ICES Journal of Marine Science, 62: 984e995 (2005)
doi:10.1016/j.icesjms.2005.03.010
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. 2005. 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 2005 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 30 May 2004; accepted 20 March 2005.
N. Gorska: Institute of Oceanology of Polish Academy of Sciences, ul. Powstańców
Warszawy 55, PL-81-712 Sopot, Poland. E. Ona and R. Korneliussen: Institute of Marine
Research, PO Box 1870, 5817 Bergen, Norway. Correspondence to N. Gorska: tel: C48 58
551 72 81; fax: C48 58 551 21 30; e-mail: [email protected].
Introduction
Acoustic surveys are widely used for the stock assessment
of many pelagic fish species (MacLennan, 1990). 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,
2000; Reeder et al., 2004). Numerical modelling of sound
backscattering by fish (see the review presented by Horne
and Clay, 1998; Reeder et al., 2004) and controlled
accurate laboratory measurements (Sun et al., 1985; Nash
et al., 1987; Barr, 2001; Reeder et al., 2004) 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
1054-3139/$30.00
mackerel made at 18, 38, 70, 120, 200, and 364 kHz
demonstrated that backscatter intensities at different
frequencies (i.e. frequency response) have a specific
pattern. Korneliussen and Ona (2002) found that the
frequency response for mackerel was flat at 18, 38, and
120 kHz and then increased in intensity towards 200 kHz.
Unpublished backscatter measurements made in net pens
and at sea confirmed frequency-independent backscatter at
18, 38, and 70 kHz, and stronger backscatter at 200 kHz. In
the measurement series, backscatter intensities at 120 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
Ó 2005 International Council for the Exploration of the Sea. Published by Elsevier Ltd. All rights reserved.
Acoustic backscattering by Atlantic mackerel
the source of most backscatter per se (Reeder et al., 2004).
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, 2000) 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, 2000)
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, 2000), some derivations have been made to apply
it to mackerel. A solution has been obtained for mackerel
backscattering length, fflbsc, normalized by length, lfl.
f flbsc =lfl Zða0 kÞG gfl ; hfl F kfl ; a0 ; b; ‘‘aðxÞ’’ ;
ð1Þ
985
where the F and G functions can be written as
ð1
J1 ðgf ðuÞ cos bÞ
FZ du f ðuÞ exp iguefl sin b
;
cos b
0
ð2Þ
and
2 gfl h2fl gfl
GZ
:
4gfl
ð3Þ
Here the parameter g is expressed as g Z 2(kfla0). Ratio
values of f(u) Z a(u)/a0 describe the variability of the
cross-sectional radius of mackerel body a(u), normalized by
the maximum radius a0, along the longitudinal axis of the
fish (see Figure 1a). The variable u denotes the x-variable
along the axis, normalized by fish length lfl (u Z x/lfl). 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
2002, 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 140e160, 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)
986
N. Gorska et al.
β
incident wave
x
lfl
2a0
(a)
incident wave
γmax
a
(b)
γ
ρc
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
N
l sin D X
f bsc Z
3m sin hm eihm ð1Þm ;
p D mZ0
ðStraight cylinderÞ
or
ð4Þ
ð
N
l gmax X
f bsc Z
dg
3m sin hm eihm ð 1Þm
pgmax 0
mZ0
ikaeb ð1 cos gÞ
ðUniformly bent cylinderÞ: ð5Þ
exp
gmax
The scattering phase angles hm, dm, Fm, am, bm, xm
involved in modal sum coefficients are functions of acoustic
frequency, cylinder cross-section radius, sound-speed
contrasts for compressional (hcom) and shear (hsh) waves
Acoustic backscattering by Atlantic mackerel
(i.e. parameters x, x1, x2), and the density contrast (g) inside
the backbone. They can be expressed as:
tan hm Zðtan dm ðxÞÞðtan Fm Ctan am ðxÞÞ=
ðtan Fm Ctan bm ðxÞÞ;
ð6Þ
tan dm ðxÞZ Jm ðxÞ=Nm ðxÞ;
ð7Þ
tan am ðxÞZ xJ#m ðxÞ=Jm ðxÞ;
ð8Þ
tan bm ðxÞZ xN#m ðxÞ=Nm ðxÞ;
ð9Þ
tan Fm Z 1=g tan xm ðx1 ; x2 Þ;
ð10Þ
987
efl 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 eb,
obtained in mackerel morphology studies conducted during
October 2002 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 10% in June to
25e30% in October, which is in agreement with a typical
fat content of 5.5% for mackerel landed in Norway in June
2002, increasing to 20% in JulyeSeptember (personal
communication with the Norwegian Directorate of Fisheries).
and
tan am ðx1 Þ=ðtan am ðx1 ÞC1Þ m2 = tan am ðx2 ÞCm2 12x22
x22
:
tan xm ðx1 ; x2 ÞZ 2 tan am ðx1 ÞCm2 12x22 =ðtan am ðx1 ÞC1Þ m2 ½tan am ðx2 ÞC1= tan am ðx2 ÞCm2 12x22
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, 2003). 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, ka0 values from 0 to 40
were used for fish body and ka values from 0 to 2.5 for fish
backbones. These ranges were based on the results of
a large number of body-size measurements (Korneliussen
et al., 2003) and mackerel-morphology studies conducted
during October 2002 on the RV ‘‘G. O. Sars’’ (2) and
October 2003 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 gfl, g and hfl,
hcom, hsh) 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 lfl, are
indicated in the vertical and horizontal axes. Aspect-ratio
values for fish flesh were efl Z 10.54 for lean fish and
ð11Þ
Accounting for the fat content varying from 5% to 30%,
and the empirical relationship between mackerel fat content
Ff and the density contrast gfl, gfl Z 1.03 0.094Ff, found
in our own measurements, a range of variability in density
contrast of 1.002e1.025 is considered appropriate for
mackerel flesh. According to Sigfusson et al. (2001), the
contrast varies between 1.002 and 1.025. The measured soundspeed contrast in mackerel flesh hfl varied around 1.025
both in our own measurements and in Sigfusson et al.
(2001), where it was found to be hfl Z 1.034 0.125Ff at
25(C. The measured density contrast of backbone
1.10 G 0.05 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 0.1e1.0 for shear waves and 1.3e2.0 for
compressional waves.
Length and orientation statistical distributions
Based on samples from trawl and purse-seine catches, mean
total body lengths of 30e40 cm with a standard deviation
of 10% 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
988
N. Gorska et al.
(a)
(b)
0.2
a(x) / lfl
0
0
0.2
0.4
0.6
0.8
1
x / lfl
-0.2
(c)
Figure 2. Mackerel (thick fish (a) and lean fish (b)). The digitized outer boundary of the mackerel body for different classes of
mackerel (c).
normalized backscattering length, fflbsc/lfl (Equations
(1)e(3)).
The ka0-dependencies of reduced target strength,
RTS Z 10 log(sflbsc/l2fl), are presented in Figure 3 by grey
and black lines for lean and thick mackerel, respectively. The
density contrast gfl Z 1.03 was used in both calculations at
normal incidence (i.e. b Z 0). 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
ka0 is defined mainly by the difference in speed of sound (hfl),
and it is proportional to hfl. The density contrast influences
only the value of the RTS, not the periodicity of its
oscillations. For the density contrast variation from 1 to
1.03, 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 ka0. There is a minimum
in the pattern for both curves at b Z 0 (Figure 4a). While
the minimum in the mackerel-directivity pattern at b Z 0 is
surprising, it is consistent with the shape (minimum) of the
ka0-dependence, as indicated by solid arrows in Figure 3, at
ka0 Z 2.5 and ka0 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 0 for curves in Figure
4b are explained by the shape of the ka0-dependence at
ka0 Z 1.5 (ka0 of the first maximum of RTS) and
ka0 Z 11.45 (ka0 is close to the ka0 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
Acoustic backscattering by Atlantic mackerel
-30
0
5
10
15
20
k a 0 25
989
(a)
-30
lean mackerel
broad mackerel
-45
-30
-15
0
15
30
45
-35
-40
2.5
12.15
-40
-45
-50
-50
-55
-60
-60
-65
-70
-70
-75
RTS
-80
-80
R TS
Figure 3. A comparison of the ka0-dependencies for the flesh of
mackerel of different geometrical shapes. Maximum dorsal
incidence. Calculation parameters: sound-speed of 1.025, density
contrast of 1.03 and aspect ratios of 8.5 and 10.54 for thick and
lean mackerel, respectively. The arrows indicate the values of
ka0-parameter, for which the calculations of the directivity
pattern, shown in Figure 4, were made.
by the ka0 parameter. For the larger ka0, 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 10 log(sflbsc)
(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: 0.3, 0.5, and 0.9 e from
top to bottom. The sound-speed contrast of compressional
waves does not impact the width of the peaks of resonances
(b)
-30
-45
-30
-15
0
15
30
45
-40
-50
-60
-70
-80
RTS
1.5
11.45
-90
Figure 4. Directivity pattern for flesh of thick mackerel. The
corresponding values of ka0 are shown in the legend. The
calculations were performed for thick mackerel with an aspect ratio
of 8.5, sound-speed contrast of 1.025 and density contrast of 1.03.
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/(2rc)
0.1 and 0.2, respectively. Only the magnitude of the
(a)
-40
-45
-30
-15
0
β [deg.]
15
-45
30
45
18kHz
-50
-55
-60
-65
-70
-75
TS [dB]
-80
(b)
-40
-45
-30
-15
0
β [deg.]
15
45
30
38 kHz
-45
-50
-55
-60
-65
-70
-75
TS [dB]
-80
(c)
-40
-45
-30
-15
0
15
30
β [deg.]
45
-50
-60
-70
-80
-90
120 kHz
TS [dB]
-100
(d)
-40
-45
-30
0
-15
15
30 β [deg.]
45
-50
-60
-70
-80
200 kHz
-90
-100
TS[dB]
Figure 5. The directivity pattern of thick mackerel at frequencies of 18 (a), 38 (b), 120 (c), and 200 kHz (d). The calculations were made
for thick mackerel of total length 40 cm and aspect ratio 8.5, sound-speed contrast of 1.025, and density contrast of 1.03.
Acoustic backscattering by Atlantic mackerel
0.1
0.6
1.1
1.6
2.1
2.6
0
0
0.1
0.6
1.1
1.6
991
2.1
2.6
ka
ka
-10
-10
-20
-20
-30
-30
-40
RTS (dB)
-50
-60
sound speed contrast
for shear waves 0.3
RTS (dB)
-40
-50
sound speed contrast
for shear waves 0.5
-60
0
0.1
0.6
1.1
1.6
2.1
2.6
0
0.1
0.6
1.1
1.6
2.1
2.6
ka
ka
-10
-10
-20
-20
-30
-30
RTS (dB)
-50
-50
sound speed contrast for
shear waves 0.7
-60
RTS (dB)
-40
-40
sound speed contrast
for shear waves 0.9
-60
Figure 6. Backscattering by the backbone of individual mackerel. Sensitivity to the sound-speed contrast of shear waves. Maximum dorsal
incidence (q Z 0). The sound-speed contrast of compressional wave of 1.5, density contrast of 1.1, and aspect ratio 80 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 ka0 depends only on the soundspeed contrast: period is proportional to hfl.
(ii) the features of the directivity pattern of a fish body
are highly dependent on the ka0 parameter: the ka0value in regard to the ka0 of maxima and minima of
the ka0-dependence of RTS is important. Minimum
(maximum) of the directivity pattern is observed
for the normal incidence at the ka0 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.
992
N. Gorska et al.
-10
-20
0.6
1.1
1.6
2.1
ka
RTS (dB)
0.1
0
-30
-40
-50
sound speed contrast for shear waves 0.3
(a)
-60
0.1
0
-20
1.1
1.6
2.1
ka
RTS (dB)
-10
0.6
-30
-40
-50
sound speed contrast for shear waves 0.5
(b)
-60
0.1
0
0.6
1.1
1.6
2.1
ka
-20
-40
1.3
-60
RTS (dB)
1.5
-80
2.0
sound speed contrast for shear waves 0.9
-100
(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 0). Different plots are obtained for various sound-speed contrast of shear
waves, presented in plots 0.3 (a), 0.5 (b), 0.9 (c). The density contrast 1.1 and aspect ratio 80 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.
133657/120), and the European project SIMFAMI (Grant
No. Q5RS-2001-02054).
Acoustic backscattering by Atlantic mackerel
0.1
0
0.6
1.1
1.6
993
2.1
ka
-10
-20
-30
-40
1.1
1.3
1.5
RTS (dB)
-50
-60
Figure 8. Backscattering by the backbone of individual mackerel. Influence of the density contrast. Maximum dorsal incidence (q Z 0).
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 0.3, and an aspect ratio of 80.
-10
0.6
1.1
1.6
2.1
2.6
ka
RTS (dB)
0.1
0
-20
-30
-40
-50
-60
-70
straight cylinder
bent cylinder 0.1
bent cylinder 0.2
-80
Figure 9. Backscattering by the backbone of individual mackerel. The influence of the geometrical shape of backbone. Maximum dorsal
incidence (q Z 0). The calculations were made for a sound-speed contrast for shear waves of 0.3, compressional waves of 1.3, density
contrast of 1.1, and aspect ratio 100. For the bent-cylinder backbone the values of l/(2rc) parameter are shown in the legend.
994
N. Gorska et al.
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List of symbols
2
sflbsc Zjf flbsc j
2
sbbsc Zjf bbsc j
fl
f bsc
f bbsc
lfl
l
a(x)
a0
a
efl Z lfl/a0
eb Z l/a
rc
gmax
hfl Z cfl/c
hcom Z ccom/c
hsh Z csh/c
c
cfl
ccom
csh
gfl Z rfl/r
r
rfl
g Z r1/r
r1
f
k Z 2pf/c
kfl Z 2pf/cfl
kcom Z 2pf/ccom
ksh Z2pf/csh
x
x1
x2
3m
hm, dm, fm,
am, bm, xm
Jm ( )
Nm( )
Backscattering cross-section of flesh
Backscattering cross-section of backbone
Backscattering length of flesh
Backscattering length of backbone
Length of body (from the top of the
head to the start of tail) Figure 1a
Length of backbone
Cross-sectional radius of body, variable
along the longitudinal axis Figure 1a
Maximum of a(x)
Radius of backbone cylinder
Aspect ratio of body of fish
Aspect ratio of backbone cylinder
Radius of curvature of the axis of
backbone bent cylinder
l/(2rc)
Sound-speed contrast in fish flesh
Sound-speed contrast of compressional
wave in backbone
Sound-speed contrast of shear wave in
backbone
Speed of sound in surrounding seawater
Speed of sound in flesh
Speed of compressional sound wave in
backbone
Speed of shear sound wave in backbone
Density contrast of fish flesh
Density of surrounding water
Density of fish flesh
Density contrast of backbone
Density of backbone
Carrier frequency of sound
Acoustic wavenumber in surrounding
seawater
Acoustic wavenumber in flesh
Acoustic wavenumber of compressional
wave in backbone
Acoustic wavenumber of shear wave in
backbone
ka cos q
kcoma
ksha
Neumann’s number: 30 Z 1, 3mO0 Z 2
Scattering phase angles
Bessel function of the first kind of order m
Bessel function of the second kind of
order m
Acoustic backscattering by Atlantic mackerel
Jm# ( ), Nm# ( )
b
Derivative with respect to argument of
Jm( ) or Nm( )
Angle between the direction of
incidence and the normal to the
longitudinal axis of the fish Figure 1a
q
D
Ff
995
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