Source motion detection, estimation, and ...

Source motion detection, estimation, and compensation for underwater acoustics inversion by wideband ambiguity lag-Doppler filtering Nicolas F. Josso,a) Cornel Ioana, and Je´roˆme I. Mars Grenoble Image Parole Signal Automatique-lab/De´partement Image et Signal, Grenoble Institute of Technology, GIT, 961 rue de la Houille Blanche, 38402 Grenoble, France

Ce´dric Gervaise De´veloppement Te´chnologies Nouvelles Laboratory, EA3876, Ecole Nationale Supe´rieure d’Inge´nieurs des Te´chniques de l’Armement, Universite´ Europe´enne de Bretagne, 2 rue Francois Verny, 29806 Brest Cedex, France

(Received 23 February 2010; revised 20 September 2010; accepted 27 September 2010) Acoustic channel properties in a shallow water environment with moving source and receiver are difficult to investigate. In fact, when the source-receiver relative position changes, the underwater environment causes multipath and Doppler scale changes on the transmitted signal over low-tomedium frequencies (300 Hz–20 kHz). This is the result of a combination of multiple paths propagation, source and receiver motions, as well as sea surface motion or water column fast changes. This paper investigates underwater acoustic channel properties in a shallow water (up to 150 m depth) and moving source-receiver conditions using extracted time-scale features of the propagation channel model for low-to-medium frequencies. An average impulse response of one transmission is estimated using the physical characteristics of propagation and the wideband ambiguity plane. Since a different Doppler scale should be considered for each propagating signal, a time-warping filtering method is proposed to estimate the channel time delay and Doppler scale attributes for each propagating path. The proposed method enables the estimation of motion-compensated impulse responses, where different Doppler scaling factors are considered for the different time delays. It was validated for channel profiles using real data from the BASE’07 experiment conducted by the North Atlantic Treaty Organization Undersea Research Center in the shallow water environment of the Malta Plateau, South Sicily. This paper provides a contribution to many field applications including passive ocean tomography with unknown natural sources position and movement. Another example is active ocean tomography where sources motion enables to rapidly cover one operational area for rapid environmental assessment and hydrophones may be drifting in order to avoid additional flow noise. C 2010 Acoustical Society of America. [DOI: 10.1121/1.3504709] V PACS number(s): 43.30.Pc, 43.60.Mn, 43.60.Pt, 43.30.Cq [EJS]

I. INTRODUCTION

Acoustic propagation channels can be characterized by an impulse response (IR), which is of potential interest for a great number of underwater acoustic applications such as communications, sonar detection and localization, marine mammal monitoring, etc. The IR estimate is also useful for geoacoustic inversion using matched impulse response (MIR) techniques.1 The use of IR time domain for inversion and localization was first proposed by Porter in Ref. 2. Commonly, IR is estimated by computing the so-called matched filtering,3 where the received signal is correlated with the transmitted signal. In an ideal additive white-noise background, the matched filtering method correlates the received signal with the time-delayed version of the transmitted signal. This method, based on the maximum likelihood estimator, gives very good IR estimates in the case of static scenarii but fails when the factor motion is included in the source-channela)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

3416

J. Acoust. Soc. Am. 128 (6), December 2010

Pages: 3416–3425

receiver configuration.4 In this case, the propagating signal is subject to undesirable distortion effects due to the time-varying nature of the environment and to the relative motion existing between the source and the receiver. For the low-tomedium frequency domain (300 Hz–20 kHz), distortions can be characterized as time delays from multipath propagation and as wideband Doppler scaling effects.5,6 If transmitter and receiver motions are well monitored, methods such as matched-field-processing7 or MIR can give good results, but both involves high computational costs, in particular in the case of wideband signals. Furthermore, when the relative motion between the receiver and the transmitter is unknown, such methods cannot be applied directly or without any additional information. Channel distortion such as time delays multipath and wideband Doppler scaling effects have been described using a linear time-varying channel model formulation in Refs. 8– 10. In these descriptions, the channel is represented as a continuously varying wideband spreading function directly related to the physical nature of the distortions in the channel in terms of amplitude, time delay, and scaling effect. A discrete multipath-scale system characterization was proposed

0001-4966/2010/128(6)/3416/10/$25.00

C 2010 Acoustical Society of America V

Downloaded 18 Sep 2012 to 193.48.255.141. Redistribution subject to ASA license or copyright; see http://asadl.org/terms

in Refs. 11 and 12 and has been applied to wideband transient and communication signals in Refs. 13 and 14. Although this discrete channel model is appropriate for the estimation of underwater time-varying wideband channels, the authors admit that the processing is computationally intensive. As an alternative, in this paper, we propose to correlate the received signal with a family of reference signals that represent all possible receptions. The set of reference signals accounts for all possible velocities, which are admitted for the configuration and the multipath propagation effects due to the environment. Other possible solutions were proposed by Qian and Chen15 and Mallat and Zhang16 who introduced a matching pursuit algorithm on transitory signals, which adaptively decomposes any signal into a linear combination of best-matched basis functions that are selected from a dictionary of Gabor atoms. Zou et al.17 also extended some earlier results on steadymotion based Dopplerlet transform to estimate the range and speed of a moving source. In a previous work, we analyzed the wideband ambiguity function (WAF) and proposed a uniform motion compensation,4 which estimated a motioncompensated IR under the condition that the depth of the propagation channel can be neglected when compared to its length (i.e., for long-range propagation and shallow water environment). Under this assumption, the Doppler scaling effect is considered to remain constant for all propagation paths. The uniform motion compensation shows its limitations for high order propagation paths and short-range configurations that may be encountered in passive ocean tomography. In this paper, we introduce a non-uniform motioncompensated IR characterization that estimates a different Doppler scaling effect for each propagation path. It applies to short-range as well as to long-range propagation scenarii in a shallow water environment. This method is based on an iterative process that estimates the scaling factor and the time delay associated with each propagation path on the WAF. It permits one to filter out the corresponding signal using lagDoppler warping filtering tools. The motion-compensated IR is estimated as a piecewise-defined function where each filtered propagation path contributions are computed separately. Correlation receiver performance in terms of delay and Doppler can be described with the WAF. However, when the transmitted signal is narrowband, the conventional formulation of the ambiguity function is more appropriate. Both narrowband ambiguity function and WAF are fully described and formulated for active sonar systems in Ref. 18. The motion effects, which represent a compression in time for approaching sources and an expansion for receding sources, are approximated as simple carrier-frequency shifts of the transmitted signal. Hence, the correlation receiver has a reference set of signals composed by time-delayed and carrierfrequency-shifted versions of the transmitted signal. When the ratio bandwidth-central frequency becomes higher than 0.1, the narrowband approximation is no longer valid and the signal can be considered as wideband. The studied signals have very low central frequencies (around 1300 Hz) and high bandwidths (around 2000 Hz) resulting in a ratio bandwidth-central frequency of almost 1.55. Such signals are very interesting because they allow both long-range propagation and good multipath resolution. J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010

In this paper, we propose a new IR estimation method that considers a different apparent speed for each propagating ray and that was based on our previous work on the WAF.4 Our method consists of an iterative motion and time delay estimation for each propagation path in the wideband ambiguity plane using a projection-filtering-inversion framework. Projections and inversions are processed with warping operators that enable accurate filtering of the detected propagation paths. This work is a contribution to the fields of communications, sonar detection and localization, passive ocean tomography,19 and marine mammal monitoring systems when the relative motion between the receiver and the transmitter is unknown. The paper is organized as follows. The WAF and the lag-Doppler warping filtering are described in Sec. II for underwater acoustic signals undergoing multipath and Doppler scaling effects. Applications of the proposed IR estimation are presented on a simulated Pekeris waveguide in Sec. III and on a real data set from the BASE’07 experiments in Sec. IV. Section V contains our discussion and conclusions.

II. THE WIDEBAND AMBIGUITY LAG-DOPPLER FILTERING

In order to estimate the propagation time and the Doppler scale factor associated with the signal received for each ray, we propose to use the wideband ambiguity plane. Time delay and Doppler scaling factors are estimated by cross-correlating the received signal with a set of reference signals.4,18 Our new IR estimation method proposes the use of time-warping filtering tools in order to iteratively estimate the propagation parameters of each ray. We estimate the parameters of the time-warping filtering from the WAF of the received signal.

A. Multipath and Doppler scale changes on underwater acoustic signals

Not many works have been reported on the problem of solving the resulting wave equation for a moving source in an acoustic waveguide. Guthrie et al.,20 Hawker,21 and Lim and Ozard22 obtained expressions for the acoustics field using normal theory and considering sources moving radially or horizontally. Flanagan et al.,23 Clark et al.,24 and more recently Josso et al.4 formulated the moving source problem in terms of ray theory, where each ray path has a different Doppler shift according to its angle of emission. Solutions are mostly expressed in terms of reception time, which corresponds to the time at which the signal is received. For a narrowband signal whose bandwidth is much less than its central frequency, the Doppler effect can be assumed to be a frequency shift. However, for underwater acoustic signals, this assumption does not hold because the signal bandwidth is comparable to the central frequency. As a result, transmitted signals over underwater acoustic channels, in the 500 Hz–20 kHz medium-to-high frequency band, will undergo Doppler scalings as well as multipath distortions. The (noiseless) output signal s(t) of such a wideband channel Josso et al.: Wideband ambiguity lag-Doppler filtering

Downloaded 18 Sep 2012 to 193.48.255.141. Redistribution subject to ASA license or copyright; see http://asadl.org/terms

3417

can be modeled as a linear combination of time-shifted and Doppler-scaled versions of the input signal e(t),4 sðtÞ ¼

N X

1=2

ai gi eððt  si Þgi Þ;

(1)

i

where N is the number of different propagating paths considered in the acoustic propagation channel, si corresponds to the propagation time associated with the ith path, and gi is the scaling factor associated with the propagating ray number i. The narrowband ambiguity function and WAF are fully described and formulated for active sonar systems in Ref. 18. In Ref. 4, the WAF is analytically analyzed when linear frequency modulation (LFM) signals are transmitted through a wideband time-varying channel with a moving source. Assuming that the transmitted signal is known, the propagation time and the Doppler scaling factor associated with each ray can be estimated by cross-correlating the received signal with a set of reference signals. If the transmitted signal verifies the wideband approximation with the ratio bandwidth over central frequency being higher than 0.1, then the set of reference signals is derived from Eq. (1). Hence the set of reference signals is composed of time delayed and Dopplerscaled versions of the emitted signal for the range of time delays and speeds expected.4,18 The WAF depends on s and v because of its dependency from g and is computed by Rðs; gÞ ¼

ð1

sðt þ sÞg1=2 eT ðgtÞdt;

(2)

1

where T denotes the complex conjugation, s(t) the received signal, e(t) the transmission, s the time delay, and g the time scaling factor due to the motion of the source. The complete formulation of the WAF is obtained by using Eq. (1) in Eq. (2), which yields Rðs; gÞ ¼

X i

ai ðggi Þ

1=2

ð1 1

eðgi ðt þ s  si ÞÞeT ðgtÞdt:

(3)

The WAF is often represented as the square magnitude of Eq. (3). Local maxima of the WAF are reached for each ray. For the ith ray, the WAF gives a local maximum if the reference signal and the analyzed signal match in time delay and in Doppler scaling factor.4 It is assumed that the smallest time difference between two consecutive arrivals is larger than the inverse of the transmitted signal time-bandwidth product. This ensures that each peak of the correlation can be detected and that the propagation time and the apparent speed of the ith ray can be estimated once a local maximum is detected. If these conditions are not met, the interferences potentially occurring between the local maxima are studied.19 B. Time-warping filtering

The aim of this section is to provide an efficient filter to separate different groups of arrivals. As the received signals for the different groups of arrivals have different nonstationary patterns, a time-varying filter is supposed to be used. Since the different groups of arrivals are usually not 3418

J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010

separated in time-frequency representations, time-frequency based filters cannot be used directly. Signal processing methods are often based on a change of the representation space resulting from projections of the original space into another space that shows new properties or that highlights characteristics of interest. An interesting class of unitary projection is the class of time-warping operators.25–27 Hence we propose to use a time-frequency based filtering that operates in the warped-time-frequency domain in which the ray families can be separated. Our method is based on a projection-processing-inversion framework, where the projection consists in a time warping that transforms the received signal for one ray family into a sine function. The sine function is then filtered out with classical bandpass filtering, and the signal is projected back with inverse warping operators. The time-warping principle was introduced in Refs. 25, 28, and 29 and is based on a transformation of the time axis with non-linear mapping. Time warping can be done by warping the original time axis t with a potentially non-linear warping function w(t). Warping functions are usually chosen to reveal the properties of interest in the time-warped domain. We introduce the class of warping operators W w using the warping function w(t) and following the relation:26   1 dwðtÞ > 0 : eðtÞ ! W wðtÞ eðtÞ ; W w ; wðtÞ 2 C ; dt

(4)

where eðtÞ 2 L2 ðRÞ is a square-integrable signal, and C1 is the class of derivable functions. The warping transformation operator is an unitary and linear transformation of the signal e(t) into the warped domain defined by   dwðtÞ1=2  eðwðtÞÞ: W w eðtÞ ¼  dt 

(5)

The warping operator W w is unitary as it preserves both the transformed signal energy and the inner product. Let us consider e(t) is a non-stationary, mono-component signal defined by eðtÞ ¼ expð2jpfe uðtÞÞ;

(6)

where j is the standard imaginary unit, and u(t) is the signal phase. For real application and analytical signals, we can assume that uðtÞ 2 C1 and duðtÞ=dt > 0. We define u1(t) as the inverse function of u(t) verifying uðu1 ðtÞÞ ¼ u1 ðuðtÞÞ ¼ t; 8t:

(7)

Supposing that u1 ðtÞ 2 C1 and du1 ðtÞ=dt > 0; the projection of the signal e(t) on the time-warped space defined by W u1 follows  1 1=2 du ðtÞ  expð2jpfe uðu1 ðtÞÞÞ W u1 eðtÞ ¼  dt   1 1=2 du ðtÞ  expð2jpfe tÞ: ¼  dt 

(8)

Josso et al.: Wideband ambiguity lag-Doppler filtering

Downloaded 18 Sep 2012 to 193.48.255.141. Redistribution subject to ASA license or copyright; see http://asadl.org/terms

FIG. 1. (Color online) Schematic illustration of the WALF general principle. For simplicity, we consider here three propagation paths with their associated signals: e1, e2, and e3 . The WAF absolute maximum is illustrated with a circle while filtering is represented with dashed lines.

Equation (8) states that if the phase is a known continuous and invertible function, any signal e(t) can then be projected in a time-warped space where it becomes a sine function. The resulting sine function can be accurately filtered out with classical time-frequency bandpass filtering. The filtering residue can be projected back to the initial time domain with the inverse warping operator W u . In fact, it can be shown that W w is the inverse warping operator of W w1 ,   dwðtÞ >0 ; W w1 OW w ¼ W w OW w1 ¼ I d; w 2 C1 ; dt

where (sn, gm) are chosen to represent the appropriate range of time delays and scale changes, and e(t) is the transmitted signal. The steps of the WALF iterative algorithm are summarized as follows. First we initialize the process by defining p0(t) ¼ r(t) where r(t) is the received signal. Then, at the ith iteration, i ¼ 1, 2, …, M  1, the projection of the residue pi(t) onto every dictionary element gðm;nÞ ðtÞ 2 D is computed to obtain ðm;nÞ Ki

ðm;nÞ

¼ hpi ; g

D

i¼

ð þ1

pi ðtÞðgðm;nÞ ðtÞÞT dt;

(11)

1

(9) where O is the composition operation in the operators domain, and I d is the identity operator. C. The warping lag-Doppler filtering

In this section, we introduce a new warping ambiguity lag-Doppler filtering (WALF) method. The WALF is based on a projection-filtering-inversion framework in which the projection and the inversion are processed with warping operators. The necessary warping functions will be derived from the WAF defined in Eq. (3). The general principle of the WALF algorithm is presented in Fig. 1. We define a dictionary of signals D composed of signals representing the signal received for each ray introduced in Eq. (1). The dictionary signals g(m,n)(t) are defined by ðm;nÞ

g

pffiffiffiffiffiffi ðtÞ ¼ gm eððt  sn Þgm Þ;

gm 6¼ 0;

J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010

(10)

where T denotes the complex conjugation. It is interesting to ðm;nÞ equals Ri(sn, gm) where Ri is pi(t) WAF. This note that Ki implies that the WAF properties apply to the Ki computed from D. We know that the local maxima of the WAF are reached for each ray. For the ith ray, the WAF attains a local maximum if the reference signal and the analyzed signal match in time delay and in Doppler scaling factor.4 More precisely, the absolute maximum is reached when the reference signal matches the received signal for the most energetic propagation path. Hence the selected dictionary atom for the ith path, gi(t), is the one that maximizes the magnitude of the projection ðm;nÞ

gi ðtÞ ¼ argmax jKi

j:

(12)

gðm;nÞ ðtÞ2D

For real applications, we can assume that uðtÞ 2 C1 and duðtÞ=dt > 0, where u(t) represents e(t) phase, and e(t) is the Josso et al.: Wideband ambiguity lag-Doppler filtering

Downloaded 18 Sep 2012 to 193.48.255.141. Redistribution subject to ASA license or copyright; see http://asadl.org/terms

3419

analytical version of the transmitted signal. As e(t) is assumed to be known, we can consider that u(t) is known too. We introduce ui(t) representing gi(t) phase and following ui ðtÞ ¼ uððt  sn Þgm Þ;

(13)

where (sn, gm) are estimated from Eq. (12). pi(t) is then projected on the time warped domain with the time-warping , operator W u1 i Qi ðtÞ ¼ W u1 Pi ðtÞ: i

(15)

where F represents the bandpass filter that removes the sine function received for the ith ray family after time warping. The projection is then inverted to work in the original time domain and to obtain the input signal of the next WALF iteration Piþ1 ðtÞ ¼ W ui Ui ðtÞ:

ðm;nÞ

g

(16)

(17)

where Td is the duration of transmission, and W is the bandwidth of the LFM signal. As the scale-change parameter is affected by the source velocity, the velocity parameter sampling rate is chosen as 1 dv ¼ VD : 2 3420

(18)

J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010

(19)

where vm ¼ Vmin þ

m ; 2VD

sn ¼ n=fs ;

(20)

fs is the sampling frequency, and the integers m and n satisfy m 2 ½1; ðVmax  Vmin Þ 2VD ;

n 2 ½1; fs Tt :

(21)

Another implementation issue concerns the bandwidth of the bandpass filter introduced in Eq. (17). The bandpass filter bandwidth cannot be analytically expressed for any signal. In this part, we explain how it can be obtained if LFM signals are transmitted. From Eqs. (1) and (6), the signal received for the ith propagation path can be expressed as follows: 1=2

si ðtÞ ¼ ai gi expð2jpfe ui ðtÞÞ;

(22)

where fe is a normalization constant, ui represents the phase of the analytical signal received for the ith propagation path and from Eq. (1) it verifies ui ðtÞ ¼ uððt  si Þgi Þ:

Issues on the implementation of the WALF algorithm are presented in this section for its use on underwater acoustic applications. The first step is to compute the dictionary D for the appropriate range of time delays and scale changes. The atom dictionary D needs to represent signals computed as in Eq. (10). In Ref. 11, a complete and discrete time-scale characterization of wideband time-varying systems was presented, from which the expected time delay and scale-change parameters, needed here, can be obtained. Another approach is to consider the Doppler tolerance of known signals and then decide on the range of the scale-change parameter according to this tolerance. For example, if a linear frequencymodulated chirp signal is transmitted, the Doppler tolerance (i.e., half-power contour) is given by Refs. 4, 30, and 31, 2610 knots; Td W

 t  sn ðtÞ ¼ s vm ; ð1  vm Þ1=2 1  c c

D. WALF implementation

VD ¼ 6



1

(14)

transforms the received signal The warping operator W u1 i for the ith ray family into a sine function in order to filter it out easily. Because of the propagation properties, uj 6¼ ui ; 8 j 6¼ i; j ¼ 0; 1; …; M  1, and only the signal received for the ith ray family is transformed into a sine function. The sine function corresponding to the signal received for the ith ray family is then filtered out in the time-frequency plane with a classical bandpass filter Ui ðtÞ ¼ F Qi ðtÞ;

Let us assume that the expected velocities are bounded in v [ [Vmin, Vmax] and the expected time delays are bounded in s [ [0, Tt], where Tt is the time delay spread of the channel. Then, the atoms in the dictionary D are obtained by

(23)

When LFM signals are transmitted, the phase of the analytical signal received for propagation paths is a second order polynomial that follows ui ðtÞ ¼ Ai t2 þ Bi t þ C

(24)

and Ai ¼

g2i k ; 2fe

Bi ¼

f0 gi  kg2i si ; fe

Ci ¼

wi ; fe

(25)

where f0 is the starting frequency of the LFM signal, k is its chirp rate, and wi is the value of ui at the origin of time. The signal instantaneous frequency (IF) is defined as the derivative of its phase with respect to time, IFi ðtÞ ¼ fe

dui ðtÞ : dt

(26)

The studied signals are non-constant analytical signals so their IFs are strictly positive. Hence u1 i exists and is unique for any propagating path; its expression is given by

u1 i ðtÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðB2i Þ2  Ai ðCi  tÞ Ai



Bi : 2Ai

(27)

Josso et al.: Wideband ambiguity lag-Doppler filtering

Downloaded 18 Sep 2012 to 193.48.255.141. Redistribution subject to ASA license or copyright; see http://asadl.org/terms

FIG. 2. Spectrogram of the received LFM signal transmitted by the towed source. This time-frequency representation shows the effects of the multipath underwater propagation such as the appearance of time-delayed echoes.

FIG. 3. Wideband ambiguity plane of a simulated multipath propagation. The circle represents the position of the WAF theoretical absolute maximum while the cross illustrates its estimate.

Since the warping function needed in Eq. (14) is known now, we analyze the IF separation between two time warped received signal in order to determine the bandpass filter bandwidth. After calculation, when j = i, the IF of the signal received for the jth ray and time warped with the warping function u1 i verifies

process based on the ray theory for signals transmitted from a moving source4 and considering transmission as well as reception time to compute the different received rays. In order to be close to the realistic shallow water scenario analyzed in Sec. IV, we chose appropriate parameters for the Pekeris waveguide model. Specifically, the water column had a constant sound speed of 1520 m/s, a density of 1000 kg/m3, and a depth of 135 m. The simulated sea bottom was a flat sandy mud bottom with a sound velocity of 1550 m/s and a density of 1700 kg/m3. For clarity purpose, the simulations presented here are noise-free and illustrate the acoustical propagation properties of a signal. The WALF algorithm performances in noisy environments will be studied on a real data set from BASE’07 campaign in Sec. IV. The transmitter depth is 24 m, the hydrophone depth is 90 m, and the range between the transmitter and the receiver is 500 m. The simulated source motion is rectilinear and constant at 5 m/s. The transmitted signal is an LFM with a central frequency of 1300 Hz, a 2 kHz bandwidth, and a duration of 4 s. The transmitted signal’s bandwidth is very large compared with the signal’s central frequency, showing a ratio of 1.55. This corresponds to the signal transmitted during the BASE’07 campaign and analyzed in Sec. IV. Figure 2 represents the spectrogram of the received signal for the simulated underwater propagation with a Pekeris waveguide. Even in this simplistic scenario, the arrivals are not separated on the time-frequency representation. This justifies that time-frequency based filters should not be used directly. The WAF of the received signal is represented in Fig. 3, where each arrival is characterized by a sweep-like shape which broadens with the distance between the reference and the simulated speed.4 The cross (Fig. 3) represents the theoretical absolute maximum position and the circle its estimate. From Fig. 3, it can be seen that the position of the absolute maximum is well estimated. This enables a correct estimation of the warping function used for the first iteration of the WALF. Figure 4 represents the time-frequency representation of the time warped signal received after underwater propagation where the dashed lines illustrate the bandpass filtering operation. For the simulated scenario having the above parameters, the maximum bandpass filter bandwidth is computed from Eq. (30),

 Bi Aj Ai þ Bj fe Aj fe ffi: W u1 FIj ðtÞ ¼  q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 i Ai 2 B2i Ai ðCi  tÞ

(28)

When j ¼ i, the signal received for the ith propagation path is transformed into a pure frequency signal, and its IF is given by FIi ðtÞ ¼ fe : W u1 i

(29)

pffi As illustrated in Fig. 4, the term in 1= t quickly becomes negligible compared with Aj fe=Ai term in Eq. (28). Finally, for the ith iteration of the process, the maximum bandpass filter bandwidth depends on the ratio of the compression factors. Using Eqs. (25), (28), and (29) it can be defined as Bmax ¼ min 2fe 1 

g2j g2i

!! ;

i 6¼ j:

(30)

As the WALF is a recursive algorithm, the residual energy can be used as the algorithms stopping criterion. If the signal-to-noise ratio (SNR) of the signal is known, then the WALF can stop iterating when the ratio of the signal energy to the residual energy reaches the SNR. Other possible stopping criteria include the rate of decrease of the residual energy or a fixed number of iterations based on some prior knowledge on the range of values of the channel parameters. III. SIMULATION WITH A PEKERIS WAVEGUIDE UNDERWATER CHANNEL MODEL

The proposed warping lag-Doppler filtering method was applied to underwater propagation signals that were simulated using the Pekeris waveguide channel model.32 We developed a software that simulates the propagation J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010

Josso et al.: Wideband ambiguity lag-Doppler filtering

Downloaded 18 Sep 2012 to 193.48.255.141. Redistribution subject to ASA license or copyright; see http://asadl.org/terms

3421

FIG. 4. (Color online) Time-frequency representation of the time warped signal received after underwater propagation. The dashed lines illustrate the bandpass filter used to filter out the sine function.

Bmax ¼ 7  102 Hz;

(31)

where the normalization parameter used for computation is fe ¼ 980 Hz, as illustrated in Fig. 4. As expected, each ray family is well separated in the time-warped domain and, if an LFM signal is transmitted, its shape respects Eq. (28). The signal obtained after the first ray extraction is represented in the ambiguity plane in Fig. 5. The bottom panel illustrates the WAF of the signal obtained after the filtering operation for the first WALF iteration while the top panel is the WAF of the first extracted ray. It can be observed that the first path is removed without modifying other parts of the signal, and that the second ray detection becomes easier. The final results of the WALF method are presented in Figs. 6 and 7. Figure 6 is the WAF representation of the

FIG. 5. (Color online) WAF of the signal obtained after the filtering operation of the first iteration (bottom panel) and WAF of the first extracted ray (top panel). The cross illustrates the estimate the first arrival position. 3422

J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010

FIG. 6. (Color online) WAF of the received signal where the crosses represent the theoretical local maxima and the circles their estimated positions. The solid curve represents the WALF method (one compression factor by delay), the dashed line illustrates the uniform motion compensation (one compression factor for all delays), and the dotted line stands for the IR estimation without including motion compensation in the WAF.

received signals where the WALF method and other IR estimation methods are illustrated. The lag and Doppler scaling factors estimated at each WALF iteration are represented with circles while their theoretical positions are the crosses. Our new adaptive speed compensation method for IR estimation consists in computing each filtered path contributions separately after each WALF iteration, as illustrated in Fig. 1. The global IR estimate is then obtained by summing the contributions of the different paths. Graphically, our IR estimation method can be seen as keeping the values of the ambiguity function along the solid line linking each estimated local maximum, as illustrated in Fig. 6. This is equivalent to considering a different Doppler scaling factor for each time delay. Hence, a different Doppler scaling factor is considered for each propagation path and one Doppler scaling factor is estimated at each iteration of the WALF in Fig. 6. The dashed line illustrates the uniform motion compensation4 and the dotted line illustrates the IR estimation without motion compensation in the WAF. In the uniform motion compensation, the Doppler scaling factor is assumed to be constant, which is equivalent to estimating the IR by keeping the values of the WAF along the line at constant speed while passing through its absolute maximum.

FIG. 7. (Color online) IR estimated with different methods. The crosses represent the theoretical time delays and amplitudes of the simulated propagation channel. The solid curve is the motion-compensated IR obtained with the WALF method, the dashed line illustrates the uniform motion compensation, and the dotted line stands for the IR estimated without motion compensation. Josso et al.: Wideband ambiguity lag-Doppler filtering

Downloaded 18 Sep 2012 to 193.48.255.141. Redistribution subject to ASA license or copyright; see http://asadl.org/terms

tion, each propagation path is retrieved as a sharp peak, with a time delay and an amplitude very close to its theoretical values. It should be noticed that the Doppler scaling factors are very close for the two first propagation paths. This explains that for these paths, the IR estimated with the uniform motion compensation (dashed line) is very close to the one estimated with the WALF. However, the source-receiver separation being small, Doppler scaling factors are not the same for all the propagation paths and in this case, only the WALF methods gives satisfying results. The IR estimated without considering the source-receiver motion (dotted line) does not give any satisfying result. FIG. 8. Spectrogram of the LFM signal transmitted by the towed source. The effects of the multipath underwater propagation such as the apparition of time-delayed echoes can be seen on this time-frequency representation. The faint and steeper LFM signals present in this time-frequency representation are not part of the transmitted signal and illustrate some saturation effects due to the relatively small source-receiver separation.

Figure 7 represents the estimated IR time series coming from different methods (WALF, uniform compensation, and zero motion) and the IR theoretical values. The different IR estimations are obtained from the WAF presented in Fig. 6. The IR estimated with the WALF method (solid line) is very close to the theoretical values of the actual propagation channel. With the WALF motion-compensated IR estima-

IV. APPLICATION TO UNDERWATER ACOUSTIC DATA FROM THE BASE’07 EXPERIMENT A. Experiment description

The BASE’07 experiment was jointly conducted by the North Atlantic Treaty Organization Undersea Research Center (NURC), the Applied Research Laboratory (ARL), the Forschungsanstalt der Bundeswehr fu¨r Wasserschall und Geophysik (FWG), and the Service Hydrographique et Oce´anographique de la Marine (SHOM). Two additional days of measurements were also conducted by the SHOM to collect data for geoacoustic inversion testing. Results based on the BASE’07 experiment can be found in Refs. 4, 6, and 33. The

FIG. 9. (Color online) Towed source and hydrophones’ positions as a function of time and evolution of the source-receiver separation over time for the analyzed day of BASE’07 campaign. J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010

Josso et al.: Wideband ambiguity lag-Doppler filtering

Downloaded 18 Sep 2012 to 193.48.255.141. Redistribution subject to ASA license or copyright; see http://asadl.org/terms

3423

FIG. 10. (Color online) Wideband ambiguity function of the received signal. The circles represent the local maxima detected with the WALF, and the solid line illustrates how the IR is computed from the WAF.

real data we used were collected from a shallow water environment on the Malta Plateau. An LFM signal, illustrated in Fig. 8, was transmitted by a source moving rectilinearly, at constant speed (2–12 knots) and with a source-receiver separation ranging from 750 to 25 000 m. The transmitted LFM signal had a 2 kHz bandwidth, a central frequency of 1.3 kHz, and a duration of 4 s. B. Results

This section presents the results obtained by the WALF method for one short-range transmission. This transmission was chosen because of its small source-receiver separation (765 m) and its average relative speed (1.4 m/s) for underwater acoustic applications. A higher relative speed with a very small source-receiver separation, like in the example of Sec. III, for example, would have shown more impressive results but such a scenario was not available from the BASE’07 campaign. In the presented results, the source depth was 65 m, the hydrophone depth was 90 m, the sourcereceiver separation was 765 m, and the depth of the propagation channel was 135 m (Fig. 9). Figure 10 presents the results obtained from the WALF method with the WAF of the received signal as a back-

ground. The circles illustrate the positions of the detected local maxima in the WAF, and the solid line is the line linking them. From Fig. 10, it can be seen that the apparent speed estimated for each ray family differs as is declination varies. The results obtained on the real data can be compared with the results obtained from the simulated Pekeris waveguide in Fig. 6. However, the apparent speed evolution is not as important because the relative speed of the source and the receiver is not very high and the source-receiver separation is slightly bigger. The first ray speed estimate is v1 ¼ 1.35 m/s, which is coherent with the towed source speed of v ¼ 1.4 m/s. From Fig. 10, it can be seen that as expected, rays having higher time delays have a lower apparent speed. Hence the speed parameters estimation leads to coherent estimations using the real data from BASE’07. As any iterative method, the WALF requires a stopping criterion. The SNR based stopping criterion is well adapted to real data experiments. This stopping criterion stops the WALF iterating when the ratio of the signal energy to the residual energy reaches the SNR. The estimated motionless IR obtained with the WALF method is shown in the bottom panel of Fig. 11. In a realistic case, and contrary to the simulated Pekeris waveguide results introduced in Fig. 7, each arrival can be seen as a family of rays in which many propagation paths with different amplitudes arrive within a short time delay. The estimated motionless IR is compared with the IR computed with classical matched filtering in the top panel of Fig. 11 as well as a simulated IR obtained from a Pekeris waveguide having parameters close to the data recorded in situ. The simulated Pekeris waveguide has a constant propagation speed of 1530 m/s and is composed of a flat sandy mud bottom with a sound speed of 1550 m/s, a density of 1700 kg/m3, and a 135 m deep propagation channel. The WALF method gives a motionless IR estimation with sharp peaks for each ray family that are very close to the simulated Pekeris waveguide in time delay and amplitudes. Hence, the WALF motion-compensated IR estimation give good results for real signal applications, and this method is well adapted to real signal processing. As shown in

FIG. 11. (Color online) IR estimations with the WALF method (bottom panel, solid line) and without taking motion into account (top panel, dashed line). The crosses represent the IR simulated with a Pekeris waveguide with parameters close to the one recorded in situ.

3424

J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010

Josso et al.: Wideband ambiguity lag-Doppler filtering

Downloaded 18 Sep 2012 to 193.48.255.141. Redistribution subject to ASA license or copyright; see http://asadl.org/terms

Fig. 11, even with a low source-receiver relative motion, the propagation channel cannot be well estimated if the motion is not taken into account. It can be noticed that the simulated Pekeris waveguide and the motionless IR estimation do not match very well for the third ray family. This is most likely due to the fact that the Pekeris waveguide model is not the real propagation waveguide and can thus not perfectly match a complex and real environment. V. CONCLUSIONS

This paper introduces a new approach for the characterization of a signal propagating over an underwater channel and taking into account the relative movement between receiver and transmitter. The signal characterization is used to develop a new and general motion-compensated IR estimation. The WALF method consists of the use of time-warping filtering that enables the estimation of the time delay and the relative speed for each propagating signal. It is based on an iterative and model-based estimation and filtering of the received signal components. The WALF method is applied to the estimation of motion-compensated IR where different Doppler scaling factors are considered for the different time delays. The resulting motion-compensated IR estimation was successfully evaluated using simulated data as well as data from the BASE’07 experiment (SHOM, South of Sicily, 2007). This method can be used for active or passive tomography in cases in which the source velocity is not well monitored or unknown. ACKNOWLEDGMENTS

This work was supported by De´le´gation Ge´ne´rale pour l’Armement (DGA) under SHOM research, Grant No. N07CR0001. We would like to thank A. Papandreou-Suppappola and J. Zhang (School of Electrical, Computer and Energy Engineering, Arizona State University) for fruitful discussions. 1

Z.-H. Michalopoulou, “Matched-impulse-response processing for shallowwater localization and geoacoustic inversion,” J. Acoust. Soc. Am. 108, 2082–2090 (2000). 2 M. B. Porter, S. M. Jesus, Y. Ste´phan, E. Coelho, and X. De´moulin, “Single phone source tracking in a variable environment,” in Proceedings of the European Conference on Underwater Acoustics, Rome, Italy (1998). 3 M.-I. Taroudakis and G.-N. Makrakis, Inverse Problems in Underwater Acoustics (Springer-Verlag, New York, 2001), pp. 1–215. 4 N. F. Josso, C. Ioana, J. I. Mars, C. Gervaise, and Y. Ste´phan, “On the consideration of motion effects in the computation of impulse response for underwater acoustics inversion,” J. Acoust. Soc. Am. 126, 1739–1751 (2009). 5 N. F. Josso, C. Ioana, C. Gervaise, Y. Stephan, and J. I. Mars, “Motion effect modeling in multipath configuration using warping based lagDoppler filtering,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Paris, France (2009), pp. 2301–2304. 6 N. Josso, J. Zhang, A. Papandreou-Suppappola, C. Ioana, J. Mars, C. Gervaise, and Y. Ste´phan, “On the characterization of time-scale underwater acoustic signals using matching pursuit decomposition,” in Proceedings of MTS/IEEE Conference OCEANS’09, Biloxi, MS (2009), pp. 1–6. 7 A. Baggeroer, W. Kuperman, and P. Mikhalevsky, “An overview of matched field methods in ocean acoustics,” IEEE J. Ocean. Eng. 18, 401–424 (1993).

J. Acoust. Soc. Am., Vol. 128, No. 6, December 2010

8

L. Sibul, L. Weiss, and T. Dixon, “Characterization of stochastic propagation and scattering via Gabor and wavelet transforms,” J. Comput. Acoust. 2, 345–369 (1994). 9 R. Shenoy and T. Park, “Wideband ambiguity functions and affine Wigner distributions,” EURASIP J. Signal Process. 41, 339–363 (1995). 10 B. Iem, A. Papandreou-Suppappola, and G. Boudreaux-Bartels, “Wideband Weyl symbols for dispersive time-varying processing of systems and random signals,” IEEE Trans. Signal Process. 50, 1077–1090 (2002). 11 Y. Jiang and A. Papandreou-Suppappola, “Discrete time-frequency models of generalized dispersive systems,” IEEE Int. Conf. Acoust., Speech, Signal Process. 3, 349–352 (2006). 12 S. Rickard, Time-frequency and Time-scale Representations of Doubly Spread Channels (Princeton University, Princeton, NJ, 2003), pp. 1–110. 13 N. F. Josso, J. J. Zhang, A. Papandreou-Suppappola, C. Ioana, C. Gervaise, Y. Stephan, and J. I. Mars, “Wideband discrete transformation of acoustic signals in underwater environments,” in Proceedings of Asilomar IEEE Conference on Signals, Systems, and Computers, Asilomar, CA (2009), pp. 118–122. 14 N. F. Josso, J. J. Zhang, D. Fertonani, A. Papandreou-Suppappola, and T. M. Duman, “Time-varying wideband underwater acoustic channel estimation for OFDM communications,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Dallas, TX (2010), pp. 5626–5629. 15 S. Qian and D. Chen, “Signal representation using adaptive normalized Gaussian functions,” Signal Process. 36, 1–11 (1994). 16 S. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process. 41, 3397–3415 (1993). 17 H. Zou, Y. Chen, J. Zhu, Q. Dai, G. Wu, and Y. Li, “Steady-motionbased dopplerlet transform: Application to the estimation of range and speed of a moving sound source,” IEEE. J. Ocean. Eng. 29, 887–905 (2004). 18 J. Hermand and W. Roderick, “Delay-doppler resolution performance of large time-bandwidth-product linear FM signals in a multipath ocean environment,” J. Acoust. Soc. Am. 84, 1709–1727 (1988). 19 C. Gervaise, S. Vallez, C. Ioana, Y. Stephan, and Y. Simard, “Passive acoustic tomography: New concepts and applications using marine mammals: A review,” J. Mar. Biol. Assoc. UK 87, 5–10 (2007). 20 A. N. Guthrie, R. M. Fitzgerald, D. A. Nutile, and J. D. Shaffer, “Longrange low-frequency CW propagation in the deep ocean: Antiguanewfoundland,” J. Acoust. Soc. Am. 56, 58–69 (1974). 21 K. E. Hawker, “A normal mode theory of acoustic doppler effects in the oceanic waveguide,” J. Acoust. Soc. Am. 65, 675–681 (1979). 22 P. H. Lim and J. M. Ozard, “On the underwater acoustic field of a moving point source. I. Range-independent environment,” J. Acoust. Soc. Am. 95, 131–137 (1994). 23 R. P. Flanagan, N. L. Weinberg, and J. G. Clark, “Coherent analysis of ray propagation with moving source and fixed receiver,” J. Acoust. Soc. Am. 56, 1673–1680 (1974). 24 J. G. Clark, R. P. Flanagan, and N. L. Weinberg, “Multipath acoustic propagation with a moving source in a bounded deep ocean channel,” J. Acoust. Soc. Am. 60, 1274–1284 (1976). 25 R. Baraniuk and D. Jones, “Unitary equivalence: A new twist on signal processing,” IEEE Trans. Signal Process. 43, 2269–2282 (1995). 26 A. Jarrot, C. Ioana, and A. Quinquis, “Toward the use of the time warping principle with discrete time sequences,” J. Comput. Acad. Pub. 2, 49–55 (2007). 27 G. Le Touze´, B. Nicolas, J. I. Mars, and J.-L. Lacoume, “Matched representations and filters for guided waves,” IEEE Trans. Signal Process. 57, 1783–1795 (2009). 28 R. Altes, “Wide-band, proportional-bandwidth Wigner-Ville analysis,” IEEE Trans. Acoust., Speech, Signal Process. 38, 1005–1012 (1990). 29 A. Papandreou, F. Hlawatsch, and G. F. Boudreaux-Bartels, “The hyperbolic class of quadratic time-frequency representations Part I: Constant-Q warping, the hyperbolic paradigm, properties, and members,” IEEE Trans. Signal Process. 41, 3425–3444 (1993). 30 B. Harris and S. Kramer, “Asymptotic evaluation of the ambiguity functions of high-gain fm matched filter sonar systems,” Proc. IEEE 56, 2149– 2157 (1968). 31 S. Kramer, “Doppler and acceleration tolerances of high-gain, wideband linear FM correlation sonars,” Proc. IEEE 55, 627–636 (1967). 32 C. L. Pekeris, “Theory of propagation of explosive sound in shallow water,” in Propagation of Sound in the Ocean, Memoir 27 (Geological Society of America, New York, 1948), pp. 1–117. 33 G. Theuillon and Y. Ste´phan, “Geoacoustic characterization of the seafloor from a subbottom profiler applied to the BASE’07 experiment,” J. Acoust. Soc. Am. 123, 3108 (2008).

Josso et al.: Wideband ambiguity lag-Doppler filtering

Downloaded 18 Sep 2012 to 193.48.255.141. Redistribution subject to ASA license or copyright; see http://asadl.org/terms

3425