© 2006 International Council for the Exploration of the Sea
Using multi-angle scattered sound to size fish swimbladders
Marine Physical Laboratory, Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 92093-0238, USA
*Correspondence to J. S. Jaffe: tel: +1 858 5346101; fax: +1 858 5347641. e-mail: jules{at}mpl.ucsd.edu.
Common current practice in fisheries acoustics is to use sound that has been backscattered at 180° in order to infer parameters of individual or aggregations of animals. This article proposes that there is interesting information that can be obtained by processing scatter from other observation angles. Using a simple one-dimensional model of scatter from a fish swimbladder, an expression is derived that predicts the location of the nulls of the scattered sound as a function of transmit angle, observation angle, and tilt. The model was used retrospectively to compute the size of a swimbladder from an existing data set (that of Foote, K. G. 1985. Rather-high-frequency sound scattering by swimbladdered fish. Journal of the Acoustical Society of America, 78: 688–700), with good agreement. In order to pursue the development of a pragmatic collection system, a method is suggested that uses a single transmitter with multiple receivers. The locations of the receivers can be determined using a design methodology that considers bandwidth, centre frequency, and the size of the intended object. The method ensures that a spatially unaliased backscattered waveform can be measured over a specified sampling interval. The technique is illustrated with a practical example that uses a small number of receivers, placed in the backscattered hemisphere.
Keywords: acoustic diffraction, multi-angle sound, swimbladder estimation
Received 15 June 2005; accepted 25 April 2006.
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Introduction |
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 Top Introduction Theoretical backgroundEstimating swimbladder size from...Extension to a single-look,...Discussion and conclusionsReferences |
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A crucial need in understanding the status of any ecosystem is knowledge of abundance that includes size frequency information, or demographics, of the population. The use of remote sensing tools such as acoustics has many advantages for obtaining information about aquatic ecosystems. However, active sonar systems would be more valuable if reliable information could be obtained in judging various characteristics of the animal population. Therefore, if sonar systems could not only count animals but also more accurately estimate their size than currently possible, it would be of great advantage.
Advanced echosounding techniques have included the development of both dual- and split-beam methods (Ehrenberg, 1979). The more recent use of multibeam systems permits, in addition to recording individual responses, collection of information about fish school shape (Gerlotto et al., 1999). Multibeam systems have also been used for judging the in situ behaviour of marine organisms (Jaffe et al., 1995, 1998, 1999; Genin et al., 2005). Over many decades of observation, the utilization of echosounding techniques has progressed to a point where routine observations are made in order to provide reliable estimates of standing stocks of many commercially valuable species (MacLennan and Simmonds, 1992).
One of the greatest problems in estimating animal abundance is the strong dependence of individual or group backscatter cross-section on animal orientation. Therefore, for example, McClatchie et al. (1996) state that "Information on fish orientation has lagged behind the development of models to estimate target strength from fish swimbladders, despite fish tilt being an important variable influencing target strength." As pointed out by McQuinn and Winger (2003), animal orientation can systematically alter the amount of reflected energy, which in turn will lead to a bias in estimated abundance.
Recent work on the problem highlights the importance and potential benefits of generating more accurate estimates of the size distribution of marine populations. In one case, Stanton et al. (2003) demonstrated that extremely wideband sound could be used to resolve echoes from the front and back of targets, and hence to estimate target "thickness". Size estimates have also been performed through temporal analysis of echo signal length (Burwen et al., 2003).
Here, I demonstrate that multi-angle scatter can be used to estimate both orientation and size via the systematic dependence of scatter upon animal orientation. The theory, and its success in inferring target size, is demonstrated through reconciling a previously published data set of multiple aspect dependent measurements of a single target (Foote, 1985). An extension of the method is proposed which uses a set of transducers to sense the scattered sound at multiple angles and to obtain, from a single look, the size and orientation of the animal.
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Theoretical background |
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TopIntroductionTheoretical backgroundEstimating swimbladder size from...Extension to a single-look,...Discussion and conclusionsReferences |
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The underlying theory for the idea can be understood from examining the relationship between the sound that has been scattered from an object, in this case an animal, and its measurement at some substantial distance. The basic geometry is illustrated in Figure 1, which depicts an acoustic wave incident on a three-dimensional object. The object can be viewed as a secondary source that re-radiates acoustic energy. Although the details of the physics are somewhat beyond the intended scope of this article, under a certain range of circumstances, the acoustic radiation emanating from this object can be approximated by either a single two-dimensional object or a set of two-dimensional objects whose aspect-dependent scatter has a straightforward relationship with the true three-dimensional object (Kak and Slaney, 2001). The latter case forms the basis of tomography. The figure is meant to illustrate the case where a single two-dimensional object will suffice. In this case, the object can be represented by a plane, here shown to be a shaded two-dimensional representation of the true three-dimensional object. The reflected acoustic radiation is then propagated over a distance z that is large enough for the receiver to be in the "far field". This is ordinarily met if z > d2 /
, where d is the length of the object and is the wavelength of the sound. In addition, we require that there are no scatterers or sources in the intervening medium.
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Under the above circumstances, the well-known relationship between the complex valued source, U(
, 
), and its far-field waveform U(x, y; z), the amplitude and phase of the acoustic wave field at z, can be computed as (Goodman, 1996)
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| (1) |
. Note that the time-dependence of the acoustic field has been eliminated, and also that the equation describes the amplitude and phase relationship between a source and its far-field pattern for only a single wavelength. If non-monochromatic sound is used, the contributions from the individual frequencies can be linearly superposed.
The main advantage of this expression is that, neglecting the phase factors preceding the integral, the measured complex radiation pattern at range z: U(x, y, z) can be viewed as a Fourier Transform of the complex radiation pattern of the source U(, ). Here, the Fourier coefficients are viewed at spatial frequencies fx = x / z, and fy = y / z. Note that herein the subscript refers to spatial, not temporal, frequency. A simple model for the animal can then potentially be used to infer its spatial features, given the sound field recorded at (x,y,z).
As a hypothetical example, assume that an animal can be regarded as consisting of two scattering functions: (1) a constant value object of length L whose value is determined by the acoustic contrast of the animal; and (2) a modulation that is superimposed upon this value. If (2) is small relative to (1), the scattered pattern will mostly resemble (1). On the other hand, if (2) is large compared with (1), the scattered pattern can be more complex. Interestingly though, because the object is in some sense the spatial spectrum of the complex waveform measured at z; U(x, y, z), and it is of finite extent, the Fourier Transform of the measured waveform is spatially band-limited. This means that the Fourier spectrum has no values greater than a given spatial frequency; referred to as the band limit. Additionally, this ensures that values at U(x, y, z) can be interpolated from measurements at a finite number of locations (Gonzalez and Woods, 2002).
As one implementation of the theory, let us call L the "characteristic length" of the animal, the length that dominates the scatter pattern. For fish with swimbladders, this will mostly be due to the swimbladder when 8 < L/ < 36 (Medwin and Clay, 1998). The angularly dependent pattern of sound scattered from this simple target can be computed straightforwardly. Consider a simple one-dimensional model (Figure 2) of reflection by an animal equivalent to an absolutely reflective bar of length L and orientation 
tilt relative to the axis. Owing to the orientation of the bar, the reflected sound will suffer an additional change in path length of 2 tan tilt, compared with the case when the bar is simply perpendicular to the incident beam. This introduces a "phase delay" in the arriving waveform of value 2
i(2 tan tilt) / , which implies that the far-field pattern is then the Fourier Transform of the function 
integrated over the suitable limits of ±L cos tilt / 2:
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| (2) |
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| (3) |
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obs = x / z, simplifying the expression allows one to write
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| (4) |
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Estimating swimbladder size from the simple model |
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TopIntroductionTheoretical backgroundEstimating swimbladder size from...Extension to a single-look,...Discussion and conclusionsReferences |
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In order to examine the utility of this simple model, a procedure was used which refined the output of the model against an existing and well-known data set (that of Foote, 1985). In order to estimate L, the characteristic scattering length, a match is found between the locations of the innermost four nulls, two on each side, relative to the central peak of the observed data with the predicted location of the nulls, using Equation (4). This was accomplished through a simple exhaustive search that minimized the mean squared error between the predicted and the observed locations of the nulls as a function of L for each of the four frequencies in Foote's (1985) data. The nulls of this Sinc function occur when , and n is a non-zero integer. For comparison, the separation between the source and the receiver was assumed to be zero (
obs = 0), which implies that L(–2 sin tilt) / = ±n. The observed locations of the nulls were estimated by visual inspection of the published data (Foote, 1985), shown in Figure 3.
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Table 1 contains a comparison of the locations of the nulls estimated from Foote's (1985) data with those derived via the error-minimization procedure. The predicted values of L are also shown for each frequency. The table illustrates that the predicted values are somewhat close to the measured swimbladder length of 10.5 cm. Moreover, the various frequencies lead to a consistent estimate. Figure 4 contains a graph of the least-squared residual error between the observed and predicted location of the nulls as a function of L for the four frequencies. Interestingly, the lowest frequency graph seems to display a steeper minimum, whereas the highest frequency has a shallower minimum. This suggests that the use of lower frequency sonar, relative to the size of the scatterer, may be an advantage. A possible explanation for this observation is that the second derivative of the mean squared error is inversely proportional to
. Hence, lower frequencies should have a steeper null.
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Extension to a single-look, multi-angle scatter system |
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TopIntroductionTheoretical backgroundEstimating swimbladder size from...Extension to a single-look,...Discussion and conclusionsReferences |
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Although the above method produced interesting results in inferring animal bladder size when source and receiver were co-located and the animal was systematically tilted, the development of a practical system for in situ characterization requires a system that can obtain such results from a single or a very limited number of transmissions. In general, the invertability of Equation (1) is guaranteed if the complex scattered field can be obtained. If the sound scattered by the animal can be approximated by a simple model, then a set of multi-angle observations can provide the required data. A simple means of acquiring these data is to have a single transmitter and a large, almost continuous, number of receivers. Unfortunately, this scheme is impractical for a true observational system. Alternatively, a discrete number of receivers can be used in conjunction with varying wavelengths of incident sound. In addition, if all that is desired is an estimate of L, observation of the central peak and nulls of the scattered sound can be used for this purpose, as shown above. Here, I describe a sampling theorem and a strategy for selecting a set of receiver angles using the relationships between the set of available angular separations, bandwidth, and the size range of the desired characteristic lengths.
Sampling strategies
In order to explore the options available in varying both view angle and wavelength, a geometric approach is taken. As Equation (2) can be interpreted as regarding the scattered field as a superposition of spatial frequencies; fx = x / z = tan obs / , a graph (Figure 5) of vs. tan obs reveals that radial lines from the origin have a slope equal to fx. Changing and keeping tan obs constant therefore leads to sampling different values of fx. Figure 5 demonstrates the set of observable values of fx given a range of wavelengths 
centred about some value and a range of potential angular views from obs = 0 to obs = max. The design of a multi-angle system can be viewed as selecting some set of observation angles in consideration of the available central value for and the bandwidth.
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for a given angle of observation, The graph reveals that, for a fixed
, larger observation angles result in observation of a greater range of spatial frequencies. This highlights the utility of using wideband sound, with a large number of wavelengths, in order to minimize the number of receivers. In addition, no spatial information other than fx = 0 is available at obs = 0. Two sampling strategies are now envisioned under the assumption that the system is designed to view animals of equivalent length ranging from Lmin to Lmax. First, the angular sampling necessary for unaliased estimation of the scattering function is described for a monochrome system. Next, the extension to varying wavelength is considered.
The angular sampling theorem for monochrome illumination that permits unaliased reconstruction of the object of finite size can be derived by considering the distances between the nulls of the Sinc function. Returning to the discussion of the equivalent bar of length L, consider that the nulls occur when L(cos tilt tan obs – 2 sin tilt) = ±n. This, in turn, implies that
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| (5) |
. When n = 0, 
, corresponding to the peak in the scattered sound pattern. Note that: (a) the spacing between the observation angles corresponding to these nulls incurs a minimum value (for fixed tilt) when = min and L = Lmax, i.e. when / L is a minimum. In the case that tilt = 0, the sampled spatial frequencies are fx = n/L, a simple statement of the Nyquist spatial sampling theorem (Gonzalez and Woods, 2002); (b) for a given tilt angle, fx = (n / (L cos tilt) + 2e tan tilt) / ) 
n / L, indicating that the spatial frequencies are farther apart for the tilted object because tangent is a monotonically increasing function. In consideration of (a) and (b), the sampling theorem below may be stated.
Theorem An unaliased version of the scattered sound can be obtained if Equation (5) is used for the determination of the tilt angles with = min and L = Lmax. In this case, the spacing between 
and 
can be written as
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| (6) |
Although the above strategy guarantees that the sampled waveform will not be aliased, there are many, but two very good, reasons for not adopting this approach. First, the exact reconstruction of the scattering object can only be obtained when both phase and amplitude are available. Unfortunately, phase is not commonly measured in acoustics. Second, the sampling theorem assumes that a complete data set is available. Measurement of the scattered sound only over a finite range of values will result in multiplication of the observed data by an aperture whose angular extent is the range of observation angles. This will result in a convolution of the reconstructed function with the Inverse Fourier Transform of the aperture. Although various schemes that use apodization can ameliorate this effect, the end result for a small angular range of views will likely be poor.
One idea for circumventing these difficulties is to use a range of wavelengths in order to obtain a number of values of fx = tan(obs) / for a single obs. In this way, nearly continuous observation of fx can be obtained by judicious choice of the 
angles when combined with varying wavelengths. Figure 5 can be used to illustrate the basic idea. Given 
and 
, select 
so that 
. Given 
, select 
so that 
until a value of 
is obtained. This ensures that there is continuous coverage over the observed set of fx for all values 
. The starting value of 
is assumed to be arbitrary, but selecting too large a value might miss a peak whereas selecting it too small will result in many observation angles. In practice, it is good to select this value so that it is less than the Nyquist angular frequency discussed above.
Inversion example
Here, implementation of the above scheme is considered in a practical context. The example is taken from sonar under development in my group. Assuming a centre frequency of 200 kHz and 40% bandwidth (80 kHz) implies that the wavelengths are 6.25 mm 
9.375 mm. For practical purposes, this was then discretized into nine frequency bands between 160 kHz and 240 kHz (160, 170, 180, 190, 200, 210, 220, 230, and 240 kHz). Next, assume a test object of L = 10.5 cm, as in the Foote (1985) data set. The first null of the scattered sound from this object is at an angle of 4.28°, taking the centre wavelength and assuming no tilt. Therefore, assume that 
and arbitrarily select the first-look angle to be half the unaliased sampling frequency, so that 
. Implementing the above design, a set of angles is then determined so that = {0, 2.14, 3.2, 4.8, 7.18, 10.7} degrees. The sonar therefore has six receivers and one transmitter: the transmitter and the receiver at 0° are co-located. Note that the range of angles was selected so that the centre peak and the three innermost nulls can be recorded. A wider range of scattered angles also accommodates the possibility that the animal or its swimbladder is tilted (as in the Foote, 1985 data). Figure 6 shows a graph of amplitude vs. spatial frequency for this target. The figure shows that the nulls of the pattern are at spatial frequencies (mm–1) of {0.00952, 0.019, 0.0286} corresponding, to values of n = 1, 2, and 3. As 
, one can infer L easily. This yields 10.5 cm, 10.52 cm, and 10.48 cm. Because the target conforms to the model exactly, these values are close to the true value of 10.5 cm. The small discrepancy is due to the fact that the computer programme used to create the plot used linear interpolation between the sampled points (grey in Figure 6).
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Discussion and conclusions |
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TopIntroductionTheoretical backgroundEstimating swimbladder size from...Extension to a single-look,...Discussion and conclusionsReferences |
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This work has suggested the use of multi-angle scatter in order to infer animal size. In this context, both theoretical and experimental issues have been considered. The potential use of the technique is supported through the interpretation of an existing data set (that of Foote, 1985) in the context of a simple model where the swimbladder is approximated as an absolutely reflective bar of length L. The four frequencies in Foote's (1985) manuscript were used to produce an estimate with varying degrees of accuracy (10–25%). Given that the characteristic nulls of the scattered sound were inferred from a diagram in a journal manuscript, and not from the actual data, the estimates are surprisingly accurate. The reason that the swimbladder is underestimated might be because of its shape, or because the axis of the bladder and the source–receiver separation were not precisely aligned. Therefore, if the ends of the swimbladder were tapered and this resulted in less-reflected sound coming from the ends, the characteristic length of the bladder might be somewhat smaller than the measured length. Other effects such as curvature, not considered in the above model, might also result in some systematic changes. In any event, if the reason for the mismatch is geometric, perhaps a systematic correction can be used to correct the data.
Inspection of Figure 1 reveals that the ventral aspects of the animals are not as "Sinc-function like" as the dorsal ones. Therefore, the ability of this method to infer characteristic length from the ventral aspect is unknown and perhaps suspect. Fortunately, most commercial echosounders use a downward-looking sonar beam that is reflected from the dorsal aspect. However, the data also point out the potential problems in using the technique in side-looking mode. More sophisticated techniques may be needed in order to interpret this spatial information. As explained, because of the finite size of the object, the variations in the scattered (far) field cannot be arbitrarily large, but rather can be interpolated from a sufficiently close sampling interval with a special interpolation function (Gonzalez and Woods, 2002). Unfortunately, however, this does not guarantee that a good estimate of equivalent length can be obtained from a limited number of views over a small range of look-angles.
As one caveat of the method, it is necessary to isolate the recordings from single individuals. Although this may be a daunting task in high-density swarms, echo-counting can be performed in many situations of practical interest. Ranges from isolated, time-delayed reflections from single animals in conjunction with source–receiver distance define a three-dimensional ellipse where the animal must be located. Having a multitude of such ellipses allows computation of the animal's three-dimensional position. In addition, because the animal must be located inside the projected sound beam, a region of interest can be defined that restricts the animal location to that region. Nevertheless, the practical aspects of such a scheme, although theoretically possible, should be explored in all future experimental work.
Another interesting and important question concerns the extension of the method to the case where the geometry is different from that discussed here. For example, in Foote's (1985) data and the one-dimensional model used here, the long axis of the reflector is parallel to the line connecting the source and the receiver. In a real-world survey, there is no guarantee that the head-to-tail axis is parallel to the source–receiver separation. Two potential solutions to this problem exist. Given the probability density function that characterizes the probability of an animal being orientated in a given direction and the recorded, inferred lengths, an inverse transformation can be used to transform from the observed data to the set of animal lengths (Jaffe, 2005). Alternatively, a two-dimensional receiving array can be used to record the reflected data. Although the details of how this can be accomplished are not described here, the basic idea would be to employ a more sophisticated two-dimensional model that generalizes the concept of equivalent length to an equivalent two-dimensional shape that can be described by the characteristic dimensions Ltransverse and Lparallel.
As a practical aspect of the proposed methodology, the physical locations of the source and the receivers should be considered. In the case of a single-ship survey, the maximal separation between source and receiver is the ship length. However, if other types of sonar strategies are used, such as moored systems or systems that deploy the sonar via a bottom mount, the scattering angles that can be utilized could be quite large. Equation (6) implies that decreasing the wavelength of the sound relative to fixed animal length, will decrease the angular spacing of the nulls. If limited source–receiver separation is available, a bigger wavelength may prove advantageous. However, using too high a frequency results in a very shallow null, as demonstrated. The determination of an optimal frequency range for a set of animals in a certain size range is therefore an open issue.
As one generalization of the method, not considered in detail here, it may be that sophisticated models of the scatter of sound by fish could be employed to determine other interesting features of the animals. The fact that data are being collected using the sound field radiating from the animal at different spatial frequencies raises the possibility that spatial reconstruction of the animal might be possible. The idea and methods that can be used arise mainly in the area of tomography (Kak and Slaney, 2001). Perhaps in future it will be possible to perform reflection tomography on fish using sound. Reflection tomography techniques have been discussed in the literature (Lehman and Norton, 2004), but to my knowledge, their application to sensing marine organisms has not been pursued.
In summary, a simple model for swimbladder-bearing fish is proposed; it predicts the location of nulls in the far-field pattern of a radiating aperture of length L. The predicted pattern was then reconciled with a set of published data in order to estimate the animal's characteristic length L. Good agreement was found for this single case. A multi-static configuration is suggested in order to achieve similar results in a field-deployable system from a single transmission. It is conjectured that extensions of the theory to two-dimensional arrays to sense the scattered sound can potentially allow the inference of animal dimension from a sophisticated single transmit, multi-angle receive system.
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Acknowledgements |
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I thank my colleagues D. V. Holliday, Paul Roberts, and Alex De Robertis for constructive comments on the manuscript.
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References |
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TopIntroductionTheoretical backgroundEstimating swimbladder size from...Extension to a single-look,...Discussion and conclusionsReferences |
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