Rather‐high‐frequency sound scattering by swimbladdered fish

TL;DR: In this article, a new model was proposed to describe acoustic scattering by swimbladdered fish of lengths from at least 8 to 36 wavelengths, which represented a fish by an ideal pressure-release surface having the exact size and shape as the swimbladder.

Abstract: A new model describes acoustic scattering by swimbladdered fish of lengths from at least 8 to 36 wavelengths. It represents a fish by an ideal pressure‐release surface having the exact size and shape as the swimbladder. The backscattering cross section, or target strength, is computed by means of the Kirchhoff approximation. To test the model, predictions of target strengths based on swimbladder morphometries of 15 gadoids of lengths from 31.5 to 44.5 cm are compared with conventional target strength measurements on the same, surface‐adapted fish, anesthetized before acoustic measurement, and shock‐frozen immediately afterwards. Details are given of the swimbladder morphometry. In essence, this consists of slicing the frozen fish with a microtome, photographing the exposed swimbladder cross sections, digitizing the contours, and triangulating the surface between pairs of contours on adjacent, parallel planes. Theory and experiment are compared through the dorsal and ventral aspect target strength functions, their averages, and simulated probability density functions.

Summary

Introduction

• The authors goal is to design broadcast schedules that minimize the waiting time, i.e., the amount of time the client needs to wait in order to obtain the most recent update.
• Fig. 1 depicts an example of a data broadcast system the authors consider.
• Moreover, the idle time can vary for different clients.
• This correspondence focuses on the design of broadcast schedules that minimize the amount of time the client needs to wait (after it is ready) to obtain the desired update.

A. Universal Schedules

• The authors begin by observing that deterministic schedules do not behave well in the presence of an adversarial client.
• Indeed, as shown in Fig. 3(c), such a client misses the beginning of the current update and needs to wait an entire period to receive the next update, resulting in a long waiting time.
• Their goal is to design a random schedule that minimizes the worst case expected waiting time of a client, where the expectation is taken over the probability distribution governing the schedule.
• A natural approach to limit the power of the adversarial client is to bound its adaptivity to the schedule.
• Universal schedules make no assumption on the distribution of clients’ idle times.

B. Related Work

• Developing efficient data broadcast methods attracted a large body of research.
• Several studies [5]–[8], [10], [11] have focused on minimizing the worst case waiting time for arbitrary client requests.
• The basic assumption in [5]–[7] is that the server is aware of client requests and can respond to them.
• In contrast, this work focuses on the “push-based” approach, where the server proactively transmits the data over the broadcast channel.

C. Our Results

• The authors present (for the restricted adversarial clients discussed above) a universal broadcast schedule that guarantees a worst case expected waiting time of 1= p 2 ' 0:7 time units, regardless of the clients’ access patterns.
• The authors work mainly addresses adversarial clients with limited adaptively to the schedule that can be quantified by one time unit.
• One of the important characteristics of the schedule is the number of updates it sends over a period of time.
• The authors show that, under certain restrictions on the server, this is the best possible schedule.

D. Organization

• The remainder of this correspondence is organized as follows.
• In Section III, the authors focus on clients with unit adaptivity and prove their main results.
• In Section IV, the authors briefly discuss schedules for highly adaptive clients.
• Finally, conclusions are presented in Section V.

A. Random Schedules

• As mentioned in the Introduction, the authors are interested in designing universal schedules for delivering a series of data updates from a single information source over a broadcast channel.
• The length of the interval is chosen without loss of generality, as their techniques (with an appropriate scaling) can be applied for time intervals of an arbitrary length.
• In a random schedule, the interleaving times are random variables.
• The authors define the client’s waiting time as the length of the time interval between t and the beginning of the transmission of the next packet.

B. Adversarial Clients

• The authors goal is to design schedules that perform well for any client, regardless of the viewed history of the schedule.
• This allows to simplify the analysis, without any loss of generality.
• The authors say that an adversarial client is !-adaptive if its actions at time t depend only on the history V 2 V(S; t !) of the schedule S at time t !.
• Note also that the expression sup0 t<!EWT (S; t) bounds the maximum waiting time for requests placed at times t < !, when the client does not have any knowledge of the schedule’s history.
• For a real random variable X , the authors denote by FX(t) = Pr[X < t] its cumulative distribution function, by X = 1 0 (1 FX(x))dx its expectation, and by fX its probability density function (when exists).

C. Transmission Rate

• The transmission rate of a schedule S = fX1;X2; . . .g is defined to be the expected fraction of the time the channel is in use.
• Definition 4 (Transmission Rate r): Let S = fX1;X2; . . .g be a random schedule and let Rt be the expected number of packets sent in S up to time t.
• The transmission rate of S is defined to be r = lim t!1.

III. UNIVERSAL SCHEDULES FOR ! = 1

• The authors study the design of optimal universal schedules for adversarial clients with a degree of adaptivity of one time unit (i.e., ! = 1).
• The authors schedules are defined by a single random variable X .
• That is, all interleaving timesXi in their schedules are independent and have the same distribution as X .

A. Optimal Schedules

• Each random schedule in the family the authors present is associated with a parameter , which is equal to the expected value ofX , i.e., = E[X].
• 2) If event A happens, the client needs to wait X1j A t time units until the transmission of the first packet begins.

C. Optimal Schedules for Large Rates

• This corresponds to the situation where excessive slackness results in a high waiting time.
• Second, notice that the tradeoff curve the authors present has a knee phenomenon.
• That is, increasing the rate beyond 2 1+ p 2 ' 0:82 has little effect on the worst case ex- pected waiting time until r reached a value of approximately 0:95.

IV. HIGHLY ADAPTIVE ADVERSARIAL CLIENTS

• The authors discuss schedules for adversarial clients with very small degree of adaptivity !.
• For this extreme case the authors give a tight analysis of W (S; !).
• Let n be a sufficiently large integer, also known as Theorem 6.
• For a client’s request at time t 1 n , Lemma 1 implies that the expected waiting time is bounded by 1 1 4n (for n large enough).

V. CONCLUSION

• The authors defined the notion of universal schedules that guarantee low waiting time for any client, regardless of its access pattern.
• The authors studied the performance characteristics and the design of universal broadcast schedules, focusing on adversarial clients whose adaptivity is bounded by one time unit.
• Moreover, the authors have shown that this is the best possible schedule.
• For larger values of r the authors have presented a tight analysis of the tradeoff between the transmission rate and the minimum worst case expected waiting time.
• For smaller values of !, these expressions do not differ significantly from those appearing in this work and an analysis of similar nature may be performed.

Full text

Rather-high-frequency sound scattering by swimbladdered fish
Kenneth G. Foote
Institute of Marine Research, 5011 Bergen, Norway
(Received 18 December 1984; accepted for publication 15 April 1985)
A new model describes acoustic scattering by swimbladdered fish of lengths from at least 8 to 36
wavelengths. It represents a fish by an ideal pressure-release surface having the exact size and
shape as the swimbladder. The backscattering cross section, or target strength, is computed by
means of the Kirchhoffapproximation. To test the model, predictions of target strengths based on
swimbladder morphometries of 15 gadoids of lengths from 31.5 to 44.5 cm are compared with
conventional target strength measurements on the same, surface-adapted fish, anesthetized
before acoustic measurement, and shock-frozen immediately afterwards. Details are given of the
swimbladder morphometry. In essence, this consists of slicing the frozen fish with a microtome,
photographing the exposed swimbladder cross sections, digitizing the contours, and triangulating
the surface between pairs of contours on adjacent, parallel planes. Theory and experiment are
compared through the dorsal and ventral aspect target strength functions, their averages, and
simulated probability density functions.
PACS numbers: 43.20.Fn, 43.30. Dr, 43.80.Jz
INTRODUCTION
The backscattering cross section, or target strength, of
fish is a pivotal quantity in the acoustic assessment of fish
abundance. • As a result, it has been the object of numerous
and diverse studies. These have included in situ measure-
ments, controlled measurements on tethered or encaged fish,
and modeling. Without repeating Midttun's comprehensive
review, • the measurement side of target strength determina-
tion has had its successes, but generally suffers from the par-
ticularity of the measurement situation when the fish target
is known, and unknown behavioral effects when the fish tar-
get is constrained.
The alternative to fish measurement is modeling. There
apparently has been substantial interest in this, divided
between measurements on artificial models 2-•] and theoreti-
cal computations. •2-24 The number of distinct models has
been small, however, undoubtedly owing to the dominance,
if not preeminence, of the swimbladder in scattering by
fish. 11,25-27 While the individual models serve their authors'
original purposes, they are unsatisfactory for a priori deter-
minations of the target strength of commercially important
fish at the usual ultrasonic survey frequencies, above about
30 kHz.
Specifically, models based on simple geometric shapes,
e.g., sphere, prolate ellipsoid, and finite circular cylinder, are
inadequate, if only because such shapes are symmetrical
with respect to the horizontal or transverse plane, while the
general swimbladder is not. 28-38 The consequence of asym-
metry in swimbladder form is often observed in the signifi-
cant asymmetry of dorsal and ventral aspect target strength
functions of the same fish. 3%4ø Admittedly, many uses of
simple shapes have been directed to resonant or other low-
frequency scattering where the target strength is indepen-
dent of fish orientation. A rare success of a simple-shape
model at high frequencies is Kalikhman's computation of
backscattering by a 27-cm herring {Clupea harengus) at 30
kHz. 18, 22
Models based on arrays of point scatterers, as in Refs. 21
and 24, while apparently successful in simulations of echo
statistics, including the representation of behavioral effects,
are relative. As such, they depend on a posteriori knowledge
of target strength as a function of orientation for determina-
tion of the point-scattering strengths. Thus they cannot, in
themselves, predict absolute magnitudes of target strength.
For this, recourse to actual measurement is necessary.
Composite and whole-fish-body models have also been
proposed to describe scattering by swimbladdered fish.
These, together with measurements on artificial models, are
disregarded for being unnecessarily complicated.
It is the present aim to introduce a simple model for
scattering by swimbladdered fish which, at the least, is appli-
cable for rather high frequencies, with fish lengths in the
nominal range from 8 to 36 acoustic wavelengths. The new
model resembles the basic bubble-type model in representing
fish entirely by the swimbladder, which is equated to an ideal
pressure-release surface. Unlike its predecessors, however,
the swimbladder form is not approximated by a simple
shape; rather, it is assumed to have the very size and shape of
the organ, as morphometrically determined and mathemat-
ically represented by a finite-element triangulation. In keep-
ing with the intent of the model, the scattering amplitude is
computed by means of the Kirchhoff approximation, hence
without the effects of diffraction.
To demonstrate the power of the model, predictions are
made of the dorsal and ventral aspect target strength func-
tions of 15 fish, at each of four frequencies, from the respec-
tive swimbladder morphometries. These are compared with
conventional measurements performed on the same fish,
anesthetized and tethered, before shock-freezing and even-
tual anatomical measurement. The principal criteria used to
compare theory and experiment are based on the two expres-
sions of target strength most widely used in fisheries acous-
tics: the average and the probability density function. In ad-
dition to considering the computational results, limitations
688 J. Acoust. Soc. Am. 78 (2), August 1985 0001-4966/85/080688-13500.80 ¸ 1985 Acoustical Society of America 688
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of the model and challenges for it are discussed. The findings
or achievements of the model are summarized in the Conclu-
sions.
I. MODEL
A swimbladdered fish is represented entirely by its
swimbladde/'. This is assumed to be ideally pressure-releas-
ing. Sound scattering is thus described by the solution of the
scalar wave equation exterior to a soft shape, or closed sur-
face with a homogeneous Dirichlet boundary condition.
Solution is achieved by the Kirchhoff approximation. 41
Accordingly, the pressure field on the scattering surface is
equated to that which would obtain on the front side of the
same surface if there were no diffraction. The solution for the
monochromatic backscattering amplitude F due to plane-
wave ensonification of the surface $ is
F =/•, --1 • exp(2ik ß r)•(•' ß •)cos(•' ß ?)dS, (1)
where A is the acoustic wavelength, k is the wave vector in
the source or backscattering direction, • = k/k, r is the posi-
tion vector of the surface element with infinitesimal area dS,
and •F{x} is the Heaviside gtep function with values 1 for
x > 0, « for x = 0, and 0 for x < 0. Related expressions are
found in Ref. 42 for a surface of arbitrary reflectivity, and in
Refs. 43-45 for rigid surfaces.
For the present application to fish scattering, the ob-
servable quantity corresponding to F is the backscattering
cross section. Given the customary use of finite-signal wave-
forms and finite-bandwidth receivers, the operational defini-
tion of backscattering cross section •r is appropriate; name-
ly, 46
ISF 12 12 do, (2)
where S is the signal spectrum, H is the receiver frequency
response function, and co is the angular frequency, co = ck,
where c is the medium sound speed.
In the case of narrow-band S and H, or long, simple
signals and narrow-band receivers, Eq. (2) can be significant-
ly reduced. The result is the usual monochromatic back-
scattering cross section
4rrl F [2. (3)
For convenience, the backscattering cross section is also
expressed through its logarithmic measure, the target
strength. This is defined in the traditional manner, 47 al-
though with use of S1 units,
TS = 10 log cr/4rr. (4)
The target strength of the idealized perfectly reflecting
sphere of 2-m radius is thus 0 dB.
II. MATERIALS AND METHODS
The source of data on the swimbladder form was identi-
cal to that of the comparative acoustic measurements. This
was 13 pollack {Pollachiuspollachius) and two saithe {Polla-
chius oirens), part of a special sample of 20 fish'that were, in
turn, measured acoustically and then shock-frozen on 24
July 1980, during a one-day digression from the larger exper-
iment described in Ref. 48.
The acoustic measurements were performed in the in-
variable manner of the larger experiment, resembling earlier
measurements too. 39'49-51 Individual, surface-adapted fish
were anesthetized, tethered in a suspension system, and mea-
sured, under tilting, at each of four frequencies, in both dor-
sal and ventral aspects. Earlier suspicions that similar pol-
lack were not adapted to the shallow surface layer 4s are no
longer held by this author, who accepts Ona's attribution of
the extreme orientations of the encaged swimming fish to the
smallness of the cage in lateral extent. •2'53
The acoustic frequencies and pulse durations of the echo
sounder signal were measured a number of times during the
larger experiment. The result of interpolating these for the
mentioned date is shown in Table I.
Following the acoustic measurements, the basic biologi-
cal characteristics of length and weight were measured. The
still-anesthetized fish were then grasped by tongs at snout
and tail, held tautly in normal, extended, horizontal posture,
and totally immersed in a bath of alcohol maintained at a
temperature of - 50 øC by the addition of dry ice. It was
held in the same posture for the several minutes required for
thorough freezing. The fish were then tagged and stored in
an insulated box containing dry ice. Upon completing the
particular measurement series, the 20 fish were transferred
to a large freezer at the author's institute for long-term stor-
age at - 35 øC.
In February 1981, the fish were removed from the
freezer for anatomical measurement. Prior to slicing with a
microtome, with nominal 1-/•m accuracy, the fish were en-
cased in rectangular blocks of carboxymethyl cellulose
(CMC). This was accomplished by immersing the frozen fish
in a solution of CMC and water held in the microtome's
freezing frame, followed immediately by freezing of the en-
tire system by immersion in a bath of alcohol and dry ice
maintained at a temperature of - 70 øC.
The fish-encasing block was trimmed in even thick-
nesses of 200/•m until the fish was exposed. It was then
sliced in even thicknesses of 100/•m, enabling rapid changes
in swimbladder form to be detected. The slicing was per-
formed in the sagittal plane to minimize the amount of pho-
tography. The swimbladder cross sections were photo-
graphed at intervals varying from 200 to 1400/•m in order to
allow significant detail to be registered for the eventual re-
construction of the three-dimensional surface.
In the course of the slicing, four of the 20 specimens
were irremediably damaged. A fifth specimen was lost to
further work by loss of the alignment reference under pho-
tography.
TABLE I. Center frequencies and durations of pulses transmitted by four
Simrad echo sounders.
Echo Center frequency (kHz) Pulse duration (ms)
sounder Nominal Measured Nominal Measured
EK-38 38.0 38.1 0.6 0.64
EK-50 49.5 49.6 0.6 0.57
EY-M 70.0 68.4 0.6 0.60
EK-120 120.0 120.4 0.6 0.68
689 J. Acoust. Soc. Am., Vol. 78, No. 2, August 1985 Kenneth G. Foote: Scattering bY swimbladdered fish 689
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Table II. Biology of the 15 fish specimens of the investigation. The swim-
bladder data are derived from the triangulations.
Swimbladder
Fish Length Mass Surface Volume
no. Species {cm) {g) area (cm 2 ) (cm 3)
201 Pollack 31.5 195 39.03 6.88
202 Pollack 44.0 533 69.47 16.37
204 Pollack 35.5 321 52.57 10.16
205 Pollack 39.0 380 56.54 11.31
206 Pollack 35.0 287 35.46 7.83
207 Pollack 44.5 635 89.35 19.76
209 Saithe 38.5 385 48.37 10.12
213 Pollack 34.5 259 45.06 7.18
214 Pollack 39.0 406 54.21 9.82
215 ß Pollack 37.0 332 40.89 8.27
216 Pollack 36.5 343 48.79 10.46
217 Pollack 34.5 253 39.30 6.61
218 Pollack 32.5 257 35.26 6.28
219 Pollack 35.5 292 40.10 8.04
220 Saithe 38.0 406 53.87 10.57
CONVEX
NON CONVEX STAR-SHAPED
SIIIPLY- REENTRANT
llULTIPLY-REENTRANT
DISCONNECTED
The basic characteristics of the surviving 15 fish speci-
mens, the subjects of the present investigation, are shown in
Table II. The fish numbers refer to the order of acoustic
measurement in the more extensive target strength measure-
ment series performed during the experiment described in
Ref. 48.
All Of the biological measurements were performed by
Egil Ona, who devised the shock-freezing technique, in addi-
tion to another technique, for morphometric studies of the
swimbladder. 52
III. SWIMBLADDER-SURFACE TRIANGULATION
Representation of the swimbladder surface for evalua-
tion of Eq. (1), hence for realization of the model, is achieved
through a triangulation. This is performed independently
for each pair of adjacent contours. When combined with
triangulations of the end surfaces, the surface mapping is
complete.
A. Manual procedures ,
The triangulation commences with a digitization of
each sagittal cross section. This is conveniently effected by
tracing the outline, or contour, of the inner swimbladder
wall boundary on photographic print with a cursor connect-
ed to a digital computer. Prior enhancement or marking of
the contour by a knowledgeable fisheries biologist is useful, if
not necessary, to avoid problems of interpretation. Because
of the general complexity in form of gadoid swimbladders,
often characterized by a lobed structure due to lateral pro-
trusions of the swimbladder between ribs, the apparent con-
tour may not be convex. In fact, it may not be star-shaped,
and may even be disconnected. Several contour types are
illustrated in Fig. 1.
The next task in the triangulation procedure is pairing of
connected contours on adjacent, parallel planes. This may
entail cutting connected contours for matching with discon-
nected contours on the next plane, especially if lying out-
wards from the medial plane of the fish. It may also involve
FIG. 1. Illustration of contour types, derived from actual swimbladder sa-
gittal cross sections.
cutting of severely pinched connected contours to avoid un-
natural treatment as reentrants by the automatic algorithm
which effects the triangulation.
Preparation of the digitized contours for triangulation is
completed by ensuring that the points are arranged with the
same rotation sense, or handedness. For definiteness, this is
chosen to be counterclockwise.
B. Automatic algorithm
The surface between pairs of connected contours on ad-
jacent, parallel planes is now triangulated by means of the
following algorithm. The numbers of points on the two digi-
tized contours are compared. That contour with the smaller
number is called the "lesser." The points of each contour are
referred to the planar centroid of the lesser contour, and then
mapped onto the respective unit circle. This occurs by radial
projection where possible. Reentrants present the single ex-
ception; their points are mapped consecutively and evenly
onto the arc between those radially projected, bounding
points that preserve the counterclockwise order. The degen-
erate case of reentry, in which successive points have the
same radial projection, is treated as an ordinary case of
reentry. The mapping of points of a simple connected con-
tour onto the unit circle is thus homeomorphic.
The unit circle corresponding to the "greater" contour,
the "outer" circle, provides the frame of reference for the
next operation. In this, points are sought on the "inner" cir-
cle with angular locations intermediate to those of each pair
of successive points on the outer circle. Wherever there is not
at least one intermediate point on the inner circle, one is
created by interpolation.
Triangulation of the cylindrical surface between the two
unit circles, which are aligned and on parallel planes by de-
690 J. Acoust. Soc. Am., Vol. 78, No. 2, August 1985 Kenneth G. Foote: Scattering by swimbladdered fish 690
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FIG. 2. Illustration of triangulation by angular affinity. Solid circles repre-
sent points obtained by the standard homeomorphic mapping. Open circles
represent interpolated points.
finition, is now accomplished. Points on alternate circles are
joined on the basis of angular affinity, as illustrated in Fig. 2.
The original coordinates of the projected points, now
become the vertices of triangles, are restored. The coordi-
nates of interpolated points are established as the medians of
their immediate neighbors on the same, lesser contour.
The special case of end surfaces is treated in the follow-
ing manner. All contours lacking appropriate mates on adja-
cent planes are identified. Single, isolated points, called
"endpoints," are added to planes midway between the con-
tour-containing plane and nearest slice. Multiply reentrant
or severely pinched, singly reentrant contours are cut to
form a set of more simply connected contours. The planar
position of each endpoint is defined by the planar centroid of
the contour needing pairing. The surface between endpoint
and contour is triangulated by a simple connection of con-
tour points to the endpoint, which serves as the common
vertex of a system of triangles whose bases are the segments
between successive points •!ong +• -•,-*•,,,•
The algorithm concludes with determination of the me-
dian position vector, outward normal unit vector, and area
of each triangle.
C. Checking routines
G. iven the complexity of most contours and the desire to
effect the triangulations by an automatic algorithm, it is con-
venient to employ checking routines during the data analy-
sis. Examples are enumerated.
{1} Plotting of the cross sections exactly as digitized.
Hard-copy plots may be compared directly with the raw ma-
terial in the form of photographic prints.
{2) Computation of distances between adjacent points on
each contour. Summary of the distances in a histogram for
all contours of the same swimbladder enables the fineness of
digitization to be confirmed. At the same time, the unique-
ness of each digitized point can be established, thereby
avoiding having to do the same in the automatic algorithm.
{3) Superimposed plotting of matched pairs of contours.
This routine permits a final confirmation of the basic materi-
al, particularly after its undoubted transformation by align-
ment and scaling operations following the initial digitization
and plotting.
{4} Statistical analysis of triangle areas. Summary of the
areas of the finite-element triangles in a histogram is conven-
ient, although not foolproof, for confirming the working of
the algorithm. Extreme, unusual, or new geometries may
very well produce individual elements with large areas. The
same routine may confirm the absence of element areas with
negative values. These may arise in the automatic algorithm
with very small values, consistent with underflow, or lack of
precision in the floating-point operations. At the time of
their detection, such negative values should be recorded, for
later inspection, and replaced by nulls.
(5} Computation of the total swimbladder surface area
and volume. These quantities are useful for comparison with
gross estimates of the same when derived from the maximal
swimbladder dimensions and assumption ofprolate ellipsoi-
dal form.
D. Data statistics
The digitization was performed with equipment with a
nominal positioning accuracy of 10/zm. This was reckoned
to exceed the realized precision by a factor of 10 to 20.
Lengths in the resulting triangulation were therefore ex-
pressed to the nearest 0.01 cm.
The listed checking routines were exercised for each
swimbladder triangulation. The result of combining all 15
histograms of digitization-segment lengths was a slightly
skewed, nearly normal distribution with a mean of 1.25 mm
and a standard deviation of 0.38 mm. Analysis of the element
areas revealed an approximately exponential distribution
with mean of 0.72 mm 2 .
Gross swimbladder dimensions are presented in Table
III. In estimating the swimbladder surface area and volume,
the height and width of the swimbladder were averaged, and
the overall form assumed to be that of a prolate ellipsoid.
'• .... '• ...... :'•' '•'^" • .... •^-:"• • .... •'• tfia gulatio
'-•,•'--v,- ..... a n ns,
also shown in Table II, indicates the expected greater com-
plexity of the actual form.
IV. RESULTS
The fundamental computational quantity is the back-
scattering cross section, or target strength, as a function of
the ensonification conditions and fish orientation. This has
been determined systematically for each of the 15 fish speci-
mens whose swimbladder surfaces were triangulated. Com-
putations were performed in accordance with the simple fi-
nite-element, or numerical, realization of Eq. {1). The
difference in wideband and monochromatic target strengths
was found to be negligible for the several fish examined com-
paratively in this way; hence the simpler monochromatic
formula, Eq. {3}, was uniformly used in the basic theoretical
computations.
An example of the basic computations performed for
each fish is shown in Fig. 3 for fish No. 201, a 31.5-cm pol-
lack. The tilt angle is defined as the angle made by the center-
line, or imaginary line running from the root of the tail to the
tip of the upper jaw, with the horizontal plane. The sign
convention is that positive angles denote head-up orienta-
tions; negative angles, head-down orientations. Presented
with the theoretical target strength functions are the actual
measured functions. The respective correlation coefficients
are shown. These are based on the backscattering cross sec-
tion.
The result of combining the corresponding target
strength functions of each of the 15 fish is shown in Fig. 4. As
with the correlation coefficients, the domain of combination
is the backscattering cross section. The arithmetic average of
the backscattering cross section is formed, therefore, before
691
J. Acoust. Soc. Am., Vol. 78, No. 2, August 1985
Kenneth G. Foote: Scattering by swimbladdered fish
691
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TABLE lll. Gross dimensions, surface area, and volume of the swimbladder when represented as a prolate ellipsoid. Corresponding triangulation-derived
measures are presented together with the relative error.
Fish Maximal swimbladder Swimbladder surface Swimbladder volume
Fish length dimensions (cm) area (cm 2) (cm 3
no. (cm) Length Height Width Ellip. Triang. Error Ellip. Triang. Error
201 31.5 10.58 0.98 1.44 31.78 39.03 -- 0.19 7.82 6.88 0.14
218 32.5 11.00 0.98 1.36 31.92 35.26 -- 0.10 7.68 6.28 0.22
217 34.5 10.93 0.99 1.72 36.80 39.30 -- 0.06 9.75 6.61 0.47
213 34.5 9.89 1.05 1.64 33.09 45.06 -- 0.27 8.92 7.18 0.24
206 35.0 8.74 1.34 1.76 33.88 35.46 -- 0.04 10.79 7.83 0.38
219 35.5 10.98 1.04 1.46 34.07 40.10 -- 0.15 8.73 8.04 0.09
204 35.5 12.44 1.18 1.66 43.85 52.57 -- 0.17 12.76 10.16 0.26
216 36.5 11.99 1.17 1.80 44.24 48.79 -- 0.09 13.22 10.46 0.26
215 37.0 10.71 1.03 1.54 34.18 40.89 -- 0.16 8.90 8.27 0.08
220 38.0 13.27 1.18 1.68 47.07 53.87 -- 0.13 13.77 10.57 0.30
209 38.5 11.31 1.28 1.74 42.48 48.37 -- 0.12 13.19 10.12 0.30
205 39.0 13.93 1.08 1.78 49.39 56.54 -- 0.13 14.02 11.31 0.24
214 39.0 12.71 1.30 1.64 46.38 54.21 -- 0.14 14.19 9.82 0.44
202 44.0 13.00 1.40 2.62 65.15 69.47 -- 0.06 24.97 16.37 0.52
207 44.5 16.39 1.54 2.12 74.43 89.35 -- 0.17 28.02 19.76 0.42
/ OORSRL
-50 t' ' ' z '
VENTRRL
38.1 _
.,/.••• 0=•.7•6 _
I , "',:' .•'•
n- -30
-40
-50
-60
VENTRRL /
- 49.6 -I
' ' ",, C=0.79P /
• • , , W 'A• ,•'1
VENTRRL /
- ,.., 68.4 _•
.. i ½ _ ,•\• C=I•. 774 t
FIG. 3. Target strength functions of fish No.
201, a 31.5-cm pollack, distinguished by aspect
and frequency in kilohertz. The correlation co-
efficient C is given for corresponding computed
and measured functions, drawn respectively
with solid and dashed lines.
692 J. Acoust. Soc. Am., Vol. 78, No. 2, August 1985 Kenneth G. Foote: Scattering by swimbladdered fish 692
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