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

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## Abstract

**:**

## 1. Introduction

## 2. Notation and Problem Formulation

- $\delta $ indicates the direction of the path of the sound emitted by the source: if the path is toward the surface, $\delta =-1$, otherwise $\delta =+1$,
- ${n}_{B}$ is the number of bottom reflections, and
- ${n}_{S}$ is the number of surface reflections.

- Only one ray is detected by the array during the scenario; in this case, we have at each time t a measurement $m\left(t\right)$, given the path along which the wave propagates.
- Two rays (traveling on two different paths) arrive at the sensor’s antenna. In this case, the available measurement at time t is a couple of measurements, say $\left({m}_{1}\left(t\right),{m}_{2}\left(t\right)\right)$, given the two paths along which the wave propagates.

- If not, what are the ghost targets (those which could be detected at the same set of measurements $\left\{m\left(t\right),t\in \left[0,T\right]\right\}$)?
- How do we make $X$ observable or with which new information?
- Is the vector $X$ observable from the set of couples $\left\{\left({m}_{1}\left(t\right),{m}_{2}\left(t\right)\right),t\in \left[0,T\right]\right\}$?

## 3. TMA from One Ray

#### 3.1. Observability Analysis

**Theorem**

**1.**

- 1.
- If the target is broadside to the antenna, then the set of ghost targets is composed of virtual sources broadside to the antenna.
- 2.
- If the target is endfire to the antenna, the set of ghost targets is composed of virtual sources endfire to the antenna.
- 3.
- If the target has the same heading as the array (but is not endfire to it), then the set of ghost targets is composed of virtual targets with the same heading as the antenna. More precisely, the ghost image of each ghost target is moving on a cylinder whose axis is the antenna axis, and whose radius is a positive scalar $\beta $. The relative ghost target velocity is equal to $\beta $ times the target’s velocity. The initial distance between the ghost image and the center of the antenna is equal to $\beta $ times the initial distance between the target-image and the center of the antenna.
- 4.
- In any other case, for a chosen image-depth ${\zeta}_{G}$, the set of ghost targets is composed of virtual targets whose motion relative to the array is defined by ${P}_{OG}\left(t\right)=\beta {P}_{OT}\left(t\right)$ or ${P}_{OG}\left(t\right)=\beta \mathit{S}{P}_{OT}\left(t\right)$, where $\mathit{S}$ is the 2D axial symmetry around the line of the array, and $\beta $ is a positive scalar. The scalar $\beta $ is equal to $\frac{\left|{\zeta}_{OG}\right|}{\left|{\zeta}_{OT}\right|}$ if ${\zeta}_{OT}\ne 0$. If ${\zeta}_{OT}=0$ (which can happen with a direct path only), $\beta $ can have any positive value.

**Proof of Theorem 1.**

**Remark**

**1.**

- 1.
- When the source and the observer are at the same depth, and the path is direct, Theorem 1 recovers the conclusions given in [26].
- 2.
- The cases (1), (2) and (3) of Theorem 1 are “rare events”, since the events of dealing with a source in endfire, broadside or having the same heading as the antenna during the scenario occur with a probability equal to 0.However,when the target has a trajectory close to one of these special cases, the estimates will have a poor behavior.
- 3.
- For case (4), when the detected ray is not a direct path, for example, when the ray is bottom-reflected, a hypothesis about the source is sufficient to obtain one solution, corresponding to a ghost target. Indeed, if we suppose that the depth of the target is ${z}_{As}$ (whereas the true value is ${z}_{T}$), then we have $\beta =\frac{2D-\left({z}_{As}+{z}_{O}\right)}{2D-\left({z}_{T}+{z}_{O}\right)}$, whose biggest value ${\beta}_{Max}=\frac{2D-{z}_{O}}{2D-\left({z}_{T}+{z}_{O}\right)}$, and the minimum value is ${\beta}_{Min}=\frac{2D-\left({z}_{Max}+{z}_{O}\right)}{2D-\left({z}_{T}+{z}_{O}\right)}$, where ${z}_{Max}$ is the largest depth of a submarine vehicle. Typically, in deep water, $D\ge 4000$ m. A reasonable choice of ${z}_{Max}$ could be 400 m. We can then have a range of $\beta $: $\left[{\beta}_{Min},{\beta}_{Max}\right]=\left[\frac{7600-{z}_{O}}{8000-\left({z}_{T}+{z}_{O}\right)},\frac{8000-{z}_{O}}{8000-\left({z}_{T}+{z}_{O}\right)}\right]$. For instance, when the depths of the antenna and the target are, respectively, 200 and 100 m, we have $\left[{\beta}_{Min},{\beta}_{Max}\right]=\left[0.974,1.013\right]$. In this way, we bound the set of ghost targets, and we can expect that the bias induced by a wrong choice of ${z}_{As}$ is very low.
- 4.
- For case (4) again, with a direct path, if the target is not at the same depth as the antenna, $\beta =\frac{{z}_{As}-{z}_{O}}{{z}_{T}-{z}_{O}}$. Because $\beta $ is a positive number, ${z}_{As}-{z}_{O}$ has the same sign as ${z}_{T}-{z}_{O}$: if ${z}_{T}>{z}_{O}$, then ${z}_{O}<{z}_{As}\le {z}_{Max}$, and $\left[{\beta}_{Min},{\beta}_{Max}\right]=\left[0,\frac{{z}_{Max}-{z}_{O}}{{z}_{T}-{z}_{O}}\right]$; if ${z}_{T}<{z}_{O}$, then $0\le {z}_{As}<{z}_{O}$, and $\left[{\beta}_{Min},{\beta}_{Max}\right]=\left[0,\frac{{z}_{O}}{{z}_{T}-{z}_{O}}\right]$. In both cases, the range $\left[{\beta}_{Min},{\beta}_{Max}\right]$ is too wide to be useful.If the target and the antenna are at the same depth,$\beta $can take any positive value.

#### 3.2. Estimation of the Trajectory

- The measurements are collected every 4 s ($\Delta t=4$s). The scenario lasts 20 min.
- The sea bottom depth is 4000 m. The detected ray is a bottom-reflected ray.
- The assumed target depth is ${z}_{As}=200$m (whereas the true one is 100 m).
- First, the measurements have been corrupted with an additive Gaussian noise whose standard deviation is $\sigma =1.7\times {10}^{-2}$.

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

#### 4.1. Observability Analysis

**Theorem**

**2.**

#### 4.2. Estimation

#### 4.2.1. Estimation of $X$

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

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

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

#### 5.1. Observability

**Theorem**

**3.**

- 1.
- If the target is broadside to the array, then the set of ghost targets is uncountable: it is composed of all the (virtual) targets at broadside to the array.
- 2.
- If the target is endfire to the antenna, the set of ghost targets is composed of virtual sources at endfire to the antenna.
- 3.
- If the route of the antenna and the route of the target are parallel, then the set of ghost targets is uncountable: at each depth ${z}_{G}$, there are two ghost targets moving on a cylinder whose axis is the antenna axis, and the radius is a positive scalar $\beta =\sqrt{\frac{D-{z}_{G}}{D-{z}_{T}}}$. The relative ghost target velocity is equal to $\beta $ times the target’s velocity. The initial distance between the ghost image and the center of the antenna is equal to $\beta $ 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 ${P}_{OG}\left(t\right)=\mathit{S}{P}_{OT}\left(t\right)$,$\text{}{P}_{OG}\left(t\right)=\beta {P}_{OT}\left(t\right)$, and ${P}_{OG}\left(t\right)=\beta \mathit{S}{P}_{OT}\left(t\right)$, where $\mathit{S}$ is the matrix of the axial symmetry around the line of the antenna, and $\beta \triangleq \frac{D-{z}_{O}}{D-{z}_{T}}$. If the depth of the antenna is equal to the depth of the source, then there is one single ghost target given by ${P}_{OG}\left(t\right)=\mathit{S}{P}_{OT}\left(t\right)$.

**Proof of Theorem 3.**

- when the target’s depth is larger than the array’s depth, there is a ghost whose depth is smaller than the array’s depth, and vice versa.
- $\beta $, which is a positive coefficient, is equal to $\frac{D-{z}_{O}}{D-{z}_{T}}$, or 1.

**Remark**

**2.**

#### 5.2. Estimation of the Trajectory

#### 5.2.1. Estimability

- First scenario

- 2.
- Second scenario

#### 5.2.2. Monte Carlo simulations

- First scenario

- 2.
- Second scenario: Bottom depth $D=4000$ m.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A

**Proof of Theorem 2.**

- the target is broadside to the antenna,
- the target is endfire to the antenna,
- the target has the same heading as the array (but is not endfire to it),
- the other cases.

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**Figure 2.**Three examples of ray paths: the solid line represents the direct path $\left(\delta ,{n}_{B},{n}_{S}\right)=\left(+1,0,0\right)$, the dashed-dotted line represents the bottom reflected path $\left(\delta ,{n}_{B},{n}_{S}\right)=\left(+1,1,0\right)$, and the dashed line represents the bottom-surface-bottom reflected path $\left(\delta ,{n}_{B},{n}_{S}\right)=\left(+1,2,1\right)$.

**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: ${\varphi}^{D}$ and ${\varphi}^{B}$ are the elevations of the direct path and of the bottom-reflected path, respectively.

**Figure 4.**The cloud of estimated position (in green), a piece of the hyperbola (intersection of the cone of ambiguity and the plane $z={z}_{As}\left(=200\mathrm{m}\right)$, for $\sigma =1.7\times {10}^{-2}$.

**Figure 5.**The cloud of estimated position (in green) for $\sigma =1.7\times {10}^{-4}$. The cloud is no longer hyperbola-shaped. The small black segment is the 90%-confidence ellipsoid.

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

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

**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, ${z}_{As}=300$m, and ${z}_{T}=100$m. The symmetrical cloud is plotted too.

**Figure 10.**The location of the sensor array (in black), the cloud of the 500 estimates of the initial positions, and the 90%-confidence ellipse when $D=4000$ m, ${z}_{As}=300$ m, and ${z}_{As}=100$ m, together with the symmetrical cloud.

**Table 1.**Performance of the estimator of the reduced state vector when $\sigma =1.7\times {10}^{-2}$, in terms of bias, sample standard deviation and the one given by the square root of the diagonal of the CRLB.

${\mathit{X}}_{\mathit{r}}$ | Bias | ${\mathit{\sigma}}_{\mathit{s}\mathit{a}\mathit{m}\mathit{p}}$ | ${\mathit{\sigma}}_{\mathit{C}\mathit{R}\mathit{L}\mathit{B}}$ |
---|---|---|---|

5000 m | −3525 | 6962 | 13,356 |

7000 m | −2367 | 4052 | 5599 |

2.83 m/s | −1.37 | 1.81 | 4.35 |

2.83 m/s | 0.53 | 1.62 | 2.75 |

**Table 2.**Performance of the estimator of the reduced state vector with $\sigma =1.7\times {10}^{-4}$.

${\mathit{X}}_{\mathit{r}}$ | Bias | ${\mathit{\sigma}}_{\mathit{s}\mathit{a}\mathit{m}\mathit{p}}$ | ${\mathit{\sigma}}_{\mathit{C}\mathit{R}\mathit{L}\mathit{B}}$ |
---|---|---|---|

5000 m | 60.40 | 138.67 | 133.56 |

7000 m | 88.20 | 58.42 | 55.99 |

2.83 m/s | 0.043 | 0.044 | 0.044 |

2.83 m/s | 0.037 | 0.028 | 0.028 |

$\mathit{X}$ | Bias | ${\mathit{\sigma}}_{\mathit{s}\mathit{a}\mathit{m}\mathit{p}}$ | ${\mathit{\sigma}}_{\mathit{C}\mathit{R}\mathit{L}\mathit{B}}$ |
---|---|---|---|

5000 m | −44.77 | 854.72 | 868.12 |

7000 m | −68.16 | 1162.1 | 1173.60 |

100 m | 7.14 | 558.55 | 545.99 |

2.83 m/s | 0.092 | 1.67 | 1.65 |

2.83 m/s | 0.194 | 2.72 | 2.68 |

${\mathit{X}}_{\mathit{r}}$ | Bias | ${\mathit{\sigma}}_{\mathit{s}\mathit{a}\mathit{m}\mathit{p}}$ | ${\mathit{\sigma}}_{\mathit{C}\mathit{R}\mathit{L}\mathit{B}}$ |
---|---|---|---|

5000 m | 65.40 | 655.61 | 606.41 |

7000 m | 93.05 | 831.75 | 762.56 |

2.83 m/s | 0.034 | 1.71 | 1.58 |

2.83 m/s | 0.063 | 2.76 | 2.52 |

${\mathit{X}}_{\mathit{r}}$ | Bias | ${\mathit{\sigma}}_{\mathit{s}\mathit{a}\mathit{m}\mathit{p}}$ | ${\mathit{\sigma}}_{\mathit{C}\mathit{R}\mathit{L}\mathit{B}}$ |
---|---|---|---|

5000 m | 401.12 | 281.85 | 281.17 |

7000 m | 557.24 | 330.87 | 319.37 |

2.83 m/s | 0.13 | 1.58 | 1.78 |

2.83 m/s | 0.12 | 1.81 | 2.02 |

${\mathit{X}}_{\mathit{r}}$ | Bias | ${\mathit{\sigma}}_{\mathit{s}\mathit{a}\mathit{m}\mathit{p}}$ | ${\mathit{\sigma}}_{\mathit{C}\mathit{R}\mathit{L}\mathit{B}}$ |
---|---|---|---|

5000 m | 306.81 | 219.28 | 130.08 |

7000 m | 432.46 | 276.61 | 115.26 |

2.83 m/s | 0.18 | 0.74 | 0.80 |

2.83 m/s | 0.18 | 0.66 | 0.71 |

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**MDPI and ACS Style**

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

**AMA Style**

Lebon A, Perez A-C, Jauffret C, Laneuville D.
TMA from Cosines of Conical Angles Acquired by a Towed Array. *Sensors*. 2021; 21(14):4797.
https://doi.org/10.3390/s21144797

**Chicago/Turabian Style**

Lebon, Antoine, Annie-Claude Perez, Claude Jauffret, and Dann Laneuville.
2021. "TMA from Cosines of Conical Angles Acquired by a Towed Array" *Sensors* 21, no. 14: 4797.
https://doi.org/10.3390/s21144797