Tracking of acoustic source in shallow ocean using field measurements

Indian Journal of Geo-Marine Sciences Vol. XX(X), XXX 215, pp. XXX-XXX Tracking of acoustic source in shallow ocean using field measurements K. G. Nagananda & G. V. Anand Department of Electronics and Communication Engineering, PES University, Bengaluru-5685, India [E-mail: kgnagananda@pes.edu, gargeshwari.anand@gmail.com] Received 22 November 214; revised 15December 214 We investigate the problem of tracking a moving source in shallow ocean in a Bayesian framework, using acoustic field measurements which are more informative than the commonly employed bearings-only or time-delay measurements. The acoustic field measurement model is described and compared with the bearings-only measurement model. A general approach to Bayesian filtering based on the Gaussian approximation model is then presented. Within this framework, we consider the unscented Kalman filter (UKF), fifth-degree cubature Kalman filter (CKF 5 ), and quasi-monte Carlo Kalman filter (QMC-KF) algorithms that employ different numerical integration procedures. Simulation results indicate that acoustic field measurements yield a significantly lower root mean square error (RMSE) than bearings-only measurements, and that the RMSEs for QMC-KF and CKF 5 are much lower than those for UKF and extended Kalman filter (EKF). [Keywords: Acoustic field measurements, Gaussian approximation filtering, shallow ocean, source tracking] Introduction Tracking a moving underwater target is a problem of great interest in ocean surveillance. The classical approach to tracking a moving target is based on time delay measurements. Time differences in the arrival of the target echo or a target-radiated signal at different elements of a hydrophone array are measured periodically, and the position of the target is estimated from these measurements. Better tracking performance can be achieved by estimating the target bearing (azimuth) and employing the bearing estimates in the tracking algorithm. In both the approaches, the tracking problem is formulated in the Bayesian estimation framework; and the Kalman filter or one of its several extended versions is used to recursively estimate the position of a moving target from noisy measurements. Neither of the afore-mentioned approaches exploits the acoustic field measurements available at the sensor array. In this paper, we propose and investigate Bayesian filtering algorithms based on acoustic field measurements for tracking a moving acoustic source in shallow ocean. Acoustic field measurements are expected to provide greater tracking accuracy since these measurements contain more information about the source position than either the time delay measurements or the bearing-only measurements. Numerical results presented in the paper show that the root mean square errors (RMSEs) of the source position estimates provided by the proposed approach are indeed lower than the corresponding errors of the conventional approach based on bearing-only measurements. The problem of tracking the range and velocity of underwater targets from time delay measurements was addressed by Moose and Dailey 1 using a linearized polar model of the target. The use of extended Kalman filter (EKF) was eliminated, and the depth estimation problem was decoupled from the polar range estimator, making the tracking algorithm computationally less intensive. Carevic 2 proposed a modified particle filter for two-dimensional tracking in the underwater scenario using time delay measurements. The generalized Kalman filter (GKF), which characterizes the tradeoff between the costs associated with estimation error and that related to the lateral discontinuity of the estimates,

INDIAN J. MAR. SCI., VOL. XX, NO. X XXXX 215 was introduced by El-Hawary et al 3 for target tracking based on a time delay measurement model. A time varying bias term unknown to the observer was introduced into the underwater tracking problem by Katsikas et al 4, who devised a dynamical estimator using the Lainiotis multimodel partitioning theory coupled with conventional constant bias estimation algorithms. El-Hawary et al 5 proposed a robust version of EKF to tackle non-gaussian noise and nonlinearities in the model for tracking underwater targets. The bearings-only target tracking problem was considered by Nardone et al 6 and a closed form solution was obtained by devising a pseudolinear estimator robust to bias. Rao 7,8 considered the bearings-only measurement model for estimating the underwater target motion parameters and target maneuvers using the modified gain EKF. Farina 9 estimated the target position and velocity from bearings-only measurements using batch-processing of data and a recursive algorithm, and evaluated the Cramer- Rao lower bound (CRLB) of the estimate for several practical scenarios. The algorithm was shown to perform particularly well in the low signal-to-noise ratio conditions. Risticand Arulampalam 1 modeled the dynamics of a moving target by multiple switching models, assuming that the measurements were collected asynchronously from possibly multiple moving platforms. Julierand Uhlmann 11 analyzed the interactive multiple model (IMM) algorithm with EKF, UKF and multi-mode particle filters, and derived a theoretical lower bound on the performance error. A closed-form instrumental variable estimator for target tracking was proposed by Dogancay 12, who showed that the estimator's error covariance matrix approaches the CRLB for sufficiently small bearing noise as the number of measurements tends to infinity. Xu et al 13 derived the CRLB for a maneuvering target tracking algorithm in the presence of measurement biases, and devised an easy-to-implement maneuver detection method that yields a satisfactory CRLB. A kernel-based filter with an improved range parameterized method was proposed by Guo et al 14 for the bearings-only tracking problem with signal propagation time delay to recursively estimate the position of the target in the underwater scenario. The IMM version of the CKF 15 and UKF for target tracking using bearings-only measurements was analyzed by Wan et al 16, and their relative merits and demerits were examined. Dini and Mandic 17 proposed a bearing estimation algorithm, which exploits the correlation structure in the source signal, for DIFAR (directional frequency analysis and recording) sonobuoy underwater target tracking. The relative merits and demerits of UKF, EKF and the iterated EKF for target tracking were analyzed by Sadhu et al 18. In this paper, we consider the problem of tracking an acoustic source moving in a horizontal plane in a shallow horizontally stratified ocean using acoustic pressure field measurements, which are more informative than either the time-delay or the bearing-only measurements. The state space model of the problem comprises (a) a linear transition model of the target locations (states), and (b) a nonlinear measurement model relating the acoustic pressure field at a horizontal linear hydrophone array to the target location. The measurement model is based on the normal mode theory of sound propagation in the ocean 19. The measurement vector is contaminated by noise which is assumed to be Gaussian and spatially and temporally white with a known variance. We consider the following filters: UKF, CKF 5 and QMC-KF 2. In each case, the posterior distribution of the system states is approximated as Gaussian or Gaussian mixture, and the first- and secondorder moments of the state vector are updated recursively. There are no linear approximations of the nonlinear measurement equations. From the computational viewpoint, there is no calculation of the Jacobian matrix. We present the quadrature rules and describe the quasi-monte Carlo (QMC) method of numerical integration used for efficient and accurate computation of integrals involving high-dimensional functions which are repeatedly encountered in the estimation process. We then compare the performance of these algorithms under simulated conditions, using the standard performance measure, viz. RMSE. Simulation results show that acoustic field measurements yield lower RMSE than bearing-only

NAGANANDA AND ANAND: ACOUSTIC SOURCE TRACKING USING FIELD MEASUREMENTS measurements at a slightly higher computational cost. But tracking based on bearing measurements involves the hidden cost of bearing estimation at each time step. QMC-KF algorithm provides the best performance, followed by CKF 5, UKF (which is accurate only up to the 3 rd -order polynomials), and EKF; but better performance is achieved at a higher computational cost. State space models State transition model We consider a three-dimensional region with spatial coordinates given by (,, ) where denotes depth below the water surface. We assume that the target moves in the horizontal (, ) plane with constant velocity (, ).The target state vector at time is given by = [,,,, ]. The discrete time state update equation is given by = +, (1) where the state transition matrix is given by 1 1 = 1 1 1, (2) denotes the sampling interval and is the 5 1 zero-mean Gaussian noise with covariance matrix, given by 21, =.(3) In (3), and denote, respectively, the level of the power spectral density (in m 2 /s 3 ) of the corresponding process noise for the - and - coordinates and for the -coordinate. Measurement model In the case of bearing-only tracking, the bearing is estimated at regular intervals of time using a hydrophone array, and the bearing estimate is treated as the measurement at time. The bearing measurement is modeled as = tan +, (4) where is the bearing estimation error which is assumed to be zero-mean Gaussian with variance. In this paper the acoustic pressure measured by a uniform horizontal linear array of hydrophones will be treated as the dimensional measurement vector. A description of the measurement model is given below. The shallow ocean is modeled as a horizontally stratified water layer of constant depth h, density, and sound speed profile overlying a horizontally stratified fluid bottom of density and sound speed profile. The variation of its acoustic properties in the horizontal direction is assumed to be negligible in the range of interest. In the following, = 1,, and = 1,,, denote the sensor and mode indices, respectively. Consider a uniform horizontal linear array of hydrophones (acoustic pressure sensors) with inter-element spacing, and let the coordinates of the element of the array be (( 1),, ). We assume that the target radiates a narrowband signal of center frequency and complex random amplitude with mean zero and variance, and that the array is located in the far-field region of the signal source. Using the normal mode theory of sound propagation 19, the complex amplitude of the 1 acoustic pressure vector measured by the array at time is represented by the following nonlinear relation between the measurement vector and the state vector : where = +, (5) = (, ) (6)

INDIAN J. MAR. SCI., VOL. XX, NO. X XXXX 215 is the channel response function, = + is the target range, = tan is the bearing. The steering vector is given by = 1, (7) where =, (8) and ( + ) are, respectively, the complex amplitude and the complex wavenumber of the normal mode in the oceanic waveguide at frequency, is the corresponding mode eigenfunction, and is the ambient noise vector. It is assumed that is complex Gaussian with mean zero and covariance matrix, =, and that and are independent. Values of the modal eigenfunctions, wavenumbers, attenuation coefficients, and the number of modes, can be readily computed using the Kraken normal mode program devised by Porter 22. State transition equation (1), in conjunction with (2) and (3), and measurement equation (5), in conjunction with (6) (8), constitute the system of state space equations for the present problem. The transition equation is linear while the measurement equation is nonlinear. The objective of the tracking algorithms is to estimate the state vectors { ; = 1,2,.}. In the Bayesian framework, is estimated by calculating the posterior state probability density function { (, )}. Methods based on Gaussian approximation filtering for calculating (, ) recursively are described in the following sections. Gaussian approximation filtering Theory For the sake of notational convenience, we denote = {,, }. The densities of interest can be updated recursively as follows: ( ) = ( ) ( ), (9) ( ) = ( ) ( ) ( ), (1) where (9) and (1) are the prediction and filtering steps, respectively. The difficulty associated with the tracking problem is due to the fact that the measurement vector that appears in (9) and (1) is a highly nonlinear function of the state vector. In the EKF, the problem is simplified by linearizing (5). But this procedure leads to large estimation errors. In this section, we present a review of the Gaussian approximation procedure proposed by Ito et al 23, that does not require linearization. This is followed by a brief description of some methods used for numerical evaluation of the integrals that appear in the implementation of the prediction and filtering steps. The Gaussian approximation is based on the following assumptions: 1. Given, is assumed to follow a Gaussian distribution, i.e.,,, (11) with mean and covariance matrix. 2. Given, (, ) is assumed to be jointly Gaussian, which implies the following:,, (12) where the mean is given by { } = { } = ( ), (13) and the covariance matrix is given by, + { } =, + ( ). (14) From the second assumption, given = (, ), has a Gaussian distribution, i.e.,

NAGANANDA AND ANAND: ACOUSTIC SOURCE TRACKING USING FIELD MEASUREMENTS ~,, (15) where the mean are given by and the covariance matrix = {, } = +, (16) and = +,, (17) respectively, where the filter gain and, are given by =,, +,, (18), = ) = ( ), (19) with = { } = ( ), (2) and, = { } = ( ). (21) We now provide approximations for various integrals that appeared in the above discussion. Consider the evaluation of the integral of a realvalued function h. Let the dimension of the vector be denoted by. A commonly employed numerical technique is to choose a point set = {,, }, where the dimension of the point or vector, = 1,,is, and compute the weighted average of h to approximate the integral, i.e., h h, (22) where the associated weights [,1] satisfy = 1. If the function h involves a Gaussian distribution ( ;, ) with mean vector and covariance matrix, i.e., if h is of the form h = h ( ;, ), then the approximation is given by h ( ;, ) h( + ), (23) where =, and can be obtained by matrix decomposition. Using (23), we can evaluate the integrals used to compute the mean vectors and covariance matrices that appeared in the previous subsection. First, consider the following: =,, = +, (24) =,, = +, (25) where is a -dimensional vector, = 1,, and, and, are the transformed weight vectors. Finally, given the state transition matrix and the measurement function (.), we have the following approximations:,, (26), +,,, (27),, (28), {,, }, (29), {,, }. (3) A summary of the framework for tracking algorithms is given below: 1. At time =, set ~,. The procedure adopted to obtain the initial estimates and is described later.

INDIAN J. MAR. SCI., VOL. XX, NO. X XXXX 215 The following steps are implemented at time steps = 1,,. 2. Compute,, = 1,, using (24). 3. Compute and using (26) and (27), respectively. 4. Compute,, = 1,,, using (25). 5. Compute,, and, using, (28), (29) and (3), respectively. 6. Compute the filter gain using (18). 7. Compute the estimated mean vector and the covariance matrix using (16) and (17), respectively. Note: The above exposition presents a unifying filtering framework for the general nonlinear dynamical system. However, for the state transition model considered in this paper, the transition matrix defined by (2) is linear. Therefore, and can be computed without resorting to point-based numericalintegration given by (26) and (27). In this case, and are given by =, (31) = +,. (32) Point-based numerical integration Various numerical rules can be used to form the approximation in (23), depending on the choice of {(, ), = 1, }, leading to different Gaussian approximation filters. When the dimension of the vector is small, a good approximation can be obtained using different quadrature rules and sparse grids. Examples include UKF 11 CKF 15, the central difference filter (CDF) 23, the Gauss-Hermite quadrature filter (GHQF) 24, the divided difference filter (DDF) 25 and the sparse grids filter (SGF) 26. The number of points in UKF, CKF, CDF and DDF grow linearly with the dimension of the state vector, but these filters are accurate only up to the 3 rd -degree polynomials. The number of points in GHQF grows exponentially with the dimension of the state vector, while the SGF is efficient only for low-dimensional problems. In the following we briefly summarize the numerical rules used to derive UKF, CKF 5 and QMC-KF. As before, let the dimension of the state vector be denoted by. In the case of UKF, the number of points or -dimensional vectors is given by = 2. + 1. The points and the weights, = 1,, are given by = = ; = 1 + ; = 2,, + 1, + ; = + 2,, 2 + 1 (33), = 1, (34), = 2,,2 + 1 where is a tunable parameter, and is the dimensional unit vector. To obtain the CKF 5 we invoke Mysovskikh's method 27 and the moment-matching method proposed by Jiaet al. 28 to devise the fifth-degree spherical and radial rules, respectively. The minimum number of points required for the 5 th - degree cubature rule is = 2 + 1. For the sake of brevity, we only provide the final form of the 5 th -degree cubature rule. Given a dimensional state vector ~ (, ), with = and =, an integral of the form h ( ;, ) can be approximated as shown below: h ( ;, ) h() + h + 2 + h +2 1+2 1 2 +1 2 +2 2 =1 +12h +2 2+h +2 2, (35) = ; = 1,, + 1, (36) ; = 1, ( + 1 ) 2 = + :1 < + 1, (37)

NAGANANDA AND ANAND: ACOUSTIC SOURCE TRACKING USING FIELD MEASUREMENTS = ; < ; = ; >. (38) For large dimensions, Monte Carlo (MC) integration is the preferred technique to approximate the integrals more accurately. is chosen as a set of independent and uniformly distributed points [, 1], and =,. However, owing to random selection of points, the conventional MC technique tends to form gaps and clusters, and does not explore the sample space in a uniform manner. QMC is the deterministic counterpart of the MC method, where comprises more regularly distributed points ; typically, the point set is constructed using the Halton sequence, the Sobol sequence, or the Niederreiter sequence 29. The resulting QMC sequences possess the low-discrepancy property, since the method explores the sample space in a more uniform manner. A multidimensional sequence can be obtained by using several onedimensional QMC sequences. Given that the error bounds of the best known QMC sequences have the asymptotic order, which is better than associated with MC, the QMC method can approximate the integral with a smaller error than MC if the sample size is sufficiently large. A drawback of the QMC method is that analyzing the accuracy of approximation is difficult. Since the method is based on deterministic sequences, statistical procedures for error estimation are not applicable. To overcome these difficulties, randomized QMC methods have been developed with the following properties: (a) Every element of the sequence has a uniform distribution over the unit cube; (b) the lowdiscrepancy property is preserved under the randomization. The central idea was to randomize the Halton/Sobol/Niederreiter sequence used to construct the point set. Guo et al 2 provide relevant details concerned with the construction of randomized QMC sequences. For a standard Gaussian pdf, a mapping (.) given by (39) is defined to project the integration domain to the unit hypercube, thereby allowing the application of low discrepancy point sets: =. (39) Suppose that, = 1, is a sequence of randomized QMC vectors in the unit hypercube [, 1). We transform each vector into a quasi-gaussian sequence, with mean and covariance matrix as follows: a) Apply Cholesky decomposition on, i.e., =. b) Transform to using the transformation (.), i.e.,, =,, = 1,,, = 1,,. c) Set = +. The quasi-gaussian points generated as above are used to approximate the integral in (23) as follows: h ( ;, ) h. (4) The approximations presented in this subsection will be used to evaluate the integrals to obtain the mean and covariance matrices during the prediction and filtering steps of CKF 5, UKF and QMC-KF. Fig. 1 Trajectory of the target over 6 time units. Results and Discussion In this section, we present results of the simulation experiments conducted to analyze the performance of the tracking algorithms described

INDIAN J. MAR. SCI., VOL. XX, NO. X XXXX 215 in the preceding section. Results for the EKF algorithm are also presented for comparison. We first describe the measurement scenario and define the performance indices used to analyze the performance. This is followed by presentation of simulation results and the related discussion. We consider the Pekeris model 19 of ocean consisting of an isospeed water layer over an isospeed fluid half space with the following parameters: water depth h = 75 m, sound speed in water = 15 m/s, sound speed in bottom = 17 m/s, density ratio / = 1.5, attenuation in bottom =.5 /λ, where λ = / is the wavelength in water. Other simulation parameters are assigned the following values: array depth = 2 m, number of sensors = 2, inter-sensor spacing = 15 m, initial target range = 5 m, bearing = 3, depth = 4 m, frequency = 5 Hz, = =.1 /, = 1s. The true initial state vector is = [25 3, 25, 4, 4, 4 3 ]. We tracked the target over a period of 6 seconds. For the first 3 seconds, we let = = = 4 m/s, = = = 4 3 m/s, course 6 w.r.t array axis. For the next 3 seconds, we let = = = 4 3 m/s, = = = 4 m/s, course 3 w.r.t array axis. The true trajectory of the target is shown in Fig. 1. The complex wavenumbers +, mode functions, and the number of modes can be computed using the Kraken model 22. For the present choice of parameters, we have = 3. The signal-tonoise ratio (SNR) at time is defined as ~ ( 12,12) and ~ ( 12,12). The initial covariance matrix is a diagonal matrix given by (,,,, ), where = =, =, = =. SNR =, (41) where [.] denotes the expectation operator. We tracked the target with an initial SNR of 12.72 db, performing 1 Monte Carlo (MC) simulations for each tracking algorithm. For each simulation, we chose the initial state vector = [,,,, ] to be a sample of the random vector [,,,, ] whose components are independent and uniformly distributed random variables: ~ ( 1,1), ~ ( 1,1), ~ (,75), Fig. 2 Estimate of the trajectory in Fig.1 by different tracking algorithms using acoustic field measurements. (a) EKF. (b) UKF. (c) CKF 5. (d) QMC-KF. The performance of the tracking algorithms was assessed by estimating the RMSE of the

NAGANANDA AND ANAND: ACOUSTIC SOURCE TRACKING USING FIELD MEASUREMENTS estimated target position as a function of time. The estimated RMSE at time is given by RMSE = +, (42) where (, ) denotes the target position estimate in the MC run and is the number of MC simulations. Sample trajectories estimated by the EKF, UKF, CKF 5 and QMC-KF algorithms, staring from the initial position estimate (6, 4), are shown in Fig. 2. The randomly chosen initial source position (, ) is identical in all the panels of this figure. For all the algorithms, the estimation error reduces rapidly and the estimated position is fairly close to the true position after only 5-6 time steps. But the subsequent evolution of the error depends on the algorithm. The variation of RMSE with time for different algorithms is shown in Fig. 3. Results obtained using bearing-only measurements are also shown for comparison. We have used the SIM (subspace intersection method) algorithm proposed by Lakshmipathi and Anand 3 to obtain the bearing estimates from the acoustic field measurement vector for =,,599. For the 2-sensor array and 12.72 db initial SNR, the bias and RMSE of the bearing estimates are approximately.28 and.7. It is seen from Fig. 3 that, for all the algorithms, acoustic field measurements yield a much lower RMSE than bearings-only measurements. For both measurement models, the QMC-KF algorithm provides the best performance, followed by CKF 5, UKF, and EKF in that order. Fig. 4 Time-averaged RMSE of estimated (x, y)-position versus SNR for acoustic field measurements. Fig. 3 Plots of RMSE of the estimated target position versus time for different tracking algorithms using acoustic field measurements. (a) EKF. (b) UKF. (c) CKF 5. (d) QMC-KF. In Fig. 4, we plot the time-averaged (over all 6 time units) RMSE for varying values of SNR for an array of = 2 sensors. We see that at low SNRs, QMC-KF performs better than CKF 5, while at moderately high SNRs their performances are

INDIAN J. MAR. SCI., VOL. XX, NO. X XXXX 215 almost similar. However, for all values of SNR, QMC-KF and CKF 5 are superior to UKF and EKF. A similar behavior is recorded in Fig. 5, where we plot the RMSE for varying number of sensors in the array for a fixed value of SNR. In Fig. 6, we plot the performance of all the four tracking algorithms using both acoustic field and bearing-only measurements. It is seen that all the algorithms perform much better with acoustic field measurements. measurements involves the hidden cost of bearing estimation at each time step. QMC-KF algorithm provides the best performance, followed by CKF 5, UKF, and EKF, in that order. But better performance is achieved at a higher computational cost. Fig. 5 Time-averaged RMSE of estimated (x, y)-position versus the number of sensors for acoustic field measurements. Conclusion We proposed the use of acoustic pressure field measurements, instead of the commonly used bearing-only or time delay measurements, for passive target tracking in shallow ocean. The complexity of the problem is due to the nonlinearity of the measurement equation. Three Gaussian approximation filtering algorithms, viz. UKF, CKF 5, and QMC-KF, were investigated. Advanced numerical integration techniques such as quadrature rules and the QMC method were employed for efficient computation of integrals of high dimensions involving Gaussian probability density functions. Simulation results show that acoustic field measurements yield significantly lower RMSE than bearing-only measurements for all the tracking algorithms considered in this paper. For each algorithm the computational cost of using acoustic field measurements is slightly higher than that of using the bearing-only measurements. But tracking based on bearing Fig. 6 Time-averaged RMSE of estimated (x, y)-position versus SNR for bearings-only and acoustic field measurements (a) EKF. (b) UKF. (c)ckf 5. (d) QMC-KF.

NAGANANDA AND ANAND: ACOUSTIC SOURCE TRACKING USING FIELD MEASUREMENTS The study in this paper was restricted to the problem of tracking a single non-maneuvering target, since the objective was to investigate the performance of different algorithms with acoustic field measurements. Work is in progress to extend the study to the problem of tracking maneuvering targets. It is also proposed to enlarge the scope of investigation to include non-gaussian measurement noise since ocean environments with highly non-gaussian and impulsive ambient noise are often encountered in practice. Acknowledgment This work was supported by Naval Research Board, Defence R & D Organization, Ministry of Defence, Government of India, through a grant for Project no. NRB-261/SSB/11-12. References 1. Moose, R. L. and Dailey, T.,Adaptive underwater target tracking using passive multipath time-delay measurements. IEEE Trans.Acoust. Speech Sig. Process., 33 (1985): 778-787. 2. 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