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Acoustic Sampling Volume Kenneth G Acoustic sampling volume Kenneth G. Foote Instituteof Marine Research,5024 Bergen, Norway (Received5 April 1990;accepted for publication10 April 1991) Knowledgeof the acousticsampling volume is necessaryin many quantitativeapplications of acoustics.In general,the samplingvolume is not merelya characteristicof the transmitting and receivingtransducers, but alsodepends on the concentrationand scatteringproperties of the target, the kind of signalprocessing performed on the echo,and the detectionthreshold. Thesedependences are statedexplicitly in formulasfor the samplingvolume and a differential measure,the effectiveequivalent beam angle. Numerical examplesare givenfor dispersedor denseconcentrations of both point scatterersand directionalfish scatterers. Application of theory to optical and other remotesensing techniques is mentioned. PACS numbers: 43.30.Ft, 43.30. Gv, 43.30.Xm INTRODUCTION range 30-200 kHz. Consequently,a scattererthat gives a A numberofpractical uses of acoustics require knowl- strongecho when in oneorientation may givea very weak or edgeof the samplingvolume. In fluid-processingapplica- unobservableecho when in another, despitebeing in the tions these include determinations of the concentration of same position in the beam. Differencesin backscattering monodispersedscatterers and the sizedistribution and con- crosssections due to ordinary changesin tilt anglewill more- centration of polydispersedscatterers, e.g., of human red overbe slightfor smallsize-wavelength ratios and potential- bloodcells ] and othersmall particles, 2-5 such as contami- ly large for big ratios,thus nearly spanningthe rangeof ef- nants of industrial fluids. In oceanographicapplications fects intrinsic to the scatterer itself. theseinclude analogous determinations of scattererconcen- The dependenceof the samplingvolume on the back- trationand size distribution, e.g., of bubbles,6fish, 7-9 plank- scatteringcharacteristics of observedfish, in addition to ton,]ø-]3 and suspended sediment. ]4-]7 Some of thecited ap- transducerproperties, has alreadybeen recognized. ?'18-25 plicationsinvolve bistatic sonar, others monostaticsonar, Dependenceof the samplingvolume on the minimumde- but the problemis the same:describing how largethe partic- tectablesignal level or so-calleddetection threshold has also ular samplingvolume is. been recognized.However, notwithstandingseveral differ- In the caseof high signal-to-noiseratio (SNR), as with ent approachesto the problem,computations are scarce,and acousticallypowerful scatterers, the samplingvolume may there is a distinct lack of conciseor comprehensiveexpres- be identical to the available or accessiblevolume. However, sionsfor the samplingvolume or thresholdeffect. in the caseof low or marginal SNRs, the samplingvolume A notable,overtly statistical approach to a differentbut will in generalbe lessthan the high-SNR volume.The reason relatedproblem is thatby Weimerand Ehrenberg. 26For a is simplythat someechoes from relativelyweak scatterers, given threshold,the effect on a distributionof target singlyor aggregated,will lie below the detectionthreshold. strengthsis describedby an integral.This is evaluatednu- This is not an academicsituation either, for what technique mericallyfor a specificnormal target strengthdistribution in acoustics,or in sciencefor that matter, is not at some time for each of several thresholds. pushedto itslimit: smaller,larger, weaker, stronger, thinner, The presentapproach attempts to addressboth the par- denser, nearer, farther. ticular fisheriesacoustics application and the more general The problemof definingthe acousticsampling volume problemof definingthe samplingvolume. Following a de- is especiallytimely in fisheriesacoustics. Several years ago, scriptionof the first-orderrole of the samplingvolume in when the Northeast Arctic cod (Gadus morhua) was sur- two methodsof scattererdensity determination, the theory veyed during its annual migration to spawninggrounds in of the samplingvolume is developed.General issuescon- Lofoten, the fish was observedat depthsexceeding 400 m, cernedwith integrationare discussed.The volumeassociat- which is much deeperthan usual. Sincethe echo sounding ed with acousticsampling by fish is computedthrough a equipmentbegins to be severelylimited with respectto the derived differentialmeasure, the effectiveequivalent beam detectionof singlecod at this range,only a fraction of the angle,for a particularcomputational model. This is intended stockwas registered, as was later demonstratedby the catch. both to illustrate the method of numerical evaluation and to Other evidencefor the importanceof sampling-volumecon- revealsome important dependences of the samplingvolume siderationsfor codhas been presented by Ona.]8 on the underlyingscatterer characteristics. Defining the sampling volume for fish registrationis I. THEORY alsoparticularly illustrative for the generalproblem because of the involvedscattering regime. This is characterizedby Two commonmethods of determiningscatterer concen- 27 28 size-wavelengthratios of about 1-100, which are due to the tration are thoseof echocounting and echointegration. ' use of ultrasonic survey frequenciesin the approximate In the case of sufficientlydispersed scatterers, the echo 959 J. Acoust.Soc. Am. 90 (2), Pt. 1, August1991 0001-4966/91/080959-06500.80 @ 1991 AcousticalSociety of America 959 Downloaded 18 Dec 2012 to 128.128.44.26. Redistribution subject to ASA license or copyright; see http://asadl.org/terms countingmethod may be used.According to this,the scat- B. Monostatic case tererdensity p is measuredby the averagenumber of echoes The expressionfor V• in Eq. (2) is completeand unam- fir,obtained from the sampledvolume Vs per sounding, biguous.However, its incorporationin echocounting and integrationschemes, TM in theirusual monostatic forms, re- The echointegration method may be appliedin the gen- quiresadapting the equivalentbeam angle •Po, which is de- eral caseof scatterersof arbitraryconcentration. 29Accord- finedentirely in termsof the transducerbeam pattern, 31 ingly, the columnscattering coefficient Sa is obtainedby in- tegratingthe volume backscatteringcoefficient sv over an •Po=f b 2dll. (3) accessiblerange interval. The fundamentalquantity sv re- Since this applies at a constant,far-field range, and latesthe mean cumulative backscattering cross section • per d V = r2 dr d12,the solid-angle analog to Eq. (2) is soundingto the sampledvolume Vs, 12 s• = •/(4rrV•), ffH(gb•a-t)dFdfl, assuming,for the sakeof simplicityonly, negligible extinc- wherecr is thebackscattering cross section. Comparing this tion.Usually so is expressed as a functionof range,or depth, with Eq. (3), it is clearthat the effectiveequivalent beam by limiting V• by a successionof generallynarrow range angleis intervals. These two methods are well known in underwater acoustics,but possessexact analogs for thesampling of other media,whether by acousticalor opticalmeans. =f f oH(gO t)dFd. (4) In the following,the theoryof the samplingvolume is Thisquantity can, in onesense, be regarded as a generaliza- first developedwithout referenceto any particularmethod tionof the equivalentbeam angle defined in Eq. (3). How- or application.It thusencompasses the bistatic case of sepa- ever, its origin is in the conceptof samplingvolume, de- rate transmittingand receivingtransducers. The derived scribedin Eq. (2), and, when•p is multipliedby r2Ar,the expressionfor thesampling volume is then specialized to the productis equalto the samplingvolume within a spherical monostaticcase of a singletransducer or collocatedtrans- shell of infinitesimal thickness Ar. mitting and receivingtransducers. The gainfactor g in the severalequations is exemplified by two extreme, but not uncommon, monostaticSituations A. Bistatic case of detectionin the usual far field of the transducer:( 1) for a In the generalbistatic case,separate transducers are singlescatterer, g = 10- ar/5r--4,where a isthe coefficient of usedfor transmissionand reception.The respectivedirec- absorptiongiven in decibelsper meterand r is the rangein tionalcharacteristics are contained in theone-way beam pat- metersto the scatterer;and (2) for a layer of identicalscat- ternsbr and bR. A receivedecho is registeredif its strength terers,g = 10-"r/•r -2. exceedsa minimum signallevel or thresholdt. The received The detectionthreshold t hasthe same units as the prod- echostrength is expressed as the product of a gainor geomet- uctgb 2cr. At thevery threshold, detection occurs essentially ric factorg, productof transmitand receivebeam patterns on the acousticaxis, where b = 1. The scatterer,if direction- b 2, andbistatic, or differential,cross section or. al, is moreoverin its mostfavorable aspect, where cr = O'max. For a nondirectional scattererwith constant or,the sam- Here, at the maximum detectionrange, g is a minimum. pling volume Vsis a fractionof the total availableor accessi- Thus t = gminO'max' In the limit that t vanishes,or the SNR ble volume Vo: becomesvery large, V• -• Voand •p-••Po. II. INTEGRATION ISSUES Vs=fv, H(gb :cr--t)dV. (1 ) In boththe generalexpression for Vsin Eq. (2) andthe The integrandis a countingfunction: the Heavisidestep associateddifferential measure for the monostaticcase, •p in Eq. (4), the spatialintegrals are shownwithout explicit lim- function,H(x) = 0, •, 1 asx < 0, x = 0, x > 0, respectively. Thus,for echo strengths gb :crexceeding t, the contribution is its, and the Heavisidestep function H appearsin the inte- fully registered. grand.The integrationwith
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