- Research Article
- Open Access
Underwater Broadband Source Localization Based on Modal Filtering and Features Extraction
© Maciej Lopatka et al. 2010
- Received: 7 July 2009
- Accepted: 11 January 2010
- Published: 26 April 2010
Passive source localization is a crucial issue in underwater acoustics. In this paper, we focus on shallow water environment (0 to 400 m) and broadband Ultra-Low Frequency acoustic sources (1 to 100 Hz). In this configuration and at a long range, the acoustic propagation can be described by normal mode theory. The propagating signal breaks up into a series of depth-dependent modes. These modes carry information about the source position. Mode excitation factors and mode phases analysis allow, respectively, localization in depth and distance. We propose two different approaches to achieve the localization: multidimensional approach (using a horizontal array of hydrophones) based on frequency-wavenumber transform ( method) and monodimensional approach (using a single hydrophone) based on adapted spectral representation ( method). For both approaches, we propose first complete tools for modal filtering, and then depth and distance estimators. We show that adding mode sign and source spectrum informations improves considerably the localization performance in depth. The reference acoustic field needed for depth localization is simulated with the new realistic propagation modelMoctesuma. The feasibility of both approaches, and , are validated on data simulated in shallow water for different configurations. The performance of localization, in depth and distance, is very satisfactory.
- Mode Sign
- Source Spectrum
- Source Depth
- Depth Localization
- Contrast Function
Passive source localization in shallow water has attracted much attention for many years in underwater acoustics. In this environment and for Ultra-Low Frequency waves ( to Hz, denoted further ULF) classical beamforming techniques are inappropriate because they do not consider multipath propagation phenomena and ocean acoustic channel complexity. Indeed, ULF acoustic propagation in shallow water waveguides is classically based on normal mode theory . ULF band is very attractive for detection, localization, and geoacoustical parameter estimation purposes, because propagating acoustic waves are almost not affected by absorption and thus can propagate at very long ranges. In this context, mainly two approaches are used: Matched-Field Processing (denoted MFP) [2, 3] and Matched-Mode Processing (denoted MMP) [4–6]. The comparative study of both approaches is given in . Matched-Mode approach can be considered as MFP combined with modal decomposition. The main difference is that MFP operates in receiver space and MMP in mode space. Both methods require a reference acoustic field (replica field) to be compared, generally by correlation techniques (building and maximizing an objective function), with the real acoustic field recorded on receiver(s). Another alternative to perform source localization is to use time reversal  which can be seen as a broadband coherent MFP. Some experiments have been performed showing the feasibility of the method . The main drawback of the method is that a numerical backpropagation has to be computed which needs a good knowledge of the environment. As MMP is less sensitive to environmental mismatches than MFP and Time Reversal methods, this technique is more interesting for practical applications, and thus is used in our approach to estimate the source depth. The access to modes not only allows estimation of mode excitation factors for depth localization, but also gives the possibility to analyze mode phase to extract information about the source distance. As a result, in this paper depth estimation is performed using MMP on the mode excitation factors and distance estimation is achieved by mode phase analysis.
Consequently, the main issue to perform underwater localization for ULF sources in shallow water is to develop signal processing methods to accomplish modal filtering. These methods should be based on physics of wave propagation in waveguides, to be adapted to signals propagating in shallow water environment. In this context, we propose two complementary techniques to localize broadband impulsive source in depth and distance. The first method based on frequency-wavenumber transform and denoted is a multidimensional approach based on array processing. The second method based on adapted Fourier transform and denoted is a monodimensional technique used on a single hydrophone.
Traditionally, matched-mode localization was applied on vertical line arrays (VLAs), and mode excitation factors were extracted by a spatial integration of pressure field. As proposed in [10, 11], we record the signal (represented in the space: radial distance and time ) on a horizontal line array (HLA), as it is generally more practical in real applications (towing possibility, faster deployment, and stability). In this configuration, modes can be filtered in the frequency-wavenumber plane ( ), which is a two-dimensional Fourier transform of radial distance-time section (signal ).
For the mono-dimensional approach, modes cannot be filtered by conventional modal filtering techniques. As modes have nonstationary properties, the only way to filter modes is to integrate modal time-frequency characteristics  into modal filtering. The main idea is to deform a signal in such way that nonlinearities in the time-frequency plane become linear (according to the frequency domain). Consequently, the signal becomes stationary and classical filtering tools can be used to filter modes. Hereby, modal filtering in mono-dimensional configuration is done in an adapted frequency domain (Pekeris Fourier transform). The classical and adapted frequency domains are related by the unitary equivalence formalism .
After a brief presentation of modal propagation theory, we give a short description of the simulator Moctesuma-2006, which is used for simulation of acoustic replica fields and acoustic parameter computation. Then, we present details about the experimental configuration. Next, we describe mode filtering methods in the mono- and multidimensional cases to finally present estimators used for depth and distance localization. Finally, results of distance and depth localization for mono and multidimensional method are presented on simulated data.
Acoustic propagation of Ultra-Low Frequency waves in shallow water waveguide can be modeled by normal mode theory. Propagating signal at long range is composed of dispersive modes. These modes are analyzed for depth (matched-mode processing) and distance (mode phase processing) localization.
To demonstrate very succinctly the idea of localization using modes, we introduce the simplest model of oceanic waveguide—the perfect waveguide. Even if this model is a simplification of real complex waveguides, it reflects the most important waveguide phenomena: modal decomposition of the propagated signal.
where is the time, is the frequency, and satisfies the general Helmholtz equation.
This short theoretical introduction of normal mode theory made on the example of perfect waveguide exposes the principle used for source depth estimation; mode amplitudes depend on source depth by the factor .
is the phase of the source at frequency
is a phase factor due to time delay of the recorded signal;
depends on the modal function sign at the source depth it is if and if
depends on the modal function sign at the receiver depth
is a phase factor at frequency linked to the propagation distance between source and receiver.
As one can notice, modal decomposition is a very useful theory for acoustic propagation in oceanic waveguide. Indeed, MMP uses this decomposition to perform localization .
In this section we demonstrated that by having access to modes, and more precisely to their excitation factors and phases, it is possible to localize source in depth and in distance.
To perform the depth estimation using MMP, we need an acoustic model to generate replica fields. Several classical underwater acoustic propagation models exist in the literature and are used according to the seabed depth, the source range, and the frequency band. Models are based on different theories: ray theory, parabolic equation modeling, normal mode models, and spectral integral models . Among the different models we choose the numerical model Moctesuma-2006—a realistic underwater acoustic propagation simulator developed by Thales Underwater Systems . For the sake of simplicity Moctesuma-2006 will be called further Moctesuma. This model, based on normal mode theory, simulates an underwater acoustic propagation for range-dependent environments. It is well adapted to transient broadband ULF signals for shallow and deep water environments. Moreover, we choose Moctesuma as it provides the acoustic parameters of the environment (wavenumbers) and the full acoustic field (time-series) .
The transmitted transient signal is first split into narrow subbands signals through a set of bandpass filters. Each subband is associated with a central frequency for which acoustic modes are fully computed. For each mode in each subband, propagation consists in delaying the original signal. The summation is performed in the time domain, so the signal causality is necessarily satisfied. Moctesuma considers different acoustic signal phenomena such as penetration, elasticity, multiple interactions inside multilayered sea bottoms and water. The time and space structure of waves is analyzed beyond simple wavefronts and Doppler effect (moving source and/or receiver).
A set of parameters is necessary to make a simulation with Moctesuma. The first parameter group concerns a description of the environment. As it is a range-dependent model, parameters are given for each environment sector. User has to provide following environmental parameters: sea state, temperature, sound speed profile, seabed type (or precise seabed structure). The second parameter group concerns the input signal and the experiment configuration (coordinates, depths, speeds and caps of the source and the antenna, antenna's length, and sensors' number).
In our analysis, we use Moctesuma to simulate the reference acoustic field (in the MMP) and to access the acoustic parameters of the environment such as horizontal wavenumbers, group velocities, and mode excitation factors signs.
3.2. Signal Sources and Reception Configurations
The source ULF is used to validate the methods in a simple case. For a more complex situation source ULF-2 is then used in Section 6.3.
Signals radiated by source are recorded on a horizontal line array (HLA) after acoustic propagation. The HLA is m long and is composed of omnidirectional equispaced hydrophones (separated by m). The sampling frequency is Hz.
number of hydrophones (traces)
duration and frequency sampling of the recording (number of samples ).
The "real data" is obtained by adding a white bi-dimensional (in time and in space) Gaussian noise to the simulated data. Several signal-to-noise ratios (SNRs) are considered.
In this paper, source localization in depth and distance is performed either by multi-dimensional or by mono-dimensional approach. The first one will be called approach (as the method operates in the frequency-wavenumber domain ) and the second one approach (as the method is based on adapted Fourier transform). They both achieve modal filtering which is described in this section.
For the first approach, in the frequency-wavenumber plane ( ) modes are separated and thus can be filtered. In the second approach, which is theoretically more difficult as we have a single hydrophone, modes are not easily separable and thus, cannot be filtered using classical signal representations such as Fourier transform or time-frequency representation. As proposed in [12, 17], we use an adapted frequency representation in which modes are separable and consequently can be filtered.
4.1. Multi-Dimensional Approach
The consequence of this processing is a shifting of every point in the plane in such way that the spatial aliasing is canceled and the representation space of modes is much larger (greater dynamics, simpler filtering).
with a frequency dependent constant related to the source spectrum.
The amplitude of the transform for each curve (dispersive mode) depends only on the mode excitation factor modulus. We use these curves to estimate the excitation factor modulus of each mode. For a perfect waveguide model there is no frequency dependence for modal functions, which is the case in reality. Therefore, excitation factor of mode is estimated as the mean value across the frequency domain. Moreover, mode excitation factors at the bottom interface are not exactly equal to and will slightly modify the estimation of the mode amplitude at the source depth. This phenomenon does not affect the result as the same methodology is applied for the replica data.
Once representation of the signal is calculated, a mask filtering has to be applied to filter modes. The mask is a binary image (with the same size as the transform) and is used to extract a mode by a simple multiplication in the domain. The mask built for each mode should "cover" the region occupied by this mode in the plane.
where and denote, respectively, dilation sizes in wavenumber ( ) and frequency ( ) domains, and and denote, respectively, the sampling period in wavenumber and frequency domains. The first parameter determines the distances between successive masks (depends on mode number ) and the second parameter defines the distance of the mask for mode to the frequency Hz. This definition of dilation parameters makes the mask width in the frequency dimension adapted to the frequency (narrower masks at high frequencies for lower number modes and larger masks at lower frequencies for higher number modes). The dilation process is restricted by limitation that the masks for different modes must not overlap. These masks allow an efficient filtering even for higher modes which are usually more difficult to filter.
the limited HLA length-mode energy spreads around dispersion curves in the plane;
the mismatch between real and simulated environments .
4.2. Monodimensional Approach
where is the sound speed in water, is the waveguide depth, is the time, and is the distance. This relation defines temporal domain of group delay where is arrival time of the wavefront.
Note that this tool is reversible, so one can go back to the initial representation space (time or frequency).
In this short presentation of adapted transformation for the perfect waveguide we demonstrated the principal idea of this technique which consists in transformation of modes into linear structures. In our work we use a method adapted to Pekeris waveguide model, as it is a more complex model (closer to reality) taking into account the interaction with the sea bottom (described in details in ).
As the non-linear time-frequency signal structures become linear after this transformation, the signal becomes stationary (see Figures 12(b) and 13(b)). In this case, one can use classical frequency filtering tools to filter modes. The modal filtering is then done in the adapted frequency domain (Pekeris frequency) by a simple bandpass filtering defined by the user.
In Section 4 we demonstrated how extracting modes from the recorded signal by multi and mono-dimensional approaches. In this section we discuss depth and distance estimators based on modal processing. For depth estimation, we use a matched-mode technique, and for distance estimation our approach is based on mode phase analysis. For both approaches, and , we use the same estimators for localization in depth and distance. The only difference is the representation space of modal filtering: frequency-wavenumber for method and adapted Fourier spectrum for method.
5.1. Range Estimation
The range estimation is combined with the mode sign estimation; therefore we call the estimator sign-distance estimator. The estimator applied only on the real data is based on mode phase analysis and calculates a cost function based on two mode phases ( and , where ) extracted from the data. Modes are not necessarily consecutive; however their numbers and associated wavenumbers (calculated by Moctesuma) have to be known.
being difference between two mode phases at depth ; the values of this parameter are defined by (for details see Table 1):
Absolute mode sign choice rule: the absolute sign of mode depends on the absolute sign of mode and the pass function (where ).
The mode sign estimation for takes at least steps as the first mode sign is always positive and as the estimator works sequentially on mode couples.
5.2. Depth Estimation
Source depth estimation is based on Matched-Mode Processing. The principle of MMP is to compare modes in terms of excitation factors extracted from the real data with those extracted from the replica fields. The modeled acoustic field (replica) is simulated with Moctesuma. The depth estimator is based on a correlation which measures a distance between mode excitation factors estimated from real and from simulated data (for a set of investigated depths). The depth for which this correlation reaches maximum (the best matching) is chosen as the estimated source depth.
The localization performance is strongly dependent of the matching accuracy between real and simulated acoustic fields. Study of the influence of environmental and system effects on the localization performance is presented in [11, 22]. The dilation used to build masks in the plane makes the method more robust against these errors.
The maximum of indicates the estimated depth of the source.
In matched-mode localization, modes for which the function defined by (27) is calculated are theoretically unrestricted. However, in case of ULF localization, only the first modes are used. The upper mode number limit is given by the environment and existence of cut-off frequencies as the methods presented in this paper are based on broadband signal processing. In our analysis the number of used modes is between and .
Theoretical performances of depth localization for the studied environment and for all source depths are presented in Figure 14 (each vertical line corresponds to a contrast function ). The figure presents two plots: for the method without mode signs (a) and for the method with mode signs (b). The result is obtained by the application of (27) to mode excitation factors directly taken from Moctesuma simulations. For the method without mode signs, one can notice the existence of "mirror solutions" which is a line of secondary peaks intersecting with the primary peaks line indicating the true source positions. That line does not exist for the method with mode signs, as the "mirror solutions" are cancelled by adding mode signs to mode excitation factor modulus. In such way, one can remove the localization ambiguity, which is problematic especially in low signal-to-noise conditions.
5.3. Source Spectrum Estimation
where is the estimated source spectrum, is a spectral factor correcting the signal attenuation over frequency, is a number of hydrophones, and is the spectrum of the first mode on hydrophone estimated from the plane. For better performance the correction factor can be measured in the field (by recording a known broadband signal at some distance). As we do not operate on real field data, to calculate we use theoretical values of spectral attenuation (for frequency range of interest).
We present some examples of source localization using methods described in Sections 4 and 5. First, examples of localization in distance and in depth are presented using a single hydrophone, and then using a horizontal hydrophone array (HLA). Moreover, we show the interest of mode signs and source spectrum estimations in case of depth localization by and approaches. Due to limited paper's length, we do not expose here the study of the robustness of the methods against noise. These considerations have been studied in [11, 25]. We give only some most important conclusions. The simulations on source depth estimation demonstrate that to obtain the primary peak-to-secondary peak ratio of dB the signal-to-noise ratio has to be superior to dB for method and dB for method. The impact of noise on source range estimation seems to be more relevant. These considerations concern white (in time and in space) gaussian model of local (non propagating) noise.
6.1. One Hydrophone
The objective of this section is to show performance of localization method using a single hydrophone. The methods are validated for the environment and configuration described in Section 3 for a signal-to-noise ratio of dB. The distance between source and hydrophone is equal to km. Source is deployed at m of depth and the hydrophone is on the seabed.
water column depth: m;
sound speed velocity in water: m/s;
sound speed velocity in sediments: m/s;
water density: ;
sediment density: .
Within the parameters, the water column depth is a correct value, and other parameters are approximations of the real values to demonstrate robustness of the method.
Then, the method allows a filtering of modes (classic bandpass filter applied on spectral representation given in Figure 13(b)), and these modes are analyzed for distance and depth estimation.
For the distance localization an access to mode phases is essential. First, a modal filtering by is performed and then for each analyzed mode, its phase is calculated through a Fourier transform. Wavenumbers needed by the estimator defined in (19) are provided by Moctesuma.
Results of the source distance estimation for approach.
Estimated distance (km)
For depth localization an estimation of mode excitation factors is needed. First, a modal filtering is performed on real and simulated data by approach, and then for each analyzed mode, its mode excitation factor modulus is calculated as a mean over frequency. Moreover, mode signs estimated above can be used in the contrast function .
In the mono-dimensional configuration in lower signal-to-noise ratio conditions the mode sign and distance estimations can be inaccurate. Also, the depth localization performance cannot be satisfied. Therefore, we propose the multi-dimensional configuration that is more robust and efficient due to a richer information about the source and the environment recorded on the HLA.
6.2. Horizontal Line Array
This section presents results of localization in distance and in depth using approach. The objective is to show the performance of localization method. The methods are validated in the environment and configuration described in Section 3 for a signal-to-noise ratio of dB. The distance between the source and the first hydrophone of HLA is km. The source depth is m and the HLA is on the seabed.
According to the Shannon theorem and for the ULF band ( Hz) the maximal spatial sampling should be done every m. Thus, in theory we could consider every second HLA hydrophone without any information lost (as the whole HLA samples linearly the space every m). However, with a higher space sampling, better noise canceling algorithms can be implemented. What is more important, is a length of the HLA. When the length of HLA reduces, the localization performance decreases. This is provoked by a spreading of the signal in the plane which results from a not sufficiently long radial distance sampling of the modal signal . Different issues of the use of HLA are discussed in .
The first step of the method is a velocity correction which is done with the minimum value of the sound speed profile in water m/s. Then, the transform is calculated and this representation is used for mode filtering. These modes are then analyzed for distance and depth estimations.
After filtering, the phase of each mode is calculated through a Fourier transform. The wavenumbers needed by the estimator defined in (19) are provided by simulator. This estimation is applied to each hydrophone of the HLA ( estimations) .
We apply the estimator on five different mode couples: , , , , and , and research erea km with step m. The estimated distance values are given in Figure 19 and its mean values are given in Table 4. Moreover, the sign-distance estimator gives as a result mode sign. In multi-dimensional case, we dispose of estimations of mode signs for each mode couple option. For a mode number equal to , the sign-distance estimator is applied on following mode couples: (for mode ), , (for mode ), , , (for mode ), , , (for mode ), , , (for mode ), , and (for mode ) and the user has to select the couple he wants to use. This information is used here to maximize the probability of correct choice within available options for each estimation step. As the mode sign estimation is sequential it is primordial to not commit an error at the beginning to avoid its propagation. At each step (for each mode sign estimation) a series of parameters is calculated to help the user in taking the decision. For the first step these parameters are calculated once (for the couple ), for the second step we dispose of two set of parameters (for the couples and ), and for the following steps we have always three sets of parameters. These parameters are defined as follows.
where is a number of sign changes ( ). This criteria should be maximal.
where denotes a set of distance estimations and denotes a second derivative with respect to the hydrophone number. For the no-error estimation of distance the first derivative is equal to interhydrophone distance. Then, the second derivative is equal to zero as the first derivative is a constant function. This criterion allows to measure the variability of distance estimations across all hydrophones and should be minimal.
where denotes a distance estimation for hydrophone at actual step analysis ( ) and denotes the final estimation of distance from previous step ( ). This criterion allows cancel secondary peak solutions for which the first two criteria gave good results and should be obviously minimal.
Mean values of source distance estimation for multi-dimensional approach. The bin width is m.
Mean distance value (km)
For the example presented here, the mode signs were estimated on the same mode couples as distance. The estimation of signs of modes no. to is correct and the absolute signs are , , , , , and .
After modal filtering, the mode excitation factor modulus of each mode is calculated as a mean over the region. Moreover, the sign-distance estimator can be used for mode signs estimation.
6.3. Source Spectrum Issue
In Section 5.3 we described a simple method of estimation of the source spectrum. Now, we quantify the impact of this estimation on depth localization.
We use a source with flat spectrum for simulation of the replica field (source ULF-1)-common approach when unknown source.
We estimate a source spectrum by the method defined in (29) and use it to simulate the replica field.
Nevertheless, our method is designed for broadband sources. Therefore, even if spectral characteristics of the source are perfectly known, but present narrowband or comb-type structures, the localization performance decreases. The performance decrease due to nonbroadband source is higher than the gain due to acquaintance of source spectral characteristics.
In this paper we propose passive source localization in shallow water based on modal filtering and features extraction. The depth and distance of an Ultra Low Frequency source are estimated in the mono-dimensional configuration (a single hydrophone) and in the multi-dimensional configuration (a horizontal line array). The localization techniques are, respectively, based on adapted Fourier transform and frequency-wavenumber transform. In both representations modes are separable and thus can be filtered. We discuss modal filtering tools, then the localization itself is performed.
For distance estimation, we base our localization method on the analysis of mode phases. The proposed distance estimator is naturally combined with mode sign estimator. For depth localization, we use matched-mode processing, a technique that widely demonstrated its performance in a shallow water environment. The principle is based on comparison (by a contrast function) of mode excitation factors extracted from real data with a set of mode excitation factors (for simulated source depths) extracted from replica data (modeled with Moctesuma). We demonstrate that adding the mode signs to the mode excitation factor modulus improves significantly the localization performance in depth. We also propose a method of estimation of the source spectrum, which is very important for depth localization using Matched-Mode Processing.
The localization results, in depth and distance, obtained on signals simulated with Moctesuma in realistic geophysical conditions are very satisfactory and demonstrate the performance of the proposed methods.
This work was supported by Project REI 07.34.026 from the Mission pour la Recherche et l'Innovation Scientifique (MRIS) of the Delegation Generale pour l'Armement (DGA-French Departement of Defense).
- Jensen FB, Kuperman WA, Porter MB, Schmidt H: Computational Ocean Acoustics. AIP Press, New York, NY, USA; 1994.MATHGoogle Scholar
- Baggeroer AB, Kuperman WA, Mikhalevsky PN: An overview of matched field methods in ocean acoustics. IEEE Journal of Oceanic Engineering 1993, 18(4):401-424. 10.1109/48.262292View ArticleGoogle Scholar
- Fawcett JA, Yeremy ML, Chapman NR: Matched-field source localization in a range-dependent environment. The Journal of the Acoustical Society of America 1996, 99(1):272-282. 10.1121/1.414538View ArticleGoogle Scholar
- Wilson GR, Koch RA, Vidmar PJ: Matched-mode localization. The Journal of the Acoustical Society of America 1998, 104(1):156-162. 10.1121/1.423265View ArticleGoogle Scholar
- Shang EC, Clay CS, Wang YY: Passive harmonic source ranging in waveguides by using mode filter. The Journal of the Acoustical Society of America 1985, 78(1):172-175. 10.1121/1.392554View ArticleMATHGoogle Scholar
- Collison NE, Dosso SE: Regularized matched-mode processing for source localization. The Journal of the Acoustical Society of America 2000, 107(6):3089-3100. 10.1121/1.429338View ArticleGoogle Scholar
- Bogart CW, Yang TC: Comparative performance of matched-mode and matched-field localization in a range-dependent environment. The Journal of the Acoustical Society of America 1992, 92(4):2051-2068. 10.1121/1.405257View ArticleGoogle Scholar
- Kuperman WA, Hodgkiss WS, Song HC: Phase conjugation in the ocean: experimental demonstration of an acoustic time-reversal mirror. The Journal of the Acoustical Society of America 1998, 103(1):25-40. 10.1121/1.423233View ArticleGoogle Scholar
- Prada C, de Rosny J, Clorennec D, et al.: Experimental detection and focusing in shallow water by decomposition of the time reversal operator. The Journal of the Acoustical Society of America 2007, 122(2):761-768. 10.1121/1.2749442View ArticleGoogle Scholar
- Bogart CW, Yang TC: Source localization with horizontal arrays in shallow water: spatial sampling and effective aperture. The Journal of the Acoustical Society of America 1994, 96(3):1677-1686. 10.1121/1.410247View ArticleGoogle Scholar
- Nicolas B, Mars JI, Lacoume J-L: Source depth estimation using a horizontal array by matched-mode processing in the frequency-wavenumber domain. EURASIP Journal on Applied Signal Processing 2006, 2006:-16.Google Scholar
- Le Touzé G, Nicolas B, Mars J, Lacoume J-L: Matched representations and filters for guided waves. IEEE Signal Processing Letters. In pressGoogle Scholar
- Baraniuk RG, Jones DL: Unitary equivalence: new twist on signal processing. IEEE Transactions on Signal Processing 1995, 43(10):2269-2282. 10.1109/78.469861View ArticleGoogle Scholar
- Yang TC: A method of range and depth estimation by modal decomposition. The Journal of the Acoustical Society of America 1987, 82(5):1736-1745. 10.1121/1.395825View ArticleGoogle Scholar
- Cristol X, Passerieux J-M, Fattaccioli D: Modal representations of transient sound pulses in deep and shallow environments, with investigations about detailed space and time correlation of propagated waves. MAST, 2006Google Scholar
- Lopatka M, Nicolas B, Le Touzé G, et al.: Robust underwater localization of ultra low frequency sources in operational context. Proceedings of the Uncertainty Analysis in Modelling (UAM '09), 2009, Nafplion, GreeceGoogle Scholar
- Le Touzé G, Nicolas B, Mars JI, Lacoume J-L: Time-frequency representations matched to guided waves. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '06), 2006 3: 440-443.MATHGoogle Scholar
- Papandreou-Suppappola A, Murray R, Iem B-G, Boudreaux-Bartels GF: Group delay shift covariant quadratic time-frequency representations. IEEE Transactions on Signal Processing 2001, 49(11):2549-2564. 10.1109/78.960403View ArticleGoogle Scholar
- McClure M, Carin L: Matching pursuits with a wave-based dictionary. IEEE Transactions on Signal Processing 1997, 45(12):2912-2927. 10.1109/78.650250View ArticleGoogle Scholar
- Le Touzé G: localisation de source par petits fonds en UBF (1−100Hz) à l'aide d'outils temps-frequence, Ph.D. dissertation. INP, Grenoble, France; 2007.Google Scholar
- Nicolas B, Le Touzé G, Mars JI: Mode sign estimation to improve source depth estimation. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '08), 2008, Las Vegas, Nev, USA 2437-2440.Google Scholar
- Jesus SM: Normal-mode matching localization in shallow water: environmental and system effects. The Journal of the Acoustical Society of America 1991, 90(4):2034-2041. 10.1121/1.401631View ArticleGoogle Scholar
- Wilson JH, Rajan SD, Null JM: Inversion techniques and the variability of sound propagation in shallow water. IEEE Journal of Oceanic Engineering 1996, 21(4):321. 10.1109/JOE.1996.544043View ArticleGoogle Scholar
- Nicolas B, Mars J, Lacoume JL: Geoacoustical parameters estimation with impulsive and boat-noise sources. IEEE Journal of Oceanic Engineering 2003, 28(3):494-501. 10.1109/JOE.2003.816687View ArticleGoogle Scholar
- Le Touzé G, Mars JI, Lacoume J-L: Matched time-frequency representations and warping operator for modal filtering. Proceedings of the European Signal Processing Conference (EUSIPCO '06), September 2006, Florence, ItalyGoogle Scholar
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