Enhanced Kalman Filter Algorithm Using the ... - Semantic Scholar

IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 34, NO. 4, OCTOBER 2009

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Enhanced Kalman Filter Algorithm Using the Invariance Principle Chensong He, Jorge E. Quijano, and Lisa M. Zurk

Abstract—Target tracking in multistatic active sonar systems is often limited in shallow-water environments due to the high level of bottom reverberation that produces false detections. Past research has shown that these false alarms may be mitigated when complete knowledge of the environment is available for discrimination, but these methods are not robust to environmental uncertainty. Recent work has demonstrated the existence of a waveguide invariant for active sonar geometries. Since this parameter is independent of specifics of the environment, it may be used when the environment is poorly known. In this paper, the invariance extended Kalman filter (IEKF) is proposed as a new tracking algorithm that incorporates dynamic frequency information in the state vector and uses the invariance relation to improve tracker discrimination. IEKF performance is quantified with both simulated and experimental sonar data and results show that the IEKF tracks the target better than the conventional extended Kalman filter (CEKF) in the presence of false detections. Index Terms—Invariance, Kalman filter, shallow water, underwater tracking.

I. INTRODUCTION

T

HE goal of an active sonar system is to detect and track targets (e.g., submarines) present in the water column. An active sonar system consists of a transmitter and a receiver: the transmitter generates an acoustic pulse with a characteristic shape and frequency band, and the receiver listens for the return of the pulse replicas from all the potential targets in the water column. In the detection stage, the system attempts to determine which returns are due to a true target versus those which are present in a noise-only scenario. One of the most challenging operational scenarios for an active sonar is the shallow-water environment (water column is on the order of hundreds of meters in depth) because the acoustic energy is reflected off the air–water interface and the water–sediment interface (reverberation) as well as from inhomogeneities in the water column, and these returns can be mistaken for true targets. The high number of false alarms complicates the tracker logic and decreases performance.

Manuscript received August 18, 2008; revised March 11, 2009; accepted June 02, 2009. First published October 30, 2009; current version published November 25, 2009. This work was sponsored by the U.S. Office of Naval Research (ONR). Associate Editor: D. A. Abraham. C. He was with the Department of Electrical and Computer Engineering, Portland State University, Portland, OR 97201 USA. He is now with Apple, Inc., Cupertino, CA 95014 USA. J. E. Quijano and L. M. Zurk are with the Department of Electrical and Computer Engineering, Portland State University, Portland, OR 97201 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JOE.2009.2028058

One approach to mitigate this problem is to employ propagation physics to identify true target returns by predicting the propagation-imposed signature (e.g., depth-dependent mode structure). However, the success of these techniques often depends critically upon having accurate knowledge of the ocean environment, and knowledge of this nature is not commonly available. This realization has motivated interest in robust techniques which are not dependent on environmental knowledge, and one notable example is the use of the invariance principle in passive sonar. In this paper, a tracking algorithm for bistatic active sonar geometries based on the waveguide invariance principle is proposed. The invariance principle was first derived by Chuprov [1] and has been utilized in passive sonar applications [2], [3]. The waveguide invariant provides a relationship that describes the acoustic intensity of a broadband source as a function of range and frequency. For a moving source, the invariance is observed as striations of constant intensity in a time/range-frequency graph, and it can be shown that for shallow water, the slope of these striations is determined by a single scalar , which is approximately unity. In recent experiments, a similar striation structure has been observed in monostatic active sonar configurations [4], [5]. In the more complicated bistatic case, the transmitter and the receiver are located separately as shown in Fig. 1 and the transmitted signal goes through two different paths as well as targetdependent scattering before being received by the receiver. The , where and are the bistatic range is defined as ranges from the transmitter to the target and the target to the receiver, respectively. A mathematical description of the time-frequency striations observed in a general bistatic geometry (i.e., a bistatic invariant) has been proposed by Zurk et al. [6]. The invariance has also been observed in experimental and simulated reverberation data, and Goldhahn et al. [7] has used this to estimate the reverberation level to improve the probability of detection without increasing the probability of false alarms. In this paper, the bistatic invariance principle is utilized as a constraint to formulate a new tracking algorithm that improves the performance of the extended Kalman filter (EKF). The EKF consists of a state vector, which is defined by the dynamics of the target (i.e., relationship between time, velocity, and acceleration) and an observation vector, which is determined by the active sonar measurements of the position of the target (horizontal range and azimuth angle). The basic recursive algorithm of any Kalman filter performs a prediction of the future position of the target, which depends on the state vector of the current position and the dynamics of the target, and then it performs an update based on the actual measurements. Under this logic, the accuracy of the prediction step relies on the model adopted to

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For a point source located at range and depth , the acoustic at depth can be expressed using normal pressure time convention modes. Assuming an

(1)

Fig. 1. Top-down view of the bistatic active sonar network used in this paper. r r . The bistatic range r is defined as r

= +

where is the horizontal wave number for the th propagating mode that depends on the frequency is the corresponding depth-dependent mode function, and is a normalizing constant. The variable is introduced as a shorthand notation for convenience. The acoustic intensity

can be written as describe the dynamics of the target. In this paper, the accuracy of the target dynamics is improved by incorporating an additional constraint that enforces the invariance, and the new method is shown to reduce the number of false detections that a tracker must contend with. This paper is organized as follows. Section II is a review of the derivation of the invariance principle for passive and active configurations. In Section III, the algorithm of the conventional extended Kalman filter (CEKF) is summarized and the equations for the proposed invariance extended Kalman filter (IEKF) are introduced. In Section IV, active sonar data from the 2004 Deployable Experimental Multistatic Undersea Surveillance (DEMUS’04) experiment are used, and in Section V, simulated data are used to compare the IEKF performance to the CEKF.

(2) Equation (2) shows that pairs of normal modes add construc, which are also a function of tively at ranges frequency due to the wave numbers and . The location of those local maxima as a function of range and frequency results in striations of constant intensity that can be described by (3) After substitution of (2) into (3), the invariant [8]

is defined as

(4) II. INVARIANCE PRINCIPLE The concept of an invariance principle explains the spatial interference patterns resulting from the coherent addition of propagating normal modes. As described by Brekhovskikh and Lysanov [8], the propagation of broadband underwater signals is characterized by a range-frequency structure which is approximately invariant to small perturbations of underwater environmental parameters. When the acoustic intensity is plotted as a function of time (delay time is related to range for active sonar) and frequency, this structure is observed as striations whose slope is proportional to a scalar known as the invariant. There has been extensive work on application of the invariant to single-path passive data, and recent work [4], [5], [7] suggests that a similar invariant applies to the more complicated active sonar case. The active case is more complicated because acoustic energy travels over two (potentially different) propagation paths, and the observed response is influenced by the target scattering. In both cases, the equations that describe the waveguide invariant can be derived from normal mode propagation theory. Some of the experimental results from monostatic active sonar data are presented in [5].

where and represent small perturbations in range and frequency, and and are the average of the group and phase speed, respectively. For many shallow-water environments, it [8]. has been demonstrated that For a bistatic configuration, the acoustic wave travels over two different paths instead of one, and the modes interact when they scatter at the target. The coupling of incident and scattered modes at the target makes the situation more complicated than that in a single path. The normal mode expression of the bistatic scattered pressure can be written as

(5) is the depth of target, represents where the scattering matrix [9], [10], is the grazing angle correis the azimuth angle of sponding to the incoming th mode, and are the corresponding grazing the incoming wave, and and azimuth angles for the scattered mode. (Note that Doppler

HE et al.: ENHANCED KALMAN FILTER ALGORITHM USING THE INVARIANCE PRINCIPLE

shifts due to target motion are not included in the above expression.) Substitution of the bistatic intensity into (3) results in the bistatic invariant [6], which is a complex expression that depends not only on the average phase and group speed as in the passive case, but also depends on the scattering properties of the target. The invariant can be related to a moving target by defining as the discrete time at ping

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which is the a priori estimate of compute given the previous state . When the new observation at is available, the a priori estimate is updated as time

where

is known as the Kalman gain and

and which is the difference between the frequencies of two points along a chosen striation. The new form of (4) can be written as (6) , and for some cases Previous analysis indicates (e.g., in monostatic geometries where ), it can be shown . In this paper, we assume . A topic of futhat ture research is the variation of as a function of bistatic geometry and target scattering properties. If this variation is known, it could potentially be incorporated into the tracker formulation.

represents the new information or innovation [12]. In the following sections, dependence on time will be denoted in terms of the index for brevity [for example, ]. In the proposed application, the position and velocity of a moving target in a Cartesian coordinate system are estimated from active sonar measurements of bistatic range and bearing angle. More literature on tracking target using Kalman filter can be found in [13] and [14]. A. Conventional Extended Kalman Filter For the CEKF, the state vector is defined as

(9)

III. KALMAN TRACKING In this section, the details of the CEKF and the proposed IEKF trackers are presented. The Kalman filter is a widely used recursive filter that can estimate the state of a dynamic system from a series of noisy indirect measurements. The state vector is defined as a group of variables that represents quantities that must be estimated as a function of time (i.e., position, velocity, acceleration, etc.). The system is modeled with a physics-based model and the state vector is a function of the previous state affected by white Gaussian noise (WGN) [11]

(7)

where is, in general, a nonlinear, time-varying function and is the process noise and it is assumed to be WGN. The observation vector corresponds to (noisy) measurements of indirect quantities that are related as to

(8)

where is the nonlinear observation function and is a WGN process that accounts for errors introduced by the measurement based on system. The CEKF algorithm estimates and in a two-step process that consists of prediction and update. During the prediction step, (7) is used to

(10) is the position of the target in Cartesian where and represents the corcoordinates at time responding instantaneous velocity in the - and -directions. is the process noise covariance matrix and is the amount of time between pings. The transition is based on the assumption that the targets have small acceleration which is a reasonable assumption for underwater tracking. Therefore, and [17]. The measurement model relates polar observations to the state equation as [15] (11) (12) where is the bistatic range of the target is the bearing of the target as shown in Fig. 1. In the and measurement error covariance matrix , the symbols and represent the variance of range and bearing, respectively, and is the standard range-bearing covariance. The observation function is derived from the geometry of a general bistatic active sonar system as shown in (13) at the top of the next page, and are the active sonar measurements of the where and bistatic range and bearing of the target and are the locations of the receiver and the transmitter, respectively. The function is the four-quadrant arctangent function. The CEKF procedure is described in the following steps [11].

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(13)

a) Prediction: 1) Predict the state ahead (14) The state vector at time is estimated based on the filtered state at time via the transition function defined in (9). 2) Predict the error covariance (15) where

parameters and close to unity. For a broadband waveform, the invariance principle suggests a method to constrain the track hypothesis space by relating the frequency-dependent signal characteristics to physically realizable target ranges. To enhance the performance of the CEKF, an IEKF incorporates an additional parameter, frequency, in the state vector. The instantaneous frequency content of the received signal is related to the target position via the invariance relationship. The time-frequency relationship predicted by the invariance principle utilizes additional information about the target and the accuracy is expected to be enhanced. For the IEKF, the state vector is

(16) (22) The error covariance at time is estimated based on the filtered error covariance at time and this prediction is also related to the transition function . b) Update: 1) Update the Kalman gain

(17)

where an additional parameter is incorporated to represent the time-dependent frequency at ping period. The parameter represents the dynamic estimate of the frequency in hertz derived from the invariance relationship and will be discussed later in this section. The process noise covariance matrix for the IEKF is also extended to

where is the Jacobian of as shown in (18) and (19) at the bottom of the page. Then, the Kalman gain is used in the next step to correct the prediction. 2) Update the state

(20)

(23)

The filtered state vector at time is given by a combination of prediction- and observation-based update. 3) Update the error covariance matrix

where the variables , and are the variance of the process noise for the and positions and the and components of the velocity, respectively, while can be used to take into account any mismatch between the assumed and true values of for certain environment. From the definition of invariance (6)

(21) The error covariance matrix is updated in the same manner as the state vector.

(24)

B. Invariance Extended Kalman Filter As discussed in Section II, is a scalar that is assumed to be approximately invariant to perturbations in the environmental

so that expression of

is given by (25)

(18)

(19)


where

since they this work, intensity maxima are used to estimate provide the greatest signal-to-noise ratio (SNR). Further issues are discussed in Section IV-B. on the estimation of (26)

and (27) Then, under the assumption could be derived in terms of

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, an expression for

(28) Thus, for the IEKF state–space representation, the state transition is given by

(29)

In the transition, the first four variables in the state space are updated based on the same transition of the CEKF (9). The , is updated via fifth variable, the time-dependent frequency (28). The observation vector is given by (30)

(31) where (32) shown at the bottom of the page holds. The observation is similar to that in the CEKF, exwith corresponding addicept there is one more variable tional covariance terms in the measurement noise. Since is the time-dependent frequency and its value is unchanged in the mapping of polar to Cartesian coordinate, it does not depend on the measurement coordinate system and the observation for is itself. must be measured from the To implement the IEKF, spectrogram at each ping. The current method of doing this is to track the striations in the active spectrogram obtained as described in [5]. While multiple striation lines may be present, in

C. Tracker Logic The tracking algorithm includes logic-based track initiation and termination following the logic described in [17]. For each ping, it is assumed that there are several detections that pass the detection stage. Those detections are presented to the tracker as to form possible tracks. These detections the observation may contain both the targets as well as some false detections. Each of these detections is used to initiate an unconfirmed track. One unconfirmed track has several associated detections which are the best matches of predictions. Also, it is assumed that the moving target has limited speed. Thus, if the best match at ping indicates a position too far from the position of the target at , the detection is considered missed at ping . We ping use the criteria that each track is designated as confirmed if it contains at least detections and less than missed detections within pings. Specifically, a track can be: • confirmed: if contacts are associated within pings; in adjacent snapshots of the received other words, within signal, there are detections that exceed the threshold and are picked for this track; • discarded: if less than contacts are associated within pings; • terminated: if the track is confirmed and after consecutive missed detections. To examine whether a detection belongs to a track, is defined as [17]

(33) and the best match is the one that has the smallest . For the CEKF, the detection process (33) will select the one that best matches the Kalman prediction of the physical position of the target. For the IEKF, because the state vector con, the detectains the additional time-dependent frequency tion process will select the one that best matches the Kalman of the target. prediction of the position and the scans, the algorithm checks each unconfirmed For each track to see whether there are contacts associated within these scans. If there are associated contacts, then this track is confirmed. Otherwise, this track is discarded. For each confirmed track, if consecutive detections are missed, this track is terminated. Basically, the values of and determine how many false detections are tolerated in one track. A large with a small means that the tracker is tolerant of false detections and means thus more tracks are picked. A small with a large

(32)

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Fig. 2. DEMUS receiver consists of nine staves arranged at angular separation of 40 . Seven sensors are placed on each stave.

the tracker is not tolerant of too many false detections and thus less tracks are picked. IV. RESULTS USING EXPERIMENTAL DATA

to the data to compensate for the lower false detection rate due to the decay in reverberation during the echo repeater delay. Two types of false detections are added. One is time-invariant false detections which are due to fixed bottom scatterers such as rocks, sea mounts, or coral reef. These false detections are unchanged over pings and independently drawn from a uniform distribution within the interval [0 km 15 km] in -direction and [0 km 20 km] in -direction. The other is time-varying false detections which are due to nonstationary effects in the water column. These false detections are independently drawn from the same distribution but each ping represents an independent realization of the random process. For each ping, 50 time-invariant false detections and 30 time-varying false detections are added. The series of detections for each ping are the input to the CEKF and the IEKF tracking algorithms as the observation . For the IEKF, the spectrogram corresponding to the target is also required to determine the time-dependent frequency corresponding to constant intensity striations. Estimation of this frequency is discussed in Section IV-B.

A. Description of the DEMUS’04 Experiment

B. Striation Extraction From Spectrogram Data and Simulation Setup

In this paper, active sonar data from the DEMUS’04 experiment are utilized to generate a series of detections that are used with different tracking algorithms to compare performance. The geometry of one run from the DEMUS’04 data set is appropriate for invariance tracking but, as will be explained in Section IV-B, the two-minute pulse repetition interval (PRI) was too long to be able to clearly see striation patterns in the spectrograms. In this section, DEMUS’04 data are used to show improved tracking capability of the IEKF and a more comprehensive evaluation is presented in the following section using simulated data. The DEMUS’04 Joint Research Project [NATO Undersea Research Center, the U.K. Ministry of Defense (U.K. MOD), and the U.S. Office of Naval Research (ONR)] undertook a sea test in September 2004 in the Malta Plateau area of the Mediterranean Sea. A diagram of the experiment is shown in Fig. 1, in which a transmitter generates a broadband signal every 2 min and the target is a vessel carrying an echo repeater with an 8-s delay. The 8-s delay is introduced to separate the reverberation from target signature (reverberation decays during the delay time). The receiver used for the DEMUS’04 experiment has sensors arranged along the circumference of circles of varying radii. As is shown in Fig. 2, the DEMUS’04 receiver consists of nine staves with seven sensors apiece and an additional sensor at the center for a total of 64 channels. With this sensor arrangement, there are limited aperture ambiguities and the receiver can be steered in any direction from 0 to 355 . The receiver time-series data are sampled at 732.42 Hz and consist of the direct arrival from the transmitter, the scattered energy from the target, and noise (both reverberation and ambient noise). The raw time series were match-filtered and beamformed with an angle resolution of 5 . In this paper, the detections are generated by first setting a threshold according to the rolloff of the reverberation noise and then identifying signals which exceed this threshold as a detection. In addition to actual detections, false detections are added

Fig. 3(a) shows the bistatic range of the target as a function of time and Fig. 3(b) shows the spectrogram produced from the data from this geometry. This target spectrogram was generated by performing a spectral decomposition of the DEMUS’04 match-filtered output at the time-dependent target range and bearing. Target location is known from the global positioning system (GPS) data, but the location of the target return can differ from this position (e.g., due to beam migration from soundspeed mismatch). To locate the precise target return, a local search is performed jointly in bearing and range to find the high SNR target return. Then, a short-time Fourier transform (STFT) is applied on the target signal which is centered at the estimated arrival time and has a duration of 0.1 s. The final result is normalized to the maxima. Looking at Fig. 3(b), it is difficult to claim that striations are present. One possible reason is that for the DEMUS’04 experiment, the PRI is 2 min long which might be too long to produce a spectrogram in which striations can be visualized. To test this hypothesis, a simulated spectrogram is generated to compare with the experimental spectrogram shown in Fig. 3(b). To generate the simulated spectrogram, the bistatic pressure is computed using normal modes as indicated in (5), using the GPS and . The mode functions data of the target to determine and wave numbers were generated as described in [5] using the KRAKEN normal mode program [16] and the parameters shown in Table I, with PRIs of 2 min [Fig. 3(c)] and 0.5 min [Fig. 3(d)]. In the data-generated spectrogram, reverberation is observed particularly in regions of low target SNR. To make the target structure apparent, the noise and the reverberation are not included in both of the simulations (thus the simulation is expected to have more dark regions than the real data). Both the experimental data and simulated spectrograms produced with a 2-min PRI [Fig. 3(a)–(c)] have a grainy nature that makes the striations difficult to see, even in the absence of reverberation noise. It also means that standard image processing


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TABLE I MAIN EXPERIMENTAL PARAMETERS FOR THE DEMUS’04 EXPERIMENT

Fig. 3. (a) GPS bistatic range of a target track in DEMUS’04 data. The range is between 9.87 and 18.58 km. (b) Spectrogram produced from an STFT of the match-filtered DEMUS’04 data for a target whose bistatic range is shown in Fig. 3(a) with a PRI of 2 min. (c) Simulated spectrogram of the DEMUS’04 data using the range shown in Fig. 3(a) and a PRI of 2 min. Noise and reverberation are not added in this simulation. (d) The same as (c), but with a PRI of 0.5 min. One of the striations is selected by searching a local maximum at each ping from an initial frequency 3240 Hz at time t 27 min.

=

techniques for tracking the striations (e.g., Hough transform [18]) may not perform well. This may imply that consideration

of shorter PRI is a factor to consider for future active sonar systems that would utilize IEKF tracking. Alternatively, frequency estimation techniques that are not based on image processing or the design of frequency-sensitive sonar waveforms may be applicable to provide the ability to estimate dynamic spectral content. Whatever the estimation technique is used, anticipated error from this process may be included through the error covariance matrices in the Kalman filter framework and estimates with high associated error will have less corresponding impact on the tracker output. In this work, the emphasis is to evaluate the performance of the IEKF with the assumption that an adeis available. Thus, for the results in this quate estimate of section, a spectrogram is produced from KRAKEN simulations computed with a shorter 0.5-min PRI, as shown in Fig. 3(d), and this spectrogram is used to estimate the time-dependent target frequency. In a fully developed striation processing algorithm, logic would need to be included to determine where to initiate the striation estimation. In this paper, the initial frequency of a given striation is handpicked at the initial ping of this striation, and for the remaining pings, an algorithm searches on the spectrogram to follow the striation of constant intensity. This algorithm searches on a small frequency region that is centered corresponding to the previous ping. at the frequency Fig. 3(d) shows the result of applying this procedure to the striation with initial frequency 3240 Hz. As mentioned previously, 80 additional false detections are added each ping and each of these detections has a value for in the state vector. To generate for the stationary noise, the simulated spectrograms are generated with KRAKEN assuming the scatterer is at the bottom of the water channel; then . a maximum search is applied on the spectrogram to get The second type of additive noise is nonstationary noise whose location is random and it varies form ping to ping. For those false is modeled as a detections incurred by this kind of noise, uniformly distributed random variable within the bandwidth. C. Performance of the CEKF Tracker Versus the IEKF Tracker The performance of the IEKF and the CEKF on the DEMUS’04 data is illustrated in Fig. 4. It is assumed that the position of the first true detection is known and both trackers are initialized from this position. The true detections are generated by searching local maxima in the receiver data within a 2-s window centered at the location acquired by GPS. The output of the trackers shows that the IEKF yields a track that remains

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Fig. 4. CEKF and IEKF tracking results on one of the DEMUS’04 tracks. The tracker is initiated from the initiation of the real target track. The CEKF misses 21 true detections and the IEKF misses three true detections.

around the true GPS position of the target, while the CEKF tends to diverge. For this example, the CEKF missed 21 true detections within 23 pings while the IEKF missed 3 true detections. Because of the false detections, the CEKF is not able to track the target and uses false detections as measurements for the update stage of the Kalman filter; these detections are selected by the tracker logic and presented to the tracker since the positions of those detections have the smallest value [see (33)]. On the other hand, with the additional information from the time-dependent frequency, the IEKF is able to extract most of the true target detections, because false detections are not consistent with the predicted frequencies and thus give a large value. In this case, there is only one target and single-target track logic is applied. To illustrate the multiple-target tracking performance, the simulation is extended to the more complicated case where more than one target may exist. V. MULTIPLE-TARGET SIMULATION RESULTS In this section, the multiple-target tracking performance of the IEKF is illustrated with simulated data. With a multiple-target assumption, more uncertainties are introduced in the process. These additional uncertainties include the following. • The number of targets is unknown. With the single-target assumption, there is always one target (or no target) and the tracker needs to determine if new detections are associated with this target. With the multiple-target assumption, detections come from multiple targets and there are multiple choices to associate with these detections.

• More confusion from false detections. With the singletarget assumption, false detections are eliminated if the positions of these detections conflict with a certain track. With the multiple-target assumption, false detections can be connected to multiple tracks and thus they are more likely to be accepted. • Initial positions and time of targets are unknown. It is not realistic to assume that the initial positions of all the targets are known or that all the targets start to move at the same time. All the parameters used in this simulation are similar to Section IV and the PRI is 0.5 min (total time is 27 pings). The relative locations of the transmitter, the receiver, and the target are shown in Fig. 1. The performances of the IEKF and the CEKF are computed by averaging over 100 Monte Carlo realizations, in which the same target track is used with different realization of the random measurement noise/false detections drawn from the same distributions. The Monte Carlo simulation includes the following five steps. 1) Generate simulated track. The simulated target track is generated to test tracker performance. Bistatic range and bearing are calculated from the geometry. from the spectrogram. The spectrogram is 2) Obtain generated by using the KRAKEN normal mode program from the target track to compute the bistatic preswith is obsure over the band 3–5 kHz, and the value for tained by local maxima search as described earlier. 3) Generate all detections (including true detections and false detections). The true detections are produced by adding to the simulated range random measurement noise and bearing, as indicated in (11) for the CEKF and in (30) for the IEKF. The measurement noise is independently drawn from zero-mean Gaussian distributions with 500 m for the bistatic range, 2 for the bearing, and 2000 Hz for the frequency measured from the spectrogram. The false detections are generated as mentioned in the previous section. 4) Run the IEKF and the CEKF on this simulated data set. In all cases, the variance terms of the process noise in (23) are 1 m, 1 m/s, and 1 Hz. The first four terms are small due to the assumption of nearly is expected to be constant velocity of the target, while small if is truly an invariant (i.e., not strongly affected by small perturbations in the environment), but more experimental work is required to determine the low and high boundary for this parameter. 5) Repeat step 3) and step 4) 99 times and average the result over the 100 realizations. The Monte Carlo process is applied on five different simulated tracks as shown in Fig. 5. All the five tracks use the same sonar system geometry but the movement of the target varies. Also, to illustrate the performance under different noise levels, varying amounts of false detections are injected. Fig. 6(a) shows one of the simulated tracks and Fig. 6(b) shows the bistatic spectrogram obtained for this track. Fig. 6(b) obtained from also shows the time-dependent frequency the track using the estimation approach described previously; this is used as an input to the IEKF.


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Fig. 5. Five simulated tracks that are used to compare the performance of the IEKF and the CEKF.

Fig. 6. (a) Simulated track shown in a top–down plot. (b) Spectrogram of the target track in (a): the time-dependent frequency used in the IEKF is determined by a local search and it is shown with an overplotted line.

Fig. 7(a) shows an example from one realization in which the CEKF algorithm forms tracks from both the true observations and the false detections. While the true track is generated, other tracks connecting false detections are also selected. Fig. 7(b)

Fig. 7. One example of the tracking results of the CEKF and the IEKF using the DEMUS’04 experiment geometry. (a) The CEKF finds eight false tracks and misses 13 true detections. (b) The IEKF finds no false track and misses four true detections.

shows the IEKF result of the same simulated data. Due to the existence of the time-frequency constraint in the state space, the

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Fig. 8. Number of selected (a) false detections and (b) false tracks produced by the CEKF and the IEKF are shown for the five simulated target tracks in Fig. 5. Results of Monte Carlo simulation are shown plotted for a total of 70, 100, 130, and 160 false detections with 50 time-invariant false detections in each ping.

IEKF tracker gets more information about the position of the target and thus selects less false detections as compared to the CEKF. The performance of both trackers is assessed by varying the number of false detections presented to the tracker and measuring the ability to reject them. Fig. 8 shows the relative performance of the trackers for 70, 100, 130, and 160 total false detections (number of time-invariant false detections remains 50) in terms of the ability to discriminate true targets from false detections. Although the number of false detections varies, for all cases examined in this simulation and for all of the five tracks, the IEKF selects fewer false detections (shown in left plot) and fewer false tracks (show in right plot) than the CEKF. Furthermore, for both the CEKF and the IEKF, the number of selected false detections and false tracks increases while the number of false detections increases. However, the number of accepted false detections and false tracks of the IEKF increases less than the CEKF and thus the IEKF has better performance compared to the CEKF for any level of false detections.

concept of using the invariance principle to enhance tracker performance, especially in shallow water where the reverberation noise is more significant. Striation patterns in the DEMUS’04 active sonar data were not clearly observable, and this was attributed to the relatively long PRI used in the collection. The performance of the algorithms was evaluated by estimating frequency patterns from simulated spectra and injecting clutter at various levels to quantify noise performance. While this provided an assessment of the efficacy of the proposed IEKF, it did not provide a data demonstration of the striation patterns in truly bistatic geometries (Quijano and Zurk [5] showed multiple instances of active striations in monostatic active data). Performance of invariance-based algorithms such as IEKF is clearly dependent on the existence and robustness of this structure. The formulation presented in this paper accounts for in the the uncertainty of the invariant through the term process noise. This uncertainty could be in the form of an offset (i.e., the assumed value of is different from the invariant corresponding to some particular environment) or due to nonstationarity of as a result of large perturbations in the environment. In both cases, theoretical work has been conducted for the passive invariant to determine the range of possible values for [19] and its stationarity [2], but similar studies are yet to be done for the active invariant. In this paper, it has been assumed that and 1 Hz, corresponding to a nearly constant invariant. The performance of the IEKF is also determined by the error from the spectrogram, associated with the extraction of in the measurement which is included in the formulation as error covariance matrix. When the quality of a target spectrogram makes estimation difficult, large observation errors will occur and the Kalman filter will rely more heavily on the physics-based model in (7) rather than in the observations, and an improvement in the performance is expected for smaller . In any case, exploration of the effect of values of and deserves further analysis to fully determine how the performance of the IEKF degrades due to mismatch on the assumed invariant and due to inaccurate estimation of . Another area of future research is to improve the ability to track the time-frequency striations. The algorithm presented in this paper is a relatively simple single-frequency line tracker, but much more sophisticated approaches may be possible.

VI. CONCLUSION

ACKNOWLEDGMENT

In this paper, simulated and experimental sonar data have been processed and presented to compare the efficiency of the CEKF and the IEKF. Using invariance frequency information as a new variable in the state vector gives more information about the position of the target and hence the IEKF tracks the target better. In both the simulated and experimental data, the IEKF selects fewer false detections, fewer false tracks, and misses fewer true detections. The results show that with the single-target assumption, when false detections exist, the CEKF may lose the target while the IEKF tracks the target more accurately. With the multiple-target assumption, the IEKF tracks the target better and selects fewer false tracks. When the simulation is extended to higher noise level (more false detections), the IEKF performance is even better compared to the CEKF. This supports the

The authors would like to thank NATO Undersea Research Center (URC), U.K. Ministry of Defense (U.K. MOD), and the U.S. Office of Naval Research (ONR) for providing the DEMUS’04 data. This DEMUS’04 data set was made possible by the Deployable Experimental Multistatic Surveillance Joint Research Project (JRP), a collaboration between NATO URC, U.K. MOD, and the U.S. ONR. REFERENCES [1] S. D. Chuprov, “Interference structure of a sound field in a layered ocean,” in Ocean Acoustics. Current State, L. M. Brekhovskikh and I. B. Andreeva, Eds. Moscow, Russia: Nauka, 1982, pp. 71–91. [2] A. Thode, “Source ranging with minimal environmental information using a virtual receiver and waveguide invariant theory,” J. Acoust. Soc. Amer., vol. 108, pp. 1582–1594, 2000.


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Chensong He received the B.S. degree in electrical engineering from Nanjing University, Nanjing, Jiangsu, China, in 2006 and the M.S. degree in electrical and computer engineering from Portland State University, Portland, OR, in 2008. He joined the Northwest Electromagnetics and Acoustics Research Laboratory (NEAR-Lab) in 2006. During his two years at NEAR-Lab, he worked on advanced signal processing, image processing, and underwater target tracking algorithm. He is now working at Apple Inc. as a Software Engineer.

Jorge E. Quijano received the B.E. degree in electrical engineering from the Instituto Tecnológico de Costa Rica, Costa Rica, in 2001 and the M.S. degree in electrical engineering from Portland State University, Portland, OR, in 2006, under a Fulbright Scholarship. He is currently working towards the Doctorate of Philosophy in electrical engineering at Portland State University. He was awarded a Graduate Traineeship Award by the U.S. Office of Naval Research to support his ongoing research on ocean bottom scattering at midfrequencies. In 2005, he joined the Northwest Electromagnetics and Acoustics Research Laboratory, Portland, OR, as a Researcher in areas of underwater acoustics. His current research interests include deconvolution, scattering from random media, and wave propagation.

Lisa M. Zurk received the B.S. degree in computer science from the University of Massachusetts, Amherst, in 1985, the M.S. degree in electrical and computer engineering from Northeastern University, Boston, MA, in 1991, and the Ph.D. degree in electrical and computer engineering from University of Washington, Seattle, in 1995. Currently, she is an Associate Professor at the Electrical and Computer Engineering Department, Portland State University, Portland, OR, and the Director of the Northwest Electromagnetics and Acoustics Research Laboratory. Her research work includes wave propagation modeling for electromagnetics and acoustics and development of signal processing techniques for radar and sonar.