© 2006 International Council for the Exploration of the Sea
Improved parameterization of the SDWBA for estimating krill target strength
Southwest Fisheries Science Center 8604 La Jolla Shores Drive, La Jolla, CA 92037, USA
*Correspondence to S. G. Conti: tel: +1 858 534 7792; fax: +1 858 546 5652. e-mail: sconti{at}ucsd.edu.
Recently, a Stochastic Distorted Wave Born Approximation (SDWBA) model was proposed to improve target strength (TS) estimates for Antarctic krill, Euphausia superba. The krill shape is modelled by a collection of cylinders, and total sound scatter is estimated by semi-coherent summation of scatter from each element. The SDWBA model was evaluated with a generic krill shape comprising 14 cylinders and a phase variability of 
, and predictions were validated with empirical TS and total TS data at 120 kHz, and over a broad bandwidth, respectively. For general application, parameterization of the SDWBA model is improved to account explicitly for dependence among four of the model parameters: standard length of krill, number of cylinders used to describe its shape, amplitude of inter-element phase variability, and acoustic frequency. The model improvements are demonstrated, and the uncertainty in orientation distribution of krill beneath survey vessels and its ramifications on krill biomass estimates are highlighted.
Keywords: Antarctic krill, backscattering cross-section, SDWBA model, sensitivity, target strength
Received 13 September 2005; accepted 23 February 2006.
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Introduction |
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 Top Introduction MethodsResults and discussionConclusionReferences |
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Antarctic krill, Euphausia superba, are the basis of the ecosystem in the Southern Ocean and the target of a large international fishery. To support management strategies for the resource, surveys of krill dispersion and abundance are conducted using multi-frequency echosounders. Estimates of krill target strength (TS) are used to convert the total backscattered acoustic energy attributed to krill into an estimate of krill abundance (Hewitt et al., 2002). The accuracy of the biomass estimate is directly proportional to the accuracy of the TS estimate (Demer, 2004). Krill TS depends on the animal size, shape, and morphology, and the acoustic wavelength and angle of incidence.
To account for the many factors influencing krill TS, McGehee et al. (1998) proposed a physics-based model based on the distorted-wave Born approximation (DWBA; Morse and Ingard, 1968). The krill body is idealized by a string of discrete bent cylinders. Krill TS is estimated from the coherent summation of scattering from each of the cylinders. This model was validated for dorsal aspect, on the main scattering lobe, at 120 kHz, from measurements with a live krill (McGehee et al., 1998). However, there were large discrepancies between the model and data off the main axis.
Demer and Conti (2003a) introduced the Stochastic Distorted Wave Born Approximation (SDWBA) to explain the discrepancies between measurement and theory. Relative to the DWBA, the SDWBA accounts for the flexure and complexity of the krill body by incorporating a phase variability term 
j between the scattering form functions of each individual cylinder j of the body. Thus, the SDWBA provides a more realistic prediction of field measurements of sound scattering from krill. The SDWBA model was validated with the McGehee et al. (1998) TS measurements at 120 kHz (Demer and Conti, 2003a), and total target strength (TTS) measurements of Antarctic krill (Demer and Conti, 2003b) and northern krill (Conti et al., 2005).
For all these studies, the SDWBA model was evaluated using a generic krill shape (Figure 1), N0 = 14 cylinders, and a standard deviation of the phase variability 
. The selection of these parameters followed the shape and N0 parameters used in McGehee et al. (1998), and an empirical determination of 
, respectively. N, s.d., f, and the krill standard length L are dependent in their effects on the SDWBA predictions.
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To illustrate these dependences, the SDWBA was solved with the generic krill shape (McGehee et al., 1998), fattened by 40% (Demer and Conti, 2003a). The s.d.
was increased from 
to 
, in 
intervals, with f = 100 and 200 kHz and N = 14 and 22, respectively (Figure 2). At 100 and 200 kHz, the theoretical predictions remain the same on the main scattering lobe, independently of s.d.. Off the main axis, however, the TS predictions increase with increasing s.d.. As the maximum value of 
was determined from the McGehee et al. (1998) measurements made at 120 kHz, the SDWBA predictions at 100 kHz with the maximum s.d. can be viewed as the benchmark. In contrast, at 200 kHz, the theoretical predictions show unrealistic values off the main axis as s.d. becomes too large. Remarkably, maximum TS shifts from the main scattering lobe. Next, f and s.d. were held constant and N was modulated from 14 to 28 in increments of 2 (Figure 3). The SDWBA predictions are only stable if N is large relative to the ratio of fish length to wavelength, to ensure an accurate representation of the body in regard to wavelength.
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, where x varies from 0.1 to 1.0 by 0.1 (from dark to light). The dorsal aspect corresponds to a 90° incident angle
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(a), and 
(b), respectively. The number of cylinders N varies from 14 to 28 in increments of 2 (from dark to light). The dorsal aspect corresponds to a 90° incident angle These examples clearly demonstrate the need to adjust s.d.
and N vs. f and L via better parameterization of the SDWBA. Here, the SDWBA model is improved to account explicitly for the interdependence of these factors. |
Methods |
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TopIntroductionMethodsResults and discussionConclusionReferences |
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The calculations for SDWBA have been well described (McGehee et al., 1998; Demer and Conti, 2003a). The formulation is repeated here to provide a convenient reference for the subsequent model derivations.
Krill is approximated by N discretized-bent cylinders of various radii aj. In that case, the backscattering form function for the cylinder j and incident angle 
is
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| (1) |
, 
= (2 – 1) / 2, the subscript 1 denotes the ambient seawater and 2 the krill. J1 is the Bessel function of first kind of order 1, 
the position vector, |
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j:
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| (2) |
The phase variability j is obtained from a Gaussian distribution centred on 0, with standard deviation s.d., for each cylinder j along the body. Finally, the backscattering cross-section 
bs() is obtained from the average of multiple realizations of the ensembles of phase j:
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| (3) |
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| (4) |
The generic krill shape was defined by McGehee et al. (1998, standard length L0 = 38.35 mm). The width of the generic shape was increased by 40% in Demer and Conti (2003a), because freshly caught krill were fatter than the starved animals measured in McGehee et al. (Figure 1). At f0 = 120 kHz, and with N0 = 14 cylinders, 
was estimated to be 
from comparison of the SDWBA predictions with the experimental measurements. The factors N, s.d., f, and L are dependent in their effects on the SDWBA results; the model will be improved if it can account explicitly for the interdependence of these factors.
To ensure generalized utility of the SDWBA, s.d. must be expressed as a function of f, N, and L. First, as the flexure and complexity of the body are frequency-independent, the product s.d.(f)f must be constant,
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| (5) |
Also, if the frequency f or the body length L is modified, then N must be adjusted so that the spatial resolution of the body of the krill remains constant relative to wavelength 
. The ratio between wavelength and the length of each individual cylinder must remain constant:
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| (6) |
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| (7) |
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| (8) |
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| (9) |
Thus, s.d. and N can be adjusted to the desired L and f. For example, TS was estimated at f = 120, 200, 300, and 400 kHz by solving the SDWBA with a generic 40% fat krill shape, L = L0, and adjusting N and s.d. according to Equations (8) and (9) with N0 = 14 and 
. The values of the parameters with frequency are summarized in Table 1.
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were obtained from comparisons of the model with the measurements at 120 kHz from |
Results and discussion |
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TopIntroductionMethodsResults and discussionConclusionReferences |
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By applying the improved parameterization to the SDWBA model, there are no unrealistic off-axis lobes in the TS predictions (Figure 4). Correct adjustment of s.d.
provides a realistically stable and frequency-independent plateau, ca. –80 dB, off the main scattering lobe.
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Accurate application of the generalized SDWBA model still requires accurate characterization of the many parameters, particularly the orientation distribution of krill beneath a survey vessel (Demer and Conti, 2005), and density and sound-speed contrasts (g and h; Foote et al., 1990; Chu and Wiebe, 2005). The implications of TS modelling with SDWBA can be illustrated using the distribution of krill lengths and volume-backscattering strength measurements at 120 and 38 kHz (Sv120 kHz – Sv38 kHz) from an international survey of krill in the Scotia Sea (Hewitt et al., 2002). The survey is subdivided in three geographic regions (cluster C1, C2, and C3), according to the length distributions of krill. The SDWBA model was inverted in a least-squares sense to estimate an in situ distribution of orientations for each cluster individually, and for all clusters weighted by surveyed area or by krill net-catches. The mean and standard deviation of the normal distributions of orientation for the least-squares estimate ranged from 0° to 25° and 1° to 30°, respectively, by 1° steps. The distributions of orientations for clusters 1, 2, and 3 are estimated to normal distributions with a mean of 9°, 4°, and 3° and a standard deviation of 1°, 2°, and 1°, respectively (Figure 5a–c). For all clusters weighted by surveyed cluster area or krill catch demographics, the distributions of orientations are estimated to N(4°,2°) and N(11°,4°), respectively (Figure 6a, b). This last result should be compared with the previously published orientations of distributions from Kils (1981), Endo (1993), or Demer and Conti (2005). With the same data used by Demer and Conti (2005), but considering only the least-square criteria, an N(11°,4°) distribution of orientations gave a better fit to the data than the previously reported N(15°,5°) (Demer and Conti, 2005), or those reported by Kils (1981) or Endo (1993) (Figure 6b). However, two modes are predicted for the Sv120 kHz – Sv38 kHz differences. For N(15°,5°), both modes fit within the empirical distribution, but the empirical distribution is broader. In contrast, as the mean and the standard deviation of the orientation distributions decrease (e.g. for N(11°,4°) and N(4°,2°)), the predicted upper modes are not supported by the empirical data. Therefore, the last two distributions are also equivocal. One possible explanation, however, is that this second mode at higher Sv120 kHz – Sv38 kHz differences would correspond to small krill not detected at 38 kHz.
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Recognizing uncertainty in the orientation distribution, Demer and Conti (2005) provided a simplified SDWBA model for convenient estimation of Antarctic krill TS:
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| (10) |
Here, we provide new coefficients for this simplified model which account for the generalized parameters in Equations (8) and (9), and a larger range of kL, for = N(11°,4°), N(9°,1°), N(4°,2°), N(3°,1°), and N(15°,5°) (Table 2). The new simplified SDWBA can be evaluated for higher values of kL up to 200, compared with 50 in Demer and Conti (2005). The associated errors 
for the simplified SDWBA are about 1–3 dB, depending on the orientation distribution. The mean error can be reduced by increasing the order of the polynomial for the simplified SDWBA (Figure 7), but here the order of the polynomial was chosen according to Demer and Conti (2005). A comparison of krill TS vs. kL with = N(11°,4°) shows good agreement with empirical data reported in the literature (Figure 7). Note, however, that the empirical TS data do not include measurements of krill orientation, except for those from Martin-Traykovski et al. (1998). SDWBA predictions of krill TS averaged over N(4°,2°) and N(15°,5°) are, respectively, above and below those averaged with N(11°,4°). All three of these distributions of orientations suggest that krill are swimming fast beneath the surveying vessel (Kils, 1981).
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Finally, the application of the SDWBA using N(11°,4°) results in a krill biomass estimate nearly three times the previous estimate made using the Greene et al. (1991) TS model, as presented in Table 3. The estimates of biomass vary significantly with krill-orientation distribution, whereas the associated coefficients of variation remain similar. Demer and Conti (2005) emphasized that there is still significant uncertainty about krill orientation beneath survey vessels, and the temporal and spatial dependences. However, krill orientation is usually not accounted for when estimating empirical krill TS.
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Conclusion |
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TopIntroductionMethodsResults and discussionConclusionReferences |
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When applying the SDWBA model to estimate krill TS, it is necessary to adjust s.d.
vs. f, N, and L. Equations (8) and (9) are provided for this purpose. If the standard deviation of the phase 
can be estimated empirically for a shape of krill N0 L0 for frequency f0, then the appropriate N and s.d. can be determined to estimate TS vs. kL. Using Equations (8) and (9), krill biomass in the Scotia Sea could be re-evaluated to nearly three times the previously reported biomass, but these new estimates depend greatly on the krill orientation used for the model. The study of this dependence goes beyond the aim of this work, but should be investigated further to provide better estimates of krill biomass. Using survey data, the krill-orientation distribution was estimated here to either N(4°,2°) or N(11°,4°) depending on the data being averaged per surveyed cluster area or krill catch demographics, respectively. Coefficients for the simplified version of the SDWBA model are also provided for both orientation distributions. One might also investigate whether there are differences in the orientation distributions of krill beneath different survey vessels and with survey speed.
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Acknowledgements |
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We thank Kazuo Amakasu and Masahiko Furusawa, Tokyo University of Marine Science and Technology, for bringing this sensitivity of the SDWBA model to our attention, and two reviewers for their valued comments on an earlier draft of the manuscript.
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References |
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TopIntroductionMethodsResults and discussionConclusionReferences |
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