Importance of the swimbladder in acoustic scattering by
fish: A comparison of gadoid and mackerel target
strengths
Kenneth G. Foote
Department of Applied Mathematics, University of Bergen, 5014 Bergen, Norway
(Received 3 December 1979; accepted for publication 7 March 1980)
Previous determinations of the swimbladder contribution to the fish backscattering cross section have
been hindered by ignorance of the acoustic boundary conditions at the swimbladder wall. The present
study circumvents this problem by direct comparison of target strengths of three gadoid species and
mackerel- anatomically comparable fusiform fish which respectively possess and lack a swimbladder.
The relative swimbladder contribution to both maximum and averaged dorsal aspect backscattering cross
sections is shown to be approximately 90% to 95%, which is higher than most other estimates. The new
results were established for fish of 29- to 42-cm length and acoustic frequencies of 38 and 120 kHz.
PACS numbers: 43.80.Jz, 43.30.Dr, 43.30.Gv
INTRODUCTION
The importance of the swimbladder in acoustic scat-
tering by physoclistous and physostomatous fish has long
been recognized. l-l? There is conflicting evidence,
however, for the magnitude of its contribution to the
fish backscattering cross section. This may be due in
part to systematic species and frequency differencesAn
scattering properties, but undoubtedly also reflects the
variety of methods which have been applied in its deter-
mination.
Experimental studies have been essentially compara-
tive. Backscattering cross sections or target strengths
of fish with intact swimbladders have been compared
with measurements of the same fish with deflated, •"5'14
flooded, 5 or model-substituted •"4 swimbladders. Com-
parisons have also been made with measurements on
air-filled sacs with an equivalent volume l'13 and solid
swimbladder modelsf Theoretical studies have model-
led the swimbladder as a gas-filled cylinder, ?'8 spheri-
cal air bubble in water ? or in an elastic mediumf sphe-
roidal gas bubble, 12 and spherical viscoelastic shell. 13'l•
Several of these models are examined further, in the
light of measurements, in Refs. 15, 18, and' 19. Esti-
mates of the swimbladder contribution derived from
some of the cited studies ai'e presented in Table I.
The problem common to the various investigations,
which is also a cause of the differing results, is that of
preserving boundary conditions. The swimbladder is
generally aspherical l'ls'l• and cannot be approximated
by a simple geometric shape except possibly at rather
low frequencies. In addition, the swimbladder is sup-
ported unequally by the surrounding tissue. •'ø This was
observed dramatically in a recent series of radiograph-
ic observations of the swimbladder of several fish sub-
TABLE I. Estimates of the swimbladder contribution to fish backscattering cross sections derived
from earlier studies.
Frequencies Swimbladder contribution
Method Objects of acoustic comparison (kHz) (percentage) Ref.
Experiment Gutted cod of 60--75 cm length 10, 14, 30 35-70 1
and model swimbladders with
equivalent swimbladder volume
Experiment Perch of 20 cm length with 30 50 2
full and deflated swimbladders
Experiment I crappie (32 cm), I large mouth 20, 40, 20-80 5
bass (40 cm), and 2 yellowfin 50, 280
tuna (69 and 73 cm) with full
and deflated swimbladders
Experiment Rubber cylinders of lengths from 1480 30-90 8
14 to 30 acoustic wavelengths
with and without air-filled
cylindrical cavities
Theory Same rubber cylinders as above 1480 96 8
Experiment I cod (62 cm) with and without 278 20 10
its swimbladder
Experiment 140 "Funa" of lengths from about 50, 200 68 14
1 to 20 acoustic wavelengths at
each frequency with and without
their swimbladder s
2084
J. Acoust. Soc. Am. 67(6), June 1980
0001-4966/80/0620•4-06500.80
¸ 1980 Acoustical Society of America
2084
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jected to large external pressure changes. 2i
The present study attempts to preclude all considera-
tions of boundary conditions by direct comparison of
measured backscattering cross sections of gadolds and
mackerel, which respectively possess and lack a swim-
bladder, but are otherwise similar in size, shape, and,
to an extent, anatomy.
I. DATA BASE AND ANALYSIS
The data base of this study consists in Nakken and
Olsen's measurements of the dorsal aspect target
strength functions of three gadold species and mackerel
at 38 and 120 kHz. 22'23 Only those measurements cor-
responding to fish with lengths from 29 to 42 cm are
used. This length range represents the extent of the
mackerel measurements, which is more limited than
that of any of the gadold species. The numbers of avail-
able target strength functions are described by species
and frequency in Table II.
These data have been prepared for further analysis by
extraction of maximum values and by averaging of each
dorsal aspect function. The averaging proceeds accord-
ing to the model described in detail in Ref. 24 and used
elsewhere. 2•-3ø For present purposes it is sufficient to
consider ensonification of fish by a directional echo
sounder. The position and orientation of fish in the echo
sounder beam are described by probability distribution
functions which are, respectively, uniform and essen-
tially normal in tilt angie. Independence of the two dis-
tributions, which is tantamount to neglecting avoidance
reaction, 3ø'31 is also reasonable for the intended compu-
tations here.
The tilt angie distribution is defined precisely as a
normal distribution which is truncated at angles depart-
ing from the mean by three standard deviations. Empir-
ical justification for use of this distribution is presented
in Refs. 32 and 33. The mean and standard deviation of
the nontruncated distribution are chosen to be 0 and 10
deg, which are characteristic of a rather loose aggre-
gation. 28 For these parameter values the noted defi-
ciencies of some of the mackerel data in Ref. 23 are en-
tirely negligible.
Possible systematic species differences in the gadold
target strength data are ignored. The merged data are
distinguished only by target strength type and frequency.
These are compared with corresponding target strengths
for mackerel. To facilitate this comparison, the target
strengths of each set are regressed linearly on fish
length according to the prescription
TS=m log/+ b, (1)
where TS is the target strength predicted for fish of
length l, expressed in centimeters, and m and b are the
estimated regression coefficients. Evidence for the
validity of' linear regression analysis of similar target
strength data is cited in Ref. 26.
In order to determine the contribution of the swim-
bladder to the backscattering cross section, the target
strength of Eq. (1) is expressed as a cross section (•
according to the definition
TS- 10 log((•/4•r).
The echo sounder is represented by beam patterns
equivalent to that of an ideal circular piston with half-
beamwidth, or angular distance from acoustic axis to
-3 dB level, of 2.5 deg.
TABLE IT. Numbers of available and analyzed target strength functions of gadoids and mackerel
with lengths from 29 to 42 cm as distinguished by species and frequency.
(2)
The units of TS are decibels and •, square meters,
such that the idealized perfectly reflecting sphere of
2 m radius has a target strength of 0 dB.
Swimbladder contribution
The contribution of the swimbladder to the backscat-
tering cross section is defined here by the relative
quantity
1 - •/•,,
where (•l and (•2 are the respective backscattering cross
sections of gadolds and mackerel of the same length.
Since (•l and (•2 are derived from data which are intrins-
ically stochastic on the scale size of measurement, the
swimbladder contribution is specified within limits that
obtain with a given probability, say 1 -- a. If the cum-
ulative distribution function of the gadold target strength
variable yl is denoted Fl(•,l) and the probability density
function of the mackerel targei strength variable Y2 is
denoted f2(Y2), then
1 _10-•, /,0<_ 1_(•2/(•i <_1 _10-a• /io (3)
with probability 1 - c•, where di and d2 are determined
by numerical solution of the equation
{•/2, for d =d i ,
I:F,(y + d)fz(y)dy = 1 - a/2, for d =d•. (4)
Species
Numbers of target strength functions
Frequency= 38 kHz Frequency= 120 kHz
Cod 22 12
(Gadus morhua)
Saithe 17 12
(Pollachius virens )
Pollack 11 11
(Pollachius pollachius)
Mackerel 35 ' 24
(Scornbet scombrus)
2085 J. Acoust. Soc. Am., Vol. 67, No. 6, June 1980 Kenneth G. Foote: Acoustic scattering by fish 2085
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This last equation simply expresses the probability that
the difference of the two independent random variables
Yl and Y2 does not exceed the values dl or d2 .34 Accord-
ing to the plausible hypotheses on which the linear re-
gression analyses are based, each distribution is nor-
mal. For fish of length l in the interval [29,42] cm the
defining parameters of the distribution of target strength
variable y are the mean
+ (5)
and standard deviation
s•=s•.x (n-• +(l-•)7•(xi _•)••/2
ß ,
where s•.x is the standard error of the regression, x i is
the logarithm of length for a single datum, and • is the
mean of the logarithmically transformed lengths of all
n data underlying the regression.
II. RESULTS
Target strengths corresponding to the data enumerated
in Table II are presented on scatter diagrams in Figs.
1-4. The maximum dorsal aspect target strengths of
Figs. 1 and 3 were derived by simple extraction from
the data presented in Ref. 23. The target strengths of
Figs. 2 and 4 were derived from the data of the same
reference by the averaging method outlined above and
described fully in Ref. 24.
Results of regressing both the merged gadold target
strengths and mackerel target strengths on fish length
for the various data sets are described in Table III.
There the estimated standard errors of estimated re-
gression coefficients are denoted s m and s a. The stan-
dard error of the regression is denoted SE. The corre-
lation coefficient p of data is attached for reference.
The described linear regressions are shown on the fig-
ures.
The principal results of the study are shown in Fig. 5.
This is composed of four sets of figures, corresponding
to Figs. 1-4, which express the relative swimbladder
contribution as a percentage. The contribution is de-
scribed within limits that obtain with probability 0.95.
-2O
-30 -
I I I
++ +++ +
+
LEO[NO
m coo
4• 8RITHE
ß POLLRCK
+ MRCKEREL
i I I
-6025 :30 :35 40 45
LENGTH [CM!
FIG. 1. Scatter diagrams with regressions of maximum dor-
sal aspect target strengths on length for merged gadoids and
for mackerel at 38 kHz.
III. DISCUSSION
The target strength data of each of Figs. 1-4 are di-
visible into two groups with only a small "gray zone" of
possible ambiguity or overlap. The gadold data are ap-
parently homogeneous, which justifies their merging.
General systematic species differences among gadold
target strength data 26-28 are probably absent in the pre-
sent case because of the particular, narrow, length
range of the data. The mackerel data are similarly ho-
mogeneous, although more dispersed. Both the gadold
and mackerel target strength data are assumed to be
amenable to linear regression analysis, which is sup-
ported by the analysis of Table III and other computa-
tions .2s
TABLE III. Regression analyses of maximum and averaged target strengths on fish length for ga-
doid and mackerel data at 38 and 120 kHz. m and b are the estimated regression coefficients, cf.
Eq. (1); s m and sa, the corresponding standard errors; SE, the standard error of the regression;
and 9, the correlation coefficient.
Frequency
TS-type Fish (kHz) m s m b s a SE 9
Maximum Gado ids 38 27.1 5.6 --71.7 8.6 1.8 0.572
Maximum Mackerel 38 39.6 13.9 --100.7 21.5 2.8 0.445
Average G ado ids 38 23.8 5.5 -71.3 8.4 1.7 0.532
Average Mackerel 38 39.8 14.9 -106.3 23.0 3.0 0.423
Maximum Gadolds 120 27.2 7.1 -70.4 10.9 1.8 0.553
Maximum Mackerel 120 54.7 17.8 --125.0 27.6 3.3 0.547
Average Gadolds 120 22.9 6.2 -71.2 9.5 1.6 0.541
Average Mackerel 120 53.7 19.5 -130.2 30.2 3.6 0.506
2086 J. Acoust. Soc. Am., Vol. 67, No. 6, June 1980
Kenneth G. Foote' Acoustic scattering by fish 2086
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The backscattering cross sections and associated sta-
tistics derived from the regression analyses were used,
as in Eqs. (3) and (4), to determine the swimbladder
contribution to the backscattering cross section. It is
reasoned that this contribution can be estimated as the
difference in cross sections of anatomically comparable
fish, of the same length or mass, which respectively
possess and lack a swimbladder. The cross section of a
bladderless fish is thus taken to be a measure of the
cumulative scattering power of fish flesh, bone, and
other organs. While the backscattering cross section of
an individual fish is a sensitive function of its precise
composition, ?'tø'l? it is reasonable to assume that indi-
vidual variations are smoothed out through the kind of
regression analysis performed here. Because gadoids
and mackerel are approximately similar in their gross
anatomy and fusiform shapes, the difference in cross
sections may be accepted as a measure of the scattering
strength of the swimbladder. The similarity in condi-
tion factors for the mackerel and gadoids of measure-
ment 35 supports the comparison of the target strengths,
as presented in Figsø 1-4, both for fish of the same
mass and for fish of the same length.
From the several parts of Fig. 5, the swimbladder
contribution to the backscattering cross sections of ga-
doids is observed to be about 90% to 95%. This is high-
er than that of many earlier studies, cf. Table I, for
example, but is entirely consistent with Yudanov's a
i/•ostcriori assertion that the swimbladder contributes
at least 90%, and often much more, to the backscatter-
ing cross section. 36
-2O
-3O
I I I
LEGENO
COO
8R I THE
POLLRCK
HRCKEREL
I I I
-6025 30 35 40 45
LENGTH (C1'1)
FIG. 3. Scatter diagrams with regressions of maximum dor-
sal aspect target strengths on length for merged gadolds and
for mackerel at 120 kHz.
-2O
-30 -
I
LEGENO
COO
$R I THE
POLLRCK
MRCKEREL
I I
m
++
-60•5 • • •
30 35 40 45
LENGTH {CId]
FIG. 2. Scatter diagrams with regressions of averaged dorsal
aspect target strengths on length for merged gadoids and for
mackerel at 38 kHz.
-2O
-3O
-40 -
-50 -
I
%EGENO
COO
$RITHE
POLLRCK
MRCKEREL
I I
-i-
+ +
+
i i i
-60•5 30 35 40 45
LENGTH [ C1'1]
FIG. 4. Scatter diagrams with regressions of averaged dorsal
aspect target strengths on length for merged gadoids,and for
mackerel at 120 kHz.
2087 J. Acoust. Soc. Am., Vol. 67, No. 6, June 1980 Kenneth G. Foote' Acoustic scattering by fish 2087
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lOO
90
80
7o
100
90-
80
100
90-
80
100
8
i i I
TSMR X 38 KHZ
I I I
i I I
TSRv E 38 KHZ
I I
T3MR x 1•o KHZ
I I I
TSRVE
1•0 KHZ
I I
30 35
LENGTH [CM)
4O
45
FIG. 5. Percentage swimbladder contribution to maximum
and averaged dorsal aspect backscattering cross sections of
fish at 38 and 120 kHz, with 95% confidence as defined by the
data of Figs. 1-4.
Differences in results between this study and the cited
earlier studies are attributed both to differences in the
particular species and acoustic frequencies of investi-
gation and to the methods of analysis. The advantage of
the present method is that it avoids altering the basic
boundary conditions at the swimbladder wall--the inter-
face between the acoustically lossy and elastic fish
flesh and strongly reflecting gas sac of the swimblad-
der. •
In revising upwards previous estimates of the scatter-
ing contribution of the swimbladder, at least for gadoids
at typical ultrasonic survey frequencies, the present
study may provide a new impetus to acoustic modelling
of swimbladder-bearing fish. This is anticipated to be
equally applicable to physoclists and physostomes.
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