Passive Tracking of Multiple Underwater Targets in Incomplete Detection and Clutter Environment

A major advantage of the use of passive sonar in the tracking multiple underwater targets is that they can be kept covert, which reduces the risk of being attacked. However, the nonlinearity of the passive Doppler and bearing measurements, the range unobservability problem, and the complexity of data association between measurements and targets make the problem of underwater passive multiple target tracking challenging. To deal with these problems, the cardinalized probability hypothesis density (CPHD) recursion, which is based on Bayesian information theory, is developed to handle the data association uncertainty, and to acquire existing targets’ numbers and states (e.g., position and velocity). The key idea of the CPHD recursion is to simultaneously estimate the targets’ intensity and the probability distribution of the number of targets. The CPHD recursion is the first moment approximation of the Bayesian multiple targets filter, which avoids the data association procedure between the targets and measurements including clutter. The Bayesian-filter-based extended Kalman filter (EKF) is applied to deal with the nonlinear bearing and Doppler measurements. The experimental results show that the EKF-based CPHD recursion works well in the underwater passive multiple target tracking system in cluttered and noisy environments.


Introduction
The tracking of multiple underwater targets using passive sonar (e.g., bearings-only multiple-target tracking; bearing and Doppler multiple target tracking) is receiving a great deal of attention in practical defense and civil applications [1][2][3]. Passive multiple-target tracking aims to obtain the number of expected targets in the tracking space, as well as the states of the targets, from passive measurements such as bearing, bearing rate, Doppler, Doppler rate, and the time arrival deference. The greatest advantage of passive multiple target tracking systems is that passive sonar emits no signal, and thus can be kept covert when conducting the passive measurements such as bearings measurements and Doppler frequency measurements, avoiding the risk of being tracked [4]. In addition, passive sonar tracking systems are simple and small in terms of configuration, and have low maintenance costs.
The main challenges associated with passive multiple underwater target tracking are that the passively obtained information is highly nonlinear [5][6][7], the targets' range may be unobservable, and the data association uncertainty between passive measurements and targets is complicated.
The range unobservability for bearings-only target tracking means that the passive sonar sensors cannot obtain accurate range information about the targets [8][9][10][11]. The general methods to avoid the target range unobservability include the use of more than one mum entropy fuzzy based on the fire-fly and PF [33], and the distributed cross-entropybased δ-generalized labeled multi-Bernoulli (δ-GLMB) filter [34].
In this paper we investigate the tracking performance of CPHD recursion in passive underwater multiple target tracking in the two-dimensional state space under a cluttered environment, using bearing and Doppler measurements. We also consider the case of a single stationary passive sonar observer. In order to improve the tracking performance, the EKF is used to deal with the nonlinear bearing and Doppler measurements with independent Gaussian white noise. In order to avoid the range unobservability problem, in this paper we introduce nonlinear Doppler measurement. The CPHD, which can jointly estimate the state and number of targets, is used to handle the targets-to-measurements data association uncertainty in the cluttered environment.
The remainder of the paper is organized as follows. The passive multiple underwater target tracking system model using Doppler and bearing measurements is given in Section 2. Section 3 develops the recursive CPHD suitable for multiple targets tracking under a dense clutter environment. The simulation results are given in Section 4. Lastly, a summary is given in Section 5.

System Model
We assume that the number of targets is variable and unknown during the entire tracking procedure in two-dimensional Cartesian space. In this paper, we consider the dynamic nearly constant turn (CT) system model for all the targets with Gaussian process noise [35]. Some maneuvering target tracking problems are discussed in [36,37].
The state for target m at time t is in which w t,m is the target's turn rate, and x t,m , y t,m , is the target's position and velocity, where the target's position is (x t,m , y t,m ), the target's velocity is .
x t,m , . y t,m , and t = 1, 2, · · · , T is the time index. The targets to single stationary observer station (passive sensor) tracking scenario using bearing and Doppler measurements information is depicted in Figure 1.
The target transition model with process noise is given by where w t,m is the system process noise. For the nearly CT target tracking model, F t,m is a system state transition matrix which is given by in which ∆T is the sampling interval. This paper supposes the tracking process noise w t,m is Gaussian white noise with covariance δ is the acceleration noise intensity.

Measurement Model
In this paper, we assume a single stationary passive sensor located at coordinates. We assume the single stationary passive sensor can only measure bearings and Doppler information, which are both nonlinear with respect to sensor and targets' state.
As seen in Figure 1, for target m , the measurement can be modeled as

Measurement Model
In this paper, we assume a single stationary passive sensor located at the original coordinates. We assume the single stationary passive sensor can only measure the targets' bearings and Doppler information, which are both nonlinear with respect to the passive sensor and targets' state.
As seen in Figure 1, for target m, the measurement can be modeled as where h t,m is the bearing and Doppler measurement function and u t,m is measurement noise. We assume the bearing and Doppler measurement noise u t,m is Gaussian white noise with covariance where u θ,t,m is bearing noise covariance and u D,t,m is Doppler noise covariance. This paper assumes that the tracking process noise and measurement noise are independent.
The bearing and Doppler measurement information function h t,m is as follows: where θ t,m is the bearing of target m, c is the transmit speed of sound, and f 0 is the underwater targets' radiant frequency.

CPHD Recursion Based on Bayesian Theory
This section first presents the multiple target tracking situation under the RFS framework. Then, the nonlinear EKF-based CPHD recursion for bearing and Doppler multiple target tracking is developed.

RFS Formulation of Multiple Target Filtering
In time t, suppose that the targets' states are x t,1 , x t,2 , · · · x t,M(t) ∈ χ in which M(t) is the number of targets in time t. At the next time, some existing targets may be disappear, and some fresh targets may enter into the tracking space. For the bearing and Doppler measurements obtained by the passive sensor, N(t) is the number of measurements in time t. The bearing and Doppler measurements at time t are z t,1 , · · · z t,N(t) ∈ Z. The purpose of multiple target tracking is to jointly estimate the number of targets in the surveillance volume as well as their states from measurements with noise and clutter.
The collection of target states at time t can be modeled as a target RFS, The collection of bearing and Doppler measurements at time t can be modeled as a measurement random finite set, (10) in which F (χ) is the collection of all targets' random finite subsets of χ, and F (F ) is the collection of all measurement random finite subsets of Z. Both the targets' finite set X t and the bearing and Doppler measurement finite set Z t are out of order.
The key of RFS is to treat X t as the multiple targets' state, and to treat Z t as the multiple target observation.
In this paper we model the multiple targets' dynamics at time t as where X t−1 is the multiple targets' state at time t − 1, which, including the existing targets at time t − 1 and the fresh targets which appear at time t, S t|t−1 (x t−1 ) is the existing target RFS at time t, Γ t is the RFS of spontaneous births at time t.
The multiple targets' bearing and Doppler measurements, including clutter at time t, is where Θ t (x) is the bearing and Doppler measurement from target x, and K t is the RFS of clutter measurements. Let f t|t−1 (·|·) be the multiple targets' transition density. Similarly, let h t|t−1 (·|·) be the multiple targets' likelihood. The posterior density of the multiple targets' states is denoted by p t (·|Z 1:t ) The RFS formulation of multiple target Bayesian filtering propagates the targets' posterior probability using the following equations: where the integrals in Equations (13) and (14) are set integrals. In practical situations, the Bayesian recursion in Equations (13) and (14) are intractable due to the multiple integrations. Suboptimal algorithms like PHD and CPHD are used to solve the integrals problem.

CPHD Recursion
The key idea of the CPHD Bayesian filter is to jointly propagate the multiple targets' intensity equation and the cardinality probability distribution [38,39].
In order to evaluate the CPHD, we assume that: • Each expected target produces bearing and Doppler measurements independently from one another; • The targets' birth RFS is independent of the surviving target RFS; • The clutter's RFS is independent from the true targets' measurement RFS; • The prior and predicted multiple targets' RFSs are both independent and identically distributed processes.
In order to evaluate the CPHD algorithm, we denote the following: The binomial coefficient is given by C l j = l! j!(l−j)! , the permutation coefficient is given by P n j = n! (n−j)! , the inner product between α and β is α, and e 0 (Z) = 1.
Let v t|t−1 (x) denote the predicted targets' intensity at time t, and p t|t−1 (n) denote the predicted targets' cardinality at time t. Similarly, let v t|t (x) denote the targets' posterior intensity and p t|t (n) denote the targets' cardinality function at time t.
Suppose one knows that the target posterior intensity is v t−1 (x) and the target posterior cardinality is p t−1 (n); then, the predicted target intensity v t|t−1 (x) and predicted target cardinality where and where p S,t|t−1 (ζ) is the probability of target existence, γ t (·) is target birth intensity, and p Γ,t (·) is target birth cardinality. Assume that the predicted intensity v t|t−1 (x) and predicted target cardinality p t|t−1 (n) are known. Then, the updated intensity v t (x) and updated cardinality distribution p t (n) at time t are given by where and where p D,t (x) is the target detection probability, κ t (·) is the clutter intensity at time t, and p K,t (·) is the clutter cardinality distribution.
Equation (19) is a Bayesian update, γ 0 t v t|t−1 , Z t (n) is the likelihood of the multiple target observation set Z t given that there are n targets, and γ 0 t v t|t−1 , Z t , p t|t−1 is the normalizing constant.

EKF-Based CPHD Recursion
This subsection develops the EKF-based CPHD recursion to the linear Gaussian system model in (3) and the nonlinear Gaussian measurement model (bearing and Doppler measurement model) in (6).
According to Equations (3) and (6), the target model x t,m = F t,m x t,m + w t,m is a linear Gaussian model, while the bearing and Doppler measurement model z t,m = h t,m (x t,m , v t,m ) is a nonlinear Gaussian model. Due to the nonlinearities of measurement function h t,m , the targets' posterior function is nonlinear and non-Gaussian. Analogous to EKF, the linear and Gaussian CPHD recursion is extended to the nonlinear model by linearizing the nonlinear bearing and Doppler measurement function h t,m .
The linearity for the measurement function h t,m of EKF is given by Next, we will develop the linear Gaussian CPHD recursion, then extend it to the nonlinear Gaussian model of bearing and Doppler measurements.
For the linear Gaussian system and measurement model, each target and sensor measurements are presented by a linear Gaussian function, where N (·; m, P) is a Gaussian distribution whose mean is m and covariance matrix is P, and the other parameters have been defined above. The survival probability p S,t (x t−1 ) and detection probability p D,t (x) are state independent for the linear Gaussian model, that is The target birth RFS intensity for a linear Gaussian system and measurement model is given as where w   Assume the targets' posterior probability hypothesis density v t−1 (x) and the targets' posterior cardinality p t−1 (n) for a linear Gaussian system, and the bearing and Doppler measurement model are given. The posterior probability hypothesis density v t−1 (x) is as follows: Then, under the condition of linear and Gaussian assumption, the predicted target probability v t|t−1 (x) and the predicted target cardinality are also Gaussian mixture, given by where the birth random finite set intensity for target γ t (x) is given in (25), and For the linear Gaussian system and measurement model, suppose that the predicted probability hypothesis density v t|t−1 (x) and the predicted target cardinality p t|t−1 (n) at time t are given, and the predicted probability hypothesis density v t|t−1 (x) is a Gaussian mixture distribution. Suppose that the target posterior probability hypothesis density v t (x) is also Gaussian mixture at time t, and in which For the linear target model and the nonlinear bearing and Doppler measurement model in this paper, the update step for the CPHD recursion is to approximate the nonlinear bearing and Doppler measurement model. That is, predicted targets are updated by using the first order of the Taylor extension when a nonlinear composition appeared [33]. That is, Equations (46) and (47) are used instead of Equations (35) and (36), and Equation (48) is used for the calculation of (46) and (47) in which

Simulations
This section presents a simulation of a nonlinear passive underwater multiple target tracking scenario in a cluttered environment using bearing and Doppler measurements to show the tracking performance of the EKF-based CPHD recursion. The multiple targets moved with the CT model with a varying turning rate in a space of [0, 3000] m × [200,2500] m. The number of targets was time varying, and they appeared and disappeared at different times and positions. The maximum number of targets was 4.
The true state for each target is given in Figure 2. There were two crossing moving targets. The sampling period ∆t = 1 s, the total number of sampling scan is 80. Target 1 arose at the first time step and disappeared at 80 s. Target 2 appeared at 20 s and disappeared at 80 s. Target 3 appeared at 10 s and disappeared at 66 s. Target 4 appeared at 20 s and disappeared at 75 s. The true measurement of bearing and frequency for the four targets is shown in Figure 3.   The standard deviation of the system process noise was 5 m/s 2 . The stan tion values of the measurement noise for bearing and frequency were 0.5  a spectively. The clutter followed a Poisson distribution with clutter intens over the region -2 2 rad 750,795 Hz -that is, the average clutter ea 42. For simplicity, the detection probability of all targets was assumed to be th was set to D, 0.98 t p = . We assumed that all targets' survival probabilities we and they were set to , 0.99 To inspect the tracking performance of the CPHD recursion, the 1000 Monte Carlo process was calculated. The scans of the targets' estimated trajectories and true targets' tracks coordinates versus time for the EKF-based CPHD recursions and PHD recur picted in Figures 4 and 5, respectively. As seen in Figures 4 and 5, the estima x-and y-coordinates are similar to the true targets' tracks for the four targets. that the CPHD recursion performed well in tracking multiple targets using and Doppler measurements. Correspondingly, the PHD algorithm's tracki was inferior to that of the CPHD. The calculation time for the EKF-based CP four targets was 3.46 s per sample run over 80 time scans, implemented i R2016a, Intel Core i7 CPU, 16 GB. The standard deviation of the system process noise was 5 m/s 2 . The standard deviation values of the measurement noise for bearing and frequency were 0.5 • and 3 Hz, respectively. The clutter followed a Poisson distribution with clutter intensity λ c = 0.3 over the region [−π/2, π/2]rad × [750, 795]Hz-that is, the average clutter each time was 42. For simplicity, the detection probability of all targets was assumed to be the same, and was set to p D,t = 0.98. We assumed that all targets' survival probabilities were the same and they were set to p S,t = 0.99. To inspect the tracking performance of the EKF-based CPHD recursion, the 1000 Monte Carlo process was calculated.
The scans of the targets' estimated trajectories and true targets' tracks in x-and y-coordinates versus time for the EKF-based CPHD recursions and PHD recursion are depicted in Figures 4 and 5, respectively. As seen in Figures 4 and 5, the estimated tracks in x-and y-coordinates are similar to the true targets' tracks for the four targets. This means that the CPHD recursion performed well in tracking multiple targets using the bearing and Doppler measurements. Correspondingly, the PHD algorithm's tracking accuracy was inferior to that of the CPHD. The calculation time for the EKF-based CPHD for the four targets was 3.46 s per sample run over 80 time scans, implemented in MATLAB R2016a, Intel Core i7 CPU, 16 GB.      The scans of the average OSPA distance for the four targets versus time for both the EKF-based CPHD recursion and PHD recursion are shown in Figure 6. It appears that the average OSPA distance for the CPHD and the PHD were approximately 30 m and 40 m, respectively. The results also show that the OSPA distance was large when the number of targets was changing, which is normal and means that the CPHD recursion can adapt to changes in targets' cardinality. It can also be noted that the EKF-based CPHD recursion had a faster reaction to changes in the targets' cardinality distribution.
The scans of the average OSPA localization for all four targets versus time for the EKF-based CPHD recursion and PHD recursion are given in Figure 7. In terms of target localization accuracy, Figure 7 shows that the average OSPA localizations for the EKFbased CPHD recursion and PHD recursion were approximately 25 m and 35 m per target, respectively, which demonstrates that the CPHD had a better tracking accuracy performance than the PHD recursion. Figure 8 presents the Monte Carlo means of estimated cardinality for the EKF-based CPHD recursion and the true targets' cardinality. It can be seen that the EKF-based CPHD recursion could distinguish the appearance of new target and the disappearance of old The scans of the average OSPA distance for the four targets versus time for both the EKF-based CPHD recursion and PHD recursion are shown in Figure 6. It appears that the average OSPA distance for the CPHD and the PHD were approximately 30 m and 40 m, respectively. The results also show that the OSPA distance was large when the number of targets was changing, which is normal and means that the CPHD recursion can adapt to changes in targets' cardinality. It can also be noted that the EKF-based CPHD recursion had a faster reaction to changes in the targets' cardinality distribution.
The scans of the average OSPA localization for all four targets versus time for the EKF-based CPHD recursion and PHD recursion are given in Figure 7. In terms of target localization accuracy, Figure 7 shows that the average OSPA localizations for the EKF-based CPHD recursion and PHD recursion were approximately 25 m and 35 m per target, respectively, which demonstrates that the CPHD had a better tracking accuracy performance than the PHD recursion.

Conclusions
The aim of passive multiple underwater target tracking is to jointly obtain the number of targets in the tracking surveillance space as well as their states (e.g., position, velocity, acceleration) from passive sensor measurements. The greatest merits of passive sonar target tracking are that the passive sonar tracking system is simple and low cost, and can operate covertly due to the use of passive bearing and Doppler measurements. To guarantee that the target range is observable when using the passive measurements, in this paper we employ bearing and Doppler measurements (passive measurements) to track multiple nearly constant turn targets. CPHD is used to handle the data association uncertainty in clutter. The EKF is applied to solve the nonlinearity of bearing and Doppler measurements. The simulation results show that the EKF-based CPHD recursion has accurate propagation in target cardinality due to its fast response, and the estimated tracks in x-and y-coordinates are similar to the true tracks for all targets, indicating that the proposed method has little error. Additionally, the tracking OSPA is small, which means that proposed algorithm has better tracking performance for the passive multiple target tracking problem in a cluttered environment.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.

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