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## Underwater environment and pre-processing module

Side-scan sonar systems have been an invaluable tool for providing high-resolution images of large swaths (e.g. 100 meters) of the sea-floor. Side-scan systems work by emitting a pulse – called ping. The array beam is is designed to be wide in the elevation plane but narrow in the azimuth plane, so as to insonify a fine strip of the sea floor at each emission. However, the beam being wide in the elevation plane, also interacts with the sea-surface, generating interference echoes (surface reverberation). Given the state of the sea-surface, interference echoes vary from specular reflections to diffuse scattering. The acoustic-wave propagation is nearly spherical, both forwards and backwards, and points located at the same distance are received at the same moment with different DOA. On the other hand, the acoustic wave does not insonify some parts of the bottom (e.g. in presence of tall objects), thus creating “acoustic shadows”. Recently, constructors started fitting several receiving elements on side-scan systems, providing not only amplitude images of the sea-floor backscatter but also bathymetric estimates. Such systems are usually called interferometric or bathymetric side-scan sonars [11, Ch.3.3.1]. The sonar employed in this work is the EdgeTech 4600, which disposes of N = 8 half-wavelength, linearly spaced receivers. The sonar operates by emitting a linear frequency modulated pulse, where the pulse bandwidth was set to 10kHz with the carrier frequency being 540kHz. Other sonar specifications are given in [6] and Appendix G. Array processing algorithms are employed to estimate the DOA parameters i t with i = 1, · · · , nt, of nt sources of acoustic waves, directly from the phased-array signal: yt = Xnt i=1 a(i t)si t + nt.

### Post-processing module: the NNIPDA-UKF filter

This section presents the post-processing module, that is responsible with filtering multiple echoes (i.e targets) from the set of observations obtained by the pre-processor. We consider a number nt of independent targets with system states denoted as {xit }nt i=1. Since the trajectories of individual targets are considered independent, echo tracking can be conducted separately on each individual target state. For the i−th echo, the state system is given by the state-transition equation (II.5a) and the observation equation (II.5b) xit = ft(xi t−1) + vit , (II.5a) zi t = Htxit + wit , i = 1, · · · , nt, (II.5b) where the state vector xit 2 X takes values in the state space X. For the sake of generality, we consider xit as a vector of dimension dx and X Rdx. However, in sections II.4 and II.5, we will consider more specific models (e.g. x = ). The state transition function ft(·) represents an added prior, i.e. a model on the DOA trajectory. The covariance Qi t of the model noise vit controls the confidence in the model. The notation xit is kept for the sake of coherence with the tracking and dynamical-systems literature. zi t is the system noisy observation supplied by the pre-processing module, as discussed in the previous section. Ht is the observation matrix. Whenever, the observation zi t only consists in maxima locations of the spectrogram (as depicted in Eq. II.4), the observation reduces to a scalar and Ht = 1. The observation noise wit with variance Ri t depends on the acquisition parameters and the pre-processing module. The noise sequences vit and wit are independent, white random processes characterized by their first two moments. Trajectory initialization is usually done by the “two out of three” heuristic rule, in other words, two observations falling inside a given gate within three consecutive range bins are assumed to represent a target. However, automatic trajectory initialization is beyond the scope of this chapter.

The pre-processing module detects the presence of a target (i.e. an echo) having state xit , with non-unitary probability of detection pD(xit ), and estimates the DOA of the target. Conditioned on the target state, pD(xit ) is by definition pD(xit ) , Pr{zi t 2 Zt|xit }.

#### Man-made object reconstruction

Another interesting case of double target tracking is showcased in Fig. II.9 a), where the primary trajectory is the sea-bottom DOA echo and the secondary trajectory corresponds to the echo backscattered from a vertical mooring pole (also called piles or pilings). Note, that the two echoes are received simultaneously, since the two echoes are situated at the same distance (slant range) from the sonar array. This leads to the layover phenomenon in side-scan images, effectively obstructing the area in front of the pole. Pre-processing is done by MUSIC coupled with AIC and require a computation time of 0.9s. The post-filtering has a computation time of 1.3s on a 3.6GHz processor in MATLAB. Compiling the code would enable real-time survey- ready processing. The models employed to track the pole echo are the VO model with QVO = 0.1(o)2, and a RW model with QRW = 1(o)2. The sea-bottom echo was tracked with the same parameters as in the previous figures. In Fig. II.9 b) we observe the evolution of the μVO t|t and μRW t|t model probabilities. Observe the pertinence of the RW model for the start of the pole trajectory and the later switch to the VO model. Tracking the pole trajectory is stopped when its existence probability t|t, given by Eq. (A.17) and presented in II.9 c), drops below the threshold value of 0.5. GoF tests were conducted for the sea-bottom echo yielding the acceptance of the whiteness hypothesis with p−values of 0.34 and 0.18 for the Box-Pierce and the Ljung-Box tests. GoF tests for the pole echo also accept the whiteness hypothesis, with p−values of 0.76 and 0.59 for the Box-Pierce and the Ljung-Box tests.

For comparison reasons, the RW model alone is also employed to track the pole echo. However, since the RW has no preferred dynamical evolution situations occur were any clutter induced DOA estimate close to the pole DOA trajectory can cause the trajectory to be irretrievably lost, as seen in Fig. II.10 a). Again the trajectory existence probability, Fig. II.10 b) is employed to stop the tracking process.

**Table of contents :**

Abbreviations and nomenclature

Acknowledgements

R´esum´e

Contexte

Les principaux d´efis

Objectifs et r´ealisations

Plan d´etaill´e de la th`ese

Conclusions

Perspectives

**I Introduction **

**II Sonar echo tracking: a data-association approach **

II.1 Introduction: Objectives and challenges

II.2 Underwater environment and pre-processing module

II.3 Post-processing module: the NNIPDA-UKF filter

II.3.1 Prediction

II.3.2 Update

II.3.3 Trajectory quality measure

II.4 Geometrical models

II.5 Results of NNIPDA-UKF

II.5.1 Simulated data results

II.5.2 Real data results

II.6 Chapter conclusions

**III Sonar echo tracking: an intensity function approach **

III.1 Point-process formalism for target tracking

III.2 Point-process theory

III.2.1 Preliminary definitions

III.2.2 A probability measure for PP

III.2.3 RFS and set integral

III.2.4 Moment measures of point processes

III.3 Examples of finite point processes

III.3.1 Independent and identically distributed cluster (iidc)

III.3.2 Poisson process

III.4 Error metrics for multi-target tracking

III.5 The PHD filter for TWS systems

III.5.1 TWS-PHD prediction

III.5.2 TWS-PHD update

III.6 TBD-PHD filter for phased arrays: a Gaussian point-target case

III.6.1 Introduction

III.6.2 Array signal and marked multi-target process

III.6.3 Approximate PHD filter for marked multi-target process

III.6.4 Results on simulated phased-array data

III.7 Chapter conclusions

**IV Sonar echo tracking: the case of extended and impulsive targets **

IV.1 Introduction

IV.2 Impulsive Sonar-signal distributions

IV.2.1 The SIRV model

IV.2.2 The joint-SIRV model

IV.3 Phased-array signal and multi-target formalism

IV.3.1 Extended and impulsive target model

IV.3.2 Marked process for extended and impulsive target signals

IV.4 Phased-array TBD-CPHD filter

IV.4.1 Phased-array TBD-CPHD prediction

IV.4.2 Phased-array TBD-CPHD intensity update

IV.4.3 Phased-array TBD-CPHD cardinality update

IV.4.4 TBD-CPHD Monte Carlo Implementation

IV.5 Simulation Results

IV.5.1 Tracking results

IV.5.2 Angular resolution analysis

IV.6 Real sonar data results

IV.7 Chapter conclusions

**V Conclusions and perspectives **

A Integrated PDAF (IPDAF)

B Observation noise covariance estimation

C Model validation

D Proof of p-value |H0 U(0, 1)

E Quadratic and Gaussian equalities

F On the distribution of random sums

G Array narrow-band condition

H Statistical analysis of sonar signals