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 respectto the volumeelement For directionalscatterers, cr varies with orientation.To dVin Eq. (2) is understoodto be performedover the entire accountfor this in Vs,the integrationin Eq. ( 1) is alsoper- volumethat isaccessible between the range intervals of inter- formedover the rangeof orientationsdetermining the sam- est.Similarly, the integrationwith respectto the solid-angle pled valuesof cr accordingto the cumulativedistribution elementdfl in Eq. (4) is assumedto be performedover that function F. Thus accessibleat the particularrange of interest.The physically availablespace, either volume or solidangle, constitutes an upperbound to the respectiveintegrals. (2) The Heavisidestep function, or countingfunction, de- scribesthe detailsof the samplingprocess. If andonly if the Thisis tantamount to Eq. (7) in Ref.30, although with dif- receivedecho strength gb :cr exceeds the detection threshold t ferencesin nomenclature.For the caseof constantor, the isthe echo fully registered. In thecase that gb :cr = t, theecho integrationover dF yieldsunity and Eq. ( 1) results. isregistered with one-halfweight or count,corresponding to

960 J. Acoust.Soc. Am., Vol. 90, No. 2, Pt. 1, August1991 KennethG. Foote:Sampling volume 960

Downloaded 18 Dec 2012 to 128.128.44.26. Redistribution subject to ASA license or copyright; see http://asadl.org/terms the statisticalregistration of suchmarginal echoes 50% of gledependence of the dorsalaspect target strength function, the time. which are tabulatedin Ref. 35. Of these,171 apply to the For generallydirective scatterers with random orienta- gadoidscod (Gadus morhua), saithe (Pollachiusvirens), tions, echoesarising from the same scattererat the same and pollack(Pollachiuspollachius), at 38 kHz. Four subsets fixed positionin spacemay or may not be registered.In a of these functions are selected to illustrate the effect of fish favorableorientation, for example,the scatteringcross sec- length,thence directionality of scatteringpattern, on •. tion crmay be solarge that gb2or exceeds t and the echois Theseare described in TableI. The backscatteringcross sec- registered.In an unfavorableorientation at the sameposi- tion•r of fish at tilt angleO' isderived from the target strength tion,gb 2or may be less than t, andthe echo will notbe regis- valueTS (0 ') accordingto thedefinition 36 TS = 10log or/ tered. It is thereforenot generallypossible to remove the 4•r, but with use of SI units. Heavisidestep function from the integrandand imposepre- Fishbehavior. This is characterized in theusual way by a ciselimits on the volumeor solid-angleintegrals. That is, Vs normalprobability density function of tilt angleN( 0 ',So,). and • are not sharplydefined regions of space.Nonetheless, Two sets of parametersare used: (O',So,)= (0,5) and they are well-definedquantities, as describedin the several ( --4.4,16.2) deg. The empiricalbases of the two setsare integrals. describedin Refs.37 and 38, respectively.The tilt angledis- The discussedintegrals, if analyticallyintractable, are, tribution is assumed to be truncated at two standard devia- in fact,easy to computeby meansof an algorithm.The phy- tionsfrom the mean.Thus the probabilitydensity function f sically available spaceis first partitioned into small cells. Each cell is then examinedseparately with respectto the indF=fdO' isf--0.95 -• exp[- (0'- 0')2/2•, ]. received echo level relative to the detection threshold for the IV. NUMERICAL METHOD range of scattererorientations specified by the distribution function F. The several continuous variables involved in this The integrationin Eq. (4) is effectedin the following are discretizedin the usualmanner, as illustratedby the ex- way.For therange r, lessthan the maximum detection range ample in Sec. IV. Receivedecho levelsare, respectively, rmax,the equationgb 2 = t is solvedfor 0. Specifically,the countedor ignoredas gb 2or exceeds or is lessthan t. equation

III. COMPUTATIONAL MODEL FOR ACOUSTIC 2J•(ka sin 0)/ka sin0 = 10- "•rmax --r)/20 (r/rmax) q SAMPLING OF FISH is solvednumerically, where q = 1 for a singlepoint scatterer and q = 1/2 for a scatteringlayer. The solution,denoted 0r, An immediateapplication of theory is in the acoustic is then usedto limit the 0 integrationin Eq. (4), for the measurementof fishdensity. It is convenientto evaluatenu- target at r cannot be detected anywhere outside the cone mericallythe effectiveequivalent beam angle in Eq. (4). O--Or. Thisis donethroug h the followingmodel. Equation (4) is evaluatedin the followingdiscrete ver- Medium. This consistsof seawaterof salinity 35 ppt and sion: temperature5 øC. The sound speed is thus 1470 m/s. 32 At 38 kHz, therefore,the absorptioncoefficient a is 0.0106 dB/ •Pr= 26064 • 0 2• O, )sin 0, m. 33 i=l Transducer. The transducer is assumed to be circular, with half-beamwidthof 4 deg or full beamwidthbetween opposite-- 3-dB levelsof 8 deg.The beampattern thus de- •j •= = 1 Hb2(Oi)•(Xo'k•__ &max gmi•gr f•(Ok) pendsonly on the polar angle 0, and b = [2J• (ka sin 0)/ (ka sin 0) ]2, where ka = 1.61/sin(•r/45) --'23.1. Perfor- • (0•) , (5) manceof theintegration in Eq. (3) yieldsthe nominal equiv- 1 alentbeam angle •o = 0.0108 sr or -- 19.66dB. where Fishbackscattering cross section. The sourceof data con- A0=0•/n•, O•= (i- 1/2)A0, A4= sistsof measurementsby Nakkenand Olsen 34 of thetilt an- 4• = U - 1/2) A4, A0' = 4So,/n•, TABLE I. Characteristicsof four subsetsof target strengthfunctions for O•- 0'--2%, + (k- 1/2)A0', gadoidsat 38 kHz, usedin computationsfor Figs. 1-4. The minimum and maximumlengths lmi, and lmaxrefer to criteriaapplied in selectingthe sub- Z0'•= •/2 - cos-• (sin0• cos4• cos0 • - cosOi sin 0 • ). sets.The numberof includedfunctions is denoted%. The mean and standard deviationof lengths1 in each subsetare shown.Units of length are The subscriptis attached to fi andg to emphasizetheir appli- centimeters. cability at range r. In the computationsreported below, n• = 20, % = 6, andn• = 40. ,

v. RESULTS Subset /min /max ns mean s.d. The effectiveequivalent beam angle • is examinedfirst 1 5 15 27 10.7 2.3 2 15 25 29 21.5 3.0 for a singlepoint scattererand a layer of point scatterers. 3 35 45 29 39.0 3.1 Equation (4), thenceEq. (5) also,is immediatelysimpli- 4 55 65 21 59.0 3.2 fied, for the scatteringis independentof orientation;hence, the integrationover dFyields unity. Since b onlydepends on

961 J. Acoust.Soc. Am., Vol. 90, No.2, Pt. 1, August1991 KennethG. Foote:Sampling volume 961

Downloaded 18 Dec 2012 to 128.128.44.26. Redistribution subject to ASA license or copyright; see http://asadl.org/terms 0.• 012 0 0 0.2 0A 0.6 0.8 1.0 0 0.2 OA 0.6 0.8 1.0 r/r ItlQX r/rmax FIG. 1.Effective equivalent beam angle •b normalized to thenominal trans- FIG. 3. Effectiveequivalent beam angle •b, after normalization, versus range ducervalue •bo as a functionof ranger relativeto themaximum detectable r relativeto rm.•x= 400 m, asin Fig. 2, but with the tilt angledistribution rangermax = 400m forboth a singlepoint scatterer and a layerof identical N( - 4.4,16.2) deg. pointscatterers. Other parameters are specified in Sec.III.

N( - 4.4,16.2). The effectof behavioron •bis showndirectly 0, integrationover •byields 2•r. Given a maximum rangeof for gadoidsof nominal length60 cm in Fig. 4. detectability,•b is reducedto the following: VI. DISCUSSION A numberof systematicdependences expected from Eq. •r= 2•rfb2(O)H(b 2 gmin.)sinOdO.gr (4) are confirmed by the computations.To elucidatethese This, or rather its discreteversion, analogous to Eq. (5), is more strongly,the dependenceon the backscatteringcross evaluatedfor ?'max• 400 m. The results,after normalization sectioncr is essentiallyeliminated in the computationsfor to the ideal limit •bo,are presentedin Fig. 1. Fig. 1 by considerationof identical point scatterers.For What is to be remarked on here, with force for the other these,the value of the product gb 2, when compared with the computationstoo, is that an absolutecomparison of the scat- thresholdvalue t, is decisivefor determiningwhether an teringstrengths of the point scattererand layer of point scat- echostrength lies above or belowt, henceis or is not detect- terers is not undertaken.Rather, each of two problemsis ed. Sincethe so-calledgain or geometricfactor g decreases examined,where each scatterer type has its detectionthresh- with increasingrange, the maximum angle of detection, old at 400 m. Under ordinary conditions,without this con- 0 = Or in the beampattern b, alsodecreases with increasing straint,if the pointscatterers in the layerwere identical with r. This is evidentin Fig. 1. the singlepoint scatterer,the detectionthresholds would of The curvesin Figs. i-4, which are ogivesin Figs. 2-4, course be different. showthe expectedmonotonic decrease in •bwith increasing The effectof directionalityin scatteringby fish on •bis r. In addition,•b is seento vanishat the maximumrange ?'max illustratedin Figs. 2-4 for the single-scatterercase, hence andto approachthe nominaltransducer value •bo asymptoti- withg = 10- ar/Sr-4.Figure 2 appliesto thetilt angledistri- cally as r decreases. bution N(0,5); Fig. 3 applies to the distribution Another systematicdependence seen in Fig. 1 is the effect of scatterertype, singleor layer, on •b.The mechanism

0.6 0.6 ) - 0.•, ' I N(-/,A,16.2) -

0 i i i 0 0.2 oA 0.6 0.8 1.0 0 02 0A 0.6 0.8 1.0 r/rmax r/rmox

FIG. 2. Effectiveequivalent beam angle •b, after normalization,versus range FIG. 4. Comparisonof the normalizedeffective equivalent beam angle •b r relativeto rm.•x = 400 m for gadoidtarget strength functions at 38 kHz, as versusrange r relativeto rm• = 400m for gadoidtarget strength functions, describedin subsets1-3 in Table I, with respectivenominal mean lengths describedin subset4 of TableI, with nominalmean length 60 cm,as com- 10, 20, and 40 cm, assumingthe tilt angledistribution N(0,5) deg. putedfor eachof two tilt angledistributions.

962 J. Acoust.Soc. Am., Vol. 90, No. 2, Pt. 1, August1991 KennethG. Foote:Sampling volume 962

Downloaded 18 Dec 2012 to 128.128.44.26. Redistribution subject to ASA license or copyright; see http://asadl.org/terms for thisis the rangedependence ofg. For the samerange, g •, = 1-- [1 + (QA):]-', (6b) for the singlepoint scattereris lessthan g for the layer of A = r/rm•x -- rmax/r,and Q is a measureof the steepnessof identicalpoint scatterers, hence •b for the single-pointscat- tererexceeds that for the layer.If thisresult seems contrary falloffof •/•o. Thefunction •: isdetermined byfitting a Fourier series, such as finite cosine series, to the residual in the contextof overallbackscattering strength, it mustbe function rememberedthat the maximumdetection range is assumed to bethe samefor the two scatterertypes. This assumptionis •res"- •/•0- •1' (6c) artificial for, all thingsbeing equal, the layer of identical A fairly reasonableagreement can be obtainedfor a seven- scattererswould be detectedat a greaterrange than the sin- term series,for example,but valuesnear the thresholdare glescatterer would be. However, it wasnot felt necessaryto most difficult to fit, both becauseof the smallnessof the data illustrate this fact here. sampleand the sensitivityof •bnear the threshold to the exact Repetitionof the computationsin Fig. 1 for different form of the target strengthfunction. valuesof rmaxyields very similarresults. For example,for a While the modeland computationalexample are direct- singlepoint scatterer with rmax = 500 m, •bis within 1 dB of ed to acousticscattering, analogs exist in optical and other •bofor rangesless than 390 m, or 78% of rmax,while with systemsusing irradiation and scatteringfor the remotesens- rmax= 200 m, •bdeparts from •boby 1 dB at 150m, or 75% of ing of scattererconcentration or size. Acknowledgmentof rm•x.The trend is indeedsimilar; the small differenceis due the importanceof the samplingvolume is evident in the to the absorptionpart of g, whichdoes not scalewith r in the studyof a forward-scatteringsyste .m by Hirlemanet al.4ø sameway that the spreadingpart does. Applicationto sizeand concentrationmeasurements, by like Havingestablished and shownhow the effectiveequiva- forwardscattering 41 or by backscatteringlidar or laserra- lent beamangle •b varies for identicalpoint scatterers,differ- dar,42'43 is apparent. ent single-fishscatterers are nowconsidered. Comparison of the single-scatterercurve in Fig. 1 with any of the other curvesin Figs. 2-4 showsthat the effectof directionalityin •M. S. Roos,R. E. Apfel,and S.C. Wardlaw,"Application of 30-MHz fishscattering is to decrease•b below that of the point scat- acousticscattering to the studyof humanred bloodcells," J. Acoust.Soc. terer. In addition, for the same orientation distribution, the Am. 83, 1639-1644 (1988). 2L.R. Abts,R. T. Beyer,P. D. Richardson,P.M. Galletti,and K. E. Karl- largerthe scattererand moredirectional the scatteringpat- son, "Microparticle detectionwith focusedultrasound," J. Acoust. Soc. tern, the smaller•b tends to be at the samer, assumingidenti- Am. Suppl. 1 60, S53-S54 (1976). cal valuesfor rm•x.This generaltrend clearly holds in Figs.2 3L.R. Abts,R. T. Beyer,P. D. Richardson,P.M. Galletti,and K. E. Karl- and 3, exceptfor small valuesof r/rm•x in Fig. 3, where the son,"Reflections from microparticlesin a flowingliquid," J. Acoust.Soc. Am. Suppl. 1 64, S62 (1978). differencesare not significant.The correspondingcurves for 4p. L. Edwardsand J. Jarzynski,"Scattering of focusedultrasound by 60-cm fish in Fig. 4 alsodeviate somewhat from the trend, sphericalmicroparticles," J. Acoust. Soc.Am. 74, 1006-1012 (1983). but againonly to an insignificantdegree. The discrepancies 5M.S. Roos, "A techniquefor the study of acousticscattering from micro- reflectvariations in scatteringproperties, especially with re- particles,"J. Acoust. Soc.Am. 83, 770-776 (1988). 6j. y. Chapelon,P.M. Shankar,and V. L. Newhouse,"Ultrasonic mea- spectto scattererorientation, that are intrinsic to the scatter- surementof bubblecloud sizeprofiles," J. Acoust.Soc. Am. 78, 196-201 ing process,but which are not so strongas to upsetthe de- (1985). scribedgeneral trend. 7S.T. Forbesand O. Nakken, Eds., "Manual of methodsfor fisheriesre- Behavior,as described by the orientationdistribution, is sourcesurvey and appraisal. Part 2. The use of acousticinstruments for fish detection and abundanceestimation," FAO Man. Fish. Sci. 5, 1-138 alsoan importantfactor affecting •b in Eq. (4). For broadtilt (1972). angledistributions, such as that in Fig. 3, the chanceof sens- 8M. L. Peterson,C. S. Clay, andS. B. Brandt,"Acoustic estimates of fish ing lower values of backscatteringcross section is much densityand scatteringfunction," J. Acoust.Soc. Am. 60, 618-622 (1976). greaterthan for rather narrow distributions,such as that in 9j. E. Ehrenberg,"A reviewof in situ targetstrength estimation techniques,"FAO Fish. Rep. 300, 85-90 (1983). Fig. 2, hencethe generaldisplacement of the two setsof •øD.V. Holliday,"Extracting bio-physical information from the acoustic curves.This is illustrateddirectly in Fig. 4, wherethe curve signaturesof marineorganisms," in OceanicSound Scattering Prediction, for N(--4.4,16.2) lies significantlybelow that for N(0,5) edited by N. R. Andersenand B. J. Zahuranec (Plenum, New York, for the sameset of fishtarget strength functions, subset num- 1977), pp. 619-624. •R. E. Pieper,"Euphausiid distribution and biomass determined acousti- ber 4 in Table I. cally at 102 kHz," Deep-SeaRes. 26, 687-702 (1979). As describedin Sec.III, the computationspresented in •2T. K. Stanton,R. D. M. Nash, R. L. Eastwood,and R. W. Nero, "A field Figs. 1-4 are basedon an ideal circular transducerwith 4- examinationof acousticalscattering from marine organismsat 70 kHz," deg half-beamwidth,measured from the acousticaxis to IEEE J. Ocean.Eng. OE-12, 339-348 (1987). •3D.V. Holliday,R. E. Pieper,and G. S. Kleppel,"Determination of zoo- -- 3-dB level.Repetition of the computationsfor other nar- planktonsize and distribution with multifrequencyacoustic technology," row transducerbeams, with half-beamwidthsover the range J. Cons. Int. Explor. Mer 46, 52-61 (1989). 2.5-10 deg,shows no or only negligibledifferences with the 14j.R. Proni, F. C. Newman,D.C. Rona, D. E. Drake, G. A. Berberian,C. presentresults. A. Lauter, Jr., and R. L. Sellers,"On the use of acousticsfor studying suspendedoceanic sediment and for determiningthe onsetof the shallow The severalfunctions described graphically can alsobe thermocline,"Deep-Sea Res. 23, 831-837 (1976). approximated.One successfulfunction that hasbeen used to iSM. H. Orr and R. R. Hess,"Remote acoustic monitoring of naturalsu- modelgb for cod target strength functions is the following: 39 spensatedistributions, active suspensateresuspension, and slope/shelf water intrusions,"J. Geophys.Res. 83(C8), 4062-4068 (1978). •/•0 = •l-•- •2, (6a) •6A. E. Hay, "On the remoteacoustic detection of suspendedsediment at where long wavelengths,"J. Geophys.Res. 88(C12), 7525-7542 (1983).

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Downloaded 18 Dec 2012 to 128.128.44.26. Redistribution subject to ASA license or copyright; see http://asadl.org/terms 17j.Sheng and A. E. Hay, "An examinationof the sphericalscatterer ap- ence Progressin SectorScanning Sonar (Institute of Acoustics,Edin- proximationin aqueoussuspensions of sand,"J. Acoust.Soc. Am. 83, burgh, 1979), pp. 44-52. 598-610 (1988). 31œ.j. Simmonds,"A comparisonbetween measured and theoretical equiv- 18œ.Ona, "The equivalentbeam angle and its effective value when applying alent beam anglesfor sevensimilar transducers,"J. SoundVib. 97, 117-

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