Rapp. P.-v. Réun. Cons. int. Explor. Mer, 189: 35 3-36 2. 1990
Sound scattering by zooplankton
Timothy K. Stanton
Stanton. Timothy K. 1990. Sound scattering by zooplankton. - Rapp. R-v. Réun.
Cons. int. Explor. Mer, 189: 353-362.
Accurately predicting abundances of zooplankton using sonar systems requires
knowledge of the acoustical scattering properties of the animals. Im portant to the
scattering properties are the size, shape, composition, and orientation of the animal.
Recently, I have derived an analytical expression describing the scattering of sound
by deformed cylinders of finite length and demonstrated reasonable-to-excellent
agreement between the solution and scattering data involving Durai (a metal alloy
similar to aluminum), shrimp, euphausiids, and fish. The solution was derived to
apply to elongated objects and was made general enough to allow for (1) bending of
the cylinder axis, (2) tapering of its cross-section, and (3) elastic materials and
composites such as an elastic shell filled with fluid. To within certain limitations,
orientation of the animal can be taken into account. In addition, in order to facilitate
calculations of backscattering strengths, I have derived simple approximate formulas
for the sphere, prolate spheroid, straight finite cylinder, and uniformly bent finite
cylinder. In this paper, I review the methodology and results of my recent studies.
The results suggest the bent finite-length fluid cylinder as a promising model to
describe the scattering of sound by elongated animals.
Timothy K. Stanton: D epartment o f Geology and Geophysics, University o f W iscon­
sin, 1215 West Dayton Street, M adison, Wisconsin 53706, USA. Present address:
W oods Hole Oceanographic Institution, W oods Hole, Massachusetts 02543, USA.
Introduction
In order to use sonars to estimate abundance of marine
organisms such as fish and zooplankton, accurate acous­
tic scattering models must be employed. Describing the
scattering of sound by organisms in an exact manner is
extremely difficult if not impossible. The solutions are
usually approximate, empirical, or numerical. For ex­
ample, Love (1971, 1977) has published empirical for­
mulas describing backscattering cross-sections of fish
for a wide range of L/X where L is the length and X is the
acoustic wavelength. Foote (1985) has successfully nu­
merically evaluated the Helmholtz-Kirchhoff integral
(with the Kirchhoff approximation) to describe scatter­
ing from the complex swimbladders of some fish.
To date, attempts to describe scattering by elongated
zooplankton such as shrimp or euphausiids have in­
volved either the Anderson fluid-sphere model (A n­
derson, 1950) or a Love equation (Penrose and Kaye,
1979). A Love equation would, at best, describe highfrequency scattering ( L A > 1 ) . Greenlaw (1977) and
Richter (1985) have demonstrated that for ka > 1 where
k is the acoustic wavenumber and a is the equivalent
spherical radius of the animals, their backscatter data
rise above the theoretical predictions. It is not surpris­
ing that a fluid-sphere model would not adequately
23
Rapports et Procès-Verbaux
describe the scattering from elongated animals (length/
width ~ 10).
Recently, I have derived a general analytical expres­
sion describing the scattering of an incident plane wave
of sound by arbitrarily deformed cylinders of finite
length (Stanton, 1989a). The theory, involving a modal
series solution, is general enough so that the cylinders
can be arbitrarily bent (radius of curvature can vary
along the axis) and the cross-section can vary along the
axis provided the bends and rate of change of tapering
are gradual. The composition can be fluid, elastic, or a
combination such as an elastic shell filled with fluid. The
direction of the incident wave, direction of receiver
position vector, and orientation of the cylinder can all
be arbitrary within a limited range of angles so that
normal or near-normal incidence is maintained to all
tangents of the bent axis (i.e., at end-on incidence, the
solution breaks down). This work was preceded by
Stanton (1988a) and Stanton (1988b) where specific so­
lutions for the straight fluid and straight elastic finite
cylinders were derived and compared with existing data.
In addition to the general modal series solution, I
have derived simple approximate formulas describing
the backscattering of sound by the sphere, prolate sphe­
roid, straight finite cylinder, and uniformly bent finite
cylinder. This was done to help facilitate the otherwise
353
time-consuming effort in scattering predictions. The
work represents a refinement and generalization of
Johnson's (1977) original development of the “highpass” scattering model of a fluid sphere. The “highpass” model is, in essence, a “sm oothed” version of the
exact modal series solution when compared on a plot of
target strength versus size or frequency. My refinement
of Johnson’s model involves using a more physical basis
for the high-frequency portion of the solution while the
generalization extends the work to the o ther shapes
mentioned as well as taking into account effects due to
irregularities of the shape.
I have compared the straight elastic cylinder model
with scatter data involving straight cylinders of Durai
from A ndreeva and Samovol’kin (1976) and showed
excellent agreement (Stanton, 1988b). In Stanton
(1988a) 1 compared the straight fluid cylinder model to
shrimp backscatter data presented by Greenlaw (1977)
and showed that in the ka > 1 region, where scatter data
would rise above the predictions of the fluid-sphere
model (Greenlaw, 1977; Richter, 1985), this cylinder
model describes the upward trend and the period of
oscillations in this geometric scattering region. While
the functional form of the solution adhered well to the
data, the absolute value did not and required an effec­
tive “acoustic length” and “acoustic radius” of the ani­
mal to be used. This is because the animal was bent and
tapered and altered the echo accordingly. The scatter­
ing was possibly dominated by a short straight section
while the rest of the body was bent away from the sonar.
These results motivated the development of the general
theory where bending of the cylinder axis and tapering
of its cross-section can be taken into account (Stanton,
1989a). As a result, the comparison between the bentcylinder calculations and euphausiid scattering data
(again, from Greenlaw, 1977) are superior to the previ­
ous efforts. The actual measured lengths of the animals
were used in the model.
In this paper, I review the methodology and results of
my recent research. For more detail, I refer the reader
to my articles referenced above. These results suggest
the bent finite-length fluid cylinder as a promising
model to describe the scattering of sound by elongated
animals. The model can be used to improve abundance
estimates by conventional single-frequency sonar sys­
tems as well as estimates of size-frequency distributions
by inversions of multifrequency sonars (Holliday, 1977;
Pieper and Holliday, 1984; Holliday and Pieper, 1984).
where P0 is the pressure amplitude of the incident plane
wave, i = V ( —1), r is the distance from the object to the
field measurement point, k is the acoustic wavenumber
of the surrounding medium (water) ( = 2j i IX where X is
the acoustic wavelength, f(G) is the scattering ampli­
tude, and Q represents the spherical angles. The backscattering cross-section, as defined by Clay and Medwin
(1977) is
o hs »
| f ( ß
j | 2
( 2 )
where ß bs is Q evaluated in the backscatter direction.
The target strength is defined in terms of obs as
TS = 10 1ogobs.
(3)
1.1. G e n e r a l m o d a l so lu tio n
A general expression for the farfield scattered pressure
due to a deformed cylinder of finite length is derived in
Stanton (1989a). The cylinder, illustrated in Figure 1, is
deformed insofar as its axis can be bent into any irreg­
ular shape and its cross-sectional radius may vary along
the axis provided the bends and tapering are slowly
varying. The cylinder can be of any composition: fluid,
elastic, or a combination (e.g., elastic shell with fluid
interior). Furtherm ore, the incident wave, (farfield) re­
ceiver position vector, and orientation of the cylinder
are of arbitrary directions within a limited range of
angles so that normal or near-normal incidence is main-
outer boundary
of deformed
cylinder
pos
origin
cylinder axis
(fin ite length)
1. Theory
The farfield scattered pressure, pscat, due to any finite­
sized object can be written in the form
gikr
P sc,,
354
=
P .
y
f ( 0 )
( 1 )
n
in cid en t plane wave
Figure 1. Arbitrarily deformed finite cylinder and general
bistatic sonar.
tained to all tangents of the bent axis (i.e., the solution
breaks down at end-on incidence).
The scattered pressure was derived from an integral
of the volume flow per unit length of the scattered field:
Pscat = pd V (\ n r /) /rp„i (\ m=0
i x co sm <p)/
e i k W ? i - f r) .fpo, j d f
i
(4 )
1.2.
S tra ig h t fluid cylind er
In Stanton (1988a) I evaluated Equation (4) for the case
in which the cylinder is straight and composed of a
homogeneous fluid material (i.e., it does not support
shear waves). Because of its simple geometrical shape
and uniformity, the modal series inside the parentheses
in Equation (4) is constant with respect to the integra­
tion, and the integration can be performed analytically
to give
where
/ i L \ sin(A) “ e m cosm<p
bm = modal coefficients for straight-cylinder case eval­
uated at rpos and depends on boundary conditions
(see later section for examples of bm),
P„ = pressure amplitude of the incident plane wave,
k = acoustic wavenumber of surrounding medium
(water),
r = distance between cylinder and receiver,
V s = vector from origin to integration point on axis of
cylinder,
rpos = unit vector in direction of integration point on
axis of cylinder,
r, = unit vector in direction of incident plane wave,
rr = unit vector in direction of receiver,
qj = cos“ 1 (ri±.rrl) = azimuthal angle of tangent to
cylinder axis,
_
11
fj
-
(fj-r.a n )
fta n
l? i
-
(M .a n )
î
l" r
(^ r-^ ta n )
^t a n
k r
( ^ r '^ ta n )
^”tanI
j
ï ) —
In general, the integral in Equations (4) and (5) must be
evaluated numerically. Only in the simplest of cases
such as when the cylinder is straight can it be integrated
analytically. The coefficients bmdepend on the composi­
tion of the cylinder, whether it is entirely fluid, elastic,
or perhaps a combination such as having an elastic outer
shell with a fluid core. The coefficients for fluid cylin­
ders are given in Stanton (1988a), elastic originally in
Faran (1951), and later reviewed for this work in Stan­
ton (1988b), and a technique for deriving coefficients
for any composition profile is illustrated in G oodman
and Stern (1962).
s
-t
t s t
where
L
= length of cylinder,
A
= - k L ( f - f r) . r c,
1
rc = unit vector indirection of cylinder axis,
e m = Neumann number, e 0= 1 and e m>0 = 2.
J ' m(K*a)Nm(Ka)
N 'm(Ka)
Jm(K *a)J'm(Ka)
gh J ' m(Ka)
Cm = ------------------------------------------J'm(K*a)Jm(Ka)
Jm(K *a)J'm(Ka)
’
rtan = unit vector of tangent to cylinder axis. All wavenumbers in bm are multiplied by |r( x rlan| to account for
incidence not normal to the tangent.
From Equation (3), the target strength is
23*
« M
g
h
K
K*
gh
= q */q ,
— c*/c,
= k sin 0,
= k* sinØ.
0 is the angle between the direction of the incident
plane wave and the axis of the cylinder, J and N are the
Bessel functions of the first and second kind, respec­
tively, of order m, and the primes indicate derivatives
with respect to their arguments, g, c, and k are the mass
density, compressional speed of sound, and acoustic
wavenumber in the surrounding fluid, and g*, c*, and
k* are those corresponding properties of the cylinder
material. The solution in Equation (6) appears to be
valid for all ka, L > a , and near broadside incidence
(within —30° of normal incidence).
For backscattering geometries, cp = jt and A = kL
cos0. Inserting these quantities into Equations (2) and
(6) gives
/ LY
w
sin(kL cos 0)
kL cos 0
2
^
e m( - l ) m
“ o 1 + iCm
For broadside incidence (0 = 90°) the directionality
function above becomes unity:
355
sin(kL cosØ)
kL cos 0
(8 )
• I, ( 0 —>90°).
While these equations are relatively complex, some of
their limiting forms are quite simple. The low and high
frequency limits are given in Table 1.
must be taken into account. The coefficients bm in
Equation (4) are far more complex than for the fluid
cylinder. In Stanton (1988b), I evaluated Equation (4)
for the straight elastic cylinder using coefficients origi­
nally derived by Faran (1951). The form of the solution
was similar to Equation (6).
1.4. U n if o rm ly b e n t c y lin d er
1.3.
E lastic cy lin d e r
When an elastic material is involved, conversions b e­
tween compressional and shear waves at the boundary
A geometry that would resemble an elongated marine
animal better than a sphere or straight cylinder would
be a uniformly bent cylinder (Fig. 2). It is important to
Table 1. High-pass models and associated limits to modal series solutions for various objects. Scattering geometries illustrated in
Figure 3 for sphere, straight cylinder, and bent cylinder; broadside incidence for prolate spheroid (Stanton, 1989b).
^bs
(where TS = 10 logobs)
Sphere.............................................
ka 1
(fluid)
ka > 1
(rigid/fixed)
2„ u
■>
az(ka) a i.
i
k
?
All ka
High-pass model
a2(ka)4 < 4 G
—
1 + [4(ka)4
Prolate s p h e ro id ............................
J L2(ka)4 a 2c
Straight c y lin d e r..........................
, ,
„ , ,
J L2(Ka)4 a 2c s-
Bent cylinder.................................
j L2(ka)4 a 2c H 2
^ L2
,
,
,
L'-(Ka) s2
,
J pca
------’ \ ^
^
—
1 + [(¥)(ka)4a 2c]/[/?2f l
J L2(Ka)4 a 2 s2G
-----1 + [it(Ka)3 a 2J/[/? /•]
i L2(ka)4 a 2 H 2G
--------- -------------------------------1 + [L2(ka)4 a 2c H 2] / ^ / ? 2^
Definitions:
1 — gh2
cx_. = ---------3gh-
1 —g
1---------- ,
1 + 2g
1 - gh1 —g
a m. = ------ :— H------------ ,
2gh1 + g
gh - 1
n = ---------gh + 1
g = (density of body material)/(density of surrounding fluid),
h = (compressional speed of sound in body material)/(compressional speed of sound in surrounding fluid),
H = Ï + 2 (ec/L) sin[(gc/L)->],
a
= spherical radius (sphere), length of semi-m inor axis (prolate spheroid), or cylindrical radius (straight and bent cylinder),
L = total length of prolate spheroid and straight cylinder; arc length of bent cylinder,
pc = radius of curvature of axis of bent cylinder,
k = acoustic wavenumber ( = 2nIX where X is acoustic wavelength),
K = k sinO (straight cylinder),
0 = angle between direction of incident wave and axis of cylinder,
sin(kL cosØ)
s = -------------------kL cos 0
F = 2 F,(ka)Yi,
'
where
( = 1 at broadside incidence),
G = [ ] d + Aje"B'|ka- (ka)°Jl2),
i
Fh Yj, A., Bj, (ka)0,j are constants
( - 1 < Aj < 0 gives null, Aj > 0 gives peak),
(ka)0j = position of j th peak or null.
In all cases, kr S> 1 and L <t zV rX where r is distance from object to sonar (i.e ., these are farfield calculations and the lengths of the
elongated objects are much shorter than the size of the first Fresnel zone of the sonar). Also, prolate spheroid calculations are
performed in the limit of high aspect ratio [(major axis)/(minor axis) a 5] using the deformed-cylinder formulation.
356
T
Ymax
TS = 10 log
max
U N IF O R M LY
BENT
C Y L IN D E R
j
( E m -i)"
o
\ m = ()
e 2ikec(I -co sv) d y
, axis of
[ deformed
I cylinder
= constant
( ba c kscatte r)
Figure 2. Uniformly bent cylinder with radius of curvature qc.
The cylinder is shown to be bent away from the sonar.
take the bend into account as it causes Fresnel zone
interferences of the echo and possibly substantial chang­
es in target strength.
For the simple case of a constant radius of curvature
qc of the axis and a cylinder that is bent symmetrically
away from the sonar, the target strength from Equation
(5) is
(9)
where y is the angle between the radius vector at the
integration point and the radius vector at the center of
the cylinder, 2ymax is the angle subtending the cylinder,
and L is the arc-length ( = 2oc ymax). Because of the
symmetry, the integral from —ymas to + y max was reduced
to one from 0 to ymax. This is a geometry broadly similar
to the dorsal aspect of a euphausiid or shrimp.
Unlike the solutions for the simple straight-cylinder
geometries described in previous sections, bm is not
separable from the integral. Thus the integral will gen­
erally need to be performed numerically. It can be
performed analytically for extreme cases such as at low
frequencies for uniform fluid materials and at high fre­
quencies for rigid and fixed materials.
To illustrate properties of the scattering from uni­
formly bent cylinders, we compare the backscattering
cross-section from the sphere (Anderson, 1950) and
straight finite cylinder (Equation (7) of this article and
Stanton, 1988a) with the uniformly bent finite cylinder
(Equation (9) of this article and Stanton, 1989a). We
have calculated the cross-sections of both the fluid and
the rigid and fixed cases for additional comparison. The
cross-section was normalized by Jia2 for the sphere and
L2 for each cylinder plot (any “a” involving the sphere is
the spherical radius, while any “a ” involving a cylinder
is the cylindrical radius).
The plots in Figure 3 have many interesting simi­
larities and differences. All curves increase monoto-
.--
i
SPHERE
o
i
STRAIGHT
0
CYLINDER *
^
)
BE NT
CYLINDER
»
' P c
10
0
-10
-20
-Ï0
-4 0
-5 0
-6 0
-7 0
Fluid
-8 0
0.
ka
Figure 3. Comparison between (normalized) backscattering cross-sections of spheres, straight cylinders (normal incidence), and
uniformly bent cylinders (bent away from the sonar) for both the rigid and fixed (upper curves) and the fluid (lower curves) cases
(g = 1 .043 and h = 1.052 for all fluid cases; and in addition qcIL = 0.5 and L/a = 10.5 for the bent cylinder. These four values were
used to describe the scattering by preserved euphausiids). The first 20 modal terms were used in all calculations ( 0 < m < 19)
giving converged solutions.
357
nically and rapidly for ka <g 1 as (ka)4, and the transition
region between this Rayleigh scattering region and the
geometric region occurs at approximately ka = 1. The
major differences occur in the geometric scattering re­
gion (ka ä 1) where the scattering cross-sections of the
sphere and bent cylinder tend to oscillate about a con­
stant value while the cross-section from the straight
cylinder oscillates about a monotonically increasing
value (a detailed discussion of this latter effect is given
in Stanton, 1988a). The reason for this difference is that
the surfaces of the sphere and bent cylinder are curved
in two dimensions while the surface of the straight cylin­
der is curved only in one dimension. Thus the Fresnel
zone interference phenom enon is functionally different
between the case of the sphere and bent cylinder and
the case of the straight cylinder.
One might initially be concerned with the fact that the
trend of the backscattering cross-section of the straight
cylinder increases (linearly) with ka for all ka > 1, which
would imply that energy is not conserved in the limit
k a —> oo. However, the restriction to the solution for the
finite straight cylinder is that the length of the cylinder
must remain much less than the length of the first Fres­
nel zone of the transceiver. The length of the first Fres­
nel zone of a plane-wave source and point receiver is
2V(rX). As a result, if the range, r, of the receiver is
fixed as ka is increased to the point that the first Fresnel
zone is much smaller than the length of the cylinder,
then the infinite cylinder solution applies to the problem
and the amplitude of the backscattered signal remains
constant as the frequency is increased even further.
"sm oothed” version of the exact modal series solution
as the model increases smoothly and monotonically with
ka, while the modal solution oscillates in the k a > l
region (see Fig. 3 for examples of the modal solution).
Because of the usefulness of Johnson’s model, I have
refined it and extended the approach to other geo­
metries: the prolate spheroid, straight finite cylinder,
and bent cylinder. In addition, I have taken into ac­
count the possibility that the object may be either math­
ematically ideal (simple shape) or more realistic (irreg­
ular shape).
The refinement of Johnson’s approach is at high fre­
quencies where he used a term from the /ow-frequency
expansion and an empirical factor to make the model
pass near or through the peaks of the Anderson modal
solution. I have replaced that approach with the product
of the rigorously derived high-frequency expression for
the scattering by rigid and fixed objects and the plane
wave/plane interface Rayleigh reflection coefficient, R.
to account for penetrability (at least to a first approxi­
mation). As a result, the refined model now passes
through intermediate values of the oscillatory high-fre­
quency region (much like an average) and also has
potential for applying to a much wider range of materi­
als, from gas to rigid and fixed objects.
As shown in several articles (Pieper and Holliday,
1984; and Stanton 1988a, 1989a, 1989b), when applying
modal series solutions to describe the scattering by ir­
regular objects such as marine organisms, the optimum
fit requires truncating the modal series before m athe­
matical convergence occurs (i.e., the summation
oo
1.5. “ H i g h - p a s s ” m o d e ls
The modal series formulation describing the scattering
by cylinders has been shown in this paper to be com­
plex. The solutions involving other objects such as
spheres and prolate spheroids are also complex. As a
result, evaluation of backscattering levels of such simple
objects is time consuming. For simple and rapid esti­
mates of backscattering strengths I have derived the
approximate formulas given in Table 1 for the sphere,
prolate spheroid, straight finite cylinder, and bent finite
cylinder.
The “high-pass” models given in Table 1 are based on
an approach originally developed by Johnson (1977),
who heuristically combined low- and high-frequency
limits of the backscattering by fluid spheres (using A n ­
derson's 1950 formulation) to derive a simple approxi­
mate formula for the scattering at all frequencies. In his
derivation, he applied an analogy that the plots of backscattering cross-section versus ka broadly resembled the
frequency characteristics of a two-pole high-pass filter
in electrical signal theory, where the low frequencies are
greatly reduced (Rayleigh scattering region) and the
high frequencies are “passed” through the circuit (geo­
metric scattering region). The model is, in essence, a
358
M
2-2
m=0
m=0
where M is finite). Apparently, the higher modes are
severely damped in such cases. As a result, the slope of
the trend of the solution is altered in the high-frequency
region, thus necessitating an additional empirical factor
F to compensate for the difference (F = 1 for ideal scat­
te re d ). Finally, while Johnson’s model increases
smoothly and monotonically with ka, I have included
the factor G to produce any nulls or peaks that one may
wish to use for better simulation of the exact solution
(G = 1 for no nulls or peaks).
To extend the high-pass model to the prolate sphe­
roid, straight finite cylinder, and bent finite cylinder, I
applied the same approach as with the sphere to the
low- and high-frequency limits of the solutions involving
those geometries from other articles (and shown in Ta­
ble 1).
2. Comparison with existing data
In this section, I give three examples of comparisons
between the modal solution (Equaton (5)), high-pass
Figure 4. Straight elastic finite-length cylinder.
Solid line: modal backscatter solution, first 20
terms used for converged solution. Dotted
line: “high-pass" model of straight “ideal” cyl­
inder. Squares: Durai cylinder backscatter
data (broadside) from Andreeva and Samovol’kin (1976). [g = 2.8, h = 4.38, and (shear
speed of sound of Dural)/(compressional
speed of sound of water) = 2.14 from pub­
lished values.]
i
Q— Z D
DURAL
CSJ
u>
-Q
CP
o
—
modal
h i g h ' pass
a
data
models (Table 1), and scattering data. Comparisons be­
tween the modal solutions and data in Figures 4, 5, and
6 are discussed in detail in Stanton (1988b, 1988a, and
1989a, respectively) while further comparisons between
the high-pass models and those figures are given in
Stanton (1989b). The functions F a n d G that were used
in the high-pass models were determined empirically
from the modal series solutions and are given in Ta­
ble 2.
The first example. Figure 4, involves scattering by
straight elastic cylinders composed of Durai (a metal
similar to aluminum) at broadside incidence. The data
were taken from Andreeva and Samovol'kin (1976) who
also formulated a finite-cylinder scattering theory, but
by qualitative arguments. Plotted is the “reduced” tar­
get strength (where the backscattering cross-section is
normalized by the square of the length of the cylinder)
versus ka. There is excellent agreement between the
modal solution (solid) and the data. The converged
modal solution was used, i.e., the series was evaluated
to model ideal cylinders, therefore prem ature trunca­
tion of the series was not imposed (20 terms were used).
The figure also shows the “high-pass” model (dotted),
which was evaluated for ideal cylinders, to coincide with
the data and modal solution for low ka and to pass
through intermediate values at high ka. The function G
was used to produce the deep null at ka ~ 5 in the
high-pass model.
Figure 5 shows data taken by Greenlaw (1977) from
preserved shrimp over the frequency range 220 kHz to
1100 kHz. Plotted are the maximum values of his data at
each frequency which presumably correspond to broad­
side incidence. Superimposed are the modal solution for
straight fluid cylinders and the “high-pass” model eval­
uated for non-ideal straight cylinders. There appears to
be excellent agreement between the modal solution and
Table 2. Empirically determined functions, F a n d G, used in calculations of high-pass models in Figures 4, 5, and 6.
F:
G:
Durai
Straight cylinder,
simple shape
Shrimp
Straight cylinder.
irregular shape
Euphausiid
B ent cylinder,
irregular shape
1.0
1-O.995e‘ 20(ka’ 5015)
3.0(ka)°“
j_0.8e-2.4k.-2.«1
3.0 + 0.0015(ka)4; 0
l - 0 . 8 e ~ 2'5(ka" 1
359
Figure 5. Straight flu id finite-length cylinder.
Solid line: modal solution, first 2 terms used
for truncated solution. D otted line: “highpass” model of straight "non-ideal” cylinders.
Squares: shrimp backscatter data (dorsal as­
pect) from Greenlaw (1977). [g = 1.043 and
h = 1.052 as determ ined by Greenlaw.]
SHRIMP
n~^
modal
^
-20
h i g h - pass
CNJ
data
o
—
-50
-60
0
1
10
ka
data, however, only after an effective length and radius
of the cross-section were used. The effective length was
0.23 times the measured length of the animal and the
effective radius was 4 % larger than the average m ea­
sured radius. Furtherm ore, the modal series solution
was truncated to include only the first two terms. It is
apparent that a straight cylinder model of a marine
animal is not adequate to describe the scattering since
the animal is actually slightly bent (Greenlaw, 1986)
with a tapered cross-section (I demonstrated in Stanton,
1989a, that at 200 kHz the backscattering cross-section
of a 23-mm-long euphausiid will decrease by 6 dB if the
animal bends by as little as 1.4 mm at the ends).
One effect of bending is that at these high frequencies
only part of the animal is in the first Fresnel zone of the
sonar. The other Fresnel zones will tend to cancel each
other out, hence a smaller acoustic length. I hypothesize
that the trend of the straight cylinder solution adhered
so well to most data points because a short straight
section of the animal dominated the scattering while the
remainder of the animal, which is possibly bent relative
to the section, produced a substantially weaker return.
Also, since the animal is smaller at the ends than it is in
the middle, the portion of the animal that contributes to
the scattering at broadside incidence will most likely be
the larger middle section, hence the larger acoustic ra­
dius.
The third example, Figure 6, resolves many of the
360
difficulties encountered with the straight cylinder
model. Illustrated are data, also from Greenlaw, of the
scattering by euphausiids in the frequency range
63.7—1100 kHz. This time the modal solution was eval­
uated for a bent fluid cylinder. The cross-sectional ra­
dius is held constant. In this example the acoustic length
equals the measured length of the animal, hence the
length error was eliminated by including the bent shape
of the animal. However, as in Figure 5, the acoustic
radius is larger than the average radius and this time by
6 0 % . Thus it is apparent from Figures 5 and 6 that the
varying cross-section of the animal must be taken into
account.
A t the time of this writing, I have conducted prelimi­
nary studies of the scattering by bent and tapered cylin­
ders and have compared the results with G reenlaw ’s
euphausiid data. The results indicate that the average
acoustic radius in this model is closer to the average
radius of the animal. Further work on this subject has
also been completed (Wiebe et al. , 1990, and Chu et al. ,
submitted) where direct measurements of length and
radius lead to favourable predictions by the vent cylin­
der model.
Finally, while discussions in this section have concen­
trated on fluid models of marine organisms, some orga­
nisms may have gas inclusions. In that case, the fluid
model is still rigorously applicable as gas does not sup­
port shear waves. The small, but finite, density and
Figure 6. Bent flu id finite-length cylinder.
Solid line: modal solution, first 6 terms
used for truncated solution. Dashed line:
“high-pass” model of bent "non-ideal” cyl­
inders. D otted lines: euphausiid backscat­
ter data (dorsal and side aspect) from
Greenlaw (1977); upper and lower bounds
are illustrated, [g = 1.043 and h = 1.052 as
determ ined by Greenlaw and oc/L = 0.5
and L /a = 10.5 determ ined acoustically.]
EUPHAUSIID
-10
-20
h ig h - p o s s —>/
data
t/i
tf
O
( u p p e r boun d
-40
-50
data
( l o w e r bound )
-60
k — modal
-70
0.1
10
ka
speed of sound contrast of the gas is used in the modal
solution (caution: applying a zero density and/or zero
speed of sound contrast, which is commonly known as
the Dirichlet or “soft” boundary condition, will produce
erroneous results at low frequencies). I refer the reader
to Stanton (1989b) for more discussions on this and also
an example calculation where 1 apply the high-pass pro­
late spheroid model in Table 1 to fish backscatter data
from Furusawa (1988). The results, which are consistent
with Furusawa’s elegant analysis, show that it is the
length of the sw im bladder, not the fish, that dictates the
scattering level. This is another example of the impor­
tance of accurate scattering models in quantitative bio­
acoustics.
Discussion
I have demonstrated the usefulness of modelling the
scattering of sound by elongated zooplankton by use of
finite-length cylinders. I have shown that it is important
to include the effect of the length of the cylinder, radius
of curvature of its lengthwise axis, and tapering of its
cross-section. In addition, the material properties need
to be accurately known.
While this study has been revealing, the problem
requires much more work. The animals used in this
research were dead and not in their natural environ­
ment. Studies need to be conducted involving many live
animals in situ. Acoustic scattering data must be col­
lected in conjunction with simultaneous photographs so
that the acoustic models and scattering properties can
be accurately related to size, shape, and behavior (tilt
angle and radius of curvature distributions). This in­
formation, combined with sonar data, can be used to
perform accurate estimates of abundance and possibly
size-frequency distributions of the animals on a routine
basis.
A cknowledgements
I would like to thank Michelyn Hass for preparing the
manuscript for this article. This work was supported by
the US Office of Naval Research.
Appendix
Ensembles of scatterers
The models in this paper describe the deterministic
case: that is, the strength of the echo at a particular
frequency due to a single animal of a particular size,
shape, orientation, and composition. In the ocean envi361
ronment, sonars are typically used to detect many ani­
mals, either on an individual basis or in aggregations.
E ither way, to best describe the scattering of sound
from the sea, an ensemble average must be performed
on the scattering cross-section to give a “school” aver­
age. Such a process would involve a weighted average
using (multivariate) probability density functions of
size, shape, orientation, and composition. This average
cross-section, which is also convolved with the sonar
beampatterns, would then be incorporated into formu­
las to predict echo-integration values such as described
in Clay and Medwin (1977). The average backscattering
cross-section takes the general form
< o bs> = J\..J P (x ,,x 2,x3...) obs(x1,x2,x3...) dxjdxjdxj...
( A. l )
where P is the multivariate probability density function
and x,, x2, x3, ... are the parameters describing the
animals. For example, X[ could be length, x2 radius of
curvature of a bent animal, x3 pitch (tilt) angle, x4 den­
sity contrast, x5 speed of sound contrast, etc. P needs to
be determined experimentally.
Further discussions of this topic are not within the
scope of this paper; however, I refer the reader to Clay
and Medwin (1977) for the fundamentals of these con­
cepts, Stanton and Clay (1986) for a review of the
statistical nature of the echoes, Foote (1978) for the
combined effects of tilt-angle distribution and multiple
scattering on the echoes, Lytle and Maxwell (1982) and
Stanton (1983, 1984) for various degrees of multiple
scattering for possible applications to dense aggrega­
tions of zooplankton, and Stanton et al. (1987) for di­
rect applications of echo statistics and (simple) scatter­
ing theory to ensembles of zooplankton in the field
environment.
References
Anderson, V. C. 1950. Sound scattering from a fluid sphere. J.
acoust. Soc. A m ., 22: 426—431.
Andreeva, I. B., and Samovol’kin, V. G. 1976. Sound scatter­
ing by elastic cylinders of finite length. Sov. Phys. Acoust.,
22: 361-364.
Chu, D ., Stanton, T. K., and Wiebe, P H. (Submitted). Fre­
quency dependence of sound backscattering from live indi­
vidual zooplankton.
Clay, C .S ., and Medwin, H. 1977. Acoustical oceanography:
principles and applications. Wiley-Interscience, New York,
USA.
Faran. J . J . . Jr. 1951. Sound scattering by solid cylinders and
spheres. J. acoust. Soc. A m ., 23: 405—418.
Foote, K. G. 1978. Analysis of empirical observations on the
362
scattering of sound by encaged aggregations of fish. Fiskeridir. Skr. Ser. H avunders., 16: 423—456.
Foote, K. G. 1985. Rather-high-frequency sound scattered by
swimbladdered fish. J. acoust. Soc. A m ., 78: 688-700.
Furusawa, M. 1988. Prolate spheroidal models for predicting
general trends of fish target strength. J. acoust. Soc. Jpn,
(E )9: 13-24.
Goodm an, R. R., and Stern, R. 1962. Reflection and transmis­
sion of sound by elastic spherical shells. J. acoust. Soc. A m .,
338-344.
Greenlaw, C. F. 1977. Backscattering spectra of preserved zoo­
plankton. J. acoust. Soc. A m ., 62: 4 4 - 52.
Greenlaw, C. F. 1986. Private communication.
Holliday, D. V. 1977. Extracting bio-physical information from
acoustic signatures of marine organisms. In Oceanic sound
scattering prediction, pp. 619-624. Ed. by N. R. Andersen
and B .J. Zahuranec. Plenum Press, New York, USA.
Holliday, D. V., and Pieper, R. E. 1984. Acoustic volume scat­
tering: 100 kHz to 10 MHz. J. acoust. Soc. A m ., 75
(Suppl. 1): S29.
Johnson, R. K. 1977. Sound scattering from a fluid sphere
revisited. J. acoust. Soc. A m ., 61: 375-377.
Love, R. H. 1971. Dorsal-aspect target strength of an individ­
ual fish. J. acoust. Soc. A m ., 49: 816-823.
Love, R. H. 1977. Target strength of an individual fish at any
aspect. J. acoust. Soc. A m ., 62: 1397-1403.
Lytle, D. V., and Maxwell, D. R. 1982. Hydroacoustic assess­
ment in high density fish schools. FAO Fish. R ep., 300:
157-171.
Penrose, J . D . , and Kaye, G. Thomas, 1979. Acoustic target
strengths of marine organisms. J. acoust. Soc. A m ., 65:
374-380.
Pieper, R. E ., and Holliday, D .V . 1984. Acoustic m easure­
ments of zooplankton distributions in the sea. J. Cons. int.
Explor. Mer, 41: 226-238.
Richter, K. E. 1985. 1.2 MHz acoustic scattering from individ­
ual zooplankters and copepod populations. Deep-Sea Res.,
32, No. 2A: 149-161.
Stanton. T. K. 1983. Multiple scattering with applications to
fish-echo scattering. J. acoust. Soc. A m ., 73: 1164-1169.
Stanton, T. K. 1984. Effects of sound-order scattering on high
resolution sonars. J. acoust. Soc. A m ., 76: 861—866.
Stanton, T. K. 1988a. Sound scattering by cylinders of finite
length. I. Fluid cylinders. J. acoust. Soc. A m ., 83: 5 5 -6 3 .
Stanton, T. K. 1988b. Sound scattering by cylinders of finite
length. II. Elastic cylinders. J. acoust. Soc. A m .. 83: 6 4 -6 7 .
Stanton, T. K. 1989a. Sound scattering by cylinders of finite
length. III. D eform ed cylinders. J. acoust. Soc. A m ., 86:
691-705.
Stanton. T. K. 1989b. Simple approximate formulas for backscattering of sound by spherical and elongated objects. J.
acoust. Soc. A m ., 86: 1499—1510.
Stanton, T. K., and Clay, C. S. 1986. Sonar echo statistics as a
remote sensing tool: volume and seafloor. IE E E J. Ocean.
E ng., OE-11: 7 9 -9 6 .
Stanton, T. K., Nash, R .D . M., Eastwood, R . L . , and Nero,
R. W. 1987. A field examination of acoustical scattering
from marine organisms at 70 kHz. IE E E J. Ocean. E ng.,
OE-12(2): 339-348.
Wiebe, P. H ., G reene, C. H ., Stanton, T. K., and Burczynski,
J. 1990. Sound scattering by live zooplankton and micronek­
ton: Empirical studies with a dual-beam acoustical system. J.
acoust. Soc. A m ., 88: 2346-2360.