance tradeoff.

4.2.3. Estimation of the Reduced State Vector by the Conventional BOTMA

In such a scenario, the conventional BOTMA can be run by neglecting the site effect, so by imposing that cos(*φ*(*t*)) = 1, ∀*t*. The (incorrect) noise-free measurement model is then

$$\cos(a(t)) = \cos(\theta(t) - h(t)).$$

The results are plotted in Figure 8. Obviously, a huge bias appears, leading to an averaged NEES equal to 1960. More precisely, the bias on the components of the reduced state vector is - −3062.8 −2319.9 14.4 15.8 *T*, rendering the BOTMA inoperative. Clearly, the conventional BOTMA cannot be recommended for the near field. This justifies a posteriori the interest in taking the site effect and the nature of the wave ray into account, as previously pointed out in the introduction of [23].

**Figure 8.** The cloud (in green) of the 500 initial positions estimates given by the classic BOTMA together with the 90%-confidence ellipse.

#### **5. TMA from the Direct Path and the Bottom-Reflected Path**

We assume in this section that the sound wave emitted by the target travels on the direct path and the bottom-reflected path.

## *5.1. Observability*

**Theorem 3.** *Let a linear antenna and a source both be in CV motion.*

*The antenna acquires the cosines of the conical angles of the direct path and of the bottomreflected path.*


*is the antenna axis, and the radius is a positive scalar β* = *D*−*zG D*−*zT . The relative ghost target velocity is equal to β times the target's velocity. The initial distance between the ghost image and the center of the antenna is equal to β times the initial distance between the ghost image and the center of the antenna.*

*4. If the route of the antenna and the route of the target are not parallel, then there are three ghost targets whose motion relative to the antenna is POG*(*t*) = *<sup>S</sup>POT*(*t*)*, POG*(*t*) = *βPOT*(*t*)*, and POG*(*t*) = *βSPOT*(*t*)*, where S is the matrix of the axial symmetry around the line of the antenna, and β* - *D*−*zO D*−*zT . If the depth of the antenna is equal to the depth of the source, then there is one single ghost target given by POG*(*t*) = *<sup>S</sup>POT*(*t*)*.*

**Proof of Theorem 3.** With no loss of generality, we will again assume that the axis of the sensor array is pointed toward north and that the target is in the half-space where the second component *y* of any vector is positive. A convenient rotation helps us in this case. So the noise-free measurements at time *t* are *<sup>m</sup>*1(*t*) = √ *yOT*(*t*) *<sup>x</sup>*2*OT*(*t*)+*y*2*OT*(*t*)+*z*2*OT*, and

$$m\_{2}(t) = \frac{y\_{OT}(t)}{\sqrt{\chi\_{OT}^{2}(t) + y\_{OT}^{2}(t) + \left[2D - (z\_{T} + z\_{O})\right]^{2}}}$$

We have to seek a five-dimensional state vector *XG* = - *xG*(0) *yG*(0) *zG* . *xG* . *yG T* defining the trajectory of a ghost target, i.e., producing the same noise-free measurement as *X*, that is *<sup>m</sup>*1(*t*) = √ *yOG*(*t*) *<sup>x</sup>*2*OG*(*t*)+*y*2*OG*(*t*)+*z*2*OG* , and *<sup>m</sup>*2(*t*) = *yOG*(*t*) *<sup>x</sup>*2*OG*(*t*)+*y*2*OG*(*t*)+[2*D*−(*zG*+*zO*)]<sup>2</sup> .

hence satisfying the two following equalities (in time):

$$\frac{y\_{OT}(t)}{\sqrt{\mathbf{x}\_{OT}^2(t) + \mathbf{y}\_{OT}^2(t) + z\_{OT}^2}} = \frac{y\_{OG}(t)}{\sqrt{\mathbf{x}\_{OG}^2(t) + \mathbf{y}\_{OG}^2(t) + z\_{OG}^2}}\tag{6}$$

$$\frac{y\_{OT}(t)}{\sqrt{\mathbf{x}\_{OT}^{2}(t) + y\_{OT}^{2}(t) + \left[2D - (z\_{T} + z\_{O})\right]^{2}}} = \frac{y\_{OG}(t)}{\sqrt{\mathbf{x}\_{OG}^{2}(t) + y\_{OG}^{2}(t) + \left[2D - (z\_{G} + z\_{O})\right]^{2}}} \tag{7}$$

under the constraint that *zG* is in [0, *<sup>D</sup>*].

Case 1: *yOT*(*t*) is a zero function, i.e., ∀*t yOT*(*t*) = 0.

The target is broadside to the antenna, so any ghost targets will be too (see Case 1 in the proof of theorem 1).

Case 2: *yOT*(*t*) is not a zero function.

From Case 2 of the proof of theorem 1, there is a positive scalar *β* such that *yOG*(*t*) = *βyOT*(*t*).

$$\begin{array}{l} (6) \Leftrightarrow \sqrt{\mathbf{x}\_{\text{OG}}^2(t) + \mathbf{y}\_{\text{OG}}^2(t) + \mathbf{z}\_{\text{OG}}^2} = \beta \sqrt{\mathbf{x}\_{\text{OT}}^2(t) + \mathbf{y}\_{\text{OT}}^2(t) + \mathbf{z}\_{\text{OT}}^2} \\ \Leftrightarrow \left[ \mathbf{x}\_{\text{OG}}^2(t) + \mathbf{y}\_{\text{OG}}^2(t) + \mathbf{z}\_{\text{OG}}^2 \right] = \beta^2 \left[ \mathbf{x}\_{\text{OT}}^2(t) + \mathbf{y}\_{\text{OT}}^2(t) + \mathbf{z}\_{\text{OT}}^2 \right] \end{array} \tag{8}$$

$$\mathbf{x}^2(\mathcal{T}) \Leftrightarrow \left[ \mathbf{x}\_{\rm OG}^2(t) + y\_{\rm OG}^2(t) + [2D - (\mathbf{z}\_G + \mathbf{z}\_O)]^2 \right] = \beta^2 \left[ \mathbf{x}\_{\rm OT}^2(t) + y\_{\rm OT}^2(t) + [2D - (\mathbf{z}\_T + \mathbf{z}\_O)]^2 \right] \tag{9}$$

Subtracting (9) from (8), we ge<sup>t</sup> *<sup>z</sup>*2*OG* − [2*D* − (*zG* + *zO*)]<sup>2</sup> = *β*2 *<sup>z</sup>*2*OT* − [2*D* − (*zT* + *zO*)]<sup>2</sup> . Now, we simplify the expressions of these two terms:

$$z\_{OG}^2 - \left[2D - (z\_G + z\_O)\right]^2 = -4(D - z\_G)(D - z\_O)$$

$$z\_{OT}^2 - \left[2D - (z\_T + z\_O)\right]^2 = -4(D - z\_T)(D - z\_O)$$

We deduce from this that

$$\beta = \sqrt{\frac{D - z\_G}{D - z\_T}}\tag{10}$$

Note that *β* = 1 iif *zG* = *zT*.

$$\begin{array}{l} (8) \Leftrightarrow \mathbf{x}\_{\rm OG}^{2}(t) + \mathbf{y}\_{\rm OG}^{2}(t) + z\_{\rm OG}^{2} = \boldsymbol{\beta}^{2} \left[ \mathbf{x}\_{\rm OT}^{2}(t) + \mathbf{y}\_{\rm OT}^{2}(t) + z\_{\rm OT}^{2} \right] \\ \Leftrightarrow \mathbf{x}\_{\rm OG}^{2}(t) - \boldsymbol{\beta}^{2} \mathbf{x}\_{\rm OT}^{2}(t) = \boldsymbol{\beta}^{2} z\_{\rm OT}^{2} - z\_{\rm OG}^{2} \end{array} \tag{11}$$

Since *<sup>x</sup>*2*OG*(*t*) − *β*<sup>2</sup>*x*2*OT*(*t*) is a polynomial function of degree 2, (11) is equivalent to

$$\mathbf{x}\_{\rm OG}^2(0) = \boldsymbol{\beta}^2 \mathbf{x}\_{\rm OT}^2(0) + \boldsymbol{\beta}^2 z\_{\rm OT}^2 - z\_{\rm OG}^2 \tag{12}$$

$$\mathbf{x}\_{\rm OG}(0)\dot{\mathbf{x}}\_{\rm OG} = \beta^2 \mathbf{x}\_{\rm OT}(0)\dot{\mathbf{x}}\_{\rm OT} \tag{13}$$

$$
\dot{\mathbf{x}}\_{OG}^2 = \beta^2 \dot{\mathbf{x}}\_{OT}^2 \tag{14}
$$

First case .*xOT* = 0

Equation (14) implies that .*xOG* = 0.

Consequently, for any *zG* in [0, *<sup>D</sup>*], the vector *XOG* = ± *β*<sup>2</sup>*x*2*OT*(0) + *β*<sup>2</sup>*z*2*OT* − *<sup>z</sup>*2*OG βyOT*(0) *zOG* 0 *β* .*yOT T* (with *β* = *D*−*zG D*−*zT* ) defines the trajectory of a ghost target.

Second case .*xOT* = 0

Using (14), and squaring the terms of (13), we ge<sup>t</sup> *<sup>x</sup>*2*OG*(0) = *β*<sup>2</sup>*x*2*OT*(0).Reporting this in (12), we obtain finally *β*<sup>2</sup>*z*2*OT* = *<sup>z</sup>*2*OG*, i.e.,

$$
\beta^2 = \left(\frac{z\_{OG}}{z\_{OT}}\right)^2\tag{15}
$$

If *zT* = *zO*, then *zG* = *zO*. In this case, *β* = 1, and consequently *yOG*(*t*) = *yOT*(*t*) and *<sup>x</sup>*2*OG*(*t*) = *<sup>x</sup>*2*OT*(*t*) from (11). The source's trajectory is observable up to the axial symmetry around the (Oy)-axis.

Equations (10) and (15) give us *D*−*zG D*−*zT* = *zOG zOT* 2.

The unknown *zG* is hence a root of the following equation of degree 2:

(*zG* − *zO*)<sup>2</sup> − (*zT*−*zO*)<sup>2</sup> *D*−*zT* (*D* − *zG*) = 0 which can be expanded as follows: *z*2*G*+ *zG*−2*zO*+ (*zT*−*zO*)<sup>2</sup> *D*−*zT* − *<sup>D</sup>*(*zT*−*zO*)<sup>2</sup> *D*−*zT*+ *z*2*O*= 0.

 Of course, *zT* is a root of this equation. For this value, *zG* = *zT*, hence *β* = 1.

The second root (*zG* itself) is hence 2*zO* − *zT* − (*zT*−*zO*)<sup>2</sup> *D*−*zT* - *zG*. We can check readily that *zG* − *zO* = *zO* − *zT* − (*zT*−*zO*)<sup>2</sup> *D*−*zT*= (*zO*−*zT*)(*<sup>D</sup>*−*zT*)−(*zT*−*zO*)<sup>2</sup> *D*−*zT*.

 Hence, *zG*−*zO zT*−*zO* = *zO*−*D D*−*zT* (which is negative).We deduce from this that:


Therefore, we have identified three ghost targets:

the first one is defined by *XOG* = - −*xOT*(0) *yOT*(0) *zOT* − .*xOT* .*yOT T*, the second is defined by *XOG* = *βxOT*(0) *βyOT*(0) −*βzOT β* .*xOT* . *βyOT T*, and the third by *XOG* = −*βxOT*(0) *βyOT*(0) −*βzOT* −*β* .*xOT* . *βyOT T*. 

**Remark 2.** *Most of the time, the depth of a submarine vehicle is under the operational constraint: values of zT are in* [0, *zMax*] *and zMax D. For example, zMax* = 400 *m, while D* = 4000 *m. Theproofoftheprevioustheoremmustbe adaptedtothisnewconstraint.*

*First, we use the fact that the function u* → *f*(*u*) - 2*zO* − *u* − (*<sup>u</sup>*−*zO*)<sup>2</sup> *D*−*<sup>u</sup> is an involution, i.e., f*(*f*(*u*)) = *u*.

*Since f*(0) = 2*zO* − *z*20*D , f* 2*zO* − *z*20*D* = 0.

*Now the question is: what are the values of zO for which the following inequality holds:* 2*zO* − *z*20 *D* ≤ *zMax, the greatest value of zO guaranteeing that* 2*zO* − *z*20 *D*−*zT* ≤ *zMax is D* − *D* 1 − *zMax D (which is less than zMax).*

*If zO* > *D* − *D* 1 − *zMax D , then zG* > *zMax. In this case, there is a unique ghost target given by XG* = - −*xT*(0) *yT*(0) *zT* − .*xT* .*yT T*.

*If zO* ≤ *D* − *D* 1 − *zMax D , then zG* ≤ *zMax. In this case, there are three ghost targets:*

*T,*

*one is defined by XG* = - −*xT*(0) *yT*(0) *zT* − .*xT* .*yT T, the second is defined by XG* = - *βxT*(0) *βyT*(0) *f*(*zT*) *β* .*xT β* .*yT andthethirdbyXG*=- −*βxT*(0)*βyT*(0)*f*(*zT*)−*β*.*xTβ*.*T*.

 *yT Notethattheoperationalconstraintallowsustobenefitfromthefollowingrange:*

 *D*−*zMax D* ≤ *β* ≤ *D D*−*zMax . For example, when zMax* = *D*10 *,* 0.9 ≤ *β* ≤ 1.11*. Consequently, the ghost target is very close to the target of interest.*

#### *5.2. Estimation of the Trajectory*

This section is devoted to the estimation of the target's trajectory, or in other words, the estimation of *X* with *t*∗ = 0 (the first time). Before going into detail, we compute the so-called Cramér–Rao lower bound to evaluate the asymptotical performance of any unbiased estimator.

We have considered two typical scenarios. In both, the array is assumed motionless (or, more realistically, all the mobiles are referenced to it) at the depth *zO* = 200 m, and the state vector defining the target's trajectory is given by the state vector *X* = - 5000 7000 100 2.83 2.83 *T*. The standard deviation of the measurement is *σ* = 1.7 × 10−2. The total duration of the scenario is 5 min, and the sampling time is Δ*t* = 4 s; consequently, the number of measurement couples is *N* = 75.

In the first scenario, the bottom depth is *D* = 2000 m, while in the second, *D* = 4000 m. Note that in the first scenario, *β* = 0.89, and in the second one, *β* = 0.97. The ghost target is hence very close to the target of interest.

## 5.2.1. Estimability

As pointed out in Section 1, the state vector *X* is "estimable" if its asymptotical performance given by the CRLB is compatible with the physical constraints. Typically, if the minimum standard deviation defined by the square root of the third diagonal element of the CRLB (hence of the depth) is much larger than the depth, then *X* is declared nonestimable.

## 1. First scenario

For this scenario, the square root of the diagonal of the CRLB *σCRLB* = - 1.16 × 10<sup>6</sup> 1.59 × 10<sup>6</sup> 8.22 × 10<sup>5</sup> 637.9 646.1 *T*.

## 2. Second scenario

With the bottom depth, things are not much better, since *σCRLB* = - 6.59 × 10<sup>5</sup> 8.96 × 10<sup>5</sup> 9.73 × 10<sup>5</sup> 352.6 362.2 *T*.

In both cases, the minimum standard deviations are huge. We can conclude that the state vector is not estimable. Computations of minimum standard deviations were made for various scenarios; in all, the state vector is not estimable.

A palliative of this is to fix the depth of the source at an arbitrary and realistic value, say *zAs*, and compute the CRLB of the reduced state vector *Xr* - - *xT*(0) *yT*(0) .*xT* .*yT T*

when we assume that *zT* = *zAs*. For example, for *zAs* = 300 m, the minimum standard deviations are

> *σCRLB* = - 281.17 319.37 1.78 2.02 *T* for the first scenario, and *σCRLB* = - 130.1 115.3 0.80 0.71 *T* for the sec ond one.

Therefore, we propose to estimate the state vector with this hypothesis (*zAs* = 300 m). In so doing, we introduce a bias. The next subsection gives us the result of the 500 Monte Carlo simulations.

#### 5.2.2. Monte Carlo simulations

The computation of the maximum likelihood estimator (MLE) is made with the Gauss-Newton routine. No numerical issue was encountered.

1. First scenario

The performance of the MLE is summarized in Table 5. We have numerically computed the bias and the empirical standard deviation (given, respectively in the second and third column of the table). We can see that the empirical standard deviation is very close to that given by the CRLB. However, as expected, the MLE is biased (of course, there is no bias if we choose *zAs* = *zT*). In Figure 9, the 90% confidence ellipse is drawn, together with the cloud of the 500 estimates (in pink).



**Figure 9.** The location of the sensor array (in black), the cloud of the 500 estimates and the 90%- confidence ellipse when *D* = 2000 m, *zAs* = 300 m, and *zT* = 100 m. The symmetrical cloud is plotted too.

2. Second scenario: Bottom depth *D* = 4000 m.

Again, the performance is presented in Table 6. The bias of the estimator is similar to the one obtained for the first scenario. Only the empirical standard deviations of - *xT*(0) *yT*(0) *T* are larger than that computed from the CRLB. However, Figure 10 shows us that the cloud of estimates is close to the true value and not spread.


**Table 6.** Performance of the estimator.

What is remarkable is the short duration and still the very good performance (in terms of accuracy) of the result. Numerous simulations (not reported here) were performed; all confirm the correct performance of the MLE. The shortness of the scenario is crucial, because everything that we propose here works properly under the condition that the sea bottom is a plane. During a short scenario, this assumption is likely.

## **6. Conclusions**

In this paper, conical-angle TMA has been addressed, and various multipaths of sound have been taken into account. The sensor is a line array. Observability was analyzed deeply, allowing all the existing ghost targets to be identified. The main results are that, if the array detects one ray (corresponding to one path), the trajectory is not observable: the set of ghost targets is composed of trajectories that are homothetic to the trajectory of the target of interest, and their symmetrical images by the axial symmetry around the line array. If the array detects two rays (corresponding to two different paths), the number of ghost targets is reduced to three (except when the target is endfire or broadside to the antenna). When the antenna maneuvers, the target's trajectory is observable (apart from the special scenario where there is one single ghost target). Even for "observable" scenarios, the depth of the target is not estimable (its asymptotical standard deviation is huge). In these cases, we give a non-restrictive palliative that allows us to provide estimates close to the truth.

In the future, in this context, many problems remain to be faced: identification of the paths, maneuvering targets, and fusion of data collected by other sensors, as in [28]. The problem of seeking a "good" maneuver of the observer, as it was solved in a 2D environment [29–31], will be addressed in the future. Some of these problems are already under investigation.

**Author Contributions:** Conceptualization, A.L. and A.-C.P.; methodology, C.J.; software, A.L.; validation, A.L., A.-C.P., C.J. and D.L.; formal analysis, A.L.; investigation, A.L.; resources, D.L.; data curation, A.L.; writing—original draft preparation, C.J.; writing—review and editing, A.L. and C.J.; visualization, A.-C.P.; supervision, A.-C.P.; project administration, not applicable; funding acquisition, not applicable. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

 *V*2

## **Appendix A**

**Proof of Theorem 2.** The proof is made when the first leg is towards North (as previously). =

 *v* sin(*α*)

*v* cos(*α*) , *S*1

=

 −1 0 0 1 ,

$$\begin{array}{rcl} \text{Hence,} & V\_1 &=& \begin{bmatrix} 0 \\ v \end{bmatrix} / \\ \text{and } \mathbf{S}\_2 = \begin{bmatrix} -\cos(2\alpha) & \sin(2\alpha) \\ \sin(2\alpha) & \cos(2\alpha) \end{bmatrix} \end{array}$$

.Moreover, we assume *α* = *k<sup>π</sup>*.

From Theorem 1, we have to consider the four following cases for each leg:


Note that if the target is in case (1) during the first leg, then in case (2) during the second one (provided that this situation is possible), the conclusion about observability will be the same as if the target is in case (2) during the first leg, then in case (1) during the second leg. To be convinced of this, we just have to reverse the time in the equation. This remark allows us to shorten the proof.

Case 1: the target is broadside to the antenna during the first leg.

Hence, *PT*(0) = *xT*(0) 0 , and *VOT* = *cT*0 during the first leg, which implies *POT*(*t*) = *xT*(0) + *tcT* 0 for *t* ≤ *τ*. The ghost targets are also in the broadside, hence *POG*(*t*) = *xG*(0) + *tcG* 0 and *VOG* = *cG*0 for *t* ≤ *τ*.

Can the target be endfire to the antenna? If so, the target has the same heading as the antenna during the second leg or, in other words, *VT* − *V*2 = *λV*2, and *POT*(*t*), which is equal to *POT*(*t*) = *POT*(*τ*) + (*t* − *τ*)(*VT* − *<sup>V</sup>*2), is collinear with *V*2, whenever *t* ≥ *τ*. The first condition cannot be satisfied since *VT* = *cTv* , and *V*2 = ±*v*0 . There is no ghost.

We skip the case where the target is in case (3) during the second leg. This will be treated later. Therefore, we now have to consider the other cases during the second leg. There are two possibilities for the ghost targets: those whose trajectories are defined by (i) *POG*(*t*) = *βPOT*(*t*), and those whose trajectories are given by (ii) *POG*(*t*) = *βS*2*POT*(*t*), both for *t* ≥ *τ*.

The derivative of (i) is *VG* − *V*2 = *βVT* − *βV*2, hence *VG* = *βVT* + (1 − *β*)*<sup>V</sup>*2.

$$\Leftrightarrow \begin{bmatrix} \mathfrak{c}\_G \\ v \end{bmatrix} = \beta \begin{bmatrix} \mathfrak{c}\_T \\ v \end{bmatrix} + (1 - \beta) v \begin{bmatrix} \sin(\alpha) \\ \cos(\alpha) \end{bmatrix} \prime$$

which implies that (1 − *β*) cos(*α*) = 1 − *β*. Since cos(*α*) = 1, *β* = 1. There is no ghost given by (i).

The derivative of (ii) is *VG* − *V*2 = *βS*2*VT* − *βV*2, hence *VG* = *βS*2*VT* + (1 − *β*)*<sup>V</sup>*2. cos(<sup>2</sup>*α*)+*v*sin(<sup>2</sup>*α*) sin(*α*)

$$\begin{aligned} \Leftrightarrow \begin{bmatrix} \mathcal{L}\_G \\ \upsilon \end{bmatrix} &= \beta \begin{bmatrix} -c\tau \cos(2a) + \upsilon \sin(2a) \\ c\_T \sin(2a) + \upsilon \cos(2a) \end{bmatrix} + (1 - \beta)\upsilon \begin{bmatrix} \sin(a) \\ \cos(a) \end{bmatrix}, \\\ two \text{ dodues that } \ell &= \dots \end{aligned}$$

We deduce that *β* = *v cT* sin(<sup>2</sup>*α*)+*v* cos(<sup>2</sup>*α*)−*<sup>v</sup>* cos(*α*).

One ghost exists if *cT* sin(<sup>2</sup>*α*) + *v* cos(<sup>2</sup>*α*) − *v* cos(*α*) is a positive quantity. If so, we then compute *cG*. There is one ghost at most.

Case 2: the target is endfire to the antenna during the first leg. Hence, *PT*(0) = 0 *yT*(0) , and *VOT* = 0*cT* , which implies that *POT*(*t*) = 0 *yT*(0)+*tcT* for *t* ≤ *τ*. During this first leg, the ghost targets are also

 endfire to the antenna, so *POG*(*t*) = 0 *yG*(0) + *tcG* for *t* ≤ *τ*, and *VG* − *V*1 = 0*cG* .

Again, we skip the case where the target is in case (3) during the second leg. This will be treated later. So, we now have to consider the other cases during the second leg. There are two possibilities for the ghost targets: those whose trajectories are defined by (i) *POG*(*t*) = *βPOT*(*t*) and those whose trajectories are given by (ii) *POG*(*t*) = *βS*2*POT*(*t*), both for *t* ≥ *τ*.

The derivative of (i) is *VG* − *V*2 = *βVT* − *βV*2, hence *VG* = *βVT* + (1 − *β*)*<sup>V</sup>*2. ⇔ 0 *cG* + *v* = *β* 0 *cT* + *v* + (1 − *β*)*v* sin(*α*) cos(*α*) . We deduce that *β* = 1. There is no ghost.

Now, differentiating (ii) gives us *VG* − *V*2 = *βS*2*VT* − *βV*2, hence *VG* = *βS*2*VT* + (1 − *β*)*<sup>V</sup>*2.

$$\begin{aligned} \Leftrightarrow \begin{bmatrix} 0\\ c\_G + v \end{bmatrix} &= \beta (c\_T + v) \begin{bmatrix} \sin(2\alpha) \\ \cos(2\alpha) \end{bmatrix} + (1 - \beta) v \begin{bmatrix} \sin(\alpha) \\ \cos(\alpha) \end{bmatrix}, \\\Rightarrow \beta (c\_T + v) \sin(2\alpha) + (1 - \beta) v \sin(\alpha) &= 0. \end{aligned}$$

We deduce that *β* = −*v* sin(*α*) *cT* sin(<sup>2</sup>*α*)+*v* sin(<sup>2</sup>*α*)−*<sup>v</sup>* sin(*α*). One ghost exists if *cT* sin(<sup>2</sup>*α*) + *v*sin(<sup>2</sup>*α*) − *v* sin(*α*) is a negative quantity. If so, we then compute *cG*. There is one ghost at most.

Case 3: the target has the same heading as the array (but is not endfire to it)

As in case (2), *VOT* = 0*cT* , but here, the first component of *POT*(*t*) is not zero. 

Hence, *VT* = 0 *cT* + *v* , and the target cannot be endfire to the antenna during the second leg.Inthiscase,*VOG*= 0,hence*VG*= 0 .

*βcT βcT* + *v* Can the target be broadside to the antenna? The answer is positive if *V*2⊥*VT* and *VOT* is collinear to *POT*(*t*), when *t* ≥ *τ*. The first condition implies that *V*2 = ±*v*0 . Since *POT*(*t*) = *POT*(*τ*) + (*t* − *τ*)(*VT* − *<sup>V</sup>*2), the second condition is satisfied if *POT*(*τ*) is collinear to *VT* − *V*2. This is not the case when the first component of *POT*(*τ*) is zero, while the first component of *VT* − *V*2 is ±*v*.

So, we now have to consider the other cases during the second leg. There are two possibilities for the ghost targets: those whose trajectories are defined by (i) *POG*(*t*) = *βPOT*(*t*) and those whose trajectories are given by (ii) *POG*(*t*) = *βS*2*POT*(*t*), both for *t* ≥ *τ*.

The derivative of (i) is *VG* − *V*2 = *βVT* − *βV*2, hence *VG* = *βVT* + (1 − *β*)*<sup>V</sup>*2 or, in other words,

 0 *βcT* + *v* = 0 *β*(*cT* + *v*) + (1 − *β*)*v* sin(*α*) cos(*α*) . We conclude that *β* = 1, i.e., thereisnoghost.

If *POG*(*t*) = *βS*2*POT*(*t*), then *VG* = *βS*2*VT* + (1 − *β*)*<sup>V</sup>*2 0 *βcT* + *v* = *β*(*cT* + *v*) sin(<sup>2</sup>*α*) cos(<sup>2</sup>*α*) + (1 − *β*)*v* sin(*α*) cos(*α*) .

This implies that *α* = 0, which must be rejected by assumption. There is no ghost. The other cases:

In the other cases, the motion of ghost targets is defined during the first leg by when *t* ≤ *τ*,

$$P\_{\rm OG}(t) = \beta\_1 P\_{\rm OT}(t) \tag{A1}$$

$$\text{for } P\_{\text{OG}}(t) = \gamma\_1 \mathbf{S}\_1 P\_{\text{OT}}(t) \tag{A2}$$

and during the second leg by

> when *t* ≥ *τ*,

$$P\_{\rm OG}(t) = \beta\_2 P\_{\rm OT}(t) \tag{A3}$$

$$\text{for } P\_{\text{OG}}(t) = \gamma\_2 \mathbf{S}\_2 P\_{\text{OT}}(t) \tag{A4}$$

Hence, at time *τ*, the position of a ghost target is

$$P\_{\rm OG}(\tau) = \beta\_1 P\_{\rm OT}(\tau) \tag{A5}$$

$$\text{for } P\_{\text{OG}}(\pi) = \gamma\_1 \mathbf{S}\_1 P\_{OT}(\pi) \tag{A6}$$

$$\text{and } P\_{\text{OG}}(\pi) = \beta\_2 P\_{\text{OT}}(\pi) \tag{A7}$$

$$\text{For } P\_{\text{OG}}(\pi) = \gamma\_2 \mathbb{S}\_2 P\_{\text{OT}}(\pi) \tag{A8}$$

Of course, (A5) and (A6) are not compatible, and neither are (A7) and (A8).

Now, let us show that (A5) is not compatible with (A8):

Indeed, if *POG*(*τ*) = *β*1*POT*(*τ*) = *<sup>γ</sup>*2*S*2*POT*(*τ*), then

$$\frac{\beta\_1}{\gamma\_2} P\_{OT}(\tau) = \mathbf{S}\_2 P\_{OT}(\tau) \tag{A9}$$

and

of

we

obtain

Equation (A9) implies that *POT*(*τ*) is an eigenvector of *S*2, with the eigenvalue *β*1 *γ*2 . Since *β*1 *γ*2 is positive, this eigenvalue is equal to 1, i.e., *γ*2 = *β*1. Hence, *POT*(*τ*) is in the second leg.

Hence, the set of ghost targets is reduced to those whose positions at time *τ* are given by (A5) or (A6), and (A7). Now suppose that a ghost target satisfies (A6) and (A7). By the same computation, we conclude that *POT*(*τ*) is in the first leg, which is impossible since *POT*(*τ*) is in the second leg.

We have proven that (A5) and (A7) only are compatible. It follows that a ghost target verifies these two equalities (given by (A1) and (A3)):

members

of

$$\begin{array}{l} P\_{OG}(\tau) = \beta\_1 P\_{OT}(\tau) = \beta\_2 P\_{OT}(\tau). \\ \text{Hence, } \beta\_1 = \beta\_2. \\ \text{Now taking the derivative of the two} \end{array}$$

 (A1) (A3), *VG*= *β*1(*VT*− *<sup>V</sup>*1)+ *V*1= *β*1(*VT*− *<sup>V</sup>*2) + *V*2,whichisequivalentto

$$(\ddot{\beta\_1} - \dot{1})\dot{\hat{\,}(V\_2 - V\_1)\hat{\,}} = 0.$$

$$\text{Since } V\_2 \neq V\_1, \beta\_1 = 1.$$

Putting this value into (A1) or (A3), we finally ge<sup>t</sup> *PG*(*t*) = *PT*(*t*). The "ghost" is the target of interest. In conclusion, there is no ghost target. 
