Pressure and velocity cross-spectrum ofnormalmodes in low-frequency acoustic vector fi eld ofshallow water and its application

Naval Academy of Armament,Beijing 100161,China

1.Introduction

Pressure and velocity cross-spectrum ofnormalmodes in low-frequency acoustic vector fi eld ofshallow water and its application

Yun Yu*,Qing Ling,and Jiang Xu

Naval Academy of Armament,Beijing 100161,China

The pressure and horizontalparticle velocity combined descriptions in the very low frequency acoustic fi eld ofshallow water integrated with the concept of effective depth of Pekeris waveguide is proposed,especially the active componentofthe pressure and horizontal particle velocity cross-spectrum,also called horizontal complex cross acoustic intensity,when only two normal modes are trapped in the waveguide.Both the approximate theoretic analysis and the numerical results show that the sign of the horizontal complex cross acoustic intensity active component is independent of the range when vertically deployed receiving dual sensors are placed in appropriate depths,the sum of which is equal to the waveguide effective depth,so it can be used to tell whether the sound source is near the surface or underwater;while the range rate is expected to be measured by utilizing the sign distribution characteristic of the reactive component.The further robustness analysis of the depth classi fi cation algorithm shows that the existence of shear waves in semi in fi nite basement and the change of acoustic velocity pro fi les have few effects on the application ofthis method,and the seabed attenuation willlimitthe detection range,but the algorithm still has a good robustness in the valid detection range.

low frequency acoustic vector fi eld,effective depth, complex cross acoustic intensity,active component,target depth classi fi cation.

1.Introduction

With the developmentofdamping and noise reducing technology,the radiation noise of the underwatermoving platform has been substantially reduced,but not at low frequencies.In other words,the moving platform is notquiet below 100 Hz,and it still radiates strong line-spectrums, the acoustic fi eld coherence of which is very strong,so the working bands of underwater acoustic detection tend to lower frequencies.The acoustic fi eld includes the scalar fi eld and vector fi eld,and only the scalar fi eld information is utilized in traditional underwater acoustic systems. The acoustic propagation theory and the signalprocessing technology based on the scalar fi eld have been studied extensively,but not the vector fi eld.Thus,the focus of this paper is on the researches aboutthe low-frequency vector fi eld,which is the fundamentalofdetection,estimation and identi fi cation adopting vector sensors.

Sound propagation in shallow water channelis a complex problem,but the physical image can be achieved by considering simpli fi ed analytical models,which identify the mostimportantacoustic fi eld properties.A good example is the effective depth approximation model.This model was introduced by Weston to resolve the problem of phase change of the plane wave re fl ection at a fl uid/fl uid boundary of Pekeris waveguide[1].Weston proposed that each of the modes trapped in the waveguide could be associated with an effective ray,one could view the re fl ection as taking place atan imaginary pressure-release boundary located at a speci fi c depth below the actual boundary,and the additional distance traveled by the ray is equivalentto applying a phase shift to the specular re fl ection.The“effective depth”is independentof the incidence angle of the plane wave when the grazing angle is less than the critical angle,a little depends on frequency and does not change with the order of the mode.Subsequently,Chapman etal. extended the conceptby allowing the seabed to be a homogeneous elastic solid in which shearwaves existin addition to compressionalwaves[2];Zhang and Tindle developed the conceptofthe complex effective depth aiming atsound propagation attenuation[3].The advantage of this approximation is thatthe normalmode wavenumbers can be estimated easily,withoutnumericalintegration.

The stable interference structure can be observed in thelow-frequency acoustic fi eld,which has become the research hotspotin recent years.In general,the interference structure is complex and not easy to be applied directly in engineering,so it is necessary to adopt some spatialtemporaltransformation to make the information taken by the interference structure become straightforward in the transform domain,or to enhance the useful information. The spatial-temporaltransformation adopted in this paper is pressure and horizontalparticle velocity cross-spectrum, also called horizontalcomplex cross acoustic intensity processing.It is hopeful to realize source depth classi fi cation and range rate measurement utilizing the simpli fi ed interference structure in the low-frequency acoustic vector fi eld.Itis a pointin this paperto combine signalprocessing with the physicalnature of the sound fi eld.

There have been some researches on moving information extraction of target characteristics using the interference structure in the low-frequency sound fi eld in some literature.Yang et al.presented a method of target moving parameter estimation based on the analysis of spectrum interference in shallow water[4],but the method is only applicable in near-fi eld.The characteristics of the low-frequency acoustic vector fi eld in Pekeris waveguide were analyzed based on the normalmodestheory ofa point source in layered media[5],especially the interference structure of verticalacoustic intensity fl ux was concerned, and the method which can tellthe source’s speci fi ed depth by comparing with a criticaldepth was proposed,however, the criticaldepth is notsuitable enough to tellwhether the target is a surface ship or a submarine.Therefore the researches were extended,and an ideal critical depth was achieved by analyzing horizontal complex cross acoustic intensity between two transducers deployed vertically[6], and the phase angle spatialdistribution of complex acoustic intensity was discussed,butonly the Pekeris waveguide was analyzed.The straightforward interference structures of the line spectrum acoustic fi eld[7],which are the physical basis for further radial velocity estimation,were achieved by adopting two kinds of spatial temporal transformations,which are the pressure cross-spectrum reactive componentof two sensors deployed vertically and the verticalcomplex acoustic intensity active componentof a single vector sensor.

It is the objective of this paper to take the method further.The pressure and horizontal particle velocity combined descriptions in the very low frequency acoustic fi eld of shallow water integrated with the concept of effective depth are proposed,and more attention is paid to the active componentof horizontalcomplex cross acoustic intensity, especially the mechanism analysis from which the interference structure is generated.Further more,the appropriate prediction formula of deployed depths of two sensors is presented and the robustness is analyzed.

2.Characteristic of complex cross acoustic intensity and its application

The normal mode theory of a point source in layered media has been mature,different from the traditional acoustic fi eld prediction.From anotherperspective,the acoustic pressure and horizontalvelocity given herein are on fi xed receivers while the position ofthe source changes with distance and depth.Complex acoustic intensity is the expression of sound intensity in the frequency domain,and is the cross-spectrum ofthe acoustic pressure and velocity which is received atthe same pointofthe acoustic fi eld simultaneously.The cross-spectrum of the acoustic pressure and velocity received atthe same time butnotatthe same pointof the acoustic fi eld is called“complex cross acoustic intensity”,which is similar as complex acoustic intensity only in the form but not in the physical meaning.In fact,the complex cross acoustic intensity is the mapping of the interference structure in the acoustic vector fi eld by means of dualreceiver complex acoustic intensity operation in order to facilitate the application.In this paper,the main concern is on the horizontal complex cross acoustic intensity when only two normalmodes are trapped in the waveguide, whose active componentsign distribution is independentof the distance,and divides the whole sound fi eld into three sign layers,so it is useful to do the target depth classi fication.However,the reactive component sign distribution of the horizontalcomplex cross acoustic intensity changes with the distance periodically,which is expected to be applied to the range rate measurement.Thus,the following will outline the fundamental theory for the approximate analysis of these phenomena.The appropriate prediction formula of optimaldeployed depths of two sensors is presented,which provides a guidance to the application ofthe algorithm.

2.1 Fundamentaltheory

Considering the most basic waveguide and the Pekeris waveguide,the expression of the effective depth[8]is as follows:

where Hedenotes the effective depth,H is the depth of sea water,b is the density ratio of sea water and bottom medium,k1is the wavenumber in the sea water,and αc=cos?1(c1/c2)denotes the criticalgrazing angle.

When the depth of the imaginary effective water column is He,the acoustic pressure fi eld excited by the point source in the horizontally free and uniform waveguide with the pressure-release boundary condition both above and below can be described as

whereγ=π/He,z and zsare the depths of the receiver and the source respectively,and r is the horizontaldistance between the source and the receiver.is the wavenumber.According to the Euler equation[9], the horizontalparticle velocity can be written as follows:

Then the expression of horizontalcomplex cross acoustic intensity is as(4),when the depths of the pressure and horizontalvelocity are z1and z2respectively.

Pay attention to the asymptotic expressions of the Hankelfunction:

And rewrite(4)in the following form:

The fi rst term on the rightof(6)is real,which is called DC component,and it is the horizontal complex cross acoustic intensity of each mode itself and is active.In other words,as for each single order mode,it transports acoustic energy horizontally,and the form in the horizontal direction is traveling wave.However,the second term on the rightof(6)is plural,which indicates thatbecause of the interactive interference between two modes,the horizontalcomplex cross acoustic intensity consists ofboth active and reactive components.And both the real part and the imaginary part are the oscillatory components,which determine the horizontal coherence scale of the acoustic fi eld.

Therefore,the active and reactive components of the horizontal complex cross acoustic intensity can be expressed respectively as

Observed from(6)and(7),itis seen thatthe active componentcontains both the DCcomponentand the oscillatory component,while the reactive componentonly has the oscillatory component.

2.2 Principle of targetdepth classification and prediction formula

Considering the particular scenario when only two normal modes are trapped in the waveguide,and substituting(6) into(7),one gets

When the receiver depths of two sensors are fi xed as z1and z2,ifone wants to realize the targetdepth classi fi cation utilizing the active componentof horizontalcomplex cross acoustic intensity,it is desired that the sign of the active component does not change with the distance r,in other words,the sign distribution is layered horizontally,so it is easy to know from(8):

Note thatξ1≈ξ2,(9)can be simpli fi ed as

Equation(10)is needed to be rightfor any source depth, and noteγz1,γz2∈[0,π),thus(10)can be furthersimplifi ed as

In other words,we can make the sign of the horizontal complex cross acoustic intensity active component independent of the range if the sum of two sensors’depths is equalto the waveguide effective depth.Substitute(11)into (8),and rewrite(8)in the following form:

We can also know that zscorresponding to M=0 is the boundary of the three regions,at this time,if we defi ne zsas critical depths zs1and zs2,(14)can be achieved according to the reciprocity principle.

It is important to tell whether the sound source is near the surface or deep enough for passive sonar.Fuzzy decision will appear if the second boundary line is located above the actualsea depth,so zs2≥H is required,which is equalto the condition zs1≤He-H.Finally,the sources depth can be classi fi ed as two kinds,near the surface or underwater,by judging the sign of the active component and comparing itwith zs1.

The critical depth is adjustable by adjusting the positions oftwo sensors,the approximate theoreticalprediction formula of these two sensors is shown as(15)by selecting an appropriate criticaldepth zs1=h.

whereγ=π/He,and Hecan be con fi rmed by(1).

3.Numericalanalysis ofhorizontalcomplex cross acoustic intensity

Numerical analysis technology of the acoustic waveguide has been extensively researched,and the characteristic of horizontal complex cross acoustic intensity will be analyzed below using fast fi eld processing(FFP).

Example 1The pointsource radiates harmonic waves ofsingle frequency.The watercolumn is iso-velocity channel,whose depth is H=100 m,the acoustic velocity and density are c1=1 500 m/s andρ1=1 000 kg/cm3respectively.While the acoustic velocity and density of the bottom medium are c2=1 610 m/s andρ2=1 900 kg/cm3,respectively,and there is no absorbance in the bottom medium.

The cutoff frequencies of the fi rst three orders are 10.3 Hz,31.0 Hz and 51.6 Hz respectively,when the source radiation frequency is between 31.0 Hz and 51.6 Hz,there are only the fi rst and the second order normalmodes.Taking the case of the source radiation frequency of 40 Hz,as mentioned earlier,the active component’s sign distribution of horizontalcomplex cross acoustic intensity between two sensors is calculated when the sources’depth and the horizontaldistance from the sensors are changed.The results are shown in Fig.1,the abscissa denotes the range between the receiverand the source,and the ordinate denotes the depth of the source;the white denotes positive signs,and the black denotes negative signs.

Fig.1 Sign distribution of the active component of the complex cross acoustic intensity versus source’s depth and range(The frequency is 40 Hz,z1 and z2 are the recieving depths of the pressure and horizontalvelocity,respectively)

The results indicate that whether the sign of the active componentof complex cross acoustic intensity is positive ornegative in the near fi eld depends on the distance,which represents the near-fi eld characteristic of the acoustic fi eld; the sign distribution of the active component of complex cross acoustic intensity is indeed independentof the horizontaldistance in the far fi eld when two receivers deployed vertically are placed in the appropriate depth;if the depth on which the sign of the active componentchanges is defi ned as the criticaldepth,the targetdepth could be classifi ed by comparing with the criticaldepth.In this example, an appropriate criticaldepth is 30 m,if the targetdepth is less than 30 m,it is judged as a surface target,while it is judged as an underwater targetin the opposite situation.

The numerical simulation also validates the law of two sensors deploymentrevealed by(11),which is thatthe sum of two transducer depths is a constant,which is just equal to the waveguide effective depth.The sum of two sensor optimaldeployed depths in Example 1 is 125 m,while the corresponding waveguide effective depth is 131 m,so they are coincident with each other approximately.Moreover, the critical depth is adjustable by selecting differentcombinations of deployed depths.The fi rst critical depth zs1is smaller and the acoustic fi eld is divided into two sign regions when choosing the combination of the deployed depths in which the verticaldistance ofthe dualtransducer is nearer.zs1is larger and the acoustic fi eld is divided into three regions in the opposite case.The phenomenon shown in Fig.1 is inconsistent with the conclusion revealed by (13).In fact,both of them are correct,and the reason why only two sign but not three regions exist in Fig.1(c)is thatthe second criticaldepth is under the actual fl uid/fl uid boundary in which case there is no fuzzy decision to the source depth.

The corresponding reactive component sign distributions of horizontal complex cross acoustic intensity are shown in Fig.2.It indicates that the sign distribution is chaotic in the near fi eld;but it is regular in the far fi eld, and the sign changes periodically in the horizontal direction when the source depth is fi xed,while the sign hopping happens in a certain depth when the source horizontaldistance is fi xed.The horizontal variety period is de fi ned as horizontal coherence scale,which is hoped to be utilized to measure the source range rate of iso-velocity or linear movement.

Fig.2 Sign distribution of the reactive component of the complex cross acoustic intensity versus source’s depth and range(The frequency is 40 Hz,z1and z2are the recieving depths of the pressure and horizontalvelocity,respectively)

4.Robustness analysis

The characteristics of horizontal complex cross acoustic intensity in the ideal layered media have been described above.The robustness will be analyzed taking the actual ocean environmentinto account,mainly concerning the effects caused by the seabed and the acoustic velocity pro fi le. We only pay attention to the active componentof horizontalcomplex cross acoustic intensity as follows.

4.1 Effect caused by the seabed

The acoustic characteristics ofthe seabed are the important factors affecting the structure of the acoustic fi eld in the case of low-frequency and shallow water.The following will discuss the effecton the targetdepth classi fi cation algorithm mentioned above caused by the existence of shear waves and attenuation in the seabed.

Example 2The absorption exists in the seabed,the absorption coef fi cientis 0.1dB/λ,and the remaining conditions are the same as Example 1.

The corresponding numericalresults of the active componentsign distribution ofhorizontalcomplex cross acoustic intensity are shown in Fig.3.We can see that the boundary of the regions which the sign of the active component is positive or negative become inclined,but it hardly affects the target depth classi fi cation by selecting a suitable combination oftwo sensor deployed depths.The reason for the incline is that the higher the order of the normal mode,the stronger the attenuation,when the frequency is fi xed,so the positive region of the active componentbecomes biggerwith the increase ofthe distance.One can imagine thatthe incline willbe more obvious when the absorption coef fi cientincreases(such as 0.3dB/λ),which also shows that the existence of absorption in the seabed will restrictthe detection distance of this algorithm,butit is stillpracticable within the effective detection distance.

Fig.3 Sign distribution of the active component of the complex cross acoustic intensity versus source’s depth and range when there is absorption in the seabed(The absorption coefficient of the seabed is 0.1dB/λ,z1and z2are the recieving depths of the pressure and horizontalvelocity,respectively)

Example 3There are shear waves in the seabed,the velocity of the shear waves is cs=200 m/s,and the remaining conditions are the same as Example 1.

The corresponding numericalresults of the active component sign distribution of the horizontal complex cross acoustic intensity are shown in Fig.4.It is found thatthe feasibility of the algorithm is hardly affected by the shear waves by comparing Fig.4 with Fig.1.

Fig.4 Sign distribution of the active component of the complex cross acoustic intensity versus source’s depth and range when there are shear waves in the seabed(The velocity of shear waves of the seabed is Cs=200 m/s,z1and z2are the recieving depths of the pressure and horizontal velocity,respectively)

4.2 Effect caused by acoustic velocity profile

The acoustic velocity pro fi le ofthe sea watercolumn is different in different places and different seasons.The typical vertical acoustic velocity pro fi le may be classi fi ed as isothermal,positive or negative gradient,and thermocline. The relative velocity gradientof the positive gradientprofi le is usually aboutthe order of 10?5,which causes only negligible in fl uence,so we only discuss the lasttwo acoustic velocity pro fi les.

Example 4Two kinds of acoustic velocity pro fi les are shown in Fig.5,the negative gradient distribution:c10= 1 500 m/s corresponding to z=0 m and c11=1 485 m/s corresponding to z=100 m.

Fig.5 Two kinds of acoustic velocity profiles

That is to say,the relative gradient is 1×10?41/m; the thermocline:the velocity is 1 510.0 m/s corresponding to z=0 m,the velocity is 1 509.0 m/s corresponding to z=25 m,the velocity is 1 493.0 m/s corresponding to z=30 m,and the velocity is 1 492.0 m/s corresponding to z=100 m;the rest of the calculation conditions is the same as mentioned above.

The corresponding numericalresults of the active component sign distribution of the horizontal complex cross acoustic intensity are shown in Fig.6 and Fig.7,and they are coincident with each other approximately.Seen from Fig.6(a)and Fig.7(a),the boundary of the positive and negative regions vibrates when the positions of two sensors are stillas Fig.1(c),which hardly affects the decision whetherthe sound source is nearthe surface orunderwater. And the boundary is still level if the depths of these two sensors are adjusted appropriately,at the same time,the sum oftwo sensors deployed depths is stilla constant(only the constantincreases in the value).

Fig.6 Sign distribution of the active component of the complex cross acoustic intensity versus source’s depth and range when the acoustic velocity profile is the negative gradient(z1and z2are the recieving depths ofthe pressure and horizontalvelocity,respectively)

Fig.7 Sign distribution of the active component of the complex cross acoustic intensity versus source’s depth and range when the thermocline exists in the acoustic velocity profile(z1and z2are the recieving depths of the pressure and horizontalvelocity,respectively)

5.Conclusions

The sign distribution characteristics of the active componentand the reactive componentof the horizontalcomplex cross acoustic intensity between two receivers deployed vertically are analyzed in the low frequency acoustic vector fi eld of shallow water,the former is hoped to be utilized to classify the source depth,while the latter may be adopted to measure the source range rate of iso-velocity or linear movement.More attention is paid to the former here.Both the approximate theoretic analysis and numerical results show that the sign of the horizontal complex cross acoustic intensity active component is independent of the range when two receivers deployed vertically are placed in appropriate depths,the sum of which is equal to the waveguide effective depth;and the deployed depths of these two sensors are only to be determined by settingup a desired critical depth.Further more,telling whether the sound source is near the surface or underwater can be achieved by selecting a pair of appropriate deploying depths of two sensors,which is important to the coastal early-warning system,verticaltowed dual-line array,aerial sonobuoy and so on.The algorithm is simple,easy to be realized and has good robustness,but further experimentalresearches are required if the algorithm is needed to be applied in engineering.

[1]D.E.Weston.A Moire fringe analog of sound propagation in shallow water.The Journal of the Acoustical Society of America,1960,32(6):647-654.

[2]M.F.Chapman,P.D.Ward,D.D.Ellis.The effective depth of a Pekeris ocean waveguide,including shear wave effects. The Journalofthe AcousticalSociety ofAmerica,1989,85(2): 648-653.

[3]Z.Y.Zhang,C.T.Tindle.Complex effective depth of the ocean bottom.The Journal of the Acoustical Society of America,1993,93(1):205-213.

[4]J.Yang,J.Y.Hui,L.Jiang.Moving target parameter estimation using the spectrum interference oflow frequency.Journal of Harbin Institute of Technology,2007,40(3):471-474.(in Chinese)

[5]J.Y.Hui,G.C.Sun,A.B.Zhao.Normal mode acoustic intensity fl ux in Pekeris waveguide and its cross spectra signal processing.Chinese Journal of Acoustics,2008,33(4):300-304.(in Chinese)

[6]Y.Yu,J.Y.Hui,A.B.Zhao,et al.Complex acoustic intensity ofnormalmodes in Pekeris waveguide and its application. Acta Physica Scinica,2008,57(9):5742-5748.(in Chinese)

[7]Y.Yu,J.Y.Hui,K.Chen,etal.Target’s radial velocity estimation based on the interference structure of the very low frequency acoustic fi eld.Acta Armamentarii,2010,31(9):1174-1180.(in Chinese)

[8]M.J.Buchingham,E.M.Giddens.On the acoustic fi eld in a Pekeris waveguide with attenuation in the bottom halfspace.The Journalofthe AcousticalSociety ofAmerica,2006, 119(1):123-142.

[9]L.M.Brekhovskikh,Y.P.Lysanov.Fundamentals of ocean acoustics.New York:Springer Verlag,2002.

Biographies

Yun Yu was born in 1984.She received her B.S. degree in electronic and information engineering and her M.S.degree in singal and information processing from Harbin Engineering University, China,in 2006 and 2009,respectively.She received her Ph.D.degree in underwater acoustic engineering also from Harbin Engineering University,China,in 2011.She is an engineer of Naval Academy of Armament.She has issued more than 20 papers,including three indexed by SCI and eight indexed by EI.She has been working for the Naval Academy of Armament,Beijing,China,since 2011.She is mainly engaged in the research direction of underwater acoustic engineering and signaland information processing.

E-mail:yuyuntc@163.com

Qing Ling was born is 1962.He received his Ph.D.degree in underwater acoustic engineering from Habin Engineering University in 2007.He has published three monographs and has issued more than 20 papers.He is a senior researcher of Naval Academy of Armament.He is mainly engaged in the research direction of underwater acoustic engineering.

E-mail:l ing q ing@163.com

Jiang Xu was born in 1975.He received his Ph.D. degree in signal and information processing from National University of Defense Technology in 2003. He is a senior engineer of Naval Academy of Armament.He is mainly engaged in the research direction of underwater acoustic engineering.

E-mail:x u j ing@163.com

10.1109/JSEE.2015.00029

Manuscriptreceived October 22,2013.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(11404406;11374072).