Acoustic backscattering by Atlantic mackerel as being representative of fish that lack a swimbladder. Backscattering by individual fish

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1 ICES Journal of Marine Science, 62: 984e995 (25) doi:1.116/j.icesjms Acoustic backscattering by Atlantic mackerel as being representative of fish that lack a swimbladder. Backscattering by individual fish Natalia Gorska, Egil Ona, and Rolf Korneliussen Gorska, N., Ona, E., and Korneliussen, R. 25. Acoustic backscattering by Atlantic mackerel as being representative of fish that lack a swimbladder. Backscattering by individual fish. e ICES Journal of Marine Science, 62: 984e995. Developing acoustic methods for the identification of fish remains a long-term objective of fisheries acoustics. The accuracy of abundance estimation may be increased when the acoustic-scattering characteristics of the fish are known, including their expected variability and uncertainty. The modelling approach is valuable during the process of interpreting multi-frequency echograms. This paper attempts to improve the understanding of sound backscattering of fish without a swimbladder, here represented by Atlantic mackerel (Scomber scombrus). Our approach includes results from modelling as well as comparisons with field data. There will be two papers. The first is a study of the non-averaged backscattering characteristics. This initial analysis is important for the understanding of the averaged backscattering cross-section, which will be considered in the second paper. In that paper the relative importance of bones in acoustic backscattering at higher frequencies will be verified. Ó 25 International Council for the Exploration of the Sea. Published by Elsevier Ltd. All rights reserved. Keywords: backbone, fish flesh, modelling, sound backscattering by mackerel. Received 3 May 24; accepted 2 March 25. N. Gorska: Institute of Oceanology of Polish Academy of Sciences, ul. Powstańców Warszawy 55, PL Sopot, Poland. E. Ona and R. Korneliussen: Institute of Marine Research, PO Box 187, 5817 Bergen, Norway. Correspondence to N. Gorska: tel: C ; fax: C ; gorska@iopan.gda.pl. Introduction Acoustic surveys are widely used for the stock assessment of many pelagic fish species (MacLennan, 199). A thorough understanding of the mechanisms of sound scattering by fish, including the understanding of the contribution of the various anatomical features to the overall backscattering, is required to improve present acoustic methods of fish species identification (Horne, 2; Reeder et al., 24). Numerical modelling of sound backscattering by fish (see the review presented by Horne and Clay, 1998; Reeder et al., 24) and controlled accurate laboratory measurements (Sun et al., 1985; Nash et al., 1987; Barr, 21; Reeder et al., 24) have been carried out to gain more knowledge of the process by which sound is scattered by selected fish species. The paper stems from the need for proper assessment from acoustic backscattering data of the abundance of the economically important fish species, Atlantic mackerel (Scomber scombrus). Multi-frequency measurements on mackerel made at 18, 38, 7, 12, 2, and 364 khz demonstrated that backscatter intensities at different frequencies (i.e. frequency response) have a specific pattern. Korneliussen and Ona (22) found that the frequency response for mackerel was flat at 18, 38, and 12 khz and then increased in intensity towards 2 khz. Unpublished backscatter measurements made in net pens and at sea confirmed frequency-independent backscatter at 18, 38, and 7 khz, and stronger backscatter at 2 khz. In the measurement series, backscatter intensities at 12 khz were more variable. In some series it was similar to that at 38 khz, and in others the increase in intensity lay anywhere in the range up to double the intensity at 38 khz. To gain an understanding of this peculiar frequency response and to study its stability, we wanted to see if this particular frequency spectrum was consistent for Atlantic mackerel and could be used for acoustic identification. In order to begin to answer this question the backscattering needed investigation. Most modelling to date has been done for fish with swimbladders where that organ is /$3. Ó 25 International Council for the Exploration of the Sea. Published by Elsevier Ltd. All rights reserved.

2 Acoustic backscattering by Atlantic mackerel 985 the source of most backscatter per se (Reeder et al., 24). Backscattering by other anatomical features, which can be important for sound incidence, are not normal to the surface of swimbladder, as well as for fish without swimbladders, are still not well known. In this paper the need to understand the impact of the different anatomical components on total backscattering and the factors which could modify mackerel target strength is addressed. Furthermore, the contribution of mackerel body and backbone to the overall backscattering characteristics across a selected frequency range is examined. The effects of orientation and the morphological condition of the fish are also considered. The modelling was done using the Distorted Wave, Born-Approximation (DWBA) (Chu et al., 1993; Stanton et al., 1993, 1998; Stanton and Chu, 2) and the Modal-Based, Deformed-Cylinder Model (MB-DCM) (Stanton, 1988a, b, 1989). A high-resolution morphology of flesh, based on measurements of the mackerel body, was considered. To study mackerel-backbone backscatter, straight and uniformly bent cylinders were used to describe its shape. The typical observed backscattering-frequency responses for mackerel are then explained in theoretical terms. The use of the frequency response in mackerel identification is justified. The final outcome of the study should be to explain the frequency response of mackerel shoals. The frequency response is defined by the averaged backscattering crosssection of mackerel at different frequencies. However, for a better understanding of the averaged characteristics the modelling has to be done first at an individual level, i.e. for the backscattering cross-section before averaging takes place. The modelling section has, therefore, been divided into two aspects; first, backscattering by individual mackerel (this paper), and second, average backscattering by mackerel (in a second paper). Material and methods Main backscattering equation: backscattering by fish flesh The DWBA-based, deformed-cylinder model (Chu et al., 1993; Stanton et al., 1993, 1998; Stanton and Chu, 2) has been used to describe backscatter by mackerel flesh. This model is assumed to be applicable because: (i) mackerel flesh has material properties that are similar, within a few per cent, to those of the surrounding water, so mackerel can be referred to as a weak scattering target; and (ii) the mackerel body has a cross-section that can be described, at first order, as circular. Using the analytical solution (Equation (6) from Stanton and Chu, 2), some derivations have been made to apply it to mackerel. A solution has been obtained for mackerel backscattering length, f fl bsc, normalized by length, l fl. f fl bsc =l flzða kþg ÿ g fl ; h fl F ÿ kfl ; a ; b; aðxþ ; ð1þ where the F and G functions can be written as FZ and ð 1 du fðuþ exp ÿ igue fl sin b J 1 ðgfðuþ cos bþ ; ð2þ cos b GZ 2 ÿ g flh 2 fl ÿ g fl 4g fl : ð3þ Here the parameter g is expressed as g Z 2(k fl a ). Ratio values of f(u) Z a(u)/a describe the variability of the cross-sectional radius of mackerel body a(u), normalized by the maximum radius a, along the longitudinal axis of the fish (see Figure 1a). The variable u denotes the x-variable along the axis, normalized by fish length l fl (u Z x/l fl ). The symbol a(x) in the argument of the F function in Equation (1) refers to the sensitivity of the function to the shape of the mackerel body, or more precisely, to the dependence of the normalized radius f(u) on u. Equations (1)e(3) incorporate the fact that the crosssectional radius varies only along the axis of the body. Although fat content is known to differ from dorsal to anterior (i.e. back to stomach), sufficient data on the properties of mackerel-body components are not available and have been treated as being homogeneous. Backscatter from the backbone The Modal-Based, Deformed-Cylinder Model (Stanton, 1988a, b, 1989) was used to study backscattering by the backbone. This model is applicable to elongated deformed cylinders with large aspect ratios and where the direction of incidence and scattering is normal or near-normal to the tangent of the axis of the cylinder. The measurements, made on the mackerel cruise by RV G. O. Sars, October 22, demonstrated the near-circularity of backbone crosssection and the large backbone elongation: the aspect ratio of backbone, i.e. the length/radius ratio of the backbone, can reach values of 14e16, so justifying the use of the model. The backbone was modelled in two different ways: (i) as an elastic, solid, straight cylinder of uniform composition (see Figure 1b); and (ii) as an elastic, solid, uniformly bent cylinder of constant radius of curvature of its axis, constant cross-section radius, and constant composition (see Figure 1c). These simple geometric shapes were chosen because approximate analyticalenumerical MB-DCM solutions have been obtained in similar cases (cf. Stanton, 1988b, 1989). Equation (7) of Stanton (1988b) was used to model the backscattering length of a straight cylinder. Equation (8) of Stanton (1989) was employed to derive solutions for the backscattering length of a bent cylinder. The modal coefficients, defined by Equation (1) of Stanton (1988b)

3 986 N. Gorska et al. β incident wave x 2a l fl (a) incident wave γ max a (b) γ ρ c θ incident wave (c) Figure 1. Backscattering geometry. Backscattering by fish flesh (a), by backbone, modelled as straight (b), and bent cylinders (c). or by Equations (22)e(25) of Faran (1951), were used in both cases. The backscattering length of the backbone can be expressed by f bsc Z ÿl ð gmax dg XN pg max mz ikaeb ð1 ÿ cos gþ exp g max 3 m sin h m e ÿih m ðÿ1þ m ðuniformly bent cylinderþ: ð5þ f bsc Z ÿl p sin D X N 3 m sin h D m e ÿih m ðÿ1þ m ; mz ðstraight cylinderþ or ð4þ The scattering phase angles h m, d m, F m, a m, b m, x m involved in modal sum coefficients are functions of acoustic frequency, cylinder cross-section radius, sound-speed contrasts for compressional (h com ) and shear (h sh ) waves

4 Acoustic backscattering by Atlantic mackerel 987 (i.e. parameters x, x 1,x 2 ), and the density contrast (g) inside the backbone. They can be expressed as: tan h m Zðtan d m ðxþþðtan F m Ctan a m ðxþþ= ðtan F m Ctan b m ðxþþ; tan d m ðxþz ÿ J m ðxþ=n m ðxþ; tan a m ðxþz ÿ xj # m ðxþ=j mðxþ; tan b m ðxþz ÿ xn # m ðxþ=n mðxþ; tan F m Z ÿ 1=g tan x m ðx 1 ; x 2 Þ; and ð6þ ð7þ ð8þ ð9þ ð1þ e fl Z 8.5 for thick fish. We assumed independence of the aspect ratio of the fish length for the two classes. Simple geometric shapes (e.g. straight and uniformly bent cylinders) were considered when modelling the backbone. Backbone dimensions (a and l) and the aspect ratio e b, obtained in mackerel morphology studies conducted during October 22 on the RV G. O. Sars (2) were used in the computations. The information on the change in angle between the main axis of the body and the backbone may be important for the study. A 3( angle was measured between the sagittal axis of the body and the backbone. There are few data on sound-speed and density contrasts for Atlantic mackerel. According to Lockwood (1988), the fat content of mackerel varies from 1% in June to 25e3% in October, which is in agreement with a typical fat content of 5.5% for mackerel landed in Norway in June 22, increasing to 2% in JulyeSeptember (personal communication with the Norwegian Directorate of Fisheries). tan x m ðx 1 ; x 2 ÞZ ÿ x2 2 2 tan a m ðx 1 Þ=ðtan a m ðx 1 ÞC1Þÿm 2 = ÿ tan a m ðx 2 ÞCm 2 ÿ 1 2 x2 2 ÿ tan am ðx 1 ÞCm 2 ÿ 1 2 x2 2 =ðtan am ðx 1 ÞC1Þÿ m 2 ½tan a m ðx 2 ÞC1Š= : ð11þ tan a m ðx 2 ÞCm 2 ÿ x2 Modelling parameters Acoustic backscattering by fish is a complex function of the geometrical shape of various body components, the properties of their materials, the orientation of the fish in space, and the acoustic frequency (Horne, 23). In the modelling process, we considered the following factors. Size and frequency To include a full range of frequencies and lengths of mackerel bodies and backbones, ka values from to 4 were used for fish body and ka values from to 2.5 for fish backbones. These ranges were based on the results of a large number of body-size measurements (Korneliussen et al., 23) and mackerel-morphology studies conducted during October 22 on the RV G. O. Sars (2) and October 23 on the RV G.O. Sars (3). Animal morphology The digitizing of fish-body morphology needs to include the acoustic properties of the flesh and backbone (the spatial, three-dimensional distribution of the contrasts g fl, g and h fl, h com,h sh ) and the three-dimensional shape. Two classes of mackerel, differing in body shape, were chosen for analysis e thick and lean mackerel. Examples of these classes are presented in Figure 2a, b. In Figure 2c, the digitizing of the outer boundary of mackerel bodies is shown using light and dark grey lines. The radius of the mackerel body and the x coordinate, both normalized by the fish length l fl, are indicated in the vertical and horizontal axes. Aspect-ratio values for fish flesh were e fl Z 1.54 for lean fish and Accounting for the fat content varying from 5% to 3%, and the empirical relationship between mackerel fat content F f and the density contrast g fl,g fl Z 1.3 ÿ.94f f, found in our own measurements, a range of variability in density contrast of 1.2e1.25 is considered appropriate for mackerel flesh. According to Sigfusson et al. (21), the contrast varies between 1.2 and The measured soundspeed contrast in mackerel flesh h fl varied around 1.25 both in our own measurements and in Sigfusson et al. (21), where it was found to be h fl Z 1.34 ÿ.125f f at 25(C. The measured density contrast of backbone 1.1 G.5 was used in the computations. Sound-speed contrast measurements for shear and compressional waves were not performed. Given the lack of available information on these two parameters, we assumed soundspeed contrasts of.1e1. for shear waves and 1.3e2. for compressional waves. Length and orientation statistical distributions Based on samples from trawl and purse-seine catches, mean total body lengths of 3e4 cm with a standard deviation of 1% were used in the analysis. Given the lack of mackerel tilt-angle measurements, we assumed a narrow distribution of orientations, based on visual observations of behaviour in net pens. Results and discussion Sensitivity analysis for mackerel flesh The possible effect of changes in mackerel morphology on backscattering was analysed using expressions for the

5 988 N. Gorska et al. (a) (b).2 a(x) / l fl x / l fl -.2 Figure 2. Mackerel (thick fish (a) and lean fish (b)). The digitized outer boundary of the mackerel body for different classes of mackerel (c). (c) normalized backscattering length, f fl bsc /l fl (Equations (1)e(3)). The ka -dependencies of reduced target strength, RTS Z 1 log(s fl bsc /l 2 fl ), are presented in Figure 3 by grey and black lines for lean and thick mackerel, respectively. The density contrast g fl Z 1.3 was used in both calculations at normal incidence (i.e. b Z ). Comparison of the two curves in the figure demonstrates that the reduced target strength is slightly dependent on the geometrical shape of the mackerel body. The reduced target strength is sensitive to the soundspeed contrast, which influences both the amplitude of the oscillations and their period. The period of oscillations over ka is defined mainly by the difference in speed of sound (h fl ), and it is proportional to h fl. The density contrast influences only the value of the RTS, not the periodicity of its oscillations. For the density contrast variation from 1 to 1.3, the reduced target strength increases 6e12 db, depending on the value of the sound-speed contrast. Sensitivity analysis of the orientation dependence (Figure 4) shows that the shape of the directivity pattern depends strongly on the value of ka. There is a minimum in the pattern for both curves at b Z (Figure 4a). While the minimum in the mackerel-directivity pattern at b Z is surprising, it is consistent with the shape (minimum) of the ka -dependence, as indicated by solid arrows in Figure 3, at ka Z 2.5 and ka Z 12.15, at which the calculations were made. The possibility of the minimum in the directivity pattern at normal dorsal incidence is also supported by the results of earlier measurements (Foote and Nakken, 1978). A maximum and local minimum at b Z for curves in Figure 4b are explained by the shape of the ka -dependence at ka Z 1.5 (ka of the first maximum of RTS) and ka Z (ka is close to the ka of seventh maximum of RTS), indicated by dotted arrows in Figure 3. Comparison between dotted and solid curves in Figure 4 demonstrates that the width of the lobes of the directivity pattern is controlled

6 Acoustic backscattering by Atlantic mackerel ka 25 lean mackerel broad mackerel (a) RTS RTS Figure 3. A comparison of the ka -dependencies for the flesh of mackerel of different geometrical shapes. Maximum dorsal incidence. Calculation parameters: sound-speed of 1.25, density contrast of 1.3 and aspect ratios of 8.5 and 1.54 for thick and lean mackerel, respectively. The arrows indicate the values of ka -parameter, for which the calculations of the directivity pattern, shown in Figure 4, were made. (b) by the ka parameter. For the larger ka, the individual lobe width is smaller and number of lobes is larger. The calculations of the orientation dependence of the target strength of mackerel flesh TS Z 1 log(s fl bsc ) (Figure 5) show that the dependence is more complex for higher frequencies. The number of the directivity-pattern lobes increases and the width of an individual lobe decreases with frequency. Sensitivity analysis for mackerel backbone The analysis demonstrates (Figure 6) that the resonance and anti-resonance structure (the ka of maxima (resonances) and minima (anti-resonances) of the target strength, the frequency of the occurrence of resonances and antiresonances, the width of the resonance and anti-resonance peaks and their amplitudes) are all extremely sensitive to the sound-speed contrast of shear waves in backbone. For the larger contrast the resonances and anti-resonances are more frequent and narrow, and their amplitude is higher. The similar qualitative dependences are observed for sound backscattering by elastic solid spheres (Hickling, 1962). The dependence of the reduced target strength on ka is illustrated for various sound-speed contrasts of compressional waves in Figure 7. Individual plots refer to different sound-speed contrasts of shear waves:.3,.5, and.9 e from top to bottom. The sound-speed contrast of compressional waves does not impact the width of the peaks of resonances RTS Figure 4. Directivity pattern for flesh of thick mackerel. The corresponding values of ka are shown in the legend. The calculations were performed for thick mackerel with an aspect ratio of 8.5, sound-speed contrast of 1.25 and density contrast of 1.3. and anti-resonances and their positions over ka-axis, but influences the value of the reduced target strength of backbone. The value increases with the contrast. The impact depends on the ka parameter. Figure 8 illustrates the sensitivity of the reduced target strength of backbone to the backbone-density contrast. The density contrast does not influence the resonance and antiresonance structure of the curves and defines only the magnitude of the reduced target strength. The magnitude increases with increasing density contrast. The sensitivity of backscatter to the backbone shape is demonstrated in Figure 9. The three curves shown refer to various backbone geometries viz. from top to bottom, straight cylinder and bent cylinders with ratios of l/(2r c ).1 and.2, respectively. Only the magnitude of the

7 (a) β [deg.] kHz TS [db] (b) β [deg.] khz TS [db] (c) β [deg.] -9 TS [db] 12 khz (d) β [deg.] -9 2 khz TS[dB] Figure 5. The directivity pattern of thick mackerel at frequencies of 18 (a), 38 (b), 12 (c), and 2 khz (d). The calculations were made for thick mackerel of total length 4 cm and aspect ratio 8.5, sound-speed contrast of 1.25, and density contrast of 1.3.

8 Acoustic backscattering by Atlantic mackerel ka ka sound speed contrast for shear waves.3 sound speed contrast for shear waves ka ka sound speed contrast for sound speed contrast shear waves.7 for shear waves.9 Figure 6. Backscattering by the backbone of individual mackerel. Sensitivity to the sound-speed contrast of shear waves. Maximum dorsal incidence (q Z ). The sound-speed contrast of compressional wave of 1.5, density contrast of 1.1, and aspect ratio 8 were taken for the calculations, which were made for a backbone modelled as a straight cylinder. reduced target strength is sensitive to the shape of the backbone. The more curved the backbone, the smaller the level. Conclusions A model has been developed to describe sound backscattering by Atlantic mackerel flesh and backbone. It has been shown that: (i) in the case of normal sound incidence, the value of RTS is defined mainly by the sound-speed and density contrasts, and to a lesser extent by the geometrical shape of the body, while the periodicity of its oscillations over ka depends only on the soundspeed contrast: period is proportional to h fl. (ii) the features of the directivity pattern of a fish body are highly dependent on the ka parameter: the ka - value in regard to the ka of maxima and minima of the ka -dependence of RTS is important. Minimum (maximum) of the directivity pattern is observed for the normal incidence at the ka of minima (maxima). (iii) the ka of maxima (resonances) and minima (antiresonances) of the backbone target strength, the frequency of the occurrence of resonances and antiresonances, the width of the resonance and antiresonance peaks, and their amplitudes are mainly defined by the sound-speed contrast of shear waves.

9 992 N. Gorska et al ka sound speed contrast for shear waves.3 (a) ka sound speed contrast for shear waves.5 (b) ka sound speed contrast for shear waves.9 (c) Figure 7. Backscattering by the backbone of individual mackerel. Sensitivity to the sound-speed contrast of compressional waves, which are indicated in the legend. Maximum dorsal incidence (q Z ). Different plots are obtained for various sound-speed contrast of shear waves, presented in plots.3 (a),.5 (b),.9 (c). The density contrast 1.1 and aspect ratio 8 were used for the calculations that were made for a backbone modelled as a straight cylinder Sound-speed contrasts of compressional waves and density contrast influence the value of the target strength of fish backbone. The value is also sensitive to the degree of curvature of the cylinder backbone. The results obtained for individual mackerel will be useful in the analysis of the averaged backscattering cross-section of mackerel aggregations and thus in the explanation of the observed frequency response. Since the lack of a swimbladder is the main acoustic feature of mackerel, modelling and measurements of sound backscattering by mackerel can increase our knowledge of scattering from other pelagic species that do not have swimbladders. Moreover, since the swimbladder of physostomous fish is compressed during descent, there is a possibility that backscatter from physostomous fish at depth is closer to that of mackerel. Acknowledgements This work has been partially supported by the Institute of Oceanology, Polish Academy of Sciences (sponsor programme 2.7), the Research Council of Norway (Grant No /12), and the European project SIMFAMI (Grant No. Q5RS1-254).

10 Acoustic backscattering by Atlantic mackerel ka Figure 8. Backscattering by the backbone of individual mackerel. Influence of the density contrast. Maximum dorsal incidence (q Z ). The values are shown in the legend. The calculations were made for a backbone modelled as a straight cylinder with a sound-speed contrast for compressional waves of 1.5 and shear waves of.3, and an aspect ratio of ka straight cylinder bent cylinder.1 bent cylinder.2 Figure 9. Backscattering by the backbone of individual mackerel. The influence of the geometrical shape of backbone. Maximum dorsal incidence (q Z ). The calculations were made for a sound-speed contrast for shear waves of.3, compressional waves of 1.3, density contrast of 1.1, and aspect ratio 1. For the bent-cylinder backbone the values of l/(2r c ) parameter are shown in the legend.

11 994 N. Gorska et al. References Barr, R. 21. A design study of an acoustic system suitable for differentiating between orange roughy and other New Zealand deep-water species. Journal of the Acoustical Society of America, 19: 164e178. Chu, D., Foote, K. G., and Stanton, T. K Further analysis of target-strength measurements of Antarctic krill at 38 and 12 khz. Comparison with deformed-cylinder model and inference of orientation distribution. Journal of the Acoustical Society of America, 93: 2985e2988. Faran, J. J Sound scattering by solid cylinders and spheres. Journal of the Acoustical Society of America, 23: 45e418. Foote, K. G., and Nakken, O Dorsal-aspect, target-strength functions of six fishes at two ultrasonic frequencies. Fisken og Havet Series B, 1978(3): 1e96. Hickling, R Analysis of echoes from solid elastic sphere in water. Journal of the Acoustical Society of America, 34: 1582e1592. Horne, J. K. 2. Acoustic approaches to remote species identification: a review. Fisheries Oceanography, 9: 356e371. Horne, J. K. 23. The influence of ontogeny, physiology, and behaviour on the target strength of walleye pollock (Theragra chalcogramma). ICES Journal of Marine Science, 6: 163e174. Horne, J. K., and Clay, C. S Sonar systems and aquatic organisms: matching equipment and model parameters. Canadian Journal of Fisheries and Aquatic Sciences, 55: 1296e136. Korneliussen, R. J., and Ona, E. 22. An operational system for processing and visualizing multi-frequency acoustic data. ICES Journal of Marine Science, 59: 293e313. Korneliussen, R., Skagen, D. W., Slotte, A., and Knutsen, T. 23. Cruise summary report of survey, pp. (In Norwegian). Lockwood, S. J The Mackerel. Its Biology, Assessment and the Management of a Fishery. Fishing News Books Ltd., Oxford. MacLennan, D. N Acoustical measurement of fish abundance. Journal of the Acoustical Society of America, 87: 1e15. Nash, R. D. M., Sun, Y., and Clay, C. S High-resolution acoustic structure of fish. Journal du Conseil International pour l Exploration de la Mer, 43: 23e37. Reeder, D. B., Jech, J. M., and Stanton, T. K. 24. Broadband acoustic backscatter and high-resolution morphology of fish: measurement and modelling. Journal of the Acoustical Society of America, 116: 747e761. Sigfusson, H., Decker, E. A., and McClements, D. J. 21. Ultrasonic characterization of Atlantic mackerel (Scomber scombrus). Food Research International, 34: 15e23. Stanton, T. K. 1988a. Sound scattering by cylinders of finite length. I. Fluid cylinders. Journal of the Acoustical Society of America, 83: 55e63. Stanton, T. K. 1988b. Sound scattering by cylinders of finite length. II. Elastic cylinders. Journal of the Acoustical Society of America, 83: 64e67. Stanton, T. K Sound scattering by cylinders of finite length. III. Deformed cylinders. Journal of the Acoustical Society of America, 86: 691e75. Stanton, T. K., and Chu, D. 2. Review and recommendations for the modelling of acoustic scattering by fluid-like, elongated zooplankton: euphausiids and copepods. ICES Journal of Marine Science, 57: 793e87. Stanton, T. K., Chu, D., and Wiebe, P. H Sound scattering by several zooplankton groups. II. Scattering models. Journal of the Acoustical Society of America, 13: 236e253. Stanton, T. K., Chu, D., Wiebe, P. H., and Clay, C. S Average echoes from randomly oriented, random-length finite cylinders: zooplankton models. Journal of the Acoustical Society of America, 94: 3463e3472. Sun, Y., Nash, R., and Clay, C. S Acoustic measurements of the anatomy of fish at 22 khz. Journal of the Acoustical Society of America, 78: 1772e1776. List of symbols s fl fl bsczjf bsc j2 Backscattering cross-section of flesh s b bsc Zjf b bsc j2 Backscattering cross-section of backbone f fl bsc Backscattering length of flesh f b bsc Backscattering length of backbone l fl Length of body (from the top of the head to the start of tail) Figure 1a l Length of backbone a(x) Cross-sectional radius of body, variable along the longitudinal axis Figure 1a a Maximum of a(x) a Radius of backbone cylinder e fl Z l fl /a Aspect ratio of body of fish e b Z l/a Aspect ratio of backbone cylinder r c Radius of curvature of the axis of backbone bent cylinder g max l/(2r c ) h fl Z c fl /c Sound-speed contrast in fish flesh h com Z c com /c Sound-speed contrast of compressional wave in backbone h sh Z c sh /c Sound-speed contrast of shear wave in backbone c Speed of sound in surrounding seawater c fl Speed of sound in flesh c com Speed of compressional sound wave in backbone c sh Speed of shear sound wave in backbone g fl Z r fl /r Density contrast of fish flesh r Density of surrounding water r fl Density of fish flesh g Z r 1 /r Density contrast of backbone r 1 Density of backbone f Carrier frequency of sound k Z 2pf/c Acoustic wavenumber in surrounding seawater k fl Z 2pf/c fl Acoustic wavenumber in flesh k com Z 2pf/c com Acoustic wavenumber of compressional wave in backbone k sh Z2pf/c sh Acoustic wavenumber of shear wave in backbone x ka cos q x 1 k com a x 2 k sh a 3 m Neumann s number: 3 Z 1, 3 mo Z 2 h m, d m, f m, a m, b m, x m Scattering phase angles J m ( ) Bessel function of the first kind of order m N m ( ) Bessel function of the second kind of order m

12 Acoustic backscattering by Atlantic mackerel 995 J m # (),N m # ( ) Derivative with respect to argument of J m ( ) or N m () b Angle between the direction of incidence and the normal to the longitudinal axis of the fish Figure 1a q D F f Angle between the direction of incidence and the normal to the backbone longitudinal axis of the fish Figure 1b kl sin q Fat fraction of total weight