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Research Letters in Signal Processing
Volume 2009, Article ID 257564, 4 pages
http://dx.doi.org/10.1155/2009/257564
Research Letter

Bearing and Range Estimation Algorithm for Buried Object in Underwater Acoustics

Institut Fresnel (UMR CNRS 6133), Faculté des Sciences et Techniques de Saint Jérôme, 13397 Marseille cedex 20, France

Received 28 June 2009; Accepted 19 July 2009

Academic Editor: Miguel Lagunas

Copyright © 2009 Dong Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A new algorithm which associates (Multiple Signal Classification) MUSIC with acoustic scattering model for bearing and range estimation is proposed. This algorithm takes into account the reflection and the refraction of wave in the interface of water-sediment in underwater acoustics. A new directional vector, which contains the Direction-Of-Arrival (DOA) of objects and objects-sensors distances, is used in MUSIC algorithm instead of classical model. The influence of the depth of buried objects is discussed. Finally, the numerical results are given in the case of buried cylindrical shells.

1. Introduction

The main target of Array processing is to estimate the bearing and range of sources or objects radiating in a medium of propagation [1]. MUSIC is one of the commonly used high resolution algorithms for DOA estimation. It uses the orthogonality property between the signal subspace and the noise subspace to localize sources [2].

In this letter, we propose a new algorithm for bearing and range estimation of buried objects in underwater acoustics. The approach is based on MUSIC combined with the acoustic scattering model [3, 4]. We consider the reflection and the refraction of wave at water-sediment interface. This method develops a new source steering vector including the information of bearing and range of buried objects. The vector is used in MUSIC algorithm instead of the classical plane wave model [5]. The attenuation in the sediment is distinct for the objects buried deep, so we discuss the influence of the depth of buried objects. The proposed algorithm is evaluated by numerical simulations in the case of buried cylindrical shells.

The remainder of the letter is organized as follows. Section 2 summarizes the problem formulation. In Section 3, the scattering acoustic model of generating the received signals is discussed. An algorithm is proposed in Section 4. Next, the influence of the depth of buried objects is presented in Section 5. Finally some numerical results are addressed in Sections 5 and 6 conclude the paper.

Throughout the paper, lowercase boldface letters represent vectors, uppercase boldface letters represent matrices, and lower and uppercase letters represent scalars. The symbol “𝑇” is used for transpose operation. The superscript “+” is used to denote complex conjugate transpose and denotes the 𝐿2 norm for complex vectors.

2. Problem Formulation

Consider a linear array of 𝑁 sensors receives the signals scattered from 𝐾 objects (𝐾<𝑁). The received signal can be grouped into a vector 𝐱(𝑓𝑙) written as 𝐱𝑓𝑙𝑓=𝐀𝑙𝐬𝑓𝑙𝑓+𝐛𝑙,(1) where 𝐀(𝑓𝑙) is the transfer matrix, 𝐬(𝑓𝑙) is the vector of signal, and 𝐛(𝑓𝑙) is the vector of additive Gaussian noise.

The wavefront is assumed to be plane when the objects are far from the array. We use MUSIC algorithm to estimate the angle 𝜃 of the plane wave associated with the objects 1MUSIC(𝜃)=𝐚(𝜃)+𝐕𝐛𝐕𝐛+𝐚(𝜃),(2) where 𝐚(𝜃)=[1,𝑒𝑗𝜑,,𝑒(𝑁1)𝑗𝜑]𝑇 is the steering vector and 𝜑=2𝜋𝑓(𝑑sin(𝜃)/𝑐), 𝐕𝐛 is the the matrix of eigenvectors spanned by the noise subspace, 𝑐 is the sound speed, 𝑑 is the distance of sensors, and 𝑗 is the complex operator.

3. Scattering Acoustic Model: To Generate the Received Signals

We assume an object is buried in the sediment (𝜃𝑘1,𝑟𝑘1) associated to the first sensor of the array. An incident plane wave propagating in the water reaches the interface with 𝜃inc (see Figure 1). A reflected plane wave is generated in the water and a refracted plane wave is propagated in the sediment. So the array receives three components [6]:

257564.fig.001
Figure 1: Geometry configuration of the buried object.
(i)the incident plane wave,(ii)the reflecting plane wave,(iii)the transmitted plane wave diffused by the object.

The pressures in the water and the sediment are given by five unknown parameters 𝜃𝑘11, 𝑟𝑘11, 𝜃𝑘12, 𝑟𝑘12, and the depth of buried object 𝑦𝑐 based on 𝜃𝑘1 and 𝑟𝑘1 (see Figure 1): 𝑦𝑐=𝑟𝑘1𝜃cos𝑘1𝜃,𝑘12𝑐=arcsin2𝑐1𝜃sininc,𝑟𝑘12=𝑟𝑘1𝜃cos𝑘1𝑐cosarcsin2/𝑐1𝜃sininc,𝜃𝑘11𝑟=arctan𝑘1𝜃cos𝑘1𝑟𝑘12𝜃cos𝑘12𝑟𝑘1𝜃sin𝑘1𝑟𝑘12𝜃sin𝑘12,𝑟𝑘11=[],cosarctan(𝑄)(3) where 𝑄 denotes (𝑟𝑘1cos(𝜃𝑘1)𝑟𝑘12cos(𝜃𝑘12))/(𝑟𝑘1sin(𝜃𝑘1)𝑟𝑘12sin(𝜃𝑘12)).

We consider the case of infinitely long elastic cylinder shell. The first sensor of the array 𝑃totcyl receives the acoustic pressure components as follows: 𝑃totcyl𝜃𝑘1,𝑟𝑘1=𝑃inwater𝜃𝑘1,𝑟𝑘1+𝑃refwater𝜃𝑘1,𝑟𝑘1+𝑃dicyl𝜃𝑘1,𝑟𝑘1,(4) where 𝑃inwater is the pressure incident in the water: 𝑃inwater𝜃𝑘1,𝑟𝑘1=𝑒𝑗𝑘1𝑟(𝑘1𝜃sin𝑘1𝜃sininc+cos(𝜃inc))(5)𝑃refwater is the pressure reflected by the sediment-water interface: 𝑃refwater𝜃𝑘1,𝑟𝑘1=𝑅𝑒𝑗𝑘1𝑟𝑘1𝜃sin𝑘1𝜃sininc𝜃cosinc,(6) where 𝑅 is the reflection coefficient of the interface.

𝑃𝑡dicyl(𝜃𝑘1,𝑟𝑘1) is the diffused acoustic pressure wave transmitted in the water [7]: 𝑃𝑡dicyl𝜃𝑘1,𝑟𝑘1=+𝑚=𝜉𝐓𝑐𝐈𝐃𝑐1𝜓𝑡cyl,(7) where 𝐈 is the identity matrix, 𝐃𝑐 is a linear operator, 𝐓𝑐 is the transition diagonal matrix, 𝜓𝑡cyl is the vector of transmitted wave, and 𝜉=[𝜉1,𝜉2,,𝜉𝑚] is defined by 𝜉𝑚=𝑇water-sed𝜃inc𝑒𝑗𝑘2𝑦𝑐𝜃cos𝑘11𝑗𝑚𝑒𝑗𝑚𝜋𝜃𝑘11,(8) where 𝑇water-sed is the transmission coefficient.

4. Algorithm for Bearing and Range Estimation of Buried Objects

(1)Find an initial estimation of 𝜃, 𝑟, and the number of objects 𝐾 by the beam forming method.(2)Fill the matrice 𝐀(𝑓𝑙)=[𝐚(𝑓𝑙,𝜃1,𝑟1),,𝐚(𝑓𝑙,𝜃𝑘,𝑟𝑘)] and the components are filled with cylindrical scattering model 𝐚(𝑓𝑙,𝜃𝑘,𝑟𝑘)=[𝑃totcyl(𝑓𝑙,𝜃𝑘1,𝑟𝑘1),,𝑃totcyl(𝑓𝑙,𝜃𝑘𝑁,𝑟𝑘𝑁)]𝑇 for 𝑘=1,2,,𝐾. The first vector is given by (4). The other 𝑃totcyl(𝑓𝑙,𝜃𝑘𝑖,𝑟𝑘𝑖) for 𝑖=2,,𝑁 associated with the 𝑖th sensor can be formed by (see Figure 1) 𝑟𝑘𝑖=𝑟2𝑘𝑖1+𝑑22𝑟𝑘𝑖1𝜋𝑑cos2+𝜃𝑘𝑖1,𝜃𝑘𝑖𝜋=2+cos1𝑑2+𝑟2𝑘𝑖𝑟2𝑘𝑖12𝑟𝑘𝑖1𝑑,𝑖=2,,𝑁.(9)(3)Estimate the spectral matrix Γ(𝑓𝑙)=𝐸[𝐱(𝑓𝑙)𝐱+(𝑓𝑙)].(4)Calculate sources spectral matrices by 𝚪𝐬𝑓𝑙=𝐀+𝑓𝑙𝐀𝑓𝑙1𝐀+𝑓𝑙×𝚪𝑓𝑙𝜎2𝑓𝑙𝐈𝐀𝑓𝑙𝐀+𝑓𝑙𝐀𝑓𝑙1,(10) where 𝐈 is the identity matrix and 𝜎2 is the noise variance.(5)Compute the average of the spectral matrices 𝚪𝑠𝑓0=1𝐿𝐿𝑙=1𝚪𝐬𝑓𝑙,1𝑙𝐿,(11) where 𝐿 represents the number of frequencies and 𝑓0 is the center frequency of the spectrum of the received signals.(6)Caltulate Γ(𝑓0)=𝐀(𝑓0)Γ𝐬(𝑓0)𝐀+(𝑓0).(7)Use the eigenvectors 𝐓(𝑓0,𝑓𝑙)=𝐕(𝑓0)𝐕+(𝑓𝑙) to obtain the focusing operator, where 𝐕(𝑓0) and 𝐕(𝑓𝑙) are, respectively, the eigenvector matrices of Γ(𝑓0) and Γ(𝑓𝑙).(8)Calculate the focused spectral matrix: 𝚪𝑓0=1𝐿𝐿𝑙=1𝐓𝑓0,𝑓𝑙𝚪𝑓𝑙𝐓+𝑓0,𝑓𝑙.(12)(9)Estimate the number of objects by AIC or MDL [8].

Calculate the spatial spectrum of MUSIC method for estimating bearing and range of buried objects: 𝑓MUSIC0,𝜃𝑘,𝑟𝑘=1𝐚𝑓0,𝜃𝑘,𝑟𝑘+𝐕𝐛𝑓0,(13) where 𝐕𝐛(𝑓0) is the eigenvector matrix of noise subspace.

5. Influence of the Depth of Buried Objects

In sandy sediment, the attenuating effect of suspended material is negligible. Conversely, the attenuating effect of the sediment is significant. It can be reported in dB/cm/kHz since the examination of attenuation yielded a linear dependency with frequency. The attenuation coefficient of common sand [9] is 𝛼att=0.5 dB/cm/kHz.

6. Numerical Results

The parameters of the simulations are defined as follows: 𝑑 is 0.002 m, 𝑁 is 10, the frequency band is [200, 300] kHz, and the signal frequency 𝑓 is 250 kHz. The wave speed in the water 𝑐1 is 1500 m/s and in the sediment 𝑐2 is 1700 m/s. The water density 𝐷1 is 1000 kg/m3, the sediment density 𝐷2 is 1500 kg/m3. The incidence angle 𝜃inc is 60°.

The array is placed in the water with the hight =0.1 m. The variance of the noise 𝜎2 is 100 and SNR is 30 dB. As shown in Figure 2, the white points coordinate two cylindrical shells (30°, 0.35 m) and (49°, 0.16 m).

257564.fig.002
Figure 2: Localization of cylindrical shells.

When the cylindrical shell is deeply buried (20°, 0.3 m), we vary the interface of water-sediment for each 𝑦𝑐. The signal may be unworkable if the object is buried deep. Furthermore, we evaluate statistically the influence of the depth of buried objects by Standard Deviation: Std=1𝐾𝐾𝑘=1𝑋exp𝑋est2𝑘,(14) where 𝑋 is the bearing 𝜃 or the range 𝑟 and 𝐾=100.

The results obtained (see Figures 3 and 4) show that the algorithm is not efficient when the object is buried deeper than 0.25 m.

257564.fig.003
Figure 3: Std between expected and estimated bearings.
257564.fig.004
Figure 4: Std between expected and estimated ranges.

7. Conclusion

In this paper, we propose a new algorithm based on MUSIC associated with acoustic scattering model for bearing and range estimation of buried objects. There is an analogy of the water-sediment interface by combining with the reflection and the refraction of wave in the model. A new directional vector, including the information for bearing and range estimation, is employed instead of the plane wave model in the MUSIC algorithm. The results of buried cylindrical shells are significantly accurate. Then we study the influence of the depth. The results show that beyond a certain depth, the attenuation becomes too large and therefore the objects cannot be detected or located neither.

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