Page 1

A numerical study of the role of the tragus in the big brown bat

Rolf Mu ¨ller

The Maersk Institute, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark

?Received 30 April 2004; revised 15 September 2004; accepted 19 September 2004?

A comprehensive characterization of the spatial sensitivity of an outer ear from a big brown bat

?Eptesicus fuscus? has been obtained using numerical methods and visualization techniques. Pinna

shape information was acquired through x-ray microtomography. It was used to set up a

finite-element model of diffraction from which directivities were predicted by virtue of forward

wave-field projections based on a Kirchhoff integral formulation. Digital shape manipulation was

used to study the role of the tragus in detailed numerical experiments. The relative position between

tragus and pinna aperture was found to control the strength of an extensive asymmetric sidelobe

which points in a frequency-dependent direction.An upright tragus position resulted in the strongest

sidelobe sensitivity. Using a bootstrap validation paradigm, the results were found to be robust

against small perturbations of the finite-element mesh boundaries. Furthermore, it was established

that a major aspect of the tragus effect ?position dependence? can be studied in a simple shape

model, an obliquely truncated horn augmented by a flap representing the tragus. In the simulated

wave field around the outer-ear structure, strong correlates of the tragus rotation were identified,

which provide a direct link to the underlying physical mechanism. © 2004 Acoustical Society of

America. ?DOI: 10.1121/1.1815133?

PACS numbers: 43.80.Ka, 43.64.Ha ?JMS?

Pages: 3701–3712

I. INTRODUCTION

Bats have evolved the common anatomical layout of a

mammal’s outer ear to yield prominent structural features

which presumably serve special functions in biosonar sens-

ing. One such feature is the tragus, a frontal prominence,

which is highly conspicuous in many bat species. Previous

work ?Lawrence and Simmons, 1982; Wotton et al., 1995;

Wotton and Jenison, 1997? has characterized the sensitivity

of the outer ear of the big brown bat ?Eptesicus fuscus? as a

function of direction ?directivity? by presenting data on im-

pulse responses or transfer functions. Such results have typi-

cally been obtained over a limited angular range and with

limited angular resolution. Directivity gain as a function of

both of the two angles ?azimuth and elevation? which char-

acterize a direction in three-dimensional space has been

shown for two frequencies only ?Wotton and Jenison, 1997?.

Similar limitations apply to work performed on other bat

species ?Grinnell and Schnitzler, 1977; Jen and Chen, 1988;

Coles et al., 1989; Obrist et al., 1993; Fuzessery, 1996; Fir-

zlaff and Schuller, 2003?. Despite limitations in the scope of

the data obtained so far, the research on the big brown bat

has revealed interesting functional properties relevant to

source localization in elevation. In particular, it has demon-

strated the importance of the tragus for elevation discrimina-

tion by comparing the performance of animals with natural

tragus position to an experimental group for which the tragus

had been bent out of the pinna aperture ?Lawrence and Sim-

mons, 1982?. However, an overview of the spatial distribu-

tion of sensitivity with sufficiently high resolution in angle,

frequency, and tragus position has not been provided. This

prohibits putting features of individual transfer functions ob-

tained for different directions into perspective and hampers

understanding the global effect of the tragus underlying these

properties. Furthermore, since prior data have been limited to

providing black-box system descriptions, they cannot yield

definitive insights into the physical mechanisms, which so

far have not been observed directly.

Here, a different methodological approach is introduced

to this problem, which addresses the above-mentioned short-

comings and hence is capable of yielding a comprehensive

picture of the role of the tragus: Based on x-ray computer

tomographic cross sections, an accurate high-resolution rep-

resentation of the shape of an outer ear can be generated.

Such a representation can be used for a numerical finite-

element simulation study of the diffraction effects taking

place around the ear surface. The far-field directional sensi-

tivity arising from the diffraction effects can also be pre-

dicted by projecting the wave field outward from the bound-

aries of the finite-element model’s computational domain.

This approach offers three key advantages over physical

measurements: First, directivity patterns can be estimated

with a high angular and frequency resolution, since the time

needed to obtain a sensitivity gain estimate is much shorter

than in any physical setup with moving parts. Second, the

digital representation of the pinna shape can be manipulated

in a well-defined way, drawing from the wide range of avail-

able computer graphics methods. Because the efficiency ad-

vantage of numerical evaluation applies to the manipulated

shapes as well, the effects of the manipulations can be stud-

ied using high-resolution directivity estimates to describe the

functional properties of the outer ear. Third, estimates of the

wave field in the volume surrounding the pinna surface,

where all diffraction effects take place, are generated as part

of the method and can be evaluated to obtain insights into the

physical mechanisms behind changes in the directivity pat-

tern. These advantages outweigh disadvantages given by the

approximations and simplifying assumptions made by the

numerical model. While these approximations represent

3701J. Acoust. Soc. Am. 116 (6), December 20040001-4966/2004/116(6)/3701/12/$20.00© 2004 Acoustical Society of America

Page 2

sources of error absent in physical measurement, it should be

noted that physical measurements have their own sources of

error.

In the results presented below, a numerical approach will

be used to

?i?

obtain a comprehensive spatial picture of the effect

the tragus has on the sensitivity of the outer ear;

understand the role of tragus position ?rotation?;

gauge the shape specificity of the spatial sensitivity

patterns; and

characterize the underlying wave-field diffraction ef-

fects.

?ii?

?iii?

?iv?

II. METHODS

A. Shape measurement

An outer-ear sample was taken from a big brown bat

?Eptesicus fuscus? carcass, which had been preserved by

cooling. The outer ear was severed from the head with the

concha and part of the ear canal left attached. It was mounted

?glued to a cylindrical holder? in a life-like manner and sub-

jected to x-ray microtomography in a Skyscan 1072 scanner.

X-ray prefiltering was applied to reduce beam-hardening ar-

tifacts in the reconstruction. Shadow images of 1024?1024

pixels with a pixel size of 19.35 ?m were collected over a

half-circle ?180°? of target rotation performed with 0.9° reso-

lution. Cross sections of the ear shape were reconstructed

usingFeldkamp’scone-beam

implemented in software provided by Skyscan. Cross-section

images ?gray scale, 8-bit resolution? were manually postpro-

cessed to remove noise and obvious artifacts which were

clearly separated from the ear cross section in the images. A

pixel classification into the two categories ‘‘tissue’’and ‘‘air’’

was performed by manual adjustment of a gray-value thresh-

old. Stacking of the resulting binary cross-section images

produced a three-dimensional voxel-array representation of

the shape. It was processed in two ways: for visualization of

the shape, a triangular mesh approximation of the boundary

surface between ear tissue and air was generated using the

marching cubes algorithm ?Lorensen and Cline, 1987?; these

meshes were postprocessed by low-pass filtering and deci-

mation ?Schroeder et al., 1992?. A rendering of the final

shape representation produced by this method is shown in

Fig. 1.

Visualization and rendering was performed using func-

tions provided by the VISUALIZATION TOOLKIT software li-

brary ?VTK, Schroeder et al., 2003?. A mesh of finite ele-

ments filling the air around the structure was generated by a

custom algorithm which evaluated all possible positions for

the placement of finite elements in a lattice of cubic elements

of uniform size. An element was placed at a given position

only if all eight corner voxels for this position had been

classified as air. The bias of this decision criterion against

placing air elements was adjusted at the voxel classification

level, where thresholding was preceded by a three-

dimensional smoothing operation using an isotropic Gauss-

ian kernel of adjustable bandwidth.

reconstructionalgorithm

B. Shape manipulation

The generated ear shape was subjected to two different

kinds of manipulation: tragus rotations and shape distortions.

Tragus rotation was performed in order to characterize the

influence which the relative position of tragus and ear aper-

ture exerts on the directivity. To define the operation, a

cuboid was placed manually in the voxel array to enclose the

portion of the tragus to be rotated. The cuboid was posi-

tioned to include the entire freestanding length of the tragus.

The classification of each source voxel ?tissue or air? inside

the cuboid volume was then transferred to a target voxel,

which was found by rounding the rotated position of the

source voxel to the resolution of the voxel grid. The classi-

fication of all source voxels was set to air, unless they were

the target voxel for another portion of the rotated structure. A

morphological closing operation was performed after the

transfer of the voxel classifications in order to fill any holes

produced by the rounding of the target positions in the voxel

grid and thereby ensure that the rotated structure was again a

homogeneous volume. Rotations were always about an axis

placed at the proximal end of the freestanding tragus portion.

In the original mounting of the ear, the tragus was placed

upright; this position—shown in Fig. 1—is referred to as

‘‘0°.’’Tragus rotation was in the direction outwards from the

pinna aperture. It was performed in steps of 15°, spanning a

total of 60° from the original position ?Fig. 2?.

Shape distortions were performed to assess the robust-

ness of the obtained results with regard to any source of

shape error or variability by means of a perturbation para-

FIG. 1. Tomographic shape reconstruction. Shown is a rendering of a

smoothed, triangular surface mesh ?119 287 triangles? approximating the

boundary between tissue and air.

FIG. 2. Shape renderings for tragus rotations performed on the ear shape

representation rendered in Fig. 1. The difference in rotation angle between

neighboring shapes is 15°; the rotation angle of the upright tragus ?left? is

defined as 0°.

3702 J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004Rolf Mu ¨ller: Role of the tragus

Page 3

digm: the voxel array representation was subjected to ran-

dom distortions by translating each voxel according to a ran-

dom displacement vector. The three Cartesian components of

the displacement vectors were drawn independently from a

zero-mean Gaussian distribution. A strong spatial smoothing

of the directions was performed in order to prevent discon-

tinuities in the displacement field which would have resulted

in breaking the continuous volume describing the ear tissue.

The smoothing operation was carried out using a three-

dimensional, isotropic Gaussian smoothing kernel with

15-mm standard deviation applied to each Cartesian vector

component. This is equivalent to computing a weighted vec-

tor average. The resulting spatially averaged displacement

vectors were normalized in length and then scaled with a set

of displacement magnitudes drawn independently from a

zero-mean Gaussian distribution. The standard deviation of

the displacement magnitude distribution was the parameter

used to control the amount of shape distortion. The field of

displacement magnitudes was spatially smoothed with a ker-

nel identical to the one used for the Cartesian components of

the displacement directions. Displacements were applied to

the voxel array in the same manner as the tragus rotations

?see Fig. 3?.

C. Generation of a simple model shape

In order to investigate to what extent the observed prop-

erties are specific to the bat ear shape, a simple model shape

was studied in parallel with the bat ear. This model shape is

based on the obliquely truncated horn introduced by Fletcher

and Thwaites ?1988? ?see also Fletcher, 1992? as an idealized

model for the mammalian pinna. In its original form the

model does not account for a tragus, however. Therefore, the

model was augmented here by addition of a frontal flap in-

tended to mimic the tragus. The overall dimensions of the

model shape were adjusted to match the natural ear shape

approximately ?Fig. 4?. The obliquely truncated horn is,

however, strictly bilateral symmetric, whereas the natural ear

shape strongly deviates from bilateral symmetry.

A detailed description of the augmented obliquely trun-

cated horn model, along with a discussion of other possible

features, can be found in Mu ¨ller and Hallam ?2004?. Here,

the obliquely truncated horn was subjected to the same tra-

gus rotations as the natural ear shape ?Fig. 2?. The shape

model was implemented using Boolean combinations of im-

plicit functions describing cones ?for the inner and outer sur-

face?, cutting planes ?for the outer-ear aperture and trunca-

tion of pinna and ear canal?, and cylinders for modeling an

ear-canal stump. Binary ?1-bit resolution? cross-section im-

ages were generated by evaluating the Boolean combinations

of the implicit functions and thresholding their function val-

ues. The binary cross-section images were then fed into the

same processing chain as the tomographic cross sections in

order to generate shape visualizations ?see Fig. 4? and finite-

element meshes.

D. Finite-element model

The finite-element simulation used cube elements ?all

edges of equal length, eight nodes per element? to approxi-

mate the air volume surrounding the outer ear. The air was

assigned a density of 1.205 kg/m3and a bulk modulus of

1.41767?105Pa; this corresponds to a sound speed of 343

m/s. The cube mesh was generated from the voxel represen-

tation of the shape as described in Sec. IIA. Element edge

length was set to eight voxels, corresponding to 154.8 ?m.

Thus, for the shortest considered wavelength ??4.6 mm at

75 kHz?, ?30 elements were placed within one wavelength.

A cuboidal computational domain around the ear shape was

meshed with finite elements, leaving a minimum distance of

several elements between the ear surface and the outer do-

main boundary on all faces but the bottom face, where the

foot plate of the ear shape was placed right on the domain

boundary. Where the outer boundary of the computational

domain was made up of elements representing air, it was

covered with two-dimensional square-shaped absorbing

boundary elements, which approximated reflection-free out-

ward propagation ?Enquist and Majda, 1977?. The ear sur-

face was assumed to be perfectly reflecting.

The finite-element model simulated an acoustical source

placed in the proximal ear-canal cross section, treating the

ear as a loudspeaker. This approach is valid to determine the

ear’s properties as a receiver by virtue of the reciprocity

principle ?Pierce, 1981?. It offers the methodological advan-

tage that a single simulation run gives predictions for the

field on all faces of the computational boundary, from which

FIG. 3. Example realizations of shapes distorted by random displacement

fields. Shown are the boundaries of the finite-element mesh, in which the

cubic shape of the individual elements is clearly visible. The parameter ?0,

0.12, 0.23, 0.46 mm? is the standard deviation of the displacement magni-

tude distribution.

FIG. 4. Comparison of the simple shape model ?obliquely truncated horn

augmented by a tragus-like flap? with the bat ear shape.

3703 J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004Rolf Mu ¨ller: Role of the tragus

Page 4

the directional sensitivity of the ear shape can be inferred for

all directions as described below. If instead a planar wave

impinging from a particular direction were to be simulated, a

separate simulation would have to be run for each direction.

In order to test the importance of the type of simulated

source on the measured system properties, two different

sources were tested and the results compared: a point source,

where a time-varying boundary condition on pressure was

placed on one node near the center of the proximal ear-canal

cross section only, and a piston source, where all nodes lying

in the cross section were subjected to the same time-varying

boundary condition. Taking advantage of the linearity as-

sumption, the signals emitted by the simulated sources were

given lattice spectra, which allowed for testing system be-

havior for an entire set of frequencies of interest in a single

simulation. The frequency components present in the signals

were spaced 500 Hz apart from 25 to 75 kHz; this covers

most of the frequency band of the emission of the big brown

bat as reported by Hartley and Suthers ?1989? ?25 to 90

kHz?.

Assembly of the master stiffness matrix from the finite-

element mesh and computation of a solution via a precondi-

tioned conjugate gradient method was performed using the

COUPLED FIELD SIMULATION ?CFS??? program developed by

the Department of Sensor Technology at the University of

Erlangen-Nu ¨rnberg, Germany.

E. Directivity estimation and visualization

The wave fields generated by diffraction from shapes

with as little symmetry as the shapes considered here will in

general show a joint dependence on all three spatial coordi-

nates and frequency. However, in the far field, the dependen-

cies on distance and angle separate and the wave-field am-

plitude P can be expressed as

P??,?,r,??

→

r?8a2/?

D??,?,??1

re?jkr,

?1?

where D is the directivity, which depends on direction ?two

angles, azimuth ? and elevation ?? and frequency ? only

?Urick, 1983?. The second term ?(1/r)e?jkr? is the expres-

sion for the wave field of a point source; it depends only on

distance r and wave number k, which is proportional to fre-

quency. Hence, the directivity D is a complete far-field sys-

tem description which contains all the information specific to

a particular source. In the following, a normalized version of

D is presented in order to treat the influence of direction

separately from overall gain. The lower boundary for the far

field given in Eq. ?1? is r?8(a2/?), where a is the charac-

teristic dimension of the source ?here the ear? and ? is the

wavelength. The height of the ear shape is ?18 mm; hence,

it may be assumed that a?9 mm. The smallest wavelength

considered here is ?4.6 mm ??75 kHz?. Consequently, the

far-field boundary is at r?14.4cm and—at the chosen reso-

lution of the finite element mesh—?1000 elements would

have to placed along each dimension to reach it. In three

dimensions, the size of the finite-element problem would be-

come a concern in terms of computation resources and accu-

mulation of numerical errors, if the computation were to be

extended significantly beyond the far-field boundary.

To avoid these problems, the wave fields were forward-

projected from the boundary faces of the computational do-

main, which were themselves still inside the near field.

Sound propagation into a space free of any diffracting ob-

jects ?and sources? can be described by a Kirchhoff integral

formulation ?Jackson, 1999?, which was used here to per-

form the forward projection of the wave field ?Ramahi,

1997?: In the frequency domain, the wave field ??x? at po-

sition x outside the boundary surface S can be expressed as

4??

S

R

??x???

1

ejkR

n•????jk?1?

j

kR?

R

R??ds,

?2?

where R is the vector between the surface element ds and

the position x, n the surface normal, ? in the argument of the

integral the field value on S, and k the wave number. The

product n•?????/?n is the derivative of the field ? with

respect to the surface normal n. In order to obtain a predic-

tion of the directivity, the positions x were chosen on a hemi-

sphere which was placed over the ear. The radius of the

hemisphere was chosen large enough to satisfy the condition

given in Eq. ?1?. Because all positions on the surface of the

hemisphere differ only in direction and not in distance, the

normalized field amplitudes ??x? can be taken as estimates

of the directivity D.

The transformation from the time-domain finite-element

results to the frequency-domain representation used in Eq.

?2? as well as for the directivities was performed using the

Goertzel algorithm ?Mitra, 2001?. Since directivity is a sta-

tionary system characterization, data points containing tran-

sients generated at the start of the time-domain simulation

were discarded prior to this transformation.

The directivity results were visualized using Lambert’s

azimuthal equal-area projection ?Weisstein, 2002?. The

equal-area property of this projection is desirable for the vi-

sualization of directivities, because the relative cross-

sectional area of lobes in the beam patterns can be compared

in the map. At the same time, the projection keeps distortions

of lobe shape close to the theoretical minimum, even if a

large portion of the directivity ?up to a hemisphere? is shown

in a single map.

The combination of directivity and spherical spreading

losses, like the geometrical attenuation included in Eq. ?1?,

results in the formation of three-dimensional ‘‘sound

beams:’’ in the directions where the directivity has a ?local?

maximum, sensitivity falls below a given threshold at a

larger distance than for the directions of the adjacent ?local?

minima. These sound beams were visualized by rendering of

iso-sensitivity surfaces. Since directivity is also a function of

frequency, there is a different isosurface for each frequency.

Isosurfaces for a set of linearly spaced frequencies were ren-

dered with different gray levels and overlaid in one image to

allow for an easy comparison of the sound beam patterns.

The isosurfaces were generated by evaluating the prod-

uct of directivity and spherical spreading given in Eq. ?1?.

Absorption as a source of spreading loss was not considered

for the sake of simplicity, since only the effect of the

frequency-dependent effect of directivity was to be visual-

3704 J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004Rolf Mu ¨ller: Role of the tragus

Page 5

ized in a three-dimensional way. To this end, it suffices that

the directivity is multiplied with a term which is a monotone

decreasing function of distance and independent of direction.

III. RESULTS

A. Robustness

1. Influence of the excitation source

A comparison of directivity results which were not nor-

malized for each frequency ?not shown? indicated a strong

effect of the chosen excitation source ?piston and point

source?. In the normalized directivities ?Fig. 5?, however,

only minor differences were found and the overall shape of

the directivities was the same for both sources.

Therefore, the choice of the source affected primarily

the overall sensitivity as a function of frequency, but it had

only a negligible effect on how this sensitivity depended on

direction for any given frequency.

2. Influence of random shape distortions

The influence of shape distortion on directivity estimates

was tested over a range of values for the standard deviation

of the magnitude of the random displacements. The tested

values resulted in distortions ranging from very slight to ma-

jor ?Fig. 3?. Directivity patterns estimated for example real-

izations ?Fig. 6? were found to be consistent with the notion

that—in general—larger shape distortions can be expected to

result in larger changes in the directivity patterns. Small

shape distortions ?standard deviation?0.1 mm, i.e., ?2/3 of

a finite-element edge length; see Fig. 6? did not alter the

pattern of the major lobes in the directivity; the location,

shape, and relative weight of these features were largely pre-

served. Larger shape distortions did change the overall lay-

out of the directivity pattern, although some common fea-

tures were identifiable even for the largest shape distortion

shown here ?standard deviation 0.46 mm; see Fig. 3 for a

rendering of the tested realization?. From the examples ex-

amined, it was not apparent that the degree to which shape

distortions changed the directivity pattern was dependent on

frequency. Hence, it cannot be said that within the studied

frequency band ?25–75 kHz? low-frequency directivities

were more or less susceptible to shape distortion than high

frequencies.

B. Tragus rotation and directivities

Rotation of the tragus had a clear, systematic effect on

the directivity pattern in the medium frequency range stud-

ied. Between ?35 and ?50 kHz, the strength—and to some

extent the location and shape—of an extensive, asymmetric

?one-sided? sidelobe was dependent on tragus rotation. In the

upright tragus position ?0°?, the sidelobe had the highest rela-

tive sensitivity; it was gradually attenuated as the tragus was

rotated out of the pinna aperture ?see Figs. 7 and 8?.

While tragus rotation did result in a large number of

small changes to the directivity, the effect on the asymmetric

sidelobe stands out by virtue of its magnitude and systematic

nature. For frequencies below 35 kHz ?e.g., 30 kHz in Fig.

7?, no sidelobe is formed, and the effect of tragus rotation on

the directivity pattern is minor. For frequencies above 50

kHz, sidelobes are common, but none of them was found to

show a systematic dependence on tragus rotation. The results

for 53 kHz in Fig. 7 are an example of a strong sidelobe in

this frequency band; like the broad main lobe at 30 kHz, it is

largely unaffected by the tragus rotation. At a higher fre-

quency resolution ?Fig. 8?, it is evident that the orientation of

the asymmetric sidelobe depends on frequency in a system-

atic fashion: the sidelobe shifts to higher elevations as fre-

quency increases.

C. Wave field correlates

In the wave fields simulated inside the computational

domain of the finite-element model effects were found which

showed a systematic dependence on tragus rotation. This de-

pendence paralleled the effects seen in the directivities. The

effects were particularly obvious in the phase component of

monochromatic wave fields. In the frequency band of the

asymmetric sidelobe, rotation of the tragus caused systematic

FIG. 5. Comparison of directivity pattern estimates for two different sources

of excitation ?point and piston source, columns?. The directivities are nor-

malized individually for each of the frequencies ?rows? shown. The graphs

show Lambert’s azimuthal equal-area projections of the directivity.

3705 J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004Rolf Mu ¨ller: Role of the tragus