DoubleCapon and doubleMUSICAL for arrival separation and observable estimation in an acoustic waveguide
 Grégoire Le Touzé^{1}Email author,
 Barbara Nicolas^{1},
 Jérôme I Mars^{1},
 Philippe Roux^{2} and
 Benoit Oudompheng^{1}
https://doi.org/10.1186/168761802012187
© Le Touzéet al.; licensee Springer. 2012
Received: 17 February 2012
Accepted: 9 August 2012
Published: 30 August 2012
Abstract
Abstract
Recent developments in shallow water ocean acoustic tomography propose the use of an original configuration composed of two sourcereceiver vertical arrays and wideband sources. The recording space thus has three dimensions, with two spatial dimensions and the frequency dimension. Using this recording space, it is possible to build a threedimensional (3D) estimation space that gives access to the three observables associated with the acoustic arrivals: the direction of departure, the direction of arrivals, and the time of arrival. The main interest of this 3D estimation space is its capability for the separation of acoustic arrivals that usually interfere in the recording space, due to multipath propagation. A 3D estimator called double beamforming has already been developed, although it has limited resolution. In this study, the new 3D highresolution estimators of double Capon and double MUSICAL are proposed to achieve this task. The ocean acoustic tomography configuration allows a single recording realization to estimate the crossspectral data matrix, which is necessary to build highresolution estimators. 3D smoothing techniques are thus proposed to increase the rank of the matrix. The estimators developed are validated on real data recorded in an ultrasonic tank, and their detection performances are compared to existing 2D and 3D methods.
Keywords
Introduction
Estimation of sound speed variations in the ocean that is based on a linearized model and using acoustic waves is known as oceanic acoustic tomography (OAT) [1]. This tomography process is classically divided into three steps: first, estimation of observables extracted from the signal, then the building of a forward model that links these observables and the sound speed variations, and finally the inversion of this problem using the extracted observables. In this study, we are interested in the first step, i.e. the observable extraction and estimation.
We focus here on shallow water environments, as typically a coastal environment, that can be modeled as a waveguide. These environments are a subject of major interest in the ocean science community since they are the place where many physical phenomena occur, such as mixing layers, streams, tides, human influence and pollution. However, unfortunately, they are also complex environments where the acoustic propagation is multipath, due to reflections from the waveguide boundaries. These different paths lead to arrivals that interfere in the traditional recording space, and also in the traditional estimation spaces (e.g. the direction of arrival [DOA] space). The arrival separation, and consequently the observable estimation step, is thus difficult to achieve.
Several methods have been developed to improve this difficult task, traditionally using vertical line arrays (VLAs). These can be divided into two groups [2]: separation methods and highresolution methods. This first group, the separation methods (which include beamforming [BF] techniques), were first proposed by [3], who developed an matched filter to estimate the arrival times and amplitudes in a noisy signal. Then improvements were proposed based on the use of more adapted signals [4, 5]. An excellent review of BF methods is presented in [6]. In this separation group of methods, adaptive BF (including Capon) has been extensively studied in signal processing [6]. The other group of methods, the highresolution methods (or subspacebased methods), include the classical MUSIC [7] or ESPRIT [8]. The asymptotical separation power of these methods is not limited by the experimental conditions, such as by signal frequency, array length or signaltonoise ratio. These methods use eigenvector decomposition of the crossspectral density matrix. They were first proposed to estimate the DOA (see [9]) and arrival times [10], both separately and then jointly [11]. However, these methods have limitations, due to the proximity of arrivals in the estimation space, which are onedimensional (1D) or 2D, depending on the experimental configuration.
To overcome this difficulty, we are interested in an original experimental configuration that is composed of two VLAs in the water column: a source array and a receiver array [12]. The arrivals are thus characterized by three observables: their direction of departure (DOD), their DOA, and their time of arrival (TOA). A threedimensional (3D) estimator is required to transform the 3D recording space (receiverfrequency) into the 3D estimation space (DOADODTOA). To date, this estimation has been achieved with double BF (DBF) [12, 13] on the source and receiver arrays. This method has resolution limitations due to the limited size of the arrays and to the source signal. This drawback is particularly problematic in the separation of the first arrivals that correspond to the shortest TOAs, which are close in the three dimensions of the estimation space.
To be able to provide a better separation of the arrivals, we propose here two methods for the improvement of the resolution in the DOADODTOA space. These methods are generalizations of the traditional Capon and MUSIC methods to the 3D configuration.
This report is organized as follows: the context and signal model that correspond to the experimental configuration are first presented. Then, the conventional estimations methods of BF, Capon and MUSIC are briefly recalled. The double Capon (DCapon) and double MUSICAL (DMUSICAL) methods are then considered. The implementation issues of these methods are discussed, and 3D smoothing of the crossspectral data matrix is introduced. We show then that similar results can be obtained using these two methods. Finally, we illustrate the performance of these methods, compared to the existing methods, on real data recorded in an ultrasonic tank. This experimental environment reproduces oceanic acoustic propagation at a small scale.
Context
Configuration
Consider a shallow water environment. The experimental configuration is composed of two arrays: a VLA of N_{ e }regularly spaced sources, and a VLA of M_{ r }regularly spaced receivers. For the sake of simplicity, the intersensor distance is Δon both arrays. We consider that N_{ e }and M_{ r }are odd, and that the reference source (respectively the reference receiver) is located at the middle of the source array: index m_{ref} = (N_{ e } + 1)/2 (respectively at the middle of the receiver array: index m_{ref} = (M_{ r } + 1)/2). The source signal is broadband, and the propagated signals are recorded on F frequency bins that cover the frequency band. In the experiment, the recording space thus has three dimensions: sourcereceiverfrequency. The recorded data contain the whole transfer matrices between each source and each receiver in the frequencydomain, and finally form the data cube X(N_{ e } × M_{ r } × F). For a given signal emitted by the source array, the propagation is multipath in the waveguide, and it will lead to several plane wave arrivals on the receiver array. Note that in our configuration, the horizontal distance between the arrays is much larger than their lengths by at least a factor of 10. The plane wave approximation is thus realistic. Each arrival p, corresponding to a given raypath, is characterized by its three observables:

the DOA ${\theta}_{p}^{r}$, which is also known as the reception angle: the angle between the raypath direction at the receiver array and the normal to the receiver array;

the DOD ${\theta}_{p}^{e}$, which is also known as the emission angle: the angle between the raypath direction at the source array and the normal to the source array;

the TOA T_{ p }, which is the travel time between the reference source and the reference receiver.
Note that in this article “source” designates the emitting sensor. To avoid ambiguity, the elementary contribution that we want to detect and estimate the parameters for, and which corresponds to a raypath in the waveguide, is designated by “arrival” (and not by “source” as it can be classically designated in array processing).
Motivation of 3D detection
Detection and estimation of arrivals are classically achieved in 1D and 2D configurations. In the 1D case, narrowband signals recorded on a receiver array lead to a DOA estimation space [7, 14, 15]. In the 2D case, two configurations are studied: broadband signals recorded on a receiver array and a DOA–TOA estimation space [11], or narrowband signals emitted by a source array and recorded on a receiver array and a DOADOD estimation space [16]. The resolution is limited by the size of the arrays and by the signal central frequency in DOA and DOD, and by the signal bandwidth in TOA (cf. Section “Conventional estimation methods”).
When combined with a receive array and broadband signals, the source array adds a third dimension in the recording space, and consequently a new dimension in the estimation space: the DOD. For arrivals with different DODs, the use of an estimator that includes the DOD dimension can better detect and estimate these arrivals. This improvement will be particularly efficient for arrivals close in the DOA and TOA dimensions, but far in the DOD dimension.
Signal model
where $\tau \left({\theta}_{p}^{e}\right)=\Delta sin\left({\theta}_{p}^{e}\right)/v$ (respectively $\tau \left({\theta}_{p}^{r}\right)=\Delta sin\left({\theta}_{p}^{r}\right)/v$) are the delays associated with the DOD (respectively the DOA), assuming a constant sound speed v at the arrays, a_{ p }is the amplitude of the p^{th} arrival, b_{m,n,ν}is the noise contribution that is generally considered uncorrelated in space and frequency, and s_{ ν }is the source spectrum, which will be assumed to be known. Note that the number of arrivals, P, is typically around 10.
Experimental constraints
We want to achieve arrival separation and observable estimation for a time evolving medium. It is thus a problem to record the different realizations at different times, because the medium and its sound speed might change between the two realizations. Consequently, we can only consider a single realization to perform the observable estimation. This constraint must be taken into account for the estimation of the crossspectral matrix (cf. Section “Implementation”).
Conventional estimation methods
We briefly recall the 1D conventional estimation methods: BF, Capon and MUSIC, in the general context of array processing. The specific experimental constraints, and particularly the smoothing issues, are discussed in Section “Implementation”.
where ${\widehat{\mathbf{R}}}_{1D}$ is the estimated covariance data matrix. The resolution of the BF is directly linked to the size l = (M_{ r }−1)Δ of the array: two sources with DOAs closer than ${\theta}_{\text{min}}^{r}\approx \lambda /l$ with the wavelength λ at the central frequency, will not be separated by BF.
Two types of techniques have been developed to overcome those limitations:
As for BF, the Capon estimator principle is to project the signal on steering vectors [15]. Capon steering vectors are calculated adaptively, so that they minimize the power contributed by the noise and by any signals coming from other directions than θ^{ r }, while maintaining a unitary gain in the direction of interest θ^{ r }. The Capon algorithm (which is also known as the minimum variance distortionless response) has already been successfully applied in underwater acoustics [17].
MUSIC is a subspacebased method [7, 14]: the recording space is divided into a signal subspace and a noise subspace. This subspace division is achieved using eigenvalue decomposition (EVD) of ${\widehat{\mathbf{R}}}_{1D}$. The signal subspace is spanned by the L eigenvectors corresponding to the L maximum eigenvalues. The noise subspace is spanned by the other M_{ r }−L eigenvectors. Finally, the estimator is the inverse of the projection of the signal on the noise subspace. For uncorrelated arrival amplitudes with spatially white noise, the MUSIC estimator is unbiased and its resolution is not limited.
DCapon and DMUSIC
The 3D model and estimators are considered in this section, in the general context of array processing. The specific experimental constraints and particularly the smoothing issues, are discussed in Section “Implementation”.
Data model
The noise long vector b is built in the same way, starting from the noise contributions b_{n,m,ν}.
where A = E{aa^{ H }} is the arrival amplitude covariance matrix (P×P) and B = E{b b^{ H }} is the 3D crossspectral noise matrix (N_{ e }M_{ r }F × N_{ e }M_{ r }F).
Note on 3D steering vectors $\mathbf{d}({\theta}_{p}^{r},{\theta}_{p}^{e},{T}_{p})$

two distinct steering vectors are independent (noncolinear), and the correspondence between a set of parameters and a 3D steering vector is thus unique;

two distinct steering vectors are never orthogonal;

given N = N_{ e }M_{ r }F as the dimension of the recording space, n ≤ N distinct steering vectors form a free family. N steering vectors thus forming a base;

assuming P ≤ N, the steering vectors linked to the signal thus form a free family and spanning a space of dimension P. The dimension of the signal subspace in R is thus equal to the rank of A. Consequently, the rank of R depends on the correlation degree between the P arrival amplitudes.
DCapon
Capon has already been extended to the 2D context [18]. The proposed DCapon method consists of extending the conventional Capon method to the 3D OAT context. We create 3D Capon steering vectors g(θ^{ r },θ^{ e },T) which minimize the power contributed by noise and any signal coming from other ‘directions’ than (θ^{ r },θ^{ e },T), while maintaining a unitary gain in the direction of interest (θ^{ r },θ^{,e}T).
where $\widehat{\mathbf{R}}$ is the estimated crossspectral data matrix and d the theoretical steering vectors built using Equations 9 and 10. The estimation of the crossspectral matrix will be discussed in Section “Implementation”.
 (1)
the recorded data to build $\widehat{\mathbf{R}}$;
 (2)
the source spectrum s _{ ν };
 (3)
the environment information Δ and v, to calculate τ(θ ^{ e }) and τ(θ ^{ r }), and thus the steering vectors d(θ ^{ r },θ ^{ e },T).
DMUSICAL
The MUSIC algorithm has already been extended to the 2D configuration by Gounon et al. in a large band context: MUSICAL (MUSIC Active Large Band) estimates the 2D (TOADOD) observables, starting from the 2D recording space: receiverfrequency. Recently, a 2D MUSIC was developed to estimate conjointly the DOD and DOA [16]. In the same way, we develop a 3D MUSIC estimator, which we call DMUSICAL, to extend the conventional MUSIC method.
where $\mathbf{V}=[{\mathbf{v}}_{1},\dots ,{\mathbf{v}}_{{N}_{e}{M}_{r}F}]$ is a N_{ e }M_{ r }F × N_{ e }M_{ r }F matrix that contains the eigenvectors v_{ i }, and ⋀ is a N_{ e }M_{ r }F × N_{ e }M_{ r }F diagonal matrix that contains the N_{ e }M_{ r }F eigenvalues λ_{ i }.
As $\widehat{\mathbf{R}}$ is a normal matrix ($\widehat{\mathbf{R}}{\widehat{\mathbf{R}}}^{H}={\widehat{\mathbf{R}}}^{H}\widehat{\mathbf{R}}$), eigenvectors are orthogonal (< v_{ i }.v_{ j }> = 0 ∀ {i,j}i ≠ j). Selecting the L largest eigenvalues and their associated eigenvectors, we span a ‘signal’ subspace, the N_{ e }M_{ r }F−L others are spanning the ‘noise’ subspace. These two subspaces are orthogonal. The signal projector (respectively the noise projector) is deduced from the L first (respectively the N_{ e }M_{ r }F − L last) eigenvectors: ${\mathbf{\pi}}^{s}=\sum _{i=1}^{L}{\mathbf{v}}_{i}{\mathbf{v}}_{i}^{H}$ (resp. ${\mathbf{\pi}}^{n}=\sum _{i=L+1}^{{N}_{e}{M}_{r}F}{\mathbf{v}}_{i}{\mathbf{v}}_{i}^{H}$).
Note that a 3D MUSIC method has already been developed for multicomponent seismic signals [19]. The third multicomponent dimension does not correspond to the same issue, and the DOD estimation cannot be achieved.
The DMUSICAL implementation needs the same information as the DCapon (cf. Section “DCapon”). Assuming that A is the full rank, and that the noise is white in the three dimensions and for variance ${\sigma}_{n}^{2}$, we choose L = P and the signal is completely represented by the signal subspace composed of the P first eigenvalues. The noise subspace contains only noise contributions. In this case, the DMUSICAL estimation is unbiased.
In our practical case, arrival amplitudes are correlated, which means that A is not full rank. The implementation of DCapon and DMUSICAL thus needs preprocessing, which is detailed in the following Sections “Smoothing issue” and “Diagonal loading and estimation of L”.
Implementation
In this section, the implementation of the proposed DCapon and DMUSICAL algorithms to our OAT context is discussed.
Smoothing issue
The Capon and MUSIC detection methods need the signal subspace to be correctly represented to be efficient. This means the arrivals must be uncorrelated, or at least not fully correlated, to generate a signal subspace of dimension P. Equivalently, the amplitude covariance matrix A must be full rank. To achieve this, classical methods assume that the arrivals are statistically uncorrelated, and estimate the $\widehat{\mathbf{R}}$ crossspectral data matrix by averaging a great number of realizations.
In our context, this type of estimation is not possible, for the two following reasons:

As explained in Section “Experimental constraints”, only one realization can be considered to perform the observable estimation. $\widehat{\mathbf{R}}$ must be determined using a single data realization x.

Moreover, even assuming a nonevolving medium, the arrival amplitudes remain correlated. Indeed, the arrivals are induced by different raypaths that result from the acoustic propagation. The correlation degree between the arrival amplitudes a_{ p }is thus determined by the propagation and not by the emitted source signals. The amplitude vector a is constant between realizations up to a multiplying factor. The P arrival amplitudes are thus fully correlated.
Considering those two issues, the rank of $\widehat{\mathbf{R}}$ will be 1 if it is estimated classically. Caponlike or MUSIClike methods are thus equivalent to the BF method.
To avoid this problem, a 3D smoothing method of the matrix is developed. Smoothing methods are used to increase the rank of $\widehat{\mathbf{R}}$. They were developed in 1D configurations [20, 21] and then extended to 2D [22]. The principle is to divide the array into K different subarrays with the same characteristics (size, sampling). Each subarray induces a signal x_{ k }. The diversity in realization is replaced by a diversity in subarray. The estimated matrix $\widehat{\mathbf{R}}$ is the mean of the matrices ${\widehat{\mathbf{R}}}_{k}$.
Note that the size of $\widehat{\mathbf{R}}$ is now ${N}_{e}^{s}{M}_{r}^{s}{F}^{s}\times {N}_{e}^{s}{M}_{r}^{s}{F}^{s}$. The numbers of subarrays K_{ e }, K_{ r } and K_{ f }depend on the configuration and on the arrival number P. In each direction, the choice of the subarray size depends on the size of the original array and on the resolution we want to obtain in the corresponding estimation dimension (DOD for the source array, DOA for the receive array, and TOA for the frequency). The aim of the smoothing is to increase the number of significant eigenvalues of $\widehat{\mathbf{R}}$, so that they represent accurately the signal subspace. This objective is achieved when the eigen structure of the smoothed matrix (i.e. the repartition of its eigenvalues) is stable with K. We empirically observe that K must be chosen so as to be a lot larger than P. This number is a priori not precisely known, but depending on the environment knowledge, we can approximately estimate it. We thus increase the number K to achieve this objective.
The 3D smoothing is a preprocessing that is necessary to achieve DCapon or DMUSICAL in our context. However, it has two limitations:

Assuming a perfectly plane wave in the 1D configuration, two different signals corresponding to two different subarrays are equal up to a multiplying factor. The amplitude of this factor is 1, and its phase depends on the delay between the subarrays and on the considered plane wave. These can thus be seen as two realizations of the same signal with different amplitudes.Considering now P plane waves, the 1D smoothing leads to vectors of arrival amplitudes a_{ k }that are different for each subarray k. Finally, the estimated amplitude covariance matrix $\widehat{\mathbf{A}}=1/K\sum {\mathbf{a}}_{k}{\mathbf{a}}_{k}^{H}$ is non singular [21] if K ≥ P. The estimated $\widehat{\mathbf{R}}$ matrix has thus a rank equal to P. In the 3D case, two contributions of the same plane wave on two different 3D subarrays are not equal up to a multiplying factor. This produces a bias in the estimation of $\widehat{\mathbf{R}}$, which prevents the 3D methods from having an unlimited resolution. However, this bias is not important in practical cases compared to the resolution needed.

As we smooth only one realization instead of averaging several realizations, we cannot recover the statistical characteristics of the additional noise. Consequently, noise must be considered as a deterministic element. The rank of $\widehat{\mathbf{R}}$ cannot exceed K and the K eigenvectors that correspond to the K first eigenvalues span a subspace in which the whole data are present, including the signal, and also the noise part and the bias induced by the smoothing.
Diagonal loading and estimation of L
where $\widehat{\mathbf{R}}(i,i)$ is the i^{th} element of the diagonal of $\widehat{\mathbf{R}}$. A natural choice is to take σ as small as possible, so that the condition ‘${\widehat{\mathbf{R}}}^{C}$ invertible’ is verified on the computational platform. As we will see, this choice does not generally give the best result. Recent methods have estimated the optimum diagonal loading [23, 24], but they are not adapted to our context.
DMUSICAL requires the estimation of the signal subspace dimension L to be achieved. For an unbiased estimation of $\widehat{\mathbf{R}}$, L would be equal to P, the number of expected arrivals. However the smoothing processing biases the estimation of $\widehat{\mathbf{R}}$. A classical L estimator based on the statistical properties of the noise [14, 25] can thus not be applied. Moreover, empirically, we observe that the bias introduced by the 3D smoothing leads to a difference between P and the number L of significant eigenvectors that actually spanned the signal subspace. This number L is larger than P. The determination of L is thus made, starting from the decreasing curve of eigenvalues calculated with the EVD. The eigenvectors that correspond to all of the significant eigenvalues (chosen with a threshold corresponding to 0.5% of the first eigenvalue) are selected and span the signal subspace, whereas all the other ones span the noise subspace. Note that L is thus lower than K (in practice, a lot smaller) and larger than P.
Computational cost
 1.
$\widehat{\mathbf{R}}$ estimation by smoothing;
 2.
$\widehat{\mathbf{R}}$ inversion;
 3.
projection of steering vectors on ${\widehat{\mathbf{R}}}^{1}$ (cf. Equation 14).
 1.
$\widehat{\mathbf{R}}$ estimation by smoothing;
 2.
EVD;
 3.
projection of steering vectors on π ^{ n }(cf. Equation 16).
The smoothing (point 1) is common for the two methods. A natural way to compute $\widehat{\mathbf{R}}$ is to follow the equation formulation: extract the k subarrays x_{ k }from the data, compute the corresponding crossspectral data matrix ${\widehat{\mathbf{R}}}_{k}$, loop this step on all of the subarrays k = 1,…,K, and finally mean the ${\widehat{\mathbf{R}}}_{k}$. An alternative way consists of building the matrix X^{ s } = [x_{1},…,x_{ K }] and noting that $\widehat{\mathbf{R}}={\mathbf{X}}^{s}{{\mathbf{X}}^{s}}^{H}/K$. This matrix product is less computationally expensive than the loop of the natural computation.
Knowing that π^{ n } = I−π^{ s }, the projection step for DMUSICAL (point 3) can be considerably sped up, noting that the size of the signal subspace L (which is of the same order of magnitude as the number of arrivals P; typically around 10) is much smaller than the size of the noise subspace ${N}_{e}^{s}{M}_{r}^{s}{F}^{s}L$ (which depends on the size of the arrays and the smoothing parameters, typically > 100).
 1.
Calculation of X ^{ s }(already calculated for smoothing processing) and T.
 2.
Calculation of the M first ${\lambda}_{i}^{\prime}={\lambda}_{i}$ and ${\mathbf{v}}_{i}^{\prime}$T.
 3.
Estimate L from the behavior of the decreasing eigenvalues and then deducing the L first eigenvectors v _{ i }of R.
Optimization gain of the DMUSICAL algorithm for the experimental data (see values of parameters in Table 2 )
Gain  

Optimized smoothing vs. conventional smoothing  ×12.5 
M first eigenvalues vs. conventional EVD  ×2.3 
M first eigenvalues on T vs. Conventional EVD  ×2.8 
Optimized projection vs. conventional projection  ×11.3 
Optimized DMUSICAL vs. DMUSICAL  ×11.2 
Values of the parameters for DMUSICAL in the experimental data
P  N _{ e } M _{ r } F  K  ${\mathit{N}}_{\mathit{e}}^{\mathit{s}}{\mathit{M}}_{\mathit{r}}^{\mathit{s}}{\mathit{F}}^{\mathit{s}}$  M  L  ProjS 

7  1815  225  343  30  16  7956780 
Result
Detection performances on real tank data
DMUSICAL has previously been validated on simulated data [26]. We apply DMUSICAL and DCapon to real data recorded in an ultrasonic tank that reproduces the oceanic acoustical propagation at a small scale.
 1.
As expected, the 3D estimation methods (DBF, DCapon and DMUSICAL) have better performances than the 2D ones (2D BF, 2D Capon and MUSICAL), comparing for instance Figure 5 with Figure 7, or Figure 6 with Figures 8 and 9.
 2.
The adapted and highresolution methods (i.e. Caponlike and MUSIClike methods) have better performances than the conventional BF ones, comparing for instance Figure 5 with Figure 6, or Figure 7 with Figures 8 and 9.
A general conclusion is that DMUSICAL and DCapon have better detection performances than all of the existing 2D and 3D methods.
DCapon and DMUSICAL comparison
This observation can been explained as follows. Under these conditions, the noise projector π^{ n } is spanned by the $B={N}_{e}^{s}{M}_{r}^{s}{F}^{s}K$ last null eigenvalues, and it can be shown that ${\widehat{\mathbf{R}}}^{C1}$ is dominated by the π^{ n }/σ^{2} term [9]. DCapon thus converges on DMUSICAL up to a multiplying factor (1/σ).
To compare the performances for other values of L and σ, we focus on the first arrivals, and particularly on the two first arrivals, which are the hardest to detect (Figure 7). We introduce the contrast C to compare the performances. This is defined by the ratio between the amplitude of the weakest peak and the amplitude of the saddle point between the two arrivals. In the previous configuration (L = K and SNR^{ C }= 150 dB), all of the arrivals are detected, but the two first arrivals have weak contrast (C = 1.16 for both DMUSICAL and DCapon). The contrast between the two first arrivals increases for DMUSICAL, from L = K to L = 16 (C = 2.5), and then decreases from L = 16 to L = 7, and the two first arrivals are not detected any more for L < 7. The DCapon results are similar: the contrast between the two first arrivals on DCapon increases from SNR^{ C }= 150 to SNR^{ C }= 29 dB (C = 2), then decreases from SNR^{ C }= 29 to SNR^{ C }= 23 dB, and the two first arrivals are not detected any more for SNR^{ C }< 23 dB. However, DMUSICAL and DCapon are no longer quasiequal for L < K and SNR^{ C }< 150 dB. This empirical experiment leads thus to the following conclusion: the detection performances for DMUSICAL when L decreases are similar to those for DCapon when σ increases.
 1.
the choice of L for DMUSICAL is achieved by taking into account the eigenvalue decrease. On the contrary, we have no indication for the choice of σ in DCapon;
 2.
the processing cost is lower for the inversion of ${\widehat{\mathbf{R}}}^{C}$ than for the EVD;
 3.
the projection step is faster for DMUSICAL than for DCapon, and the cost difference is generally large because $L<<{N}_{e}^{s}{M}_{r}^{s}{F}^{s}$.
Points 1 and 3 give priority to DMUSICAL, while point 2 gives priority to DCapon. As the projection step (point 3) represents the most important part of the processing cost (depending on the projection domain), we generally choose DMUSICAL to achieve the detection in real applications.
Conclusion
In this study, the DMUSICAL and DCapon 3D detection and estimation methods have been proposed for arrival separation in an OAT context. Starting from the 3D recording space receiversourcefrequency, they estimate observables in the 3D estimation space DOADODTOA. DMUSICAL and DCapon extend the highresolution MUSIC method and adaptive Capon method to the 3D configuration, respectively.
Smoothing issues linked to the OAT context and implementation issues have been discussed here. The methods have been validated on real data recorded in an ultrasonic tank. These methods have better detection performances than the 2D methods (2D BF, 2D Capon and MUSICAL) and than DBF. We have also shown that DCapon and DMUSICAL finally give similar performances.
Future work will concern the estimation performances of these new methods and their use in OAT experiments. As OAT has already been achieved using DBF to estimate the TOA [28], it will be particularly interesting to apply the tomography process with DCapon and DMUSICAL to estimate the TOA.
Declarations
Acknowledgements
This work was supported by the French ANR Agency (Grant ANR 2010 JCJC 030601).
Authors’ Affiliations
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