Simplified Models of the
Dolphin Echolocation Emission System
(This is an unfinished draft of a paper that readers
may find interesting enough to comment upon.
Please send questions, comments, or suggestions to the author, James
Aroyan, at jaroyan@cruzio.com.)
All rights reserved by James L. Aroyan, 2002. This material may not be published or distributed
without the express consent of the author.
Educational and/or personal use is encouraged – please provide feedback
if possible.
Introduction
Can simple
acoustic models produce signals having most of the observed characteristics of
the echolocation clicks emitted by dolphins?
Perhaps surprisingly, the answer is yes.
A simple model corresponding
to the acoustic behavior of the dolphin forehead is presented below as both a research
and a didactic tool. Movies of
simulated sound propagation are used to illustrate the behavior of the
individual and combined model components.
Only three model components are assumed here: 1) a source mechanism of broadband pulses; 2) one or more damped resonators; and 3)
a projector of the signal
produced by the first two components.
Simple models should be considered qualitative in comparison to more
sophisticated modeling techniques.
Nevertheless, they serve many purposes including clarification of the
mechanisms underlying dolphin biosonar and as a means for exploring how
biological tissues interact from an acoustical standpoint. The goal of the current modeling is to
illustrate that forehead anatomical variations between odontocete species
suggest model parameter (with resulting signal) shifts that correlate with
reported differences in signals. A
further goal is to find physical explanations for some poorly understood
echolocation signal features including:
·
The
rapid rise and ‘ringing’ decay of the pressure waveforms that are
characteristic of the echolocation signals of several odontocete species.
·
The
observed differences in the number of pressure oscillations within the overall
signal envelope for several species.
·
Correlations
between signal source level and modal frequency composition observed in the bottlenose
dolphin, the beluga, and the false killer whale.
·
A
precursor (or “forerunner”) signal feature.
·
Delayed
and attenuated signal reverberation features.
·
Aspects
of the far field distributions of odontocete echolocation signals.
Experimental versions of simplified models can be
easily built, and may serve as ideal laboratory projects for intermediate
bioacoustics courses.
It is worth emphasizing that the connections discussed here between
dolphin anatomy, echolocation signal analysis and acoustical modeling are
certainly not new. Many of the references
cited in this paper contain far more advanced discussions of dolphin anatomy,
echolocation signal analysis, and dolphin bioacoustic modeling. To researchers knowledgeable in both
acoustics and biology, most of the material covered here may seem obvious. However, to the best of my knowledge, no one
has to date combined these areas in an introductory fashion that accurately
communicates the most basic and crucial connections. In addition, even the so-called “experts” in the field (some of
whom have no understanding of acoustics!!) do not appear to have appreciated
the extent to which simple models can explain the observed characteristics of
dolphin echolocation signals.
To make the
current topic accessible to the widest possible audience, this paper begins
with an overview of the relevant anatomy and a brief description of dolphin
echolocation signal characteristics.
Overview of Dolphin Forehead Anatomy
This section
describes some of the tissues in the dolphin forehead that are thought to play
important roles in sound production and emission. For more detailed descriptions of dolphin head and nose anatomy
the reader may wish to consult Lawrence and Schevill (1956), Schenkkan (1973),
Mead (1975), Green et al. (1980), Heyning (1989), or Cranford et al. (1996).
Figure 1 diagrams selected
tissues in a parasagittal slice in a plane lying slightly to the right of the
midline of the common dolphin forehead.
This diagram includes the skull and jaw bone (gray hatched), the nasal
air sacs (red), the right MLDB complex (represented by green arrows), the nasal
plug (np), the epiglottic spout (es) of the larynx (blue), and the melon
tissues (yellow gradient). Other
tissues, including the muscle and connective tissue surrounding many of the
labeled forehead tissues, are not shown.
Figure 1. Diagram of selected dolphin head tissues. [Adapted
from Aroyan 1990.]
Among other characteristics, all cetacean skulls exhibit an evolutionary migration of the nares backwards from the tip of the snout (the typical location in most mammals) to tilt dorsally toward the vertex of the skull. The cranial vault is foreshortened and concave, and the maxillary/premaxillary region elongated. Figure 2 is a photograph of a common dolphin skull (without the lower jaw). In the common dolphin, the upper nares and the premaxillary shelves combine to form a roughly semi-parabolic depression of the upper skull surface. The common dolphin skull also exhibits a slight displacement of the midline suture to the left, indicating a directional asymmetry that may mirror related soft tissue asymmetries (the degree of skull and soft-tissue asymmetry varies considerably among different odontocetes).
Figure 2. Photograph of common dolphin skull (without
lower jaw). The left and right narial
passageways through the skull and the supranarial depression are evident.
Figure 3. Diagram of dolphin skull and nasal air
sacs. PS – premaxillary sacs (yellow);
VS – vestibular sacs (pink); NS – nasofrontal (or tubular) sacs (green); BH –
blowhole. Also colored are the
spiracular cavity (blue); the skull and upper jaw (light blue); and the lower
jaw (light purple). [Adapted from Au
2000; originally adapted from Purves and Pilleri 1983.]
Above the bony
nares sit a number of air sacs that form part of the nasal complex leading up
to the blowhole. To better illustrate
the 3D structure of the dolphin’s nasal air sacs, Figure 3 diagrams the air
sacs in the inflated condition associated with the sound production cycle (see
below). Note that the MLDB complexes
are located along the dorsal-lateral margins of the spiracular cavity (colored
blue in Figure 3), just below the floor of the vestibular sacs. The nasofrontal sacs wrap around the
spiracular cavity on either side, and the premaxillary sacs cover the
premaxillary shelves of the skull. The
detailed geometry of the nasal sacs varies between species. In the harbor porpoise, for example, the
vestibular sacs have prominent connective tissue ridges along their floors, and
the relative size of these sacs is larger than in most delphinids (Amundin and
Cranford 1980). The commerson’s dolphin
(Cephalorhynchus commersonii) and
particularly the franciscana (Pontoporia
blainvillei) also exhibit relatively large vestibular sacs.
Dolphins and perhaps all odontocetes are capable of creating substantial air pressure (roughly 0.5-1.0 atmosphere above ambient) within their bony nares (Ridgway et al. 1980; Ridgway and Carder 1988), compressed by the large muscles that pull the larynx up toward the narial cavity of the skull. This pressure may be regulated at the top of the narial cavity by the muscular nasal plugs. Several studies (Norris et al. 1971; Dormer 1979; Ridgway et al. 1980; Ridgway and Carder 1988) describe a phonation cycle in which air is pressurized within the bony nares and passed upward into the nasal sac system during production of echolocation clicks and whistles. Air need not be expelled from the blowhole during this cycle – instead, it collects in the distensible vestibular sacs during sound production and is returned to the nares afterward to repeat the cycle.
Figure 4. Visualizations of common dolphin CT data in
forehead region. (A) Visualization of
skull and outer melon isosurfaces (portions of the skin surface and nasal sacs
are also visible). (B) Visualization of
skull and skin isosurfaces. [Reproduced
from Aroyan 1996.]
The fatty melon
tissues of the dolphin forehead are shown in Figure 1 as a yellow
gradient-filled region. These tissues
actually consist of a layered topology of lipids that are chemically distinct
from blubber and other body fats.[1] The melon rests on a concave pad of
connective tissue and musculature above the bony rostrum of the skull. To better illustrate the 3D geometry of the
melon tissue within the forehead of the common dolphin, Figure 4 reproduces two
CT data visualizations of the skull, melon, and skin isosurfaces from Aroyan
(1996).
Velocity
measurements over the melon of a bottlenose by Norris and Harvey (1974)
revealed a low-velocity core surrounded by higher-velocities that grade into
the surrounding muscle and connective tissues.
Similar melon structures in other dolphins and porpoises are supported
by a growing list of species’ CT and/or MRI scans (Cranford et al. 1996). Wood (1964) was the first of many researchers
to suggest that the melon might focus and channel sound generated within the
forehead tissues and acoustically couple it to seawater. Computer simulation studies (Aroyan 1990,
1996, 2000; Aroyan et al. 1992) have verified that all of these acoustic
behaviors do indeed occur in common dolphin tissue models based on the melon
velocity distributions measured by Norris and Harvey (1974).
Some Characteristics of Delphinid Echolocation
Signals
Continuing our
introduction, we now briefly describe the echolocation signals of dolphins and
some other odontocetes. For more
detailed information and further references on the characteristics of dolphin
echolocation signals, readers are encouraged to consult Au (1993) and Au
(2000).
The echolocation
signals of odontocete cetaceans have many common features as well as characteristics
that differ between species. It is
thought that all odontocetes produce short duration (50-400 usec), intense
(SL=150-227 dB re 1 uP), and high frequency (12-140 kHz peak frequency) acoustic
pulses. The specific ranges within
which particular species’ signal parameters fall have been contrasted by many
researchers – significant differences do exist in peak frequency, spectral
composition, source level, duration, and envelope shape. Some species emit signals having relatively
fixed characteristics, making stereotypical descriptions seem appropriate. However, other species appear capable of
varying their signal characteristics, and individual animals have been observed
to exercise considerable control over their emitted signals. Hence, some flexibility is called for in
attempts to categorize species by signal type (or vice versa). It should also be kept in mind that most of
the available data on odontocete echolocation signals has been obtained from
only a small handful of species.
As an example of the echolocation signals of one delphinid, Figure 5 illustrates a click train emitted by an Atlantic bottlenose dolphin performing a target detection task. The frequency spectrum versus time is plotted on the left and the individual click waveforms are displayed on the right. Peak frequencies for most of the pulses in this click train fall roughly in the 110-120 kHz range, although other components are also visible. The broadband character of these pulses is reflected in –3 dB bandwidths of roughly 60-80 kHz. On average, these signals last only 50-60 ms, a surprisingly short duration considering their biological origin. At a velocity of 1500 m/s in seawater, sound travels 7.5 cm in 50 ms.
Figure 5. Signals and spectra of a typical sonar click
train emitted by a bottlenose dolphin (Tursiops truncatus) performing a
target detection task in Kaneohe Bay. The
individual click waveforms are displayed in the upper diagram (with
peak-to-peak SL re 1mPa, peak frequency, and time of click occurrence relative to first
click) and the corresponding frequency spectra vs. time of occurrence are
plotted in the lower diagram.
[Reproduced from Au 1993.]
Examples of sonar signals emitted by a harbor porpoise (Phocoena phocoena) and by a Commerson’s dolphin (Cephalorhynchus commersonii) are provided in Figure 6. Figures 7 & 8 illustrate variations in signals recorded from a bottlenose dolphin (Tursiops truncatus), a beluga or white whale (Delphinapterus leucas), and from a false killer whale (Pseudorca crassidens). These examples fall into two broad categories. The first category consists of higher frequency, narrow bandwidth, and low intensity signals emitted by smaller porpoises and dolphins that do not whistle, exemplified by the signals in Figure 6. The second consists of generally lower frequency, broader bandwidth, and higher intensity signals emitted by larger delphinids and other toothed whales that are capable of whistling (Au 2000), exemplified by the signals in Figures 7 & 8. These categories are not rigid, however, as there is at least one whistling species that also emits high frequency, narrow band, and relatively low intensity signals. Of course, one expects small animals to eat smaller prey, and higher frequency signals enable detection of smaller prey.
Figure 6. (A) Example of an echolocation signal from
the harbor porpoise (Phocoena phocoena). (B) Example of an echolocation signal from
the Commerson’s dolphin (Cephalorhynchus commersonii). The spectra are plotted below the
waveforms. [Signal data from Kamminga
and Wiersma 1981. Figure reproduced
from Au 2000.]
Figure 7. Examples of echolocation signals used by Tursiops
truncatus in Kaneohe Bay and in a tank.
The spectra are plotted below the pressure waveforms. The dashed curve in the bottom plot is the
spectrum of the signal measured in the tank.
[Reproduced from Au 2000.]
Figure 8. (A) Example of Delphinapterus leucas
echolocation signals in San Diego Bay and Kaneohe Bay (Au et al. 1985). (B) Examples of Pseudorca crassidens
echolocation signals. The waveforms are
shown on the left and the corresponding spectra on the right. SL is the averaged peak-to-peak source level
in dB re. 1mPa (Au et al. 1995).
[Reproduced from Au 2000.]
From an
acoustics standpoint, perhaps the most important thing to note in Figures 5-8
is that all of the signals illustrated appear to be pulses of varied bandwidths
that have been filtered by damped resonators.
We shall discuss the structures in the dolphin forehead that potentially
generate such pulses and cause resonant filtering in the modeling sections of
this paper. The acoustical interactions
of these structures may indeed provide simple explanations for the different
categories of echolocation signals noted above.
The echolocation
signal of the Commerson’s dolphin in Figure 6B clearly contains delayed and
attenuated signal reverberation features.
The delayed signal features in this case create a rippling of the signal
spectrum that splits the main spectral peak into three sub-peaks. Careful inspection of Figure 6A also
suggests the presence of reverberation features in the signal of the harbor
porpoise. The configurations of the
soft tissue and air sacs within the left and right nasal passageways of these
animals may provide a simple explanation for these reverberation features.
The variability of odontocete echolocation signals is exemplified by
Figures 7 & 8 – clearly some animals are able to vary the characteristics
of their emitted signals through a wide range.[2] For example, the two signals produced by a
bottlenose dolphin in Figure 7 have quite different peak frequencies,
bandwidths, and intensities.
Nevertheless, some highly significant correlations are present in Figures
5, 7, & 8. It is clear that the
spectra in Figures 7 & 8 exhibit bimodal frequency distributions, as do
several of the pulses in the click train of Figure 5. Bottlenose dolphin clicks often have peak frequencies in either
the 50-60 kHz or the 110-120 kHz ranges.
A remarkable series of P.
crassidens signals transitioning between a lower frequency mode (~40 kHz)
and a higher frequency mode (~110 kHz) is shown in Figure 8B. These variations in modal frequency
composition are correlated with signal source level. Au (2000) suggests that “the response of the sound generator may
be determined by the intensity of the driving force that eventually causes an
echolocation click to be produced.”
Most pneumatically driven acoustic sources do shift to higher
frequencies as they are driven harder, and there is no reason to expect the
dolphin’s click generator to be different in this regard.
It is crucial to note that if the dolphin’s click generator is capable
of producing impulses of progressively higher frequency as the driving air pressure
(and muscular tension) is increased, and if this source is located in soft
tissues of the nasal passageways that are partially surrounded by air sacs,
then the modal characteristics of the spectra in Figures 7 & 8 may simply
result from a variable source being filtered by the resonances of a soft tissue
resonator (plus a projective filter element).
This seems especially apparent in the examples of Figure 8, where it is
easy to imagine a relatively fixed bimodal resonator spectrum being driven by a
click source that increases its peak frequency with amplitude. Similarly, the narrow bandwidth and
reverberant signals of the harbor porpoise and Commerson’s dolphin in Figure 6
may result from more completely air-bounded soft tissue (left and right) cavities
that have a higher frequency resonance spectrum (plus a projective filter
element). The remainder of this paper
explores the implications of such a three-component model of the dolphin
echolocation emission system.
Simplified Acoustic Model of the Dolphin Forehead
We now consider a simplified acoustical model of the delphinid
forehead. This model was proposed over
six years ago (Aroyan 1996) but unfortunately not well understood. The model combines three components: 1) a
source mechanism of broadband pulses; 2) a damped resonator; and 3) a projector
of the signal produced by the first two components.
The source of acoustic pulses is here assumed to correspond to the
right MLDB tissue complex (represented by green arrows in Figure 1). The MLDB complexes are located along the
dorsal-lateral margins of the spiracular cavity (colored blue in Figure 3),
just beneath the floor of the vestibular sacs.
The physical mechanism responsible for producing broadband pulses is
thought to be pneumatically driven soft tissue oscillations (Cranford et al.
1996; Dubrovsky and Giro 2002). The
supranarial depression of the skull and nasal air sacs that partially surround
the MLDB soft tissues are assumed to effectively create a damped resonator,
causing a resonant filtering of the source spectrum (Aroyan 1996). Finally, a projector of the signal produced
by the source and resonator is considered to result from the forward-reflecting
portions of the skull and nasal air sacs in combination with the refractive
forehead soft tissues (including the melon).
For the purposes of model construction, simplified acoustic versions of
the forehead tissues labeled in Figure 1 can be obtained by imagining an axis
drawn from the MLDB through the melon core.
Figure 9 illustrates a cylindrically symmetric acoustic representation
of these tissues. The air sacs partially
enclosing the MLDB source region have been simplified to a single spherical sac
having gaps through which some of the incident broadband pulse energy
escapes. The projector tissues are represented
in Figure 9 as including a parabolic air sacs (or bone) reflector and a
refractive soft tissue lens.
Figure 9. Simplified
cylindrically symmetric version of model of dolphin forehead tissues.
In the current discussion, it is not critical that the interfaces creating resonances (such as air sacs partially enclosing the source region) be entirely separate from the interfaces contributing to forward reflection. These components are sketched here as being separate in order to clarify the operating mechanisms. Also, it is not crucial whether the shape of the reflector is exactly parabolic (the supranarial depression of the skull, for example, seen in Figure 2, appears approximately semi-parabolic). Finally, although the projector tissues are shown in Figure 9 as consisting of both reflective air sacs (or bone) and refractive forehead soft tissues, it is not crucial to the current argument that there be separate parts to this projective component. For example, the melon could be eliminated from a still-further simplified model, as sketched in Figure 10.
Figure 10. Further simplified sketch of emission model
components.
In the following sections, we briefly discuss each of the three assumed
components, using movies of simulated wave propagation to illustrate their
behavior.
1) Broadband Click Source Component
The anatomical location of the tissues responsible for generating
echolocation clicks has in the past been a subject of controversy. Some authors have argued that, as with terrestrial
mammals, dolphins produce sounds in their larynx. Nevertheless, almost all experimental studies with live dolphins
have implicated the region of the nasal passageways as the origin of
echolocation pulses in dolphins. These
include cineradiographic studies (Norris et al. 1971, and Dormer 1974, 1979),
pulsed ultrasonic imaging (Mackay 1980), ultrasonic Doppler-shift motion
detection (Mackay and Liaw 1981), pressure sensing and electromyographic
studies (Ridgway et al. 1980), and direct palpation during sound production
(Amundin 1990). Two recent studies have
narrowed the location of the source tissues down to the upper nasal passages,
just beneath the floor of the vestibular sacs (see Figure 3). Aroyan (1996) simulated acoustic propagation
within 3D forehead tissue models of D.
delphis using techniques analogous to those applied in seismology to locate
the epicenters of earthquakes. A close
clustering of focal centers for forward-directed beams was found that best
supports the right MLDB complex as the source of echolocation clicks in this
dolphin (Aroyan et al. 2000). In
addition, Cranford et al. (2000) observed motions of the MLDB complexes
correlated with click production in T.
truncatus. Although Cranford et al.
(2000) did not rigorously exclude other tissues as potential sources, their
observations suggest that both left and right MLDB complexes are capable of
generating echolocation clicks.
Closely related to source location issues are questions concerning the
exact physical mechanism by which clicks are produced. Many of the studies referenced above have suggested
source mechanisms, but few of these suggestions are based on solid acoustical
arguments. Perhaps the most convincing
work to date in this area is summarized in Dubrovsky and Giro (2002). Physical and mathematical models of a source
mechanism are described (Giro 1987; Dubrovsky and Giro 2002) in which brief
(50-80 msec), high frequency (fpeak
around 100kHz), and broadband (up to 80 kHz –3dB widths) pulses result from
surface accelerations during pneumatically driven oscillations. For the purpose of modeling the dolphin
emission system, this suggests that a small source of broadband pulses should
be considered to be the first component of the system.
At this point in our discussion, let us momentarily turn our attention to the simulation of sound propagation when exploring various models. For visualization purposes, we will use finite-difference time-domain (FDTD) wave propagation programs in which symmetry about the z-axis is assumed. This simplifying assumption does not overly restrict our ability to explore basic models. The geometry of the simulation grid is diagrammed in Figure 11. Z-axis symmetry allows us to reduce the simulation grid to a half-plane that includes the z-axis. If absorbing boundary conditions are added along all boundaries other than the symmetry axis, then only a small portion of the half-plane containing the model needs to be simulated. This approach allows us to compute sound fields rapidly and to illustrate the full 3D solutions in time with 2D grid movies. In all movies presented below, the simulation grids represent half of a 2D slice through the center of our 3D (axisymmetric) models. Height above the plane in these movies represents the acoustic pressure of the propagating wave solutions.
Figure 11. Simulation grid geometry: rotational
symmetry about the z-axis is assumed, and variable r represents distance from
the z-axis. Gaussian pulse propagation
snapshot.
As an example, the waveform in Figure 11 is a snapshot of a Gaussian pulse (Figure 14A) that was emitted from a small source (the small semi-circle on the z-axis). The full 3D solution is just this waveform rotated about the z-axis – i.e., a spherical wave spreading away from the source point. A movie of the Gaussian pulse emission by the source and its absorption by the boundary conditions is provided in Figure 12.
Figure 12. Movie of Gaussian pulse emission by the
source (click on figure to start).
In all simulations in this report, we will
use fixed length and time scales. These
scales are established by the dimensions of our model on the grid and by the
relevant wave velocity. Here we choose
152 grid increments to correspond to 5.00 cm, making 1 grid increment
equivalent to Dh @ 0.329 mm.
Waves traveling at the nominal simulation velocity cover k @0.612 grid increments per
time step. Assuming this nominal velocity
corresponds to a speed of sound in seawater of
c @ 1.53106 mm/s, then
one simulation time step corresponds to Dt = k (Dh / c) @ 0.134 ms. Given these values of Dh and Dt,
the above Gaussian pulse would then have a –3dB bandwidth of 148 kHz (74 kHz in
positive frequency) and a duration of about 4.2 ms (FWHM time-record of
pressure). The dimensions of the
simulation grid in Figure 12 are then 6.58 cm by 13.16 cm.
Now let us continue with our discussion of the source mechanism. Certain features of the frequency spectrum of a source are directly attributable to its physical characteristics. For example, if the surface area of the source is small, the low-frequency end of the spectrum is necessarily reduced by the low emission efficiency of small sources at large wavelengths. Also, at the high-frequency end, the spectral energy must be limited by the finite acceleration of real surfaces – biological tissues cannot undergo infinite accelerations. Furthermore, one would expect both the peak frequency and the maximum emitted pulse amplitude to increase with increasing surface accelerations. In the case of air pressure driving tissue oscillations, it makes physical sense to suggest that both the peak frequency fpeak and the maximum amplitude Apeak may be controlled by increasing or decreasing the driving pressure. All of these factors support the assumption of a broadband source with both fpeak and Apeak dependent on driving pressure (among other variables). Figure 13 represents a qualitative plot of such a source spectrum.
Figure 13. Qualitative plot of assumed broadband source
spectrum.
Figure 14 illustrates two types of pressure pulses and their
corresponding frequency power spectra. Figures
14A-B plot the signal and spectrum of the Gaussian pulse propagated in the
movie of Figure 12. Figures 14C-D plot
the signal and spectrum of a Gaussian-windowed sinusoid pulse. The low-frequency end of the spectrum in
Figure 14D agrees more closely with the spectral characteristics plotted
qualitatively in Figure 13 than does the low-frequency end of Figure 14B. [Figure 14D should also be compared with
Figure 6 in Dubrovsky and Giro (2002)].
In other words, for physical reasons we do not expect a small
soft-tissue source to emit pure Gaussian-type signals – a Gaussian-windowed
sinusoid represents a closer match to the expected spectral
characteristics. Nevertheless, we will
use Gaussian source signals in some of the following simulation movies because
they create propagation patterns that are often simpler to visually interpret.
Figure 14. Two signals and their associated power
spectra. (A) Gaussian signal. (B) Spectrum of signal in A. (C) Gaussian-windowed sinusoid signal. (D) Spectrum of signal in C.
The exact structure of the waveforms generated by pneumatically-driven
soft tissue motions depends on several variables, of course. The elastic properties of the tissues, the
driving air pressure and flow characteristics, and tissue size and geometry all
are expected to influence the waveform structure. From the standpoint of model construction, however, it is not
critical that we know the exact structure of the input waveform. Indeed, it may be possible to infer the
input waveform of the remarkable progression of signals shown in Figure 8B, for
example, by postulating that a fixed resonator spectrum is being driven by an
input pulse of sequentially increasing peak frequency, intensity, and bandwidth. The freedom that models provide to explore
the consequences of various assumptions is one of the principle motivations
behind their creation.
2) Damped Resonator Component
A resonant component is suggested by the many reflective structures
that partially surround the soft (source) tissues within the dolphin’s nasal
passages – acoustically these structures should create resonant interactions in
addition to forward reflections. It is
also suggested by the decay-like characteristics of odontocete echolocation
signals – these signals are strongly reminiscent of the responses of resonators
(one or more – with resonant spectra dependent on species) to broadband pulse
inputs. In the discussion below, we
first visualize the behavior of a resonator driven by an impulsive input. Next we consider a little theory for a
simple spherical (partially air-bounded) resonator in order to obtain a general
idea of its expected filtering effects.
Finally we illustrate some examples of dolphin-like signals generated by
simple resonator models driven by broadband sources of the type discussed in the
previous section.
What happens to
an outward propagating pulse if we partially surround the source region with
air spaces? This is not hard to
simulate, so let’s try it. Figure 15
illustrates a movie of the same outward propagating pulse as in Figure 12, but
this time surrounded by a few equally spaced “air sac” points on the (green)
circle surrounding the source. Instead
of the single outward propagating pulse that we saw in Figure 12, Figure 15
illustrates a decaying series of oscillations propagating away from the
resonator. Figure 16 is a plot of these
oscillations recorded at the center of the simulation grid.
Figure 15. Movie of Gaussian pulse emission by a
partially air-bounded source (click on figure to start). The resonator diameter is 2.5 mm (indicated
by green circle).
Figure 16. Plot of the signal recorded at the center of
the grid in Figure 15 movie.
What causes
these oscillations? The air-filled sacs
within soft tissue act like very good mirrors reflecting and/or scattering almost
all sound incident upon them. Depending
on the soft-tissue sound speed distribution and the geometry of the air sacs
and source, resonances (or nulls) are created by reinforcement (or
interference) of the reflections within any such enclosure. The resonator model is simply responding
with its characteristic spectrum to the Gaussian pulse that was input at its
center. As we shall see below, for some
simple enclosure and source geometries, the expected resonance spectra can be
calculated.[3] In cases involving more complicated
geometries such as the dolphin nasal air sac system, time-domain simulation is
an excellent way to determine the characteristic spectrum of the nasal air sac resonator.
As a highly
simplified theoretical example, consider the case of a completely air-bounded
spherical soft-tissue enclosure. A
completely air-bounded enclosure would approximate an undamped resonator. Taking the speed of sound within the soft
tissue volume to be c=1500m/s, and
the radius to be r=1.25cm, the first
two sets of radial modes would be:
Monopole (0th
order) Radial Modes:
The frequency of
the nth monopole radial
mode is just the nth
harmonic of f0,1=60kHz:
f0,n = n f0,1 = n (c/2r) = n
(60 kHz).
Dipole (1st
order) Radial Modes:
The dipole
radial mode does not contribute harmonics.
Instead, the first three resonant frequencies are:
f1,1 = 85.8 kHz,
f1,2 = 148 kHz, f1,3 = 208 kHz.
A
centered monopole source, however, would be expected to couple primarily into
the monopole radial modes, and the resonant structure of such an enclosed source
should be predominantly harmonic. Although
the symmetry properties of the click source mechanism in dolphins are not yet
known, the mathematical model proposed by Dubrovsky and Giro (2002) is in fact
a monopole source. For the present
purposes, we will simply assume that dolphin click source mechanism is
monopole. Figure 17 sketches the
monopole mode spectrum for an air-bounded spherical soft-tissue enclosure.
Figure 17. Sketch of the lowest order radial mode
harmonics of a completely air-bounded spherical enclosure. A harmonic series of delta functions occurs
in this case.
Of course, the nasal air sacs pictured in Figure 3 do not entirely
enclose the soft tissue source region (here assumed to be the right MLDB
complex). A partially air-bounded enclosure
creates decaying oscillations as sound “leaks out” of the enclosure. Instead of the infinitely high and narrow
delta functions sketched in Figure 17, the cavity resonances broaden and weaken
by amounts that depend on the fraction of the surface enclosed by air, its
geometry, and the resonant frequencies.
Figure 18 qualitatively indicates how this might modify the spectrum of
Figure 17.
Figure 18. Sketch of a possible spectrum of a partially
air-bounded spherical enclosure.
Note that we are discussing the response or filter spectrum of the
resonator component here, and not the combined system. We have not yet considered the filtering
effect of projecting or focusing the signal into a beam. The filtering created by the final
(projector) component will be discussed in the next section. Even if certain tissues within the dolphin
forehead contribute to more than one of the three model component behaviors
discussed here, it is still possible to model each stage in the emission
process separately – that is, as having a well-defined spectral filtering
effect. As we shall see below, linear
systems theory tells us how to combine the component filters into a complete
model (see also Appendix 1).
To conclude this section, let’s consider some specific examples of dolphin-like signals generated by simple resonator models driven by broadband source pulses. Figure 19 illustrates three different input (or driving) and output (or response) signals generated by another simple resonator model. This model consists of a circle of points surrounding the source with a velocity lower than the surrounding medium. Because of the axial symmetry of the simulation, this model corresponds physically to a thin spherical shell of lowered (perturbed) velocity. The value of the velocity perturbation in this model can be adjusted to create varying resonance strengths (or Q-values) – note that the resonance peak frequencies also vary with the velocity perturbation. Because of the spherical symmetry of this model, there is no directional variation in the emitted signal; hence we may record the resulting signal anywhere outside the resonator model.
Figure 19. Simulated responses of a simple spherical
shell resonator model to signals input at its center. (A) Input signal 1. (B)
Resonator response to input 1. (C)
Input signal 2. (D) Resonator response
to input 2. (E) Input signal 3. (F) Resonator response to input 3.
The first two response signals of this simple model (Figures 19B, 19D)
are similar to the two variations of bottlenose dolphin echolocation clicks
shown in Figure 7. The response signal
seen in Figure 19F is similar to the harbor porpoise echolocation click shown
in Figure 6. The center frequencies,
bandwidths, and phases of the input signals (Figures 19A, 19C, 19E) were
adjusted to yield responses that approximate the shapes of the recorded
echolocation clicks. The center
frequencies of signals D and F are only roughly matched to the echolocation
clicks shown in Figures 6 & 7 – to get closer matches, one needs to match
the modal frequencies of the model to the echolocation click peak frequencies
(this can be done by changing the radius of our simple spherical shell
resonator model). All three input
signals are Gabor functions with the zero-frequency component removed.[4]
It should be emphasized that a wide variety of dolphin-like response
signals are obtainable by varying the peak frequency, bandwidth, and phase of
the driving signals – this is obvious from a signal processing
perspective. In addition, varying of
source positioning within the resonator will affect its coupling into the
natural modes of the resonator – this is obvious from an acoustical
perspective. If all of these factors
are allowed to vary along with resonator model geometry, it becomes clear that
a very wide range of signals similar to dolphin echolocation clicks can be
produced. In principle, though, these
variations all result from driving simple resonator models with broadband pulse
inputs.
It is worth mentioning two additional connections between these simple
models and observed echolocation click characteristics. First, the longer decay of the signal in
Figure 19F was produced using a more resonant model (the velocity perturbation
of the spherical shell model was increased in this case). In the case of an air-bounded soft tissue
resonator, increasing the amount of air-covered surface around the source
greatly increases the strength of the cavity resonances. As mentioned in the anatomy overview, the
harbor porpoise, the commerson’s dolphin, and the franciscana are known to have
relatively larger air sacs enclosing the nasal soft tissue region. It thus makes sense that their recorded echolocation
signals exhibit significantly longer decays.
Second, the left and right sides of the nasal air sacs in some
odontocetes may actually create two coupled resonant cavities. If two resonators having similar resonances
are placed side by side and one is driven with a broadband pulse, delayed and
attenuated reverberations generally occur some time after the beginning of the
initial response. The echolocation
clicks of the Dall’s porpoise (Phocoenoides
dalli), the Commerson’s dolphin (Cephalorhynchus
commersonii), and some other odontocetes (Au 1993) exhibit delayed
components. Close examination of the
harbor porpoise echolocation click and its spectrum shown in Figure 6 indicates
that some delayed signal components are present in this species as well. Dudok van Heel (1981), Kamminga and Wiersma
(1981), and Wiersma (1982) have all suggested that these delayed low-amplitude
signal components are reverberations due to reflections from the skull and/or
nasal air sacs. In general support of
this idea, the time separation of the signal reverberations for all five
animals pictured in Figure 7.20 of Au (1993) appears to scale approximately
with their relative size. The
exponential amplitude dropoff seen in subsequent reverberations also generally
supports a coupled resonator reverberation hypothesis.
Most recordings of dolphin echolocation
clicks are measured at some distance in front of the animal that emits the
click. This means that these recordings
actually include the filtering effect of the beam-forming or focusing produced
by the forehead tissues. The next
section discusses this final filtering component of the emission system.
3) Reflective and/or
Refractive Projector Component
The signal produced by a nasal click source (and resonator) must pass
through the dolphin’s forehead tissues before entering seawater. The forward-reflective properties of the
skull and nasal sacs and the refractive properties of the melon and other
forehead soft tissues combine to produce additional filtering of the
signal. For this reason, a “projective”
element is considered to be the third and final component of the dolphin
emission model.
Beam-forming lenses or mirrors are essentially spatial filters that are
peaked in some focal direction. While
descriptions of lens and mirror behavior can be found in any elementary optics
textbook, in the discussion below we will add a couple of less well-known
properties that are relevant to modeling the dolphin echolocation system.
Let’s begin by considering the behavior of a refractive lens. The shape of the lens, the distribution of
refractive index within the lens, and the positioning of the source with
respect to the lens are obvious factors that affect refractive focusing by the
lens. In addition, for the range of
frequencies of interest to dolphin biosonar, the wavelengths are not small in
comparison to the dimensions of the refractive forehead soft tissues. Hence diffractive effects also need to be
included in an appropriate filter model (in this case, lens filtering depends
on both direction and frequency). For
common lens shapes and finite wavelengths, diffractive effects generally
produce a high-pass frequency filtering in the focal direction.
Similarly, a simple parabolic reflector constitutes a combined directional and frequency filter element. The geometrical acoustics directional pattern for a source at the focus of a true parabola would be a delta function in the focal direction. In the case of a finite aperture/wavelength ratio, however, diffraction modifies this pattern into a high-pass (or high-boost) frequency filter in the focal direction, where the knee of the boost transition frequency depends on the aperture size (and focal distance) of the parabola.
Figure 20. Qualitative plot of the directivity index of
the primary beam versus frequency for a real projector element.
Figure 20 illustrates what the high-pass filtering might look like for
a semi-parabolic reflector and/or lens element (or a model combining both)
assuming that the projector creates a primary beam. Figure 20 qualitatively plots the directivity index (DI) of this
primary beam versus frequency. Clearly,
diffraction must eliminate the peak for wavelengths much larger than the
projector. In other words, because
waves spread uniformly in all directions in the low-frequency limit, the DI
must approach zero as f®0.
Furthermore, some acoustic ‘leakage’ and coupling inefficiencies always
exist in real systems, and this limits the gain that is possible in the focal
direction. Hence directivity indexes
for real systems generally level out to yield a finite boost in the
high-frequency limit.
Three-dimensional simulations of the focal characteristics of the forehead tissues of the common dolphin do in fact produce directivity indexes that follow this type of curve. Figure 21 plots out the simulated DI values for the forward peak for two different tissue models from Aroyan (1996). Note that the range of simulated frequencies in this study spanned only a portion of the upper knee of the transition region. The most complete tissue model (the skull, melon & air sacs model) clearly creates a more focused forward beam than the skull-only model. We might also note that by around 100 kHz, the common dolphin is gaining most of the beam-forming benefit of its projector element (as modeled).
Figure 21. Plot of the simulated directivity index of
the forward peak versus frequency for two different 3D forehead tissue models
of the common dolphin (data from Aroyan 1996).
Now let’s check out the behavior of some simple projector models. First, let’s consider a parabolic reflector having the dimensions diagrammed in Figure 22. In this model, the reflector is composed of air sacs (discretely modeled pressure release points). The aperture diameter is 8.0 cm (model rotated about the symmetry axis), and the source is placed at the focus of the parabola at a distance of 1.3 cm from the vertex. Although this example may appear trivial, two non-trivial details of potential relevance to dolphin echolocation emission are discussed below.
Figure 22. Assumed geometry of parabolic reflector
model.
Figure 23 illustrates a movie of the emitted waveform when a Gaussian pulse (Figure 14A) is input into the focus of the parabolic reflector model diagrammed in Figure 22. The emitted signal was recorded (on an enlarged simulation grid) at a distance of 30 cm to the right of the focal point along the symmetry axis, and is displayed along with its power spectrum in Figure 24. The far field pattern for the emitted waveform forms at around 15 cm from the focus, so the signal in Figure 24A is representative of the emitted far field signal along the primary axis.
Figure 23. Movie of the waveform emitted by the
parabolic reflector model diagrammed in Figure 22 using a Gaussian input (click
on figure to start). The reflector
location is indicated by the blue curve, and the source (focal) point by a
small green semi-circle.
Figure 24. (A) Waveform and (B) power spectrum of the
signal emitted by the parabolic reflector model. Signal was recorded at 30 cm along model axis from focal point of
parabola.
Note first that a forerunner (or precursor) signal is seen in both the
movie and the signal waveform (Figure 24A).
This precursor is simply the portion of the source pulse that did not
strike the reflector, and therefore continues to spread (spherically if no
other elements are present) from the source point. The ratio of amplitudes of the precursor to the main signal
approaches a constant value in the far field that depends on several
factors. This precursor is not
reflected and hence is not filtered by the reflector element’s transfer
function. In our simple reflector
model, it therefore cleanly represents the original source pulse. Multiple reflections within the dolphin’s more
complicated forehead tissues, however, may mix the precursor and main signal
components (especially for longer source pulses).[5]
The second thing to note is that the spectrum of the emitted signal in
Figure 24 contains two broad resonances that look similar to the bimodal
spectra of some bottlenose dolphin and beluga clicks. In order to understand this result, it is perhaps better to
consider the transfer function linking the input and output of the reflector model
rather than its response to any one input.
Since we know both the input and the output (emitted) waveforms in this
model, we can use spectral methods to calculate the filter spectrum of the
reflector model itself (see Appendix 1).
The power spectrum of the parabolic reflector is plotted in Figure 25. The filtering produced by the parabolic reflector is mainly a treble boost spectrum (qualitatively sketched in Figure 20) with some smaller oscillations superimposed. These oscillations are due primarily to a delayed-signal component (a pure delay spectrum has much stronger peaks and nulls). The mild resonances at around 55, 110, 165, and 220 kHz result from the rippled-spectra of the pulse ‘echo’ reflected off the parabola. This can be demonstrated by gating out the precursor in the received signal (Figure 24A), then computing the reflector transfer function again. The result, shown in Figure 26, retains the treble-boost spectrum of Figure 25 but loses most of the superimposed oscillations. Note that adding a refractive lens in front of this reflector model might lower the frequency of the boost transition, but probably would not affect the resonance features of Figure 25.
Figure 25. Power spectrum of the filtering created by
the parabolic reflector model.
Figure 26. Power spectrum of the parabolic reflector
model calculated with the precursor pulse gated out of the received signal.
The rippled-spectra portion of Figure 25 appears to correlate with the even peaks of an echo spectrum having
(roughly) a 30 kHz fundamental, which is somewhat unexpected for this
model. Recall that the reflector is
modeled as a pressure release surface.
Pressure release surfaces invert the phase of reflected waveforms, so
one might have expected this reflector to create an inverted and delayed
Gaussian pulse following the precursor (ignoring other subtleties) – this does
indeed occur very close to the reflector surface (as seen in the movie). A signal consisting of pure delayed and
inverted pulses has spectral peaks at the odd
(n = 1,3,5…) harmonics of the
fundamental delay frequency. The delay
is approximately Dt = 2*a/c = ~17 ms, where a = 1.3 cm is the
parabola focal distance, and c = 1500
m/s is the assumed sound speed. This
yields an expected rippled-spectra fundamental of f = 1/(2*Dt)
= 29 kHz. A signal consisting of
delayed but non-inverted pulses has spectral peaks at the even (n = 0,2,4…) harmonics of the fundamental
delay frequency.
The main part of the emitted pulse in Figure 24A is not merely a phase-inverted copy of the
initial pulse, even though it starts out this way. The movie illustrates that a crest following the inverted pulse
begins to develop even before it has completely exited the reflector. Some energy can be seen scattering off the
end of the reflector, and the scattered wavefronts eventually contribute to the
formation of the crest. Decreasing the
aperture size of the reflector model increases the magnitude of the resonances
and also increases the frequency of the boost transition. The mild resonances in the spectra of
parabolic reflectors may be of some importance because they provide a simple
way to generate pulses with bimodal spectra similar to the recorded clicks of
bottlenose dolphins (and other odontocetes).
Driving a parabolic reflector with pulses having spectra peaked near
these resonances produces signals very similar to the clicks in Figures 6-8.
Behavior of Combined Elements
Now imagine an acoustical system combining the three elements described
above … To be completed.
Summary of Conclusions
To be completed.
Appendix on Spectral Methods
To be completed.
References:
Amundin M, Cranford T (1990) Forehead anatomy of Phocoena phocoena and Cephalorhynchus commersonii:
3-dimensional computer reconstructions with emphasis on the nasal
diverticula. In Thomas JA, Kastelein R
(eds), Sensory abilities of cetaceans :
Laboratory and field evidence. New
York: Plenum, pp. 1-18.
Aroyan JL (1990) Numerical
simulation of dolphin echolocation beam formation. M.S. thesis, University of California, Santa Cruz.
Aroyan JL (1996)
Three-dimensional numerical simulation of biosonar signal emission and
reception in the common dolphin. Ph.D.
dissertation, University of California, Santa Cruz.
Aroyan JL, McDonald MA,
Webb SC, Hildebrand JA, Clark D, Laitman JT, Reidenberg JS (2000) “Acoustic Models of Sound Production
and Propagation.” In: Au WWL, Popper
AN, Fay RR (eds), Hearing by Whales and
Dolphins. New York:
Springer-Verlag, pp. 409-469.
Aroyan JL, Cranford TW,
Kent J, Norris KS (1992) Computer modeling of acoustic beam formation in Delphinus delphis. J. Acoust. Soc. Am. 92:2539-2545.
Au WWL (1993) The Sonar of Dolphins. New York:
Springer-Verlag.
Au WWL (2000) Echolocation
in Dolphins. In: Au WWL, Popper AN, Fay RR (eds), Hearing by Whales and Dolphins. New York: Springer-Verlag, pp.
364-408.
Cranford TW (1992)
Functional Morphology of the Odontocete Forehead: Implications for Sound Generation. Ph.D. dissertation, University of
California, Santa Cruz.
Cranford TW, Amundin M,
Norris KS (1996) Functional morphology and homology in the odontocete nasal
complex: implications for sound generation.
J. Morph. 228:223-285.
Dubrovsky NA, Giro LP
(2002) Modeling of the clicks production mechanism in the dolphin. In Thomas JA, Moss C, and Vater M (eds), Echolocation in Bats and Dolphins. Univ.
Chicago Press (in press).
Dudok van Heel (1981)
Investigations on cetacean sonar III: A proposal for an ecological
classification of Cetacetes in relation to sonar. Aquat. Mamm. 8:65-68.
Fraser FC, Purves PE (1960)
Hearing in cetaceans: evolution of the accessory air sacs and the structure and
function of the outer and middle ear in recent cetaceans. Bull. British Mus. (Nat. Hist.), 7:1-140.
Giro LR, Dubrovskiy NA (1973)
On the origin of low-frequency component of the echoranging pulsed signal of
dolphins. In Ref. Dokl. 8-ya Vses.
Akust. Konf. (Proceedings of the 8th All-Union Acoustics Conference), Moscow,
1973.
Giro LR, Dubrovskiy NA
(1974) The possible role of supracranial air sacs in the formation of echo
ranging signals in dolphins. Akust.
Zh., 20(5):706-710.
Green RF, Ridgway SH, Evans
WE (1980) Functional and descriptive anatomy of the bottlenosed dolphin nasolaryngeal
system with special reference to the musculature associated with sound production.
In Busnel RG, and Fish JR (eds), Animal
Sonar Systems. New York: Plenum, pp. 199-238.
Heyning JH (1989)
Comparative facial anatomy of beaked whales (Ziphiidae) and a systematic
revision among the families of extant Odontoceti. Contributions in Science,
Nat. Hist. Mus. of Los Angeles County, 405:1-64.
Kamminga C, Wiersma H
(1981) Investigations on cetacean sonar II: Acoustical similarities and
differences in odontocete sonar signals.
Aquat. Mamm. 8(2):41-62.
Lawrence B, Schevill WE
(1956) The functional anatomy of the delphinid nose. Mus. Comp. Zool. Bull.,
114(4):103-151.
Litchfield C, Karol R,
Mullen ME, Dilger JP, Luthi B (1979) Physical factors influencing refraction of
the echolocative sound beam in Delphinid Cetaceans. Mar. Bio. 52:285-290.
Mead JG (1975) Anatomy of
the external nasal passages and facial complex in the Delphinidae (Mammalia:
Cetacea). Smithsonian Contr. Zool. 207:1-72.
Moore PWB, Powloski D
(1990) Investigation of the control of echolocation pulses in the dolphin (Tursiops truncatus). In Thomas JA, Kastelein R (eds), Sensory abilities of cetaceans : Laboratory
and field evidence. New York:
Plenum, pp. 305-316.
Morris R (1986) The
acoustic faculty of dolphins. In:
Bryden MM, Harrison R (eds), Research on
Dolphins. New York: Clarendon, pp.
369-399.
Norris KS (1964) Some
problems of echolocation in cetaceans.
In: Tavolga WN (ed), Marine
Bio-acoustics, Vol. 2. New York:
Pergamon, pp. 317-336.
Norris KS (1968) The
Evolution of Acoustic Mechanisms in Odontocete Cetaceans. In: Drake ET (ed), Evolution and Environment.
New Haven: Yale University Press, pp. 297-324.
Norris KS, Evans WE, Turner
RN (1966) Echolocation in an Atlantic bottlenose porpoise during discrimination. In Busnel RG (ed), Animal Sonar Systems: Biology and Bionics. Frascati, Italy, pp. 409-437.
Norris KS, Harvey GW (1974)
Sound transmission in the porpoise head.
J. Acoust. Soc. Am. 56:659.
Oelschlager HA (1986) Comparative
morphology and evolution of the otic region in toothed whales (Cetacea:
Mammalia). Am. J. Anat. 177:353-368.
Purves PE, Pilleri G (1983)
Echolocation in Whales and Dolphins.
Academic Press, London.
Popper AN (1980) Sound
emission and detection by delphinids.
In: Herman LM (ed) Cetacean
Behavior: Mechanisms and Functions.
New York: John Wiley & Sons, pp. 1-52.
Schenkkan EJ (1973) On the
comparative anatomy and function of the nasal tract in odontocetes (Mammalia,
Cetacea). Bijdragen tot de Dierkunde 43(1):93-112.
Varanasi U, Feldman HR,
Malins DC (1975) Molecular basis for formation of lipid sound lens in echolocating
cetaceans. Nature 255:340-343.
Varanasi U, Markey D,
Malins DC (1982) Role of isovaleroyl lipids in channeling of sound in the porpoise
melon. Chem. Phys. Lipids 31:237-244.
Wiersma H (1982)
Investigations on cetacean sonar IV: A comparison of wave shapes of odontocete
sonar signals. Aquat. Mamm. 9(2):57-67.