Comparing Kirchhoff-approximation and boundary-element models for computing gadoid target strengths

TL;DR: In computations of target strength as a function of tilt angle for each of 15 surface-adapted gadoids for which the swimbladders were earlier mapped, BEM results are in close agreement with Kirchhoff-approximation-model results at each of the same four frequencies.

Abstract: To establish the validity of the boundary-element method (BEM) for modeling scattering by swimbladder-bearing fish, the BEM is exercised in several ways. In a computation of backscattering by a 50-mm-diam spherical void in sea water at the four frequencies 38.1, 49.6, 68.4, and 120.4 kHz, agreement with the analytical solution is excellent. In computations of target strength as a function of tilt angle for each of 15 surface-adapted gadoids for which the swimbladders were earlier mapped, BEM results are in close agreement with Kirchhoff-approximation-model results at each of the same four frequencies. When averaged with respect to various tilt angle distributions and combined by regression analysis, the two models yield similar results. Comparisons with corresponding values derived from measured target strength functions of the same 15 gadoid specimens are fair, especially for the tilt angle distribution with the greatest standard deviation, namely 16°.

Summary

Introduction

• The 20 or so methods cited in a 1991 study3 have since been augmented significantly by a number of new te niques, including both empirical methods8,9 and theoretical models, especially those based on the deformed-cylin model10 and boundary-element method.11.
• These have been accompanied by novel applications, for example, to salm and trout,8,9 cod ~Gadus morhua!,12 orange roughy~.

II. KIRCHHOFF-APPROXIMATION MODEL

• In the Kirchhoff approximation, the field on the scatte ing surface is assumed to be knowna priori.
• For a swimbladder-bearing fish at rather high frequencies, wavelengths which are rather small compared to the m mum length of the swimbladder, the fish is represented b pressure-release surface conforming to the inner wall of swimbladder.
• The integration in Eq.~1! is performed numerically us ing Gauss quadrature over curvilinear surface elements which the position vectorr is interpolated quadratically from nodal values.
• A similar relations exists for eight-node quadrilateral elements using 333 or more Gauss points.
• This translates to a co tion that the element side-to-wavelength ratio should be than 2/5.

III. BOUNDARY-ELEMENT METHOD

• To develop the acoustic boundary-element meth ~BEM!, the wave equation for the pressurep is reduced to the Helmholtz form by assuming the harmonic time dep dence exp(ivt), wherev is the angular frequency in radian per second, hence¹2p1k2p50, wherek5v/c is the wave number.
• The use of the centroids, rather than the nodes, as calculation points for the normal-derivative form is found be sufficient to overcome the problem of the critical freque cies while not increasing the computational effort unduly.
• The accuracy of geometrical representa depends on the degree of undulation of the surface, bu should be noted that the quadratic interpolation allows elements to be curved.

IV. SWIMBLADDER MORPHOMETRY

• First, contours of the swimbladder in planes perpendicular to major axis of the fish, and hence perpendicular to the mic tomed sections, are determined at intervals along the m axis, by finding the points of intersection of each plane w the original digitized sections.
• The n meshes have been used in computations with the Kirchh approximation model in parallel with the BEM.
• At 120.4 kHz the nodal spacing, to satisfy thel/6 condition for accuracy of the BEM and Kirchhoff approximation model, should be less than 2.03 mm.
• The backscattered pressure from a rigid sphere has been computed u similar conditions, again with excellent agreement, wh avoiding discrepancies at the critical frequencies.

VI. COMPARISON OF MODEL COMPUTATIONS

• The target strength for an immersed void with the sha of the swimbladder shown in Fig. 1 has been computed function of tilt angle for both the dorsal and ventral aspe at each of four frequencies.
• Both the Kirchho approximation model and BEM have been examined.
• The target strength corresponding to each avera backscattering cross section, denotedTS, has been compute by substituting the value ofs̄ from Eq. ~14! in Eq. ~3!.
• The standard error of the regression has been computed each derived regression equation.

A. Model validation computations

• To validate the BEM for application to the gadoid swim bladder, a 25-mm-radius spherical void in sea water has b chosen as a test case in order to have a shape for whi rather simple analytical solution exists and whose surf area is greater than that of the largest swimbladder in data set.
• Sensitivity to axial orientation is negligible long as the maximal nodal separation does not exceedl/6.
• The Kirchhoff-approximation model is exercised with th identical meshes but performs less well than the BEM; it inherently different, as is proved by the difference in resp tive exact and analytical solutions for the two models for t pecial shape.
• Differences in the two models are also evident in t logarithmic measures presented in Table IV.
• In contrast, Kirchhoff approximation could be exercised with far mo elements than used here and thus, in principle, could be m amenable to computation at higher frequencies.

B. Swimbladder-shape-based computations

• The detailed computations of target strength as a fu tion of tilt angle are shown for a single specimen, No. 205 Fig. 3. Both the Kirchhoff-approximation model and BEM results are shown for the swimbladder as represented in 1.
• Sign cantly for this work, the Kirchhoff-approximation and BEM results are quite similar.
• The new mapping, for consistency with the BEM, conta fewer but curvilinear elements spanning the swimblad surface.
• Reveals a greates discrepancy of 0.2 dB, with median discrepancy of 0.1 d for the dorsal aspect.

C. Summary of comparisons

• Earlier validation exercises with the BEM have be upplemented by a new example, that of a spherical void cibels ents for he as ho nts ier ers which a simple analytical solution is known.
• Nonetheless, in the c of the swimbladder-shape-based computations, the Kirch J. Acoust.
• Soc. Am., Vol. 111, No. 4, April 2002 K. G. Foote and e ff approximation, when exercised with the curvilinear eleme used in the BEM, yields results that agree well with earl computations carried out using meshes with larger numb 1651D.
• Differences in predictions, as pressed through the regression coefficient in Eq.~15!, are less than 1 dB in all cases except at 38.1 kHz where greatest difference is 1.3 dB.
• There is some expectation the discrepancy might be largest at the lowest frequency the Kirchhoff approximation assumes high frequencies.

ACKNOWLEDGMENTS

• This work began with sponsorship by the Europe Commission through its RTD-program, Contract No. MAS CT95-0031~BASS!.
• This is Woods H Oceanographic Institution Contribution No. 10437. 1R. H. Love, ‘‘Measurements of fish target strength: A review,’’ Fish.

Figures

TABLE II. Backscattered pressure amplitude relative to the incident pressure amplitude in units of 131026 for a 50-mm-diam spherical void immersed in sea water, calculated at 1000 m range for the analytical expressed solution and BEM, and scaled to 1000 m for the farfield Kirchhoff approximation. The effect o amplitude of the orientation of the subdivided icosahedron used to represent the void in the BEM and Kir approximation is expressed through the parenthetical quantity, which is the maximum percentage de from the mean observed over a wide range of axial orientations. Values in italics indicate that the mode nodal separation exceedingl/6 at the specified frequency.

FIG. 2. Frequency dependence of a spherical void of radius 25 mm mersed in water of density 1025 kg/m3 and sound speed 1470 m/s, as giv by the series solution, Kirchhoff integral through the closed-form expres in Eq. ~13!, numerical Kirchhoff approximation using a surface mesh w 642 nodes, and boundary-element method using the same 642-node s mesh.

TABLE IV. Back- and forward-scattered pressure amplitudes shown in Table III, but expressed in de relative to 1-m range and incident wave amplitude.

TABLE III. Back- and forward-scattered pressure amplitudes relative to the incident pressure amplitude i of 131026 for a 50-mm-diam spherical void at 1000-m range for the analytical series-expressed solutio BEM, and scaled to 1000 m for the farfield Kirchhoff approximation. A single, fixed orientation is assume the axis of the subdivided icosahedron used to represent the void in the BEM and Kirchhoff approximatio percentage deviation relative to the analytical solution is given.

TABLE I. Properties of the 15 specimens whose swimbladder surfaces have been remapped, based on the original mapping~Ref. 23! with ntri small triangular facets, bynelem curvilinear elements withnnodesnodes. The nodal separation distance such that 99% of neighboring separations are smaller and the m nodal separation are both specified.

FIG. 3. BEM and Kirchhoff-approximation-model computations of target strength as a function of tilt angle compared against direct measurem specimen No. 205. The functions are shown for both dorsal and ventral aspects at each of four frequencies.

FIG. 1. Boundary-element mesh of the swimbladder to specimen No. 20 Ref. 23, shown in both oblique and dorsal views. The model has 1 elements and 3181 nodes. The meshed swimbladder length is 141 mm

TABLE V. Regression coefficient for the target strength–fish length relationship based on computati measurements of each of 15 specimens in dorsal aspect when averaged over four distinct distribution angleu, characterized by the meanū and standard deviationsu , abbreviated s.d. The coefficientb is that shown in Eq. ~15!. The associated standard error of the regression, SE, is also shown.

TABLE VI. Regression coefficient for the target strength–fish length relationship based on computatio measurements of each of 15 specimens in ventral aspect when averaged over four distinct distribution angleu, characterized by the meanū and standard deviationsu , abbreviated s.d. The coefficientb is that shown in Eq. ~15!. The associated standard error of the regression, SE, is also shown.

Full text

Comparing Kirchhoff-approximation and boundary-element
models for computing gadoid target strengths
Kenneth G. Foote
a)
Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543
David T. I. Francis
School of Electronic and Electrical Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT,
United Kingdom
共Received 16 April 2001; revised 4 December 2001; accepted 14 January 2002兲
To establish the validity of the boundary-element method 共BEM兲 for modeling scattering by
swimbladder-bearing fish, the BEM is exercised in several ways. In a computation of backscattering
by a 50-mm-diam spherical void in sea water at the four frequencies 38.1, 49.6, 68.4, and 120.4
kHz, agreement with the analytical solution is excellent. In computations of target strength as a
function of tilt angle for each of 15 surface-adapted gadoids for which the swimbladders were earlier
mapped, BEM results are in close agreement with Kirchhoff-approximation-model results at each of
the same four frequencies. When averaged with respect to various tilt angle distributions and
combined by regression analysis, the two models yield similar results. Comparisons with
corresponding values derived from measured target strength functions of the same 15 gadoid
specimens are fair, especially for the tilt angle distribution with the greatest standard deviation,
namely 16°. © 2002 Acoustical Society of America. 关DOI: 10.1121/1.1458939兴
PACS numbers: 43.30.Gv, 43.30.Sf 关DLB兴
I. INTRODUCTION
Knowledge of fish target strength has long been recog-
nized to be vitally important in acoustic measurements of
fish density, witnessed by the bibliographies in Refs. 1–5. In
the echo integration method, it appears through the back-
scattering cross section as a divisor of the area backscattering
coefficient or like proportional quantity.
6
In the echo count-
ing method, it appears in the expression for the acoustic sam-
pling volume.
7
There is a multiplicity of methods to determine target
strength. The 20 or so methods cited in a 1991 study
3
have
since been augmented significantly by a number of new tech-
niques, including both empirical methods
8,9
and theoretical
models, especially those based on the deformed-cylinder
model
10
and boundary-element method.
11
These have been
accompanied by novel applications, for example, to salmon
and trout,
8,9
cod 共Gadus morhua兲,
12
orange roughy 共Hop-
lostethus atlanticus兲,
13
and pollack 共Pollachius
pollachius兲.
14,15
Modeling fish target strength, in particular, offers oppor-
tunities of investigation that may otherwise be unrealizable
or exceedingly cumbersome, as in the case of orange roughy,
a fish that resides at 700–1500 m depth.
16
At the same time,
models generally have a domain of applicability that must be
defined and respected in applications.
A number of distinct scattering models have been ap-
plied to fish. These have been based principally on simple
shapes such as the sphere at low frequencies
17–19
and prolate
ellipsoid,
20–22
or the actual shape,
14,15,23–26
called ‘‘mapping’’
method by McClatchie et al.
25
Only the first two models
have exact solutions in general, thus the matter of approxi-
mation must be addressed, at least for realistic shapes at
relatively high frequencies, where the wavelength is not very
long compared to the size of significant scattering
organs.
23,24
The deformed fluid-cylinder model
10,27–29
has been very
popular and has been the object of a major study.
30
Essen-
tially, it reduces an observed shape to a series of axisymmet-
ric cylinders. The method has been realized by Clay and
Horne for Atlantic cod,
12
McClatchie and Ye for orange
roughy,
13
McClatchie et al. for barracouta 共Thyrsites atun兲,
red cod 共Pseudophycis bachus兲, and southern blue whiting
共Micromesistius australis兲,
25
and Sawada et al. for walleye
pollock 共Theragra chalcogramma兲.
22
Excepting the cited
case of Atlantic cod, each model has been based entirely on
the swimbladder as a deformed cylinder. The swimbladder is
important in the Atlantic cod model, but this includes other
parts of the fish too, again represented as finite cylinders.
12
The exact shape of the swimbladder has also been con-
sidered more directly in the Kirchhoff-approximation model
for pollack and saithe 共Pollachius virens兲,
23
walleye
pollock,
24
and southern blue whiting.
26
All of the high-frequency models cited so far are similar
in their neglect of diffraction. To remedy this, the boundary-
element method 共BEM兲
11
has begun to be applied in model-
ing scattering by swimbladder-bearing fish.
14,15
In addition to treating diffraction, the BEM allows use
of general conditions on the swimbladder boundary surface,
with explicit representation of the internal fluid. Thus the
BEM can also be used to study pressure-dependent effects,
which are otherwise precluded by the standard Kirchhoff-
approximation model. Establishing the validity of the BEM
for a pressure-release surface is important for the larger pro-
gram being introduced.
The present aim is to describe the two basic models that
a兲
Electronic mail: kfoote@whoi.edu
1644 J. Acoust. Soc. Am. 111 (4), April 2002 0001-4966/2002/111(4)/1644/11/$19.00 © 2002 Acoustical Society of America

represent the swimbladder by its actual shape, namely the
Kirchhoff-approximation model and the BEM, but both as-
suming a pressure-release boundary condition. Application
of these to historical swimbladder morphometric data is de-
scribed. Independent validation of the two methods is ad-
dressed, and computations with the two models are com-
pared.
II. KIRCHHOFF-APPROXIMATION MODEL
In the Kirchhoff approximation, the field on the scatter-
ing surface is assumed to be known a priori. For a
swimbladder-bearing fish at rather high frequencies, or
wavelengths which are rather small compared to the maxi-
mum length of the swimbladder, the fish is represented by a
pressure-release surface conforming to the inner wall of the
swimbladder.
23,24
The normal component of particle velocity
on the scattering surface is assumed to be equal to that of the
incident field on the directly insonified part of the surface,
and zero on the geometrically shadowed part of the surface.
Mathematically, the farfield backscattering amplitude in
this approximation is
f⫽ ␭
⫺ 1
冖
S
exp
共
2ik• r
兲
H
共
k
ˆ
• nˆ
兲
k
ˆ
• nˆdS, 共1兲
where ␭ is the acoustic wavelength, k is the wave vector in
the source or backscattering direction k
ˆ
⫽ k/k, r is the posi-
tion vector of the surface element with infinitesimal area dS,
nˆ is the unit normal to dS at r, and H(x) is the Heaviside
step function with values 1 for x⬎0,
1
2
for x⫽ 0, and 0 for
x⬍ 0.
The integration in Eq. 共1兲 is performed numerically us-
ing Gauss quadrature over curvilinear surface elements on
which the position vector r is interpolated quadratically from
nodal values. The integrand is evaluated at each integration,
or Gauss, point using the interpolated value of r. A good
representation of the phase, as given by the factor
exp(2ik• r), depends on the separation of these points. If the
integration were to be performed by primitive Riemann sum-
mation, then the points should be closer than about ␭/16.
However, the point-separation condition is undoubtedly re-
laxed by the use of Gauss quadrature. If the polynomial fit
assumed by Gauss quadrature is of order 2 or higher, a good
representation of the wave form should be obtained for a
point separation up to ␭/6. For seven-point quadrature on
six-node triangular elements, with nodes at the corners and
midsides of the elements, the Gauss point separation is at
most 0.8 times the nodal separation. A similar relationship
exists for eight-node quadrilateral elements using 3⫻ 3or
more Gauss points. A condition for validity of the numerical
integration, that the nodal separation should be less than ␭/5,
is therefore tentatively suggested. This translates to a condi-
tion that the element side-to-wavelength ratio should be less
than 2/5. In order to give commonality with the correspond-
ing condition in the case of the boundary-element method, to
be discussed in Sec. III, the slightly stricter ratio of 1/3 is
adopted in this paper for assessing the frequency range of
validity of a given mesh. The element meshes are described
in Sec. IV.
The backscattering cross section is
␴
⫽ 4
␲
兩
f
兩
2
. 共2兲
The target strength is the logarithmic expression of
␴
,
TS⫽ 10 log
冋
␴
4
␲
r
0
2
册
, 共3兲
where r
0
is a reference distance, assumed here to be 1 m.
III. BOUNDARY-ELEMENT METHOD
To develop the acoustic boundary-element method
共BEM兲, the wave equation for the pressure p is reduced to
the Helmholtz form by assuming the harmonic time depen-
dence exp(i
␻
t), where
␻
is the angular frequency in radians
per second, hence ⵜ
2
p⫹ k
2
p⫽ 0, where k⫽
␻
/c is the wave
number. This is rewritten in integral form, in which the pres-
sure at any point is expressed in terms of the acoustic pres-
sure and normal displacement u on the scattering surface S.
This surface is subdivided into elements, and the pressure
and displacement distributions on S are represented by dis-
crete values, p
i
and u
i
, respectively, at each node i associ-
ated with these elements. The standard Helmholtz integral
equation suffers from singularities at certain critical frequen-
cies, which are dense at high frequencies. To overcome this
problem, the integral is combined with a second integral de-
veloped from the first by differentiating with respect to the
normal direction at the surface.
31
In principle, the two equa-
tions are combined by adding the standard form evaluated at
each node of each element to a multiple
␣
of the normal-
derivative form evaluated at the centroid of that element in
the local coordinate system.
11
The resulting equation can be
written thus:
Ap⫽ Bu⫺ p
inc
⫺
␣
⳵
p
inc
⳵
n
. 共4兲
If the swimbladder is assumed to be ideally pressure-
releasing, p
i
⫽ 0 for all i, and Eq. 共4兲 can be solved directly
for the nodal normal displacements:
u⫽ B
⫺ 1
冉
p
inc
⫹
␣
⳵
p
inc
⳵
n
冊
. 共5兲
The coefficients of the matrix B are assembled from local
matrices pertaining to each element of the mesh. With the
calculation point of the Helmholtz integral taken at node i,
with position r
i
, integration over element m provides the
following coefficients in the standard formulation:
b
mn
共
1
兲
共
r
i
兲
⫽
␳␻
2
冕
S
m
N
n
共
q
兲
cos
␪
mn
G
共
r
i
,q
兲
dS
q
, 共6兲
where
␳
is the fluid density, q is the position vector of the
integration point on the element surface S
m
, G is the Green’s
function, given by G(r
i
,q)⫽ e
⫺ ik
兩
r
i
⫺ q
兩
/4
␲
兩
r
i
⫺ q
兩
, n is the
local nodal label, and N
n
(q) 共n⫽ 1,2,...,6 for triangular ele-
ments, n⫽ 1,2,...,8 for quadrilateral elements兲 are the shape
functions, which are of the standard second-order 共quadratic兲
form.
32
The factor cos
␪
mn
is included to allow for the devia-
tion
␪
mn
of the normal to the element m at local node n from
the mean normal at that node. The mean normal at a node is
1645J. Acoust. Soc. Am., Vol. 111, No. 4, April 2002 K. G. Foote and D. T. I. Francis: Comparing Kirchhoff-approximation and BEM

defined as the average of the normals at the node on all
contiguous elements weighted by the respective differential
surface area.
The normal-derivative form of the Helmholtz integral
equation, calculated at the centroids r
¯
l
of the elements, simi-
larly provides coefficients as follows:
b
mn
共
2
兲
共
r
¯
l
兲
⫽
␳␻
2
冕
S
m
N
n
共
q
兲
cos
␪
mn
⳵
G
共
r
¯
l
,q
兲
⳵
n
r
dS
q
, 共7兲
where the normal derivative is evaluated at the centroid.
These are combined with the previous coefficients by adding
a multiple of b
mn
(2)
(r
¯
l
) for all elements l on which global node
i lies, i.e.,
b
mn
共
r
i
兲
⫽ b
mn
共
1
兲
共
r
i
兲
⫹
␣
i
兺
l:i␧S
l
b
mn
共
2
兲
共
r
¯
l
兲
, 共8兲
where the combination factor
␣
i
is taken to be ⫺ i/kM
i
,
where M
i
is the number of elements meeting at node i.
11,33,34
The use of the centroids, rather than the nodes, as the
calculation points for the normal-derivative form is found to
be sufficient to overcome the problem of the critical frequen-
cies while not increasing the computational effort unduly.
11
The integrals are evaluated numerically using Gauss
quadrature.
The coefficients b
mn
(r
i
) are assembled into the global
matrix B by summing the coefficients that correspond to the
same global node, thus
B
ij
⫽
兺
m,n:C
共
m,n
兲
⫽ j
b
mn
共
r
i
兲
, 共9兲
where C(m,n) is the global node label of local node n on
element m. The source terms in Eq. 共4兲 are evaluated thus:
p
inc
共
r
i
兲
⫹
␣
i
兺
l:i␧S
⳵
p
inc
共
r
¯
l
兲
⳵
n
r
Given the solution for u from Eq. 共5兲, the scattered pres-
sure at any exterior point r is obtained from the standard
integral equation by calculating coefficients similar to
b
mn
(1)
(r
i
) but with r
i
replaced by the position vector r:
b
j
共
3
兲
共
r
兲
⫽
␳␻
2
兺
m,n:C
共
m,n
兲
⫽ j
冕
S
m
N
n
共
q
兲
cos
␪
mn
G
共
r,q
兲
dS
q
共10兲
and then
p
共
r
兲
⫽⫺b
共
3
兲
共
r
兲
"u. 共11兲
The backscattering amplitude at finite range r is
f
共
r
兲
⫽
r
兩
p
共
r
兲
兩
兩
p
inc
兩
. 共12兲
The farfield backscattering amplitude f is the limit of f (r)as
r approaches infinity. Expressions for the backscattering
cross section and target strength are derived by substituting
f(r), or f, in Eqs. 共2兲 and 共3兲, respectively.
The elements used here are quadrilaterals and triangles
of the quadratic isoparametric type, in which both the geo-
metric and acoustic quantities are interpolated from the nodal
values using quadratic shape functions, the nodes being situ-
ated at the vertices and midsides.
32
As a general guide, good
representation of the acoustic variables is obtained if the
lengths of the sides of the elements are less than one-third of
a wavelength. The accuracy of geometrical representation
depends on the degree of undulation of the surface, but it
should be noted that the quadratic interpolation allows the
elements to be curved. Further details of the formulation and
equations can be found in Ref. 11.
IV. SWIMBLADDER MORPHOMETRY
The origin of the morphometric data is a study per-
formed in 1980
35
on surface-adapted specimens of pollack
and saithe, described briefly in Table I. Each specimen was
anesthetized, tethered, and acoustically measured at each of
four frequencies, nominally 38, 50, 70, and 120 kHz, prior to
shock-freezing and microtoming in the sagittal plane, hence
TABLE I. Properties of the 15 specimens whose swimbladder surfaces have been remapped, based on the original mapping 共Ref. 23兲 with n
tri
small triangular
facets, by n
elem
curvilinear elements with n
nodes
nodes. The nodal separation distance such that 99% of neighboring separations are smaller and the maximum
nodal separation are both specified.
Fish No. Species
Length
共cm兲
Mass
共g兲
Ref. 23
n
tri
New meshes Swimbladder Nodal separation
n
elem
n
nodes
Surface area
共cm
2
兲
Volume
共cm
3
兲
99% limit
共mm兲
Max
共mm兲
201 Pollack 31.5 195 5 546 1168 3364 33.01 6.91 1.20 2.21
202 Pollack 44.0 533 9 965 1389 4041 58.83 16.33 1.37 1.77
204 Pollack 35.5 321 6 562 1078 3116 42.39 10.03 1.41 1.72
205 Pollack 39.0 380 7 171 1107 3181 45.75 11.34 1.43 1.93
206 Pollack 35.0 287 5 379 1159 3347 31.37 7.75 1.17 1.46
207 Pollack 44.5 635 8 695 1487 4363 65.24 19.15 1.34 1.61
209 Saithe 38.5 385 6 762 1501 4387 43.29 10.08 1.06 1.39
213 Pollack 34.5 259 10 192 1039 2935 34.11 7.83 1.33 1.61
214 Pollack 39.0 406 7 649 1164 3362 44.14 10.15 1.34 1.53
215 Pollack 37.0 332 5 265 1076 3092 38.89 8.75 1.34 1.74
216 Pollack 36.5 343 6 436 1062 3060 43.33 10.85 1.40 1.64
217 Pollack 34.5 253 5 500 962 2764 34.61 7.11 1.32 1.46
218 Pollack 32.5 257 4 689 1327 3879 29.75 6.27 1.00 1.39
219 Pollack 35.5 292 5 106 1039 3005 35.74 8.15 1.27 1.53
220 Saithe 38.0 406 8 968 1321 3857 44.32 10.46 1.13 1.32
1646 J. Acoust. Soc. Am., Vol. 111, No. 4, April 2002 K. G. Foote and D. T. I. Francis: Comparing Kirchhoff-approximation and BEM

parallel to the main axis, according to the method of Ona.
36
The thickness of successive photographed sections was 100
␮
m. Each swimbladder section was digitized as a set of co-
ordinates describing the outline of the swimbladder, and the
surface of each swimbladder was represented by a mesh con-
sisting of flat triangular facets.
23
Because of the use of curvilinear quadrilaterals and tri-
angles in the BEM, new meshes have been produced for each
of the specimens using a semiautomatic process. First, the
contours of the swimbladder in planes perpendicular to the
major axis of the fish, and hence perpendicular to the micro-
tomed sections, are determined at intervals along the major
axis, by finding the points of intersection of each plane with
the original digitized sections. Quadrilateral and triangular
elements are then fitted between neighboring contours.
Where required, nodes are interpolated using cubic splines.
This method allows the fineness of the mesh to be controlled
by the choice of the separation between the contours and the
nodal separation on each contour. Some manual fitting of
elements is required where the swimbladder branches into
separate lobes.
A further reason for the remapping exercise was to re-
duce the number of nodes in order to facilitate matrix opera-
tions inherent to the BEM. The resulting meshes have fewer
elements than the original triangular meshes,
23
but this is
offset by the allowance for curvature of the surface. An ex-
ample of one of the meshes is visualized in Fig. 1. The new
meshes have been used in computations with the Kirchhoff-
approximation model in parallel with the BEM.
Details of the meshes are listed in Table I. For each
mesh, the maximum distance between neighboring nodes is
shown in the final column, column 11; however, a better
indication of the degree of fineness of each mesh is given in
column 10, namely the limit of nodal spacing which is sat-
isfied by 99% of the distances between pairs of neighboring
nodes. At 120.4 kHz the nodal spacing, to satisfy the ␭/6
condition for accuracy of the BEM and Kirchhoff-
approximation model, should be less than 2.03 mm. All of
the meshes except that for specimen 201 are well within this
limit even on the basis of the maximum nodal spacing found
in the mesh. Detailed analysis of the mesh for specimen 201
reveals that the ␭/6 condition is satisfied for all but two pairs
of neighboring nodes out of 4530 such pairs.
V. INDEPENDENT VALIDATION OF MODELS
A cogent form of validation of the Kirchhoff-
approximation model is the direct comparison of model com-
putations and measurement results for the same fish speci-
mens used in the morphometry. This work is documented in
detail in Ref. 23 but in which the integration in Eq. 共1兲 is
performed by the primitive Riemann summation, with evalu-
ation of the integrand at the centroid of each triangular facet.
Validation of the BEM has already been documented for
a series of cases in which analytical solutions are available.
Three of those described in Ref. 11 are cited. 共1兲 The forward
scattered pressure for a plane wave incident on a rigid sphere
has been computed. The agreement over the ka range from 0
to 10 is excellent, without discrepancies at the critical fre-
quencies that arise in the standard formulation, which lacks
the normal component included in Eq. 共4兲. 共2兲 The backscat-
tered pressure from a rigid sphere has been computed under
similar conditions, again with excellent agreement, while
avoiding discrepancies at the critical frequencies. 共3兲 As an
illustration, the radiation impedance of a uniformly vibrating
circular piston of radius a in the end face of a cylinder of
radius 2a and height 4a over the ka range from 0 to 5 has
been computed and compared with the analytical solution for
a piston in an infinite baffle. With allowance for the differ-
ence between the two problems, the agreement is quite good.
An additional trial of the BEM has been designed spe-
cifically for the present study. A spherical void of radius 25
mm is assumed to be immersed in sea water of sound speed
1470 m/s and density 1025 kg/m
3
. The size has been chosen
for having an area of 7854 mm
2
, which is roughly 20%
greater than the area of the mesh spanning the surface of the
largest swimbladder, No. 207, as represented by 1487 ele-
ments 共4363 nodes兲, with an area of 6524 mm
2
. Meshes have
been generated by subdividing each spherical triangle of a
geodesic icosahedron into four subtriangles, subdivided
again to get a mesh of 320 elements 共642 nodes兲 and subdi-
vided once more to get a mesh of 1280 elements 共2562
nodes兲. The latter mesh has a maximum nodal separation of
2.06 mm, which is just outside the limit of 2.03 mm required
by the ␭/6 condition at 120.4 kHz.
The same example provides a trial for the Kirchhoff
approximation, since the integration in Eq. 共1兲 can be per-
formed analytically for the spherical shape:
FIG. 1. Boundary-element mesh of the swimbladder to specimen No. 205 of
Ref. 23, shown in both oblique and dorsal views. The model has 1107
elements and 3181 nodes. The meshed swimbladder length is 141 mm.
1647J. Acoust. Soc. Am., Vol. 111, No. 4, April 2002 K. G. Foote and D. T. I. Francis: Comparing Kirchhoff-approximation and BEM

f⫽
共
4k
兲
⫺ 1
兵
⫺ 1⫹ cos
共
2ka
兲
⫹ 2ka sin
共
2ka
兲
⫹ i
关
sin
共
2ka
兲
⫺ 2ka cos
共
2ka
兲
兴
其
, 共13兲
where a is the sphere radius. Comparison of the numerical
and closed-form analytic solutions at the frequencies 38.1,
49.6, 68.4, and 120.4 kHz demonstrates agreement to within
0.01 dB for the 2562-node sphere. Further exercise of the
Kirchhoff model reveals significant divergence of the nu-
merical solution from the exact solution at about 180 kHz for
the 2562-node sphere and at about 90 kHz for the 642-node
sphere. The truth of this last statement is evident in the target
strength spectrum in Fig. 2, which also compares the results
of the Kirchhoff approximation with the exact series solution
and BEM solution for the 642-node mesh. The observed de-
viation of the BEM solution from the exact series solution at
about 90 kHz corresponds to a nodal spacing of ␭/4, which is
coarser than the nominal criterion for validity of the BEM,
namely neighboring-point separations within ␭/6.
Numerical computations have been performed for both
the BEM and Kirchhoff-approximation model for each of the
two meshes. The computations have been repeated at each of
the four measurement frequencies, 38.1, 49.6, 68.4, and
120.4 kHz, hence with ka⫽ 4.07, 5.30, 7.31, and 12.87. The
results have been compared against the well-known analyti-
cal solution for scattering by a spherical void, with perfectly
soft boundary condition, in a homogeneous fluid with given
sound speed and density values.
37
In one set of computa-
tions, the effect of orientation of the axis of the meshed void
is examined by comparing the backscattered pressure ampli-
tude at infinity for the Kirchhoff-approximation model and at
1000-m range for the BEM. The results are shown in Table
II. In a second set of computations, the same backscattered
pressure amplitude for a single orientation is compared di-
rectly against the amplitude derived with the analytical solu-
tion. The results of these computations, as well as those for
the forward scattered amplitude by the analytical solution
and BEM, are presented in Table III. Corresponding target
strengths and forward-scattering strengths are presented in
Table IV.
It is noted that, for certain frequencies and mesh sizes,
the condition for the nodal spacing discussed in Secs. II and
III, namely that this should be less than ␭/6, is violated. The
results for these cases are included in Tables II–IV but are
shown in italics. The limit is only just exceeded by the finer
mesh at 120.4 kHz.
VI. COMPARISON OF MODEL COMPUTATIONS
The target strength for an immersed void with the shape
of the swimbladder shown in Fig. 1 has been computed as a
function of tilt angle for both the dorsal and ventral aspects
at each of four frequencies. Both the Kirchhoff-
approximation model and BEM have been examined. The
computational results are shown with the measured functions
in Fig. 3.
The same computations have been repeated for the
mapped swimbladder shapes of all 15 gadoid specimens
listed in Table I. In order to reduce these to manageable
proportions, the several functions have been averaged with
respect to normal distributions g(
␪
) of tilt angle
␪
, with
mean
␪
¯
and standard deviation s
␪
:
␴
¯
⫽
冕
g
共
␪
兲
␴
共
␪
兲
d
␪
冒
冕
g
共
␪
⬘
兲
d
␪
⬘
, 共14兲
where the integration has been performed over the range
关
␪
¯
⫺ 3s
␪
,
␪
¯
⫹ 3s
␪
兴
. For the measured target strength functions,
values at tilt angles outside the range 关⫺45°, 45°兴 were not
available, and for such angles the value of
␴
at the nearest
angle limit has been used.
Computations have been performed for each of four nor-
mal distributions of tilt angle. The paired values (
␪
¯
,s
␪
) are
共0°, 0°兲, 共0°, 5°兲, 共0°, 10°兲, and 共⫺4.4°, 16°兲. Because of the
effect of perspective, by which the apparent tilt angle of a
fish changes as it is observed at different positions in the
plane transverse to the acoustic axis,
38
the effective values of
s
␪
are larger than the nominal ones. The values shown above
have been adjusted for the perspectival effect for a circular
beam of 5° beamwidth measured between the half-power
points. The effective standard deviations for the four cases
are 2.5°, 5.5°, 10.2°, and 16°.
23
The target strength corresponding to each averaged
backscattering cross section, denoted
TS, has been computed
by substituting the value of
␴
¯
from Eq. 共14兲 in Eq. 共3兲. The
values of
TS have been regressed on fish length l in centi-
meters according to the regression equation,
TS⫽ 20 log l⫹ b, 共15兲
where the regression coefficient b is expressed in decibels.
The standard error of the regression has been computed for
each derived regression equation. The results are shown in
Tables V and VI.
VII. DISCUSSION
A. Model validation computations
To validate the BEM for application to the gadoid swim-
bladder, a 25-mm-radius spherical void in sea water has been
chosen as a test case in order to have a shape for which a
rather simple analytical solution exists and whose surface
area is greater than that of the largest swimbladder in the
data set. Finite-element representation of the sphere by a
subdivided icosahedron has allowed both the BEM and
Kirchhoff approximation to be computed according to Eqs.
共12兲 and 共1兲, respectively.
Because of the finiteness of the facets, there is an effect
due to axial orientation, which is indicated in Table II. The
BEM is seen to be quite accurate for the two meshes that
were chosen. Sensitivity to axial orientation is negligible as
long as the maximal nodal separation does not exceed ␭/6.
The Kirchhoff-approximation model is exercised with the
identical meshes but performs less well than the BEM; it is
inherently different, as is proved by the difference in respec-
tive exact and analytical solutions for the two models for this
special shape. The variability with orientation is notable for
the coarser mesh.
1648 J. Acoust. Soc. Am., Vol. 111, No. 4, April 2002 K. G. Foote and D. T. I. Francis: Comparing Kirchhoff-approximation and BEM

Frequently asked questions

Q1. What are the contributions in "Comparing kirchhoff-approximation and boundary-element models for computing gadoid target strengths" ?

In this paper, the boundary element method ( BEM ) was used for modeling scattering by swimbladder-bearing fish. 

Q2. What have the authors stated for future works in "Comparing kirchhoff-approximation and boundary-element models for computing gadoid target strengths" ?

Some of the effects mentioned here may be addressed in a future work.