Foote, Kenneth G.

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This paper not to be cited without prior reference to the autho~ International Council for the Exploration of the Sea .,, ' C.M. 1982/B:38 Fish Capture Committee Ref. Pelagic Fish Comm. ON MULTIPLE SCATTERING IN FISHERIES ACOUSTICS by Kenneth G. Foote Institute of Marine Research 5011 Bergen, Norway ABSTRACT The possibility of multiple scattering by dense or extended fish schools has worried some researchers for years. This concern is shown to be unfounded by reference to the larger .literature The sole, non·-behavioural effect of aggregation on fish detection and measurement is that of simple shadowing or extinction. A simple addition theorem is stated which is adequate for all fish aggregations of natural occurrence. Empirical evidence for the same is cited. RESUME: DIFFUSION MULTIPLE EN ACOUSTIQUE HALIEUTIQUE La possibilite d'une diffusion multiple dans des banes denses ou de grandes dimensions a inqui~te quelques chercheurs depuis des ann~es. Se basant sur une large recherche bibliographique, on montre que cette inqui~tude n'a pas et~ retrouv~e ailleurs. Le seul effet non-comportemental dft au grcupement des poissons est simplement celui de l'ombre ou de l'extinction acoustiques. On etablit une loi d'addition simple, applicable a tout rassemblement naturel de poissons. Des preuves empiriques sont ~galement donn~es sur ce m~me sujet. INTRODUCTION Multiple scattering is a recognized phenomenon in fisheries acoustics. The most striking manifestation of this in echo sounding is the weakening, if not actual loss of bottom signal due to the presence of large, dense schools. Since the numbers of fish and total bion1ass contained in such schools can be quite large, it is essential for the wider application of the echo integration method that the phenomenon be understood. There are at least several bodies of literature pertinent to the present discussion. That of fisheries acoustics has been freshly and fully elaborated by MacLennan a:-ndr Fo:rbes (1982) and Lytle and Maxwell (1982). All of these - 2 -authors appreciate the importance of the phenomenon; MacLennan and Forbes (1982) advocate further work, at least to disclose the limit of linearity, i.e., the limit of proportionality between integrated echo intensity and fish density. Lytle and Nax,.vell (1982) also review the literature, which they find lacking. They have, however, developed their own method for describing multiple scattering from schools of arbitrary density They additionally describe how this knowledge can be used to correct echo integrator estimates of fish density, especially for high-density schools. This involves application of a special t:ime-varied-gain function.which operates over the duration of the returning echo. Judging from the two cited works, it is reasonable to ask whether or not multiple scattering is still a problem, or in fact has been a problem in the sense of having wanted a solution. It is the purpose of this paper to answer the question by reference to the larger physics literature on multiple scattering. GERMANE LITERATURE Contrary to popular belief, the physics literature did not end, or even begin wi th Foldy (19Lj.5), seminal though his work was A few specific references with reviews and/or bibliographies useful to the discussion are Lax (1951), Twersky (1960, 1977)) Waterman and Truell (1961), a~d Burke and Twersky (1964). According to the several works, with augmentation by Lax (1.952), Twersky (1962, 1976), and Tsang et al. (1982), the problem of multiple scattering can be expressed through the question: What is the propagation constant in a region occupied by scatterers? Alternatively, how·· does the presence of scatterers change the propagation characteri.stics of the medium? WEAK SCATTERING DENSITY It is a useful exercise to compile a list of expressions for the propagation constant, distinguishing these by the doma.in or conditions of applicability. As there is general agreement, however, a common expression for the weak scattering density is ·defined. In terms of the maximum single-scatterer differential scattering cross section crmax' -2 -~ p<< 2'1TA· cr · max (l) where p is the scatterer density and A is the acoustic wavelength. That is, if the density sati~fies the criterion, then multiple scattering is negligible. To show the significance of Eq, (l) in fisheries acoustics, several cases are considered. In these, cr is approximated by the maximum backscattering · h maxl~ d · h b 4 i cross sect1on crb w en norma 1ze 1n te customary manner y n, .e., ,max a max a + b,max 4TI According to the ordinary definition of target strength TS (Urick 1975), (2) - 3 -(J b,max = 10TS/10 471' Substituting in Eq. (1), For fish whose target strength is described by the regression equation TS = m log .Q, + b where .Q, is the fish length in centimeters, and m and b are the regression coefficients, Since m is equal to 20 in most applications in ~cho surveying, (3) (4) (5) (6) (7) This equation is evidently amply fulfilled for the same anticipated applications, for the characteristic size-to-wavelength ratio generally lies in the range from l to 100 (Foote 1980), while the regression coefficient b very roughly lies in the range from -75 to - 55 dB. For an unfavorable, but still realistic combination of conditions, say A.= 5 cm, .Q, = 100 cm, and b = -55 dB, -3 p << 14100 m if multiple scattering is to be avoided. It is noted in passing that if p in Eqs. (6) and (7) is to be expressed in units of fish per cubic meter, then A must be expressed in meters, while .Q, is expressed in centimeters according to the convention underlying Eq. (5). As a further example, the empirical relationship for the maximum dorsal aspect target strength of cod (Gadus morhua) at 38kHz is substi'tuted in Eq. (6). Since m = 25.2 and b = -67.9 dB (Foote 1980), p << 1.015 107 .Q,-1 · 26 This is clearly satisfied for the applicable length range from 6.7 to 96 cm, for at the extreme lengths, the condition becomes 9.24 105 m- 3 3.23 104 m-3 for .Q, = 6.7 cm for Jt = 96 cm - 4 -ADDITION THEOREM The consequence of the fulfillment of Eq. (l) is the addition cross sections cited :tn Waterman and Truell (196.1). An expression fisheries acoustics is the second, and general addition theorem of This is tantamount to the statement that the mean echo intensity I depth z in a uniform aggregation whose upper surface is at depth z~ I z -2pcr (z-z ) = I e e l z,o theorem for of this in Foote (1982). from fish at < z is (8) where p is the fish density, cr is the mean extinction cross section, and I is the mean echo intensity dueeto the same fish of the aggregation at depth 2 ~0in the absence of higher=lying fish. Thus- the sole acoustic effect of propinquity of fish on the echo is an attenuation of both incident and backscattered waves by scattering out of the forward direction. Generalization to the case of spatially inhomogeneous fish distributions is straightforward. For cornparison, the hybrid solution of Lytle and Haxwell (19 82) could be consistent with Eq. (8), were the characteristic constants equal. This is not generally the case, however, as Lytle and Haxwell's constant is determined by empirical curve-fitting. It thus depends on simulating or observing the actual survey conditions of interest, and is not easily adapted to new conditions. EMPIRICAL EVIDENCE In addition to the theoretical demonstration above, there is a variety of empirical evidence for the validity of the simple interpretation of high-density scattering in terms of extinction. Direct evidence in the fisheries domain is provided by the expeTiment of RØttingen (1976) and its analysis (Foote 1978). Other direct evidence for the cited general addition theorem is provided by several classic experiments on sound propagation in bubbly water, viz. Carstensen and Foldy (~947), Fox et al. (1955), and Silbennan (1957). Experiments on the propagation of electromagnetic waves in concentrations of styrofoam spheres simulating density states from the gaseous to liquid have been similarly corroboratory of theory (Beard and Twersky 1960), as have been extensi.ons of this work, cf. Beard (1962). A successful theoretical development related to this, but oriented towards fisheries acoustics is described in,Prokopets (1982). SUMMARY Examination of the larger physics literature has disclosed a general condition for the fish density at which multiple scattering becomes important, cf. Eq. (1}. According to the general and specific evaluations, actual densities of natural fish aggregations never support the occurrence of multiple scattering. The sole acoustic effect of the presence of other fish on the echo from any one fish is that of simple extinction, as described briefly in Eq. (8). This is consistent with the principle of linearity as interpreted in Foote (1982)e Various empirical evidence both withi.n and outside of fisheries acoustics supports the simple scattering interpretation, hence too the adequacy of existing theory to treat all sc3ttering situations in fisheries acoustics l ,· - 5 -ACKNOWLEDGEMENT The author expresses his gratitude to N. Diner, Institut Scientifique et Technique des P~ches Maritimes, for his wil1ing trans1ation of the abstract. REFERENCES Beard, C.l. 1962. Statistics of phase quadrature components of microwave fie1d transmitted through a random medium. IRE Trans. Ant. Prop. AP-10: 721-731. Beard, C.I., and Twersky, V. 1960. Forward coherent and incoherent scattering from random volume distribution of spheres. Sylvania Electron. Defense Labs. Report EDL-E46. AD-246 677. 83 p. Burke, J.E., and Twersky, V. 1964. On scattering of waves by many bodies. J .. Research Natl. Bur. Standards 68D: 500-510. Carstensen, E.L., and Foldy" L.L. 1947. Propagation of sound through a liquid containing bubb1es. J. Acoust. Soc. Am. 19: 481-501. Foldy, L. L. 1945. The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers. Phys. Rev. 67: 107-119 (1945). Foote, K.G. 1978. Analysis of empirical observations on the scattering of sound by encaged aggregations of fish. Fislillir. Skr. Ser. HavUnders. 16: 423-456. Foote, K.G. 1980. Effect of fish behaviour on echo energy: the need for measurements of orientation distributions. J. Cons. int. Explor. Mer 39: 193-201. Foote, K.G. 1982. L.inearity of fisheries acoustics, with addition theorems. Contribution No. 43, Symposium on Fisheries Acoustics, Bergen, Norway, 21-24 June 1982. 18 p. Fox, F.E., Curley, S,R., and Larson, G.S. 1955. Phase velocity and absorption measurements in water containing air bubbles. J, Acoust. Soc. Am. 27: 534-539. Lax, M. 1951. Multiple scattering of waves. Rev. Mod. Phys. 23: 287-310. Lax, M. 1952. Multiple scattering of waves. II. The effective field in dense systems. Phys. Rev. 85: 621-629. Lytles D.W., and Maxwell, D.R. 1982. Hydroacoustic assess.ment in high density fish schools. Contribution No, 106, Symposium on Fisheries Acoustics, Bergen, Norway, 21-24 June 1982. 20 p. MacLennan, D.N., and Forbes, S.T. 1982. Fisheries acoustics: A review of general princip1es. Contribution No. l, Symposium on Fisheries Acoustics, Bergen, Norway, 21~·24 June 1982. 25 p. - 6 -Prokopets, S.A. 1982. On/the fonnation of the echo coherent component of backscattering. Contribution No. 76, Symposium on Fisheries Acoustics, Bergen, Norway, 21-24 June 1982. RØttingen, I. 1976. On the re1ation between echo intensity and fish density FiskDir. Skr. Ser. HavUnders. 16: 301-314. Si1berman, E. 1957. Sound ve1ocity and attenuation in bubb1y mixtures measured in standing wave tubes J. Acoust. Soc. Am. 29: 925-933. Tsang, L., Kong, J.A.-, and Habashy, T. 1982. Multiple scattering of acoustic waves by random distribution of discrete spherical scatterers with the quasicrysta11ine and Percus-Yevick approximation. J. Acoust. Soc. Am. 71: 552-558. Twersky, V. 1960. On mu1tiple scattering of waves. J. Research Nat1. Bur. Standards 64D: 715-730. Twersky, V. 1962. On scattering of waves by random distributions. I. Free-space scatterer formalism. J. Math. Phys. 3: 700-715v Twersky, V. 1976. Propagation parameters in random distributions of scatterers. J. d'Analyse Mathematique 30: 498-511. Twersky, V. 1977. Coherent sca1ar fie1d in pair-corre1ated random distributions of a1igned scatterers. J. Math. Phys. 18: 2468-2486. Urick, R.J. 1975. "Principles of Underwate.r Sound", Second edition, McGraw~Hill, New York. 384 p. Waterman, P.C., and True11, R. 1961. Mu1tiple scattering of waves. J. Math. Phys. 2: 512-537. 

On multiple scattering in fisheries acoustics

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On multiple scattering in fisheries acoustics

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