 Research Article
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Underwater Broadband Source Localization Based on Modal Filtering and Features Extraction
EURASIP Journal on Advances in Signal Processing volume 2010, Article number: 304103 (2010)
Abstract
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 UltraLow 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 depthdependent 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 frequencywavenumber 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.
1. Introduction
Passive source localization in shallow water has attracted much attention for many years in underwater acoustics. In this environment and for UltraLow 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 [1]. 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: MatchedField Processing (denoted MFP) [2, 3] and MatchedMode Processing (denoted MMP) [4–6]. The comparative study of both approaches is given in [7]. MatchedMode 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 [8] which can be seen as a broadband coherent MFP. Some experiments have been performed showing the feasibility of the method [9]. 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 frequencywavenumber 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, matchedmode 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 frequencywavenumber plane (), which is a twodimensional Fourier transform of radial distancetime section (signal ).
For the monodimensional 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 timefrequency characteristics [12] into modal filtering. The main idea is to deform a signal in such way that nonlinearities in the timefrequency 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 monodimensional configuration is done in an adapted frequency domain (Pekeris Fourier transform). The classical and adapted frequency domains are related by the unitary equivalence formalism [13].
After a brief presentation of modal propagation theory, we give a short description of the simulator Moctesuma2006, 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.
2. Modal Propagation and Modes
Acoustic propagation of UltraLow 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 (matchedmode 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.
The perfect waveguide model is made of a homogeneous layer of water between perfectly reflecting boundaries (a pressure release surface and a rigid sea bottom). The water layer is characterized by depth , velocity and density . We consider an omnidirectional point source located at depth and at distance radiating a signal . The acoustic pressure field received at a reception point of coordinates , where and are, respectively, horizontal distance and depth, can be expressed by
where is the time, is the frequency, and satisfies the general Helmholtz equation.
Using variable separation (in depth and in range) and boundaries conditions [1], the pressure at long range becomes a sum of modes
with a constant, the source spectrum, the modal function of mode , and the horizontal spatial frequency of mode . The spatial frequency is defined by and it corresponds to the classical wavenumber divided by In the following, for the sake of simplicity, we will call the wavenumber with its horizontal and vertical components and (instead of spatial frequency). The mode wavenumber spectrum is discrete and each mode is associated with an unique wavenumber. The amplitude is a function of the source depth :
with .
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 .
Then, let us demonstrate very shortly the principle of mode phase processing for distance estimation. Modes contain, besides depth information, also distance information about the source. This information is contained in mode phase. The phase of mode at frequency is defined by
where

(i)
is the phase of the source at frequency

(ii)
is a phase factor due to time delay of the recorded signal;

(iii)
depends on the modal function sign at the source depth it is if and if

(iv)
depends on the modal function sign at the receiver depth

(v)
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 [14].
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.
Moctesuma
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 [1]. Among the different models we choose the numerical model Moctesuma2006—a realistic underwater acoustic propagation simulator developed by Thales Underwater Systems [15]. For the sake of simplicity Moctesuma2006 will be called further Moctesuma. This model, based on normal mode theory, simulates an underwater acoustic propagation for rangedependent 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 (timeseries) [16].
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 rangedependent 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. Experimental Configuration
3.1. Environment
The analysis presented in this paper is done in a simulated shallow water environment located in the Mediterranean Sea during a summer period. The environment is range independent with a water depth of m. The sound speed profile of the water column (Figure 1) is characterized by a strong negative gradient of approximately m/s. The highest gradient is located in the first half of the water column. The seabed parameters are presented in Figure 2. Modal functions no. to of the studied environment are presented in Figure 3.
3.2. Signal Sources and Reception Configurations
In this paper we consider two impulsive broadband sources in the ULF band: the first one ULF lasts several tens of milliseconds and has a flat spectrum; the second source ULF lasts several hundreds of milliseconds and is made of four band signals (" hills" spectrum). Both signals are presented in temporal and spectral domains in Figure 4.
The source ULF is used to validate the methods in a simple case. For a more complex situation source ULF2 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.
The experimental configuration is given on Figure 5. Three different source depths are studied: m, m, and m. The horizontal distance between the source and the first sensor of HLA is equal to km. The HLA is located on the sea bottom (). The HLA can be located at any depth, but this information has to be known. In our simulations the source and the HLA are motionless.
3.3. Data
Moctesuma simulator provides a section of the pressure recorded on the HLA denoted signal for the sake of simplicity in the following. It is a sampling of the pressure field in radial distance and in time . The size of the data is defined by

(i)
number of hydrophones (traces)

(ii)
duration and frequency sampling of the recording (number of samples ).
The "real data" is obtained by adding a white bidimensional (in time and in space) Gaussian noise to the simulated data. Several signaltonoise ratios (SNRs) are considered.
4. Filtering Methods
In this paper, source localization in depth and distance is performed either by multidimensional or by monodimensional approach. The first one will be called approach (as the method operates in the frequencywavenumber 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 frequencywavenumber 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 timefrequency representation. As proposed in [12, 17], we use an adapted frequency representation in which modes are separable and consequently can be filtered.
4.1. MultiDimensional Approach
In the multidimensional case, the radial distancetime section is represented in the frequencywavenumber plane . The transformation, called transform and denoted by , is linked to via a twodimensional Fourier transform (in radial distance and in time ). The transform of the signal is defined by
To improve dynamics of modal representation and avoid spatial aliasing in the plane, a velocity correction on the section is applied before calculating transform [11] (classical preprocessing technique used in array processing). This operation consists in applying to every trace of the section a time shift, so that the direct wave (traveling with speed , equal to in the perfect waveguide) arrives at every sensor at the same time (giving an apparent infinite velocity). Let us denote the representation of the section . Then, representation of the section after velocity correction is
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).
If we consider a white broadband source radiating a transient signal in a perfect waveguide, the transform of the pressure signal (see (2)) received on the HLA, at long range, can be approximated by
with a frequency dependent constant related to the source spectrum.
The theoretical modal signal energy is located on the mode dispersion curves (the form of the mode dispersion curves is given in Figure 6). As one can notice, for each frequency the wavenumber spectrum is discrete. The representations of two data sets are given in Figure 7. The first plot shows a simulation done with signal ULF1 (flat spectrum) at depth m. The energy is spread across all the frequency band. The second plot presents a simulation with signal ULF2 (" hills" spectrum) at depth m. The representation reflects exactly the spectrum of the source signal (see Figure 4).
As the HLA is located at a known depth, values of the factors are known. For HLA located on the sea bottom and expression (7) can be rewritten as
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.
Mask Construction
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.
An initial mask of mode is created using its wavenumbers. These can be computed theoretically by a propagation model if the environmental parameters are known. In our case they are given by Moctesuma (see Figure 6). Then, the mask of mode is dilated independently in both domains (frequency and wavenumber) with the dilation parameter according to the following formula:
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 simulated environment has to be as close as possible to the real environment to achieve a good filtering of modes. Moreover, it is necessary to dilate the previous theoretical mask for two reasons:

(i)
the limited HLA lengthmode energy spreads around dispersion curves in the plane;

(ii)
the mismatch between real and simulated environments [11].
A total set of masks for the studied environment is given in Figure 8. This set contains masks which corresponds to the total number of propagating modes in this environment. An example of the mask adapted to mode no. is given in Figure 9. The mask of mode no. is built starting from its dispersion curve (see Figure 6, trace counting from right to left) and then dilated according to (9).
The energy spectrum of data in Figure 7(a) is shown in Figure 10. For each mode the energy is calculated as a mean of the region where this mode is present (after mask filtering).
In Figure 11 we present the result of the mode filtering for modes nos. , (mode with very low energy), and (the mode with the highest energy).
4.2. Monodimensional Approach
In the monodimensional configuration, classical signal representations such as Fourier transform or shorttime Fourier transform are not suited for description of a signal that can be decomposed in a sum of dispersive modes (nonstationary and nonlinear timefrequency patterns). Therefore, these techniques cannot correctly represent the signal having modal structures (see Figures 12(a) and 13(a)). This signal processing problem has attracted much interest for the last decades [13, 18, 19]. In this paper we based our monosensor approach on works [12, 17, 20]. The idea is to find a representation well adapted for modal signal structure to achieve modal filtering. The best way is to take into account the physics of oceanic waveguide propagation and build a representation adapted to the signal structure in the same way as Fourier transform is adapted to a monochromatic signal. Here, we discuss only an adapted frequency representation . However, on the same rule an adapted timefrequency representation can be constructed and used [12, 17].
The adapted processing tools are based on the combination between acoustic wave propagation (waveguide propagation law) and signal processing theory (unitary equivalence). Building of the adapted frequency representation is based primary on definition of the unitary operator of transformation adapted to guided waves. This unitary operator is linked to the dispersion law , where is the instantaneous frequency and is the time. For a perfect waveguide model and for each mode the dispersion law is defined by
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.
Starting from the dispersive relation given in 10, the instantaneous frequency is the derivative of the instantaneous phase :
with the cutoff frequency of mode defined by
is called the general dispersive function and is defined as follows:
As a result, the pressure signal of the perfect waveguide model is defined by
with the envelope evolution of mode The unitary operator of transformation should transform nonlinear mode structures in linear ones (in timefrequency domain), and thus allow the use of classical Fourier filtering techniques. In construction of such representation, the unitary equivalence formalism is used [13]. One of the unitary operators is a warping operator (applied to a signal ) defined as follows:
The function has to be derivable and biunivocal, and function has to exist. The operator is applied to the signal in order that becomes a sum of linear structures
To do so, we deduce from (16), the deformation function defined on
Finally, the unitary operator of transformation adapted to a perfect waveguide is
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 [12]).
As the nonlinear timefrequency 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.
5. Estimators
In Section 4 we demonstrated how extracting modes from the recorded signal by multi and monodimensional approaches. In this section we discuss depth and distance estimators based on modal processing. For depth estimation, we use a matchedmode 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: frequencywavenumber 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 signdistance 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.
The signdistance estimator is originally based on a work published in [20]. We present this estimator very succinctly, more details can be found in [21]. The principle is based on the definition of mode phase given in (4). To suppress the unknown parameters in this equation, the initial estimator uses the difference between two mode phases:
with

(i)
being difference between two mode phases at depth ; the values of this parameter are defined by (for details see Table 1):
(20)  (ii)
The estimation is done in sequential way; that is, superior mode signs (of order ) are estimated using inferior mode signs (of order ). The starting point is the absolute mode sign of mode no. which is always positive: For example to estimate the sign of mode one can use in theory modes As the frequency band shared by two modes ( and ) has to be maximized for the estimation performance, we propose the following rule to choose inferior mode number () for estimation of superior sign mode number ():
The sign of is known as the depth of HLA is known (in our case it is ). To estimate mode signs for real data, for each frequency the quantity is measured, and then we calculate the cost function defined as
where is the distance function defined as
with defined on a basic interval . The signdistance estimator is found by the minimization of the cost function :
where and
The signdistance estimator is calculated for possible values of : and . By minimizing , we find the searched value of (using (25), the relative sign between modes and is estimated) and also the distance. As we know the absolute sign of inferior mode (known or estimated in previous step of estimation) and the estimated value of the pass function betwen modes and , we can find the absolute sign of mode . The rule of mode sign estimation is given in Table 2.
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 MatchedMode 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 mode excitation factors are extracted by or approach on real and replica data in the same manner. They are positive as they are extracted from positive value spaces ( transform or modulus of the adapted Fourier transform). Their signs are then obtained using the signdistance estimator presented in the previous section. The mode signs are estimated only for real data. For the reference data, as the simulations are done for a set of determined source depths, mode signs are known. The combination of mode excitation factors with mode signs allows canceling secondary peaks in the correlation function. These peaks are due to "mirror solutions" of modal functions for some depths when considering only mode excitation factor modulus (see Figure 14). Therefore, adding mode sign information to mode excitation factor modulus improves significantly the performance of depth localization (shown on examples in Section 6).
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.
To compare mode excitation factors extracted from real and simulated data, a normalization based on the closure relationship is applied:
where is an excitation factor of mode . Then, the comparison between mode excitation factors extracted from the real data and replica data is made using the contrast function :
where is the number of analyzed modes
The maximum of indicates the estimated depth of the source.
In matchedmode 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 cutoff 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 signaltonoise conditions.
5.3. Source Spectrum Estimation
To perform matchedmode localization, knowledge (at least partial) of quantities such as geoacoustical parameters of the environment and spectral characteristics of the source is crucial. This results from the fact that the simulated acoustic field should be simulated in geoacoustical conditions similar (as much as possible) to the real conditions existing in the location of interest. In general, environmental parameters can be estimated using inversion methods [23, 24]. Unluckily, in passive approach the knowledge of the source spectrum remains notwithstanding problematic. As the influence of the source spectrum is relevant (see (2) and (7)), we propose an estimator of source spectrum based on the analysis of the first mode (most horizontal), as this mode is always excited. The property of distinct attenuation of signal frequencies (growing non linearly with the frequency) is taken into account by the estimator in order to improve estimation quality. The proposed estimator can be formulated as follows:
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).
6. Localization
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 peaktosecondary peak ratio of dB the signaltonoise 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 signaltonoise 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.
We first apply the deformation of the signal described in Section 4.2 for the Pekeris model. The parameters used to warp the modal signal are

(i)
water column depth: m;

(ii)
sound speed velocity in water: m/s;

(iii)
sound speed velocity in sediments: m/s;

(iv)
water density: ;

(v)
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.
6.1.1. Distance
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.
In Figure 15 we show the cost function corresponding to several distance estimations. As the distance estimator works on mode couples, we present five distance estimations for mode couples: , , , and and research area km with a step m. The estimated distances are given in Table 3 (the real distance is km). The first estimations are correct, and the last one is false, which is due to limited frequency band of mode . Moreover, the signdistance estimator provides mode signs. For this example, the mode signs were estimated on the same mode couples as distance. The estimation of mode signs no. to is correct and the absolute signs are , , , , and .
6.1.2. Depth
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 Figure 16 we show the result of depth localization. The performance is given for the methods "without mode signs" and "with mode signs." The difference in performance between two methods is relevant which results from the importance of taking into account mode signs when calculating the contrast function . The method "with mode signs" eliminates the "mirror solutions" (decrease of the contrast function from dB to dB for a secondary peak at m by adding mode signs). Figure 17 highlights also a problem that can appear when adding mode signs: if the mode sign estimation is false (here, sign of mode is false), then the localization performance decreases significantly. However, as one can choose between options when estimating mode signs (the current mode sign is estimated using three inferior modes ), the probability of this error should decrease (if the signaltonoise ratio is sufficiently high).
Due to the oscillating character of modal functions and because we consider only modal function modulus, there exist "mirror" depths which give secondary peaks in the contrast function (Figure 14(a)). To explain this fact and the reason why they disappear when integrating mode signs in the depth estimation, let us present an example. Figure 18 shows mode excitation factors extracted from simulated data at two different source depths: m and m. For these two depths the mode excitation factor moduli are almost the same (Figure 18(a)), and so the difference in the contrast function for these depths is not relevant. When the complete information about mode excitation factors (i.e., modulus and sign) is considered (Figure 18(b)), these depths become discriminated. This is due to especially modes no. , , but also .
In the monodimensional configuration in lower signaltonoise ratio conditions the mode sign and distance estimations can be inaccurate. Also, the depth localization performance cannot be satisfied. Therefore, we propose the multidimensional 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 signaltonoise 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 [16]. Different issues of the use of HLA are discussed in [10].
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.
6.2.1. Distance
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) [16].
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 signdistance estimator gives as a result mode sign. In multidimensional case, we dispose of estimations of mode signs for each mode couple option. For a mode number equal to , the signdistance 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.
Choice reliability:
where is a number of sign changes (). This criteria should be maximal.
Estimation variability:
where denotes a set of distance estimations and denotes a second derivative with respect to the hydrophone number. For the noerror 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.
Error distance estimation:
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.
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 .
6.2.2. Depth
After modal filtering, the mode excitation factor modulus of each mode is calculated as a mean over the region. Moreover, the signdistance estimator can be used for mode signs estimation.
In Figure 20 we show the result of depth localization. The performance is given for method "without mode signs" and "with mode signs." The difference between both methods is relevant which confirms the importance of taking into account mode signs when calculating the contrast function . The method "with mode signs" eliminates the "mirror solutions" (decrease of the contrast function from dB to dB for a secondary peak at m by adding mode signs).
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.
Let us consider an example of depth localization in the environment described in Section 3. The source ULF2 is located at m of depth. As we do not know the spectral properties of the localized source we consider two cases.

(i)
We use a source with flat spectrum for simulation of the replica field (source ULF1)common approach when unknown source.

(ii)
We estimate a source spectrum by the method defined in (29) and use it to simulate the replica field.
In Figure 21, we present spectrum of the source ULF2 estimated by the proposed method (compare with Figure 4(d)). The results of localization without and with source spectrum estimation are shown in Figure 22. We can note that estimating the source spectrum improves considerably the localization performance (of about dB in the example).
Nevertheless, our method is designed for broadband sources. Therefore, even if spectral characteristics of the source are perfectly known, but present narrowband or combtype structures, the localization performance decreases. The performance decrease due to nonbroadband source is higher than the gain due to acquaintance of source spectral characteristics.
7. Conclusion
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 monodimensional configuration (a single hydrophone) and in the multidimensional configuration (a horizontal line array). The localization techniques are, respectively, based on adapted Fourier transform and frequencywavenumber 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 matchedmode 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 MatchedMode 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.
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Acknowledgment
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 (DGAFrench Departement of Defense).
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Lopatka, M., Le Touzé, G., Nicolas, B. et al. Underwater Broadband Source Localization Based on Modal Filtering and Features Extraction. EURASIP J. Adv. Signal Process. 2010, 304103 (2010). https://doi.org/10.1155/2010/304103
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DOI: https://doi.org/10.1155/2010/304103