Duan et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890664]
Published Online 18 July 2014
Moving source localization with a single hydrophone using multipath time delays in the deep ocean Rui Duan, Kunde Yang,a) Yuanliang Ma, Qiulong Yang, and Hui Li Institute of Acoustic Engineering, Northwestern Polytechnical University, Xi’an 710072, China [email protected], [email protected], [email protected], [email protected], [email protected]
Abstract: Localizing a source of radial movement at moderate range using a single hydrophone can be achieved in the reliable acoustic path by tracking the time delays between the direct and surface-reflected arrivals (D-SR time delays). The problem is defined as a joint estimation of the depth, initial range, and speed of the source, which are the state parameters for the extended Kalman filter (EKF). The D-SR time delays extracted from the autocorrelation functions are the measurements for the EKF. Experimental results using pseudorandom signals show that accurate localization results are achieved by offline iteration of the EKF. C 2014 Acoustical Society of America V
PACS numbers: 43.60.Gk, 43.30.Wi [CG] Date Received: May 24, 2014 Date Accepted: July 9, 2014
1. Introduction In the deep ocean, a useful acoustic duct for moderate sound propagation is the reliable acoustic path (RAP).1,2 Below the deep sound channel, the depth at which the sound speed equals the maximum speed in the vicinity of the surface is referred to as the critical depth. RAP is the direct (D) and possibly the surface-reflected (SR) path under benign surface conditions between a receiver below the critical depth and a source at moderate range.3 A high signal-to-noise ratio (SNR) environment is the advantage RAP offers for source localization.2 Hereafter, the ocean environment that permits the propagation of RAP is referred to as the RAP environment. Single-hydrophone localization can be achieved by exploiting ocean propagation attributes such as multipath arrival time,4 modal dispersion,5 and the waveguide invariant.6 For a broadband source, the time delays between multipath arrivals can be obtained from the received signal’s autocorrelation functions. Using these time delays, the source location can be determined using a single hydrophone.4,7 However, it was observed in our experimental data that in the RAP environment, because of the large grazing angle at the ocean bottom interface, the reflection losses of bottom-interacting arrivals are large. Accordingly, the time delays, except for those between the D and SR arrivals, cannot be extracted under low SNR using the autocorrelation method. However, D-SR time delays alone are not sufficient for localization, when using a single hydrophone, which is why Nosal and Frazer8 used five hydrophones to track a sperm whale where only the D-SR time delays were exploited. In contrast, our experiment had one hydrophone and still only observed the D-SR time delays. The motivation of this study was the need to localize the source under these two constraints. Note that the source moved uniformly during our experiment, which provided the D-SR time delays versus duration. This paper demonstrates that the changing D-SR time delays contain enough information to determine source
a)
Author to whom correspondence should be addressed.
J. Acoust. Soc. Am. 136 (2), August 2014
C 2014 Acoustical Society of America EL159 V
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.235.251.160 On: Sat, 20 Dec 2014 16:59:08
Duan et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890664]
Published Online 18 July 2014
location and speed. A general overview of our method indicates that the autocorrelation functions are used to track the D-SR time delays, and then the localization problem is cast in the form of an extended Kalman filter (EKF) model.9 Gong et al.10 examined four approaches for instantaneous passive source localization using a towed horizontal array and showed that the EKF was a favorable approach for localizing moving sources. Note that the Kalman Filter family cannot be used if the nonlinearity of a model is significant. Particle filtering is an appropriate technique in dealing with nonlinear and/or non-Gaussian problems.11–13 2. Theory and simulation Some assumptions in this model must be emphasized beforehand, considering that the method is constrained by these assumptions. First, the source is moving at an almost steady horizontal speed. However, the model allows some fluctuation in speed because speed is assumed to be a random variable obeying a Gaussian distribution. Second, the source is moving in a radial direction, keeping its bearing unchanged. In practice, the source is likely to change the bearing as well as the range simultaneously. In that case, multipath time delays, besides the D-SR time delay, would be required to accomplish localization. For example, tracking sperm whales with as few as one hydrophone was achieved on the basis of the multipath time delays in the eastern Gulf of Alaska, with water depth of 460 m.4 Sperm whales transmit impulsive sounds called “clicks,” which makes the received signals ideal candidates for extracting multipath time delays. Since the broadband clicks are very short in duration (e.g. 25 ms8), the multipath arrivals can be observed using the spectrogram method.4 In contrast, the pseudorandom sequence in our experiment has a narrow bandwidth (100 Hz) and its time duration is 10 s. Therefore, using the autocorrelation method, only the D-SR time delays were obtained and can be used for the localization. Sections 2.1–2.3 present the measurement, state equation, and measurement function for the EKF model. The simulations are presented to illustrate the reasons for the localization algorithm. 2.1 Measurement for the EKF For a hydrophone below the critical depth, a source near the surface, and ranges within one-half of a convergence zone, the D, bottom reflected (BR), SR, surfacebottom-reflected (SBR), and other multipath arrivals are expected to depend on the environmental parameters. The time delays between them can be observed from the peaks of the autocorrelation function, which is defined as ð Ts 1 RðsÞ ¼ sðt þ sÞ sðtÞdt; (1) T s t¼0 where T is the duration of the selected time window. The time window must be shorter than the time of any significant change in the source range. Besides it should be much longer than the multipath time delays. An appropriate time window is several seconds. sðtÞ is the received signal in the time window. The normalized autocorrelation function is denoted as ^ ðsÞ ¼ jRðsÞ þ j H ðRðsÞÞj ; R Rð0Þ
(2)
where HðÞ is the Hilbert transform and j j is the absolute value, the numerator is the pffiffiffiffiffiffiffi envelope of RðsÞ, j equals 1, and Rð0Þ is the power of the received signal. Our simulations show the autocorrelations of the received signals when a broadband source moves horizontally toward the hydrophone head. The frequency band of the signal is 700–800 Hz, and white Gaussian noise is filtered by a bandpass filter to generate this signal. The source depth is assumed to be 110 m. A typical Munk sound speed profile and an ocean depth of 4500 m are chosen for the simulations. The
EL160 J. Acoust. Soc. Am. 136 (2), August 2014
Duan et al.: Single-hydrophone source localization
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.235.251.160 On: Sat, 20 Dec 2014 16:59:08
Duan et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890664]
Published Online 18 July 2014
sound speed, density, and attenuation coefficient of the ocean bottom are 1600 m/s, 1.6 g/cm3, and 0.14 dB/k, respectively. The ray propagation model14 is used. The normalized autocorrelation functions of signals on a hydrophone at 4200 m depth versus the horizontal range between the source and the hydrophone are shown in Figs. 1(a) and 1(b). No noise is added to the received signals in Fig. 1(a). In Fig. 1(b), the noise level at the hydrophone and the source level are constant. Therefore, the SNR increases with the decreasing source range. The SNR is defined as the average signal power in the signal bandwidth divided by the average noise power in the same bandwidth. The SNR increases from 6.7 dB at 35 km to 9 dB at 0 km. When the range is 15 km, the SNR is 0 dB. Four dark fringes due to the D and SR arrivals (D-SR fringe), the D and BR arrivals (D-BR fringe), the D and SBR arrivals (D-SBR fringe), and the SR and BR arrivals (SR-BR fringe) are observed in both Figs. 1(a) and 1(b). Two important features are as follows: (1) Because of the high bottom-reflection loss, the fringes except for D-SR are very blurry. This phenomenon is also demonstrated by the experimental results in Sec. 3. Therefore, our localization method is based on the D-SR time delays, which provide relatively robust observations. (2) From SNR 0 dB down, all the fringes are hardly identified. In other words, as long as the noise level is lower than the signal level, the D-SR fringe can be identified. In the real ocean, a positive SNR is usually required to detect a signal with a single hydrophone. Brune15 has demonstrated that as long as the noise level is not higher than the signal level, source localization based on the autocorrelation matching will not be significantly affected by the noise. 2.2 EKF model For a moving source, the horizontal range R, depth D, and speed v of the source are to be estimated. The state equation of the EKF is given by 2 3 2 32 3 Rk1jk1 Rkjk1 1 Dt 0 xkjk1 ¼ 4 vkjk1 5 ¼ 4 0 1 0 54 vk1jk1 5 ¼ Hxk1jk1 ; k ¼ 1; …; K; (3) Dkjk1 Dk1jk1 0 0 1 where x denotes the state vector, Dt is the update time increment, and H is the state transition matrix. The notation is designed to indicate the update process. For example, Rkjk1 is the predicted value at state k based on the data up to state k 1. Specifically, x0j0 and xKjK denote the initial state and the final state, respectively. Because the speed and depth are changing gradually, no dynamics are assumed in the state equation for the two parameters. Now consider the measurement equation. The D-SR time delays are the function of source location and can be expressed as ykjk1 ¼ f ðRkjk1 ; Dkjk1 Þ;
(4)
Fig. 1. Illustrations of the physical basis for the source localization method. Simulated autocorrelation functions versus range (a) in the noise-free environment and (b) in the presence of noise. (c) Distribution of D-SR time delays for different source ranges and depths.
J. Acoust. Soc. Am. 136 (2), August 2014
Duan et al.: Single-hydrophone source localization EL161
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Duan et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890664]
Published Online 18 July 2014
where ykjk1 is the predicted time delay based on the predicted source location. f ðÞ does not have an explicit expression, and in terms of the ocean environment is related only with the sound speed profile. Without knowledge of the local bottom bathymetry, f ðÞ can be calculated directly by the standard ray approach at range-depth grids. Equation (4) is nonlinear and thus the EKF method is adopted. The measurement function approximated by the first term in its Taylor series expansion12 gives the measurement matrix @f @f Fk ¼ ; ; 0 : (5) @R @D Rkjk1 ;Dkjk1 The contour curves of the D-SR time delays on a hydrophone at 4200 m are displayed in Fig. 1(c). Note that the normal directions of these curves are the same with the D-SR time delays’ gradients, which is just Fk . This property can facilitate our following analysis. Unlike the Kalman filter, the EKF’s performance depends on the local linearity of the measurement equation. Moreover, if the state’s initial estimate is incorrect, the filter may rapidly diverge for its linearization of a nonlinear system. Therefore, one method is always used to obtain an estimate of the true initial state. Then, the EKF is initialized using the estimate.10 However, it would be demonstrated in the following that the choice of the initial state makes no difference to our method’s final result. To start with, the measurement equation’s [Eq. (4)] local linearity needs to be analyzed. Figure 1(c) shows that the first term and the second term in the measurement matrix [Eq. (5)] are always negative and positive, respectively. Consequently, in an attempt to decrease the difference between the measurements and the outputs of the measurement function [Eq. (4)], the EKF at each state would progress toward a timedelay contour for the measurement corresponding to this state. For example, assume that the true initial state is 20 km in range and 110 m in depth, and the assumed initial state is 10 km and 200 m, respectively. From Fig. 1(c), the measured value of 0.03 s is smaller than the value of 0.11 s calculated using the measurement equation. According to local partial derivatives at the assumed initial state, the EKF would progress toward a smaller depth and larger range; it is the direction for the contour of 0.03 s. With the inputs of measured time delays, the EKF can converge to states after finite similar processes. Note that the states do not need to be the true states because the three unknown parameters (range, depth, and speed) may yield states similar to the true states. In Sec. 2.3, this phenomenon is analyzed in detail and an algorithm is presented to estimate the true states. 2.3 Algorithm Assume the final state of a single realization of the EKF is x1KjK , where the superscript “1” denotes the values in the first realization of the EKF. One estimate of the true initial state of the source can be obtained by 1
1
^ ; ^v 1 ; D ^ T ¼ ½R1 Td v1 ; v1 ; D1 T ; x^ 10j0 ¼ ½R KjK KjK KjK KjK 0j0 0j0 0j0
(6)
where the symbol “^” denotes the estimate from the final state and Td is the overall time duration of the recorded signal. Note that x^ 10j0 is generally not close to the true initial state. One reason is that the number of measurements may be very limited, with not enough for the EKF to converge. Another reason is that the EKF process is determined by many factors, such as the assumed initial state and values of the measurement matrix [Eq. (5)]. Therefore, although the EKF would progress toward the contours of the time delay of the measurements, it would not necessarily converge to the true states. A simulated example of the EKF process is shown in Fig. 2(a). A source at 110 m depth moves from 22 to 10 km in range at 1.9 m/s speed. The sign of the speed
EL162 J. Acoust. Soc. Am. 136 (2), August 2014
Duan et al.: Single-hydrophone source localization
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Duan et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890664]
Published Online 18 July 2014
Fig. 2. Illustrations of the EKF iteration for source localization. Processes of the iteration when the assumed initial states are (a) (26, 180, 2.6); (b) (10, 20, 2.6); and (c) (10, 180, 2.6), when the true initial states are (22, 110, 1.9). Estimated initial states versus iteration times when the true initial states are (d) (22, 50, 1.9); (e) (22, 260, 1.9); and (f) (16, 120, 1.9), when the assumed initial states are (26, 180, 2.6). The elements in parentheses are range in km, depth in m, and speed in m/s, respectively.
indicates the direction of source movement. The assumed initial state x10j0 is (26, 180, 2.6), where the elements in the parentheses indicate range in km, depth in m, and speed in m/s, respectively. The asterisks denote the source locations where the “measurements” are acquired, and the circles denote the source locations estimated by the first EKF. The final state of the EKF x1KjK is at the same time-delay contour as the true final source location, but x1KjK are greater in depth and farther in range than the true source locations. Note that the x^ 10j0 [the first point of the star line in Fig. 2(a)] is closer to the true initial state in depth. If it is used as the initial state for the second EKF (x20j0 ¼ x^ 10j0 ), and the process is denoted by the stars in Fig. 2(a), the final state of the second EKF x2KjK is closer to the true state than that of the first EKF. Therefore, the iteration can be performed until the final states estimated by two adjacent EKFs show little difference. The diamonds and squares in Fig. 2(a) show the processes of the middle and the last EKF, respectively. The final state estimated by the last EKF agrees well with the true state. Figures 2(b) and 2(c) show the EKF iterations when the assumed initial states x10j0 are (10, 20, 2.6) and (10, 180, 2.6), respectively. All the final results converge to the states close to the true state. Above all, the idea of the presented method is the series of ð^ x 10j0 ; x^ 20j0 ; …; x^ n0j0 ; …Þ converging to the true initial state of the source. Figures 2(d), 2(e), and 2(f) show x^ n0j0 versus the iteration times under different “measurements” with the assumed initial state (26, 180, 2.6). In Figs. 2(d) and 2(e), the sources move from 22 to 10 km in range, and the source depths are assumed to be 50 and 260 m, respectively. In Fig. 2(f), the source moves from 16 to 4 km in range, and the source depth is 120 m. The source speeds are 1.9 m/s. In general, the estimated initial states approach the true values as the number of the iterations increases, converging to a state close to the true initial state. 3. Experimental results and discussion The experiment was performed in the Indian Ocean around the location of (86 E, 1.6 N), where the ocean bottom is almost flat and the water depth is about 4500 m. A bottom-moored hydrophone was deployed at 4200 m depth. A ship moved away from and then back to the hydrophone. The maximum range was approximately 40 km.
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Duan et al.: Single-hydrophone source localization EL163
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Duan et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890664]
Published Online 18 July 2014
While the ship was moving back to the hydrophone, a source was towed at a depth of 110 m. The tow speed was about 1.935 m/s. The 700–800 Hz pseudorandom sequence was transmitted every 320 s. The duration of the sequence was 10 s. The autocorrelation functions of the 700–800 Hz pseudorandom sequence versus the range are shown in Fig. 3(a). The last sequence was generated at the range of 8.8 km. The SNR increases from about 2 dB at 35 km to about 9 dB at 8.8 km. Therefore, the D-SR fringe is obvious, except for the notch around 25 km. In contrast, only pieces of the D-BR and D-SBR fringes can be observed. The SR-BR fringe cannot be observed at all. When the source range is farther than 24 km, the D-SR time delays are almost constant. The time delay increases with the decreasing range almost linearly when the source range is smaller than 22 km. To verify our localization method, time delays from 22 to 10 km were selected. The extracted time delays are shown in Fig. 3(b); an outlier occurred at state 7, which was removed before analysis. Figure 3(c) shows the results after 12 iterations with the assumed initial state (26, 180, 2.6). The estimated initial states converge after six iterations. The final estimated initial state is (21.2, 110.9, 1.872), which is very close to the true values (21.35, 110, 1.935). To completely demonstrate the method’s performance, the assumed initial range is changed from 5 to 30 km, while the assumed initial depth is changed from 0 to 200 m; the final estimations for source depth and source range are shown in Figs. 3(d) and 3(e), respectively. The initial speed of the source is fixed to 2.6 m/s. Under all of the assumed initial states, the iterations converge to a state close to the true state. The experimental results demonstrate that the presented method is feasible for a rough estimation of the initial state. The experimental data clearly indicate that only the D-SR time delays are distinct on the autocorrelation function. Therefore, a method using this robust information is presented for the source localization. This method’s performance is based on certain physical properties. First, because the bottom-reflection loss is especially large when the grazing angle at the ocean bottom interface is large, the autocorrelation functions corresponding to the bottom-interacting arrivals are small. Only the D-SR time delays can be extracted from the autocorrelation functions. Second, the movement of a source can be modeled as a Gauss–Markov process. Third, the time delay is almost linear in a certain range-depth space, and thus the EKF is feasible for this condition. Furthermore, because the calculation cost of the EKF is small and the number of iterations is very limited, the localization method has a low computational cost.
Fig. 3. Localization results using the experimental data. (a) Autocorrelation functions from experimental data. (b) Extracted D-SR time delays. (c) Illustration of the convergence of the presented method. Estimated source depth (d) and range (e) under various assumed initial states.
EL164 J. Acoust. Soc. Am. 136 (2), August 2014
Duan et al.: Single-hydrophone source localization
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Duan et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890664]
Published Online 18 July 2014
Note that the distance of the source’s trace in our analysis is 12 km (from 22 to 10 km), which provides enough information to estimate the source’s initial state. The distance of course has a lower bound for estimating the source’s initial state. The lower bound may be related with the source range and depth at the start point and will be analyzed in our future work. One method to examine if the overall signal length is enough is that whatever the initial state (used to initializing our method) is the localization results are almost the same. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 11174235), the Doctorate Foundation of Northwestern Polytechnical University, China (Grant No. CX201226), and the Fundamental Research Funds for the central Universities (Grant No. 3102014JC02010301). References and links 1
A. B. Baggeroer, E. K. Scheer, K. Heaney, G. D. Spain, P. Worcester, and M. Dzieciuch, “Reliable acoustic path and convergence zone bottom interaction in the Philippine Sea 09 Experiment,” J. Acoust. Soc. Am. 128, 2385 (2010). 2 R. Duan, K. Yang, Y. Ma, and B. Lei, “Research on reliable acoustic path: Physical properties and a source localization method,” Chin. Phys. B 21, 124301 (2012). 3 R. McCargar and L. M. Zurk, “Depth-based signal separation with vertical line arrays in the deep ocean,” J. Acoust. Soc. Am. 133, EL320–EL325 (2013). 4 C. O. Tiemann, A. M. Thode, J. Straley, V. O’Connell, and K. Folkert, “Three-dimensional localization of sperm whales using a single hydrophone,” J. Acoust. Soc. Am. 120, 2355–2365 (2006). 5 W. Kuperman, G. D’Spain, and K. Heaney, “Long range source localization from single hydrophone spectrograms,” J. Acoust. Soc. Am. 109, 1935–1943 (2001). 6 C. Gervaise, B. G. Kinda, J. Bonnel, Y. Stephan, and S. Vallez, “Passive geoacoustic inversion with a single hydrophone using broadband ship noise,” J. Acoust. Soc. Am. 131, 1999–2010 (2012). 7 X. Mouy, D. Hannay, M. Zykov, and B. Martin, “Tracking of Pacific walruses in the Chukchi Sea using a single hydrophone,” J. Acoust. Soc. Am. 131, 1349–1358 (2012). 8 E.-M. Nosal and L. N. Frazer, “Track of a sperm whale from delays between direct and surface-reflected clicks,” Appl. Acoust. 67, 1187–1201 (2006). 9 M. Nørgaard, N. K. Poulsen, and O. Ravn, “New developments in state estimation for nonlinear systems,” Automatica 36, 1627–1638 (2000). 10 Z. Gong, D. D. Tran, and P. Ratilal, “Comparing passive source localization and tracking approaches with a towed horizontal receiver array in an ocean waveguide,” J. Acoust. Soc. Am. 134, 3705–3720 (2013). 11 E. A. Lehmann, “Particle filtering approach to adaptive time-delay estimation,” IEEE International Conference on Acoustics, Speech and Signal Processing (2006). 12 Z.-H. Michalopoulou and R. Jain, “Particle filtering for arrival time tracking in space and source localization,” J. Acoust. Soc. Am. 132, 3041–3052 (2012). 13 D. B. Ward, E. A. Lehmann, and R. C. Williamson, “Particle filtering algorithms for tracking an acoustic source in a reverberant environment,” IEEE Trans. Speech Audio Process. 11, 826–836 (2003). 14 M. B. Porter and H. P. Bucker, “Gaussian beam tracing for computing ocean acoustic fields,” J. Acoust. Soc. Am. 82, 1349–1359 (1987). 15 J. Brune, “Transient localization in shallow water environments,” M.S. thesis, Naval Postgraduate School, 1998.
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[http://dx.doi.org/10.1121/1.4890664]
Published Online 18 July 2014
Moving source localization with a single hydrophone using multipath time delays in the deep ocean Rui Duan, Kunde Yang,a) Yuanliang Ma, Qiulong Yang, and Hui Li Institute of Acoustic Engineering, Northwestern Polytechnical University, Xi’an 710072, China [email protected], [email protected], [email protected], [email protected], [email protected]
Abstract: Localizing a source of radial movement at moderate range using a single hydrophone can be achieved in the reliable acoustic path by tracking the time delays between the direct and surface-reflected arrivals (D-SR time delays). The problem is defined as a joint estimation of the depth, initial range, and speed of the source, which are the state parameters for the extended Kalman filter (EKF). The D-SR time delays extracted from the autocorrelation functions are the measurements for the EKF. Experimental results using pseudorandom signals show that accurate localization results are achieved by offline iteration of the EKF. C 2014 Acoustical Society of America V
PACS numbers: 43.60.Gk, 43.30.Wi [CG] Date Received: May 24, 2014 Date Accepted: July 9, 2014
1. Introduction In the deep ocean, a useful acoustic duct for moderate sound propagation is the reliable acoustic path (RAP).1,2 Below the deep sound channel, the depth at which the sound speed equals the maximum speed in the vicinity of the surface is referred to as the critical depth. RAP is the direct (D) and possibly the surface-reflected (SR) path under benign surface conditions between a receiver below the critical depth and a source at moderate range.3 A high signal-to-noise ratio (SNR) environment is the advantage RAP offers for source localization.2 Hereafter, the ocean environment that permits the propagation of RAP is referred to as the RAP environment. Single-hydrophone localization can be achieved by exploiting ocean propagation attributes such as multipath arrival time,4 modal dispersion,5 and the waveguide invariant.6 For a broadband source, the time delays between multipath arrivals can be obtained from the received signal’s autocorrelation functions. Using these time delays, the source location can be determined using a single hydrophone.4,7 However, it was observed in our experimental data that in the RAP environment, because of the large grazing angle at the ocean bottom interface, the reflection losses of bottom-interacting arrivals are large. Accordingly, the time delays, except for those between the D and SR arrivals, cannot be extracted under low SNR using the autocorrelation method. However, D-SR time delays alone are not sufficient for localization, when using a single hydrophone, which is why Nosal and Frazer8 used five hydrophones to track a sperm whale where only the D-SR time delays were exploited. In contrast, our experiment had one hydrophone and still only observed the D-SR time delays. The motivation of this study was the need to localize the source under these two constraints. Note that the source moved uniformly during our experiment, which provided the D-SR time delays versus duration. This paper demonstrates that the changing D-SR time delays contain enough information to determine source
a)
Author to whom correspondence should be addressed.
J. Acoust. Soc. Am. 136 (2), August 2014
C 2014 Acoustical Society of America EL159 V
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.235.251.160 On: Sat, 20 Dec 2014 16:59:08
Duan et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890664]
Published Online 18 July 2014
location and speed. A general overview of our method indicates that the autocorrelation functions are used to track the D-SR time delays, and then the localization problem is cast in the form of an extended Kalman filter (EKF) model.9 Gong et al.10 examined four approaches for instantaneous passive source localization using a towed horizontal array and showed that the EKF was a favorable approach for localizing moving sources. Note that the Kalman Filter family cannot be used if the nonlinearity of a model is significant. Particle filtering is an appropriate technique in dealing with nonlinear and/or non-Gaussian problems.11–13 2. Theory and simulation Some assumptions in this model must be emphasized beforehand, considering that the method is constrained by these assumptions. First, the source is moving at an almost steady horizontal speed. However, the model allows some fluctuation in speed because speed is assumed to be a random variable obeying a Gaussian distribution. Second, the source is moving in a radial direction, keeping its bearing unchanged. In practice, the source is likely to change the bearing as well as the range simultaneously. In that case, multipath time delays, besides the D-SR time delay, would be required to accomplish localization. For example, tracking sperm whales with as few as one hydrophone was achieved on the basis of the multipath time delays in the eastern Gulf of Alaska, with water depth of 460 m.4 Sperm whales transmit impulsive sounds called “clicks,” which makes the received signals ideal candidates for extracting multipath time delays. Since the broadband clicks are very short in duration (e.g. 25 ms8), the multipath arrivals can be observed using the spectrogram method.4 In contrast, the pseudorandom sequence in our experiment has a narrow bandwidth (100 Hz) and its time duration is 10 s. Therefore, using the autocorrelation method, only the D-SR time delays were obtained and can be used for the localization. Sections 2.1–2.3 present the measurement, state equation, and measurement function for the EKF model. The simulations are presented to illustrate the reasons for the localization algorithm. 2.1 Measurement for the EKF For a hydrophone below the critical depth, a source near the surface, and ranges within one-half of a convergence zone, the D, bottom reflected (BR), SR, surfacebottom-reflected (SBR), and other multipath arrivals are expected to depend on the environmental parameters. The time delays between them can be observed from the peaks of the autocorrelation function, which is defined as ð Ts 1 RðsÞ ¼ sðt þ sÞ sðtÞdt; (1) T s t¼0 where T is the duration of the selected time window. The time window must be shorter than the time of any significant change in the source range. Besides it should be much longer than the multipath time delays. An appropriate time window is several seconds. sðtÞ is the received signal in the time window. The normalized autocorrelation function is denoted as ^ ðsÞ ¼ jRðsÞ þ j H ðRðsÞÞj ; R Rð0Þ
(2)
where HðÞ is the Hilbert transform and j j is the absolute value, the numerator is the pffiffiffiffiffiffiffi envelope of RðsÞ, j equals 1, and Rð0Þ is the power of the received signal. Our simulations show the autocorrelations of the received signals when a broadband source moves horizontally toward the hydrophone head. The frequency band of the signal is 700–800 Hz, and white Gaussian noise is filtered by a bandpass filter to generate this signal. The source depth is assumed to be 110 m. A typical Munk sound speed profile and an ocean depth of 4500 m are chosen for the simulations. The
EL160 J. Acoust. Soc. Am. 136 (2), August 2014
Duan et al.: Single-hydrophone source localization
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.235.251.160 On: Sat, 20 Dec 2014 16:59:08
Duan et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890664]
Published Online 18 July 2014
sound speed, density, and attenuation coefficient of the ocean bottom are 1600 m/s, 1.6 g/cm3, and 0.14 dB/k, respectively. The ray propagation model14 is used. The normalized autocorrelation functions of signals on a hydrophone at 4200 m depth versus the horizontal range between the source and the hydrophone are shown in Figs. 1(a) and 1(b). No noise is added to the received signals in Fig. 1(a). In Fig. 1(b), the noise level at the hydrophone and the source level are constant. Therefore, the SNR increases with the decreasing source range. The SNR is defined as the average signal power in the signal bandwidth divided by the average noise power in the same bandwidth. The SNR increases from 6.7 dB at 35 km to 9 dB at 0 km. When the range is 15 km, the SNR is 0 dB. Four dark fringes due to the D and SR arrivals (D-SR fringe), the D and BR arrivals (D-BR fringe), the D and SBR arrivals (D-SBR fringe), and the SR and BR arrivals (SR-BR fringe) are observed in both Figs. 1(a) and 1(b). Two important features are as follows: (1) Because of the high bottom-reflection loss, the fringes except for D-SR are very blurry. This phenomenon is also demonstrated by the experimental results in Sec. 3. Therefore, our localization method is based on the D-SR time delays, which provide relatively robust observations. (2) From SNR 0 dB down, all the fringes are hardly identified. In other words, as long as the noise level is lower than the signal level, the D-SR fringe can be identified. In the real ocean, a positive SNR is usually required to detect a signal with a single hydrophone. Brune15 has demonstrated that as long as the noise level is not higher than the signal level, source localization based on the autocorrelation matching will not be significantly affected by the noise. 2.2 EKF model For a moving source, the horizontal range R, depth D, and speed v of the source are to be estimated. The state equation of the EKF is given by 2 3 2 32 3 Rk1jk1 Rkjk1 1 Dt 0 xkjk1 ¼ 4 vkjk1 5 ¼ 4 0 1 0 54 vk1jk1 5 ¼ Hxk1jk1 ; k ¼ 1; …; K; (3) Dkjk1 Dk1jk1 0 0 1 where x denotes the state vector, Dt is the update time increment, and H is the state transition matrix. The notation is designed to indicate the update process. For example, Rkjk1 is the predicted value at state k based on the data up to state k 1. Specifically, x0j0 and xKjK denote the initial state and the final state, respectively. Because the speed and depth are changing gradually, no dynamics are assumed in the state equation for the two parameters. Now consider the measurement equation. The D-SR time delays are the function of source location and can be expressed as ykjk1 ¼ f ðRkjk1 ; Dkjk1 Þ;
(4)
Fig. 1. Illustrations of the physical basis for the source localization method. Simulated autocorrelation functions versus range (a) in the noise-free environment and (b) in the presence of noise. (c) Distribution of D-SR time delays for different source ranges and depths.
J. Acoust. Soc. Am. 136 (2), August 2014
Duan et al.: Single-hydrophone source localization EL161
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.235.251.160 On: Sat, 20 Dec 2014 16:59:08
Duan et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890664]
Published Online 18 July 2014
where ykjk1 is the predicted time delay based on the predicted source location. f ðÞ does not have an explicit expression, and in terms of the ocean environment is related only with the sound speed profile. Without knowledge of the local bottom bathymetry, f ðÞ can be calculated directly by the standard ray approach at range-depth grids. Equation (4) is nonlinear and thus the EKF method is adopted. The measurement function approximated by the first term in its Taylor series expansion12 gives the measurement matrix @f @f Fk ¼ ; ; 0 : (5) @R @D Rkjk1 ;Dkjk1 The contour curves of the D-SR time delays on a hydrophone at 4200 m are displayed in Fig. 1(c). Note that the normal directions of these curves are the same with the D-SR time delays’ gradients, which is just Fk . This property can facilitate our following analysis. Unlike the Kalman filter, the EKF’s performance depends on the local linearity of the measurement equation. Moreover, if the state’s initial estimate is incorrect, the filter may rapidly diverge for its linearization of a nonlinear system. Therefore, one method is always used to obtain an estimate of the true initial state. Then, the EKF is initialized using the estimate.10 However, it would be demonstrated in the following that the choice of the initial state makes no difference to our method’s final result. To start with, the measurement equation’s [Eq. (4)] local linearity needs to be analyzed. Figure 1(c) shows that the first term and the second term in the measurement matrix [Eq. (5)] are always negative and positive, respectively. Consequently, in an attempt to decrease the difference between the measurements and the outputs of the measurement function [Eq. (4)], the EKF at each state would progress toward a timedelay contour for the measurement corresponding to this state. For example, assume that the true initial state is 20 km in range and 110 m in depth, and the assumed initial state is 10 km and 200 m, respectively. From Fig. 1(c), the measured value of 0.03 s is smaller than the value of 0.11 s calculated using the measurement equation. According to local partial derivatives at the assumed initial state, the EKF would progress toward a smaller depth and larger range; it is the direction for the contour of 0.03 s. With the inputs of measured time delays, the EKF can converge to states after finite similar processes. Note that the states do not need to be the true states because the three unknown parameters (range, depth, and speed) may yield states similar to the true states. In Sec. 2.3, this phenomenon is analyzed in detail and an algorithm is presented to estimate the true states. 2.3 Algorithm Assume the final state of a single realization of the EKF is x1KjK , where the superscript “1” denotes the values in the first realization of the EKF. One estimate of the true initial state of the source can be obtained by 1
1
^ ; ^v 1 ; D ^ T ¼ ½R1 Td v1 ; v1 ; D1 T ; x^ 10j0 ¼ ½R KjK KjK KjK KjK 0j0 0j0 0j0
(6)
where the symbol “^” denotes the estimate from the final state and Td is the overall time duration of the recorded signal. Note that x^ 10j0 is generally not close to the true initial state. One reason is that the number of measurements may be very limited, with not enough for the EKF to converge. Another reason is that the EKF process is determined by many factors, such as the assumed initial state and values of the measurement matrix [Eq. (5)]. Therefore, although the EKF would progress toward the contours of the time delay of the measurements, it would not necessarily converge to the true states. A simulated example of the EKF process is shown in Fig. 2(a). A source at 110 m depth moves from 22 to 10 km in range at 1.9 m/s speed. The sign of the speed
EL162 J. Acoust. Soc. Am. 136 (2), August 2014
Duan et al.: Single-hydrophone source localization
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.235.251.160 On: Sat, 20 Dec 2014 16:59:08
Duan et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890664]
Published Online 18 July 2014
Fig. 2. Illustrations of the EKF iteration for source localization. Processes of the iteration when the assumed initial states are (a) (26, 180, 2.6); (b) (10, 20, 2.6); and (c) (10, 180, 2.6), when the true initial states are (22, 110, 1.9). Estimated initial states versus iteration times when the true initial states are (d) (22, 50, 1.9); (e) (22, 260, 1.9); and (f) (16, 120, 1.9), when the assumed initial states are (26, 180, 2.6). The elements in parentheses are range in km, depth in m, and speed in m/s, respectively.
indicates the direction of source movement. The assumed initial state x10j0 is (26, 180, 2.6), where the elements in the parentheses indicate range in km, depth in m, and speed in m/s, respectively. The asterisks denote the source locations where the “measurements” are acquired, and the circles denote the source locations estimated by the first EKF. The final state of the EKF x1KjK is at the same time-delay contour as the true final source location, but x1KjK are greater in depth and farther in range than the true source locations. Note that the x^ 10j0 [the first point of the star line in Fig. 2(a)] is closer to the true initial state in depth. If it is used as the initial state for the second EKF (x20j0 ¼ x^ 10j0 ), and the process is denoted by the stars in Fig. 2(a), the final state of the second EKF x2KjK is closer to the true state than that of the first EKF. Therefore, the iteration can be performed until the final states estimated by two adjacent EKFs show little difference. The diamonds and squares in Fig. 2(a) show the processes of the middle and the last EKF, respectively. The final state estimated by the last EKF agrees well with the true state. Figures 2(b) and 2(c) show the EKF iterations when the assumed initial states x10j0 are (10, 20, 2.6) and (10, 180, 2.6), respectively. All the final results converge to the states close to the true state. Above all, the idea of the presented method is the series of ð^ x 10j0 ; x^ 20j0 ; …; x^ n0j0 ; …Þ converging to the true initial state of the source. Figures 2(d), 2(e), and 2(f) show x^ n0j0 versus the iteration times under different “measurements” with the assumed initial state (26, 180, 2.6). In Figs. 2(d) and 2(e), the sources move from 22 to 10 km in range, and the source depths are assumed to be 50 and 260 m, respectively. In Fig. 2(f), the source moves from 16 to 4 km in range, and the source depth is 120 m. The source speeds are 1.9 m/s. In general, the estimated initial states approach the true values as the number of the iterations increases, converging to a state close to the true initial state. 3. Experimental results and discussion The experiment was performed in the Indian Ocean around the location of (86 E, 1.6 N), where the ocean bottom is almost flat and the water depth is about 4500 m. A bottom-moored hydrophone was deployed at 4200 m depth. A ship moved away from and then back to the hydrophone. The maximum range was approximately 40 km.
J. Acoust. Soc. Am. 136 (2), August 2014
Duan et al.: Single-hydrophone source localization EL163
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.235.251.160 On: Sat, 20 Dec 2014 16:59:08
Duan et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890664]
Published Online 18 July 2014
While the ship was moving back to the hydrophone, a source was towed at a depth of 110 m. The tow speed was about 1.935 m/s. The 700–800 Hz pseudorandom sequence was transmitted every 320 s. The duration of the sequence was 10 s. The autocorrelation functions of the 700–800 Hz pseudorandom sequence versus the range are shown in Fig. 3(a). The last sequence was generated at the range of 8.8 km. The SNR increases from about 2 dB at 35 km to about 9 dB at 8.8 km. Therefore, the D-SR fringe is obvious, except for the notch around 25 km. In contrast, only pieces of the D-BR and D-SBR fringes can be observed. The SR-BR fringe cannot be observed at all. When the source range is farther than 24 km, the D-SR time delays are almost constant. The time delay increases with the decreasing range almost linearly when the source range is smaller than 22 km. To verify our localization method, time delays from 22 to 10 km were selected. The extracted time delays are shown in Fig. 3(b); an outlier occurred at state 7, which was removed before analysis. Figure 3(c) shows the results after 12 iterations with the assumed initial state (26, 180, 2.6). The estimated initial states converge after six iterations. The final estimated initial state is (21.2, 110.9, 1.872), which is very close to the true values (21.35, 110, 1.935). To completely demonstrate the method’s performance, the assumed initial range is changed from 5 to 30 km, while the assumed initial depth is changed from 0 to 200 m; the final estimations for source depth and source range are shown in Figs. 3(d) and 3(e), respectively. The initial speed of the source is fixed to 2.6 m/s. Under all of the assumed initial states, the iterations converge to a state close to the true state. The experimental results demonstrate that the presented method is feasible for a rough estimation of the initial state. The experimental data clearly indicate that only the D-SR time delays are distinct on the autocorrelation function. Therefore, a method using this robust information is presented for the source localization. This method’s performance is based on certain physical properties. First, because the bottom-reflection loss is especially large when the grazing angle at the ocean bottom interface is large, the autocorrelation functions corresponding to the bottom-interacting arrivals are small. Only the D-SR time delays can be extracted from the autocorrelation functions. Second, the movement of a source can be modeled as a Gauss–Markov process. Third, the time delay is almost linear in a certain range-depth space, and thus the EKF is feasible for this condition. Furthermore, because the calculation cost of the EKF is small and the number of iterations is very limited, the localization method has a low computational cost.
Fig. 3. Localization results using the experimental data. (a) Autocorrelation functions from experimental data. (b) Extracted D-SR time delays. (c) Illustration of the convergence of the presented method. Estimated source depth (d) and range (e) under various assumed initial states.
EL164 J. Acoust. Soc. Am. 136 (2), August 2014
Duan et al.: Single-hydrophone source localization
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.235.251.160 On: Sat, 20 Dec 2014 16:59:08
Duan et al.: JASA Express Letters
[http://dx.doi.org/10.1121/1.4890664]
Published Online 18 July 2014
Note that the distance of the source’s trace in our analysis is 12 km (from 22 to 10 km), which provides enough information to estimate the source’s initial state. The distance of course has a lower bound for estimating the source’s initial state. The lower bound may be related with the source range and depth at the start point and will be analyzed in our future work. One method to examine if the overall signal length is enough is that whatever the initial state (used to initializing our method) is the localization results are almost the same. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 11174235), the Doctorate Foundation of Northwestern Polytechnical University, China (Grant No. CX201226), and the Fundamental Research Funds for the central Universities (Grant No. 3102014JC02010301). References and links 1
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J. Acoust. Soc. Am. 136 (2), August 2014
Duan et al.: Single-hydrophone source localization EL165
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.235.251.160 On: Sat, 20 Dec 2014 16:59:08
