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ICES Journal of Marine Science: Journal du Conseil 2003 60(2):429-434; doi:10.1016/S1054-3139(03)00002-X
© 2003 by ICES/CIEM International Council for the Exploration of the Sea/Conseil International pour l'Exploration de la Mer
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Reconciling theoretical versus empirical target strengths of krill: effects of phase variability on the distorted-wave Born approximation

David A. Demer* and Stephane G. Conti

Southwest Fisheries Science Center P.O. Box 271 8604 La Jolla Shores Drive, La Jolla, CA 92037, USA

*Correspondence to D. A. Demer; tel: +858 546 5603; fax: +858 546 5608. e-mail: ddemer{at}ucsd.edu.

A model was recently proposed to predict the target strengths (TS) of Antarctic krill, Euphausia superba, versus incidence angle ({theta}) (Deep Sea Res. II 45(7) (1998) 1273). Based on the distorted-wave Born approximation (DWBA), the model depends on the coherent summation of scattering from elements of a discretized-bent cylinder. It was empirically validated at 120 kHz near-broadside incidence ({theta}{approx}90°), but large discrepancies were observed at other angles away from the main lobe. As the side-lobe measurements were both higher than the model predictions and above the noise floor, the authors noted that the differences were not entirely due to noise. In this study, the accuracy of the DWBA model is further explored. Results indicate that phase variability in the scatter from elements of a discretized-bent cylinder (krill model) causes a dramatic flattening in the side-lobe regions of TS({theta}), while negligibly affecting the main scattering lobe. These results are consistent with the krill TS measurements reported by McGehee et al. Thus, by accounting for phase variability in the solution of the DWBA model, a more accurate and thus practical tool is developed for predicting krill TS.

Keywords: phase variability, krill, target strength

Received 25 October 2002; accepted 19 December 2002.


   Introduction
Top
 Introduction
Methods
 Results
 Conclusions
 References
 
Because Antarctic krill, Euphausia superba, form the basis for the food web in the Southern Ocean and are the target of a large fishery, there has been much international endeavor to map their distribution and quantify their abundance (Hewitt et al., 2002). Accurate knowledge of the acoustic target strengths (TS) of individual krill is a vital component for the analyses of these surveys employing echo-integration methods. Greene et al. (1991) proposed a linear model of TS versus standard length (L), which is based on measurements (Wiebe et al., 1990) of a variety of crustacean zooplankton but not Antarctic krill. However, using TS measurements of live krill, the model was corroborated at an acoustical frEquationuency (f) of 120 kHz for a small number of L (Foote et al., 1990; Hewitt and Demer, 1991). Deemed the best available model at the time, the Antarctic Treaty Organization's Committee for the Conservation of Antarctic Marine Living Resources adopted the Greene et al. model as an international standard for estimating krill biomass (CCAMLR, 1991).

Motivated by observations that were inconsistent with the Greene et al. model's predictions, Demer and Martin (1995) highlighted the changes in TS due to variations in animal shape, morphology, and orientation, for which the model does not take account. Moreover, Demer (submitted for publication) provided evidence that the Greene et al. model has significantly reduced accuracy in the Rayleigh and Mie scattering regimes. In light of these concerns, McGehee et al. (1998) used the distorted-wave Born approximation (DWBA; Morse and Ingard, 1968) to model the TS of Antarctic krill versus f, and animal density ({rho}), sound speed (c), L, shape (S), and angle of orientation relative to the incident wave ({theta}). Using krill TS measurements at f = 120 kHz, they also validated the DWBA model on the main scattering lobe, near-broadside incidence. Unfortunately, at other angles (i.e. in the side-lobes), large discrepancies (5–50 dB) were observed between measurement and theory. The effects of noise could not entirely explain the nonconforming measurements because they were both higher than the model predictions and above the noise floor (McGehee et al., 1998). The significance of these differences was not assigned but the authors highlighted the need for accurate information about krill orientations.

Unfortunately, there is a paucity of information about the naturally occurring orientations of Antarctic krill. Possibly, the best quantitative information comes from Kils (1981), who meticulously quantified the orientation distributions of krill swimming in an aquarium. However, solving the DWBA model with the orientation distributions from Kils ({theta}=N(45.3°, 30.4°)), McGehee et al. (1998) showed expected values of krill TS to be 6–8 dB lower than that predicted by the Greene et al. model and the in situ TS measurements of krill by Hewitt and Demer (1991). This difference is highly significant because use of the DWBA model solved with Kil's krill orientation distribution would result in an increase of roughly a factor of 5 in biomass estimates relative to the ones computed with the Greene et al. model. For TS predictions to approach the in situ measurements, the DWBA must be solved with an improbably narrow distribution of krill orientations centered on normal incidence.

To explain this important inconsistency it should be noted first that the DWBA model was not successfully validated for {theta} greater than about 15–30° off normal incidence ({theta}{approx}90°) and the computations in McGehee et al. (1998) did not account for the stochastic nature of sound scattering. They made numerous measurements at other angles but the TS({theta}) was virtually constant in the side-lobes, at a level 5–10 dB above the measured noise floor (Figure 1). This analysis aims to reconcile the TS predictions by the DWBA model and the measurements of krill TS versus {theta} in McGehee et al. (1998). Success in this endeavor may improve the accuracy of the DWBA model and thus provide a more dependable tool for estimating TS of Antarctic krill for naturally occurring distributions of angles of incidence, animal shapes, sizes, and material properties, and over a range of acoustical frEquationuencies.


Figure 1
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  		Figure 1 The DWBA model (solid line), and the experimental TS of Animal 1 (dots) from McGehee et al. (1998). In the side-lobes, when the backscattered signal from the krill is about 10–15 dB above the noise floor (–95 dB), the agreement between the measurements and the DWBA theory ends.

 

   Methods
Top
 Introduction
 Methods
Results
 Conclusions
 References
 
As in McGehee et al. (1998), the DWBA (Morse and Ingard, 1968), is used to model acoustical scattering from krill which have mass density ({rho}) and sound speed (c) values close to those of the surrounding seawater medium. The backscattering form function for part j of the scatterer field is


Formula 1

(1)
where k1=2{pi}f/c is the acoustical wave number,


Formula

Formula is the position vector, {gamma}{kappa}=({rho}1c12/{rho}2c22)–1, and {gamma}{rho}={rho}2–{rho}1/{rho}2. The subscript 1 denotes the ambient seawater and 2 the krill.

Stanton et al. (1998) claimed that Equation (1) can be solved as a line-integral if the krill shape is approximated by a discretized-bent cylinder, elements noted j, having radii (aj) and positions Formula along a central line


Formula 2

(2)
where J1 is the Bessel function of the first kind of order 1, and ßtilt is the angle between the cylinder and the incident wave. Using N=15 cylinders with relative aj and Formula (Figure 2; generic krill shape and number of cylinders from McGehee et al., 1998), the form function is obtained by summing the components from each of the scattering elements


Formula 3

(3)


Figure 2
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  		Figure 2 The generic shape for E. superba from McGehee et al. (1998). Model parameters are listed in Table 3 of their article.

 
At broadside incidence, the relatively high backscattered level (see Figure 1) is mostly due to constructive interference. At other angles, the interference can be mostly destructive (McGehee et al., 1998). Thus, at angles away from normal incidence, the phases of the backscattered signals from each element are very important to the summation in Equation (3). There is variability in the phases of scatter from scattering elements of the krill because:
  1. scattering in a field with noise is a stochastic process;
  2. krill have shapes that are more complex than juxtapose cylinders of varying radii; and
  3. their bodies flex constantly as they swim.

To account for these realities in the model, phase variability is added to each element j of fbs, and a stochastic version of the DWBA is thus introduced (SDWBA). The form function accounting for phase variability in the scattered signal from each element is


Formula 4

(4)

As usual, the backscattering cross-sectional area and TS are


Formula 5

(5)
and


Formula 6

(6)
respectively. As previously mentioned, the phase variance is attributed to variations in noise, plus shape complexities, plus flexure. To quantify the minimum expected phase variability (e.g. that due to noise only), the variance of phase in Gaussian noise is described in Demer et al. (1999)


Formula 7

(7)
where SNR is the signal-to-noise power ratio. Note that the actual phase variability will be greater than this due to the generally unknowable effects of complexities in body shape, internal structure, and flexure. Thus, the expected value for target strength (E[TS({theta})]) is estimated by averaging {sigma}bs over multiple realizations of phase variability at a fixed sd[{phi}j] (Table 1). Because the krill TS data in McGehee et al. (1998) were measured using a 0.5 ms pulse duration ({tau}), E[TS({theta})] is also averaged with a 2 kHz bandpass filter (bandwidth~1/{tau}).


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  		Table 1 The standard deviation (sd) of phase versus SNR in decibels for Gaussian noise from Demer et al. (1999). As phase variability is the result of noise, complex and dynamic animal shapes, and general randomness in sound scatter, these sd are interpreted as minimum values at each SNR.

 

   Results
Top
 Introduction
 Methods
 Results
Conclusions
 References
 
Using Equations (4–7), the E[TS({theta})] is calculated for all {theta} (1° resolution) using 100 realizations of phase selected randomly from a Gaussian distribution. This is repeated for sd[{phi}j]=0.0224, 0.0707, 0.2236, and 0.7071 radians (rd) (Figure 3). For sd[{phi}j] = 0.0224 rd, the phase-noise has negligible effect on the DWBA model at any incidence angle. However, as the sd[{phi}j] is decreased to 0.0707 rd, the main lobe is unchanged while the side-lobes of TS({theta}) begin to disappear. The addition of phase variability (sd[{phi}j]=0.2236 and then 0.7071 rd) causes the TS values at angles away from the main lobe to increase dramatically to relatively constant levels more than 10 dB above the noise floor in the experiments of McGehee et al. (1998). Again, the main lobe is virtually unaffected by phase variability.


Figure 3
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  		Figure 3 The DWBA model solved with the generic krill shape, g=1.0357 and h=1.0279 (black lines), and the stochastic DWBA model (SDWBA) solved with different values of phase-variability (gray lines; sd({phi})=0.0224, 0.0707, 0.2236 and 0.7071 rd). The expected values of TS in the SDWBA are computed from {sigma}bs averaged over 100 realizations of the random phase.

 
The krill TS from Animal 1 of McGehee et al. (1998) are overlaid on the SDWBA model computed with sd[{phi}j] = 0.7071 rd (Figure 4). With or without taking into account phase variability, the matches between the TS measurements and DWBA model predictions remain unchanged on the main scattering lobe. However, in the side-lobes the SDWBA model is a much better match to the empirical TS data than the DWBA model.


Figure 4
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  		Figure 4 The SDWBA model (black line) computed with sd({phi}) = 0.7071 rd and a 2 kHz bandwidth filter. The experimental data (dots) are almost completely bounded by the ±1sd error bounds of the SDWBA model (gray lines).

 
The significance of the new SDWBA model can be appreciated by overlaying the SDWBA±1sd, DWBA, and Greene et al. models (Figure 5). SDWBA predicts krill TS that are about 10–12 dB greater than those predicted by the SDWBA. Below 210 kHz, the differences between the two models increase linearly with frEquationuency. Comparing the SDWBA model and Greene et al. model, the differences are either positive or negative, depending upon the frEquationuency. As the Greene et al. model does not account for scattering in the Rayleigh regime (Wiebe et al., 1990), the differences between these two models increase greatly at lower frEquationuencies (below about 50 kHz).


Figure 5
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  		Figure 5 A comparison (a) of the Greene et al. model, DWBA, SDWBA, and SDWBA±1sd. Differences between models are also shown (b). For rapid computation of E[TS] for the SDWBA model below 210 kHz, the linear approximation for the differences between the probabilistic SDWBA model and the deterministic DWBA model is {Delta}TS = 0.0086f + 9.7924 dB.

 

   Conclusions
Top
 Introduction
 Methods
 Results
 Conclusions
References
 
Phase variability in the scatter from elements of a discretized-bent cylinder causes a dramatic flattening or plateau in the side-lobe regions of the DWBA model, while negligibly affecting the main scattering lobe. These results are consistent with the krill TS measurements in McGehee et al. (1998). Thus, by accounting for the stochastic nature of sound scattering in the solution of the DWBA model, the SDWBA model (Figure 5) provides a more accurate tool for predicting krill TS as a function of acoustic frEquationuency, and animal size (generic shape; McGehee et al., 1998), g (Foote et al., 1990), h (Foote, 1990), and orientation (Kils, 1981). Although the SDWBA model is probabilistic and therefore predicts TS within a range of about 8 dB (E[TS(f)]±1sd), the expected values (E[TS(f)]) are approximately 5, –3, and –2 dB different from the deterministic Greene et al. model predictions at 38, 120, and 200 kHz, respectively (typical survey frEquationuencies). Before the important ramifications of these conclusions are discussed, the SDWBA model should be validated using scattering data from many krill collected over a broad-bandwidth (see Demer and Conti, in press).


   Acknowledgements
 
We are appreciative of the United States Antarctic Marine Living Resources Program, for funding this investigation. Special thanks to Professor Duncan McGehee, Cuyamaca College, for providing the krill TS data for Animal 1 and information about the pulse duration used to record the measurements in McGehee et al. (1998).


   References
Top
 Introduction
 Methods
 Results
 Conclusions
 References
 

    CCAMLR. (1991) Report of the 10th Meeting of the Scientific Committee, SC-CAMLR-X1991Hobart, Tasmania 427 pp.

    Demer D.A. (2000) An estimate of error for the CCAMLR 2000 estimate of krill biomass. CCAMLR. Special Issue Deep Sea Res (submitted for publication).

    Demer D.A. and Martin L.V. (1995) Zooplankton target strength: volumetric or areal dependence? Journal of the Acoustical Society of America 98:2 Pt. 11111–1118.[CrossRef][Web of Science]

    Demer D.A. and Conti S.G. Broadbandwidth total target strength measurements of Antarctic krill Euphausia superba from reverberation in a cavity. ICES Journal of Marine Science (in press).

    Demer D.A., Soule M.A., Hewitt R.P. (1999) A multiple-frEquationuency method for potentially improving the accuracy and precision of in situ target strength measurements. Journal of the Acoustical Society of America 105:42359–2376.[CrossRef][Web of Science]

    Foote K.G. (1990) Speed of sound in Euphausia superba. Journal of the Acoustical Society of America 87:41405–1408.[CrossRef][Web of Science]

    Foote K.G., Everson I., Watkins J.L., Bone D.G. (1990) Target strengths of Antarctic krill (Euphausia superba) at 38 and 120 kHz. Journal of the Acoustical Society of America 87:116–24.[CrossRef][Web of Science]

    Greene C.H., Stanton T.K., Wiebe P.H., McClatchie S. (1991) Acoustic estimates of Antarctic krill. Nature 349:110.[CrossRef]

    Hewitt R.P. and Demer D.A. (1991) Krill abundance. Nature 353:310.

    Hewitt R.P., Watkins J.L., Naganobu M., Tshernyshkov P., Brierley A.S., Demer D.A., Kasatkina S., Takao Y., Goss C., Malyshko A., Brandon M.A., Kawaguchi S., Siegel V., Trathan P.N., Emery J.H., Everson I., Miller D.G.M. (2002) Setting a precautionary catch limit for Antarctic krill. Oceanography 15:326–33.

    Kils U. (1981) The swimming behavior, swimming performance and energy balance of Antarctic krill, Euphausia superba. BIOMASS Scientific Series No. 3 122 pp.

    McGehee D.E., O'Driscoll R.L., Martin Traykovski L.V. (1998) Effects of orientation on acoustic scattering from Antarctic krill at 120 kHz. Deep Sea Research Part II 45:1273–1294.[CrossRef]

    Morse P.M. and Ingard K.U. (1968) Theoretical Acoustics(Princeton University Press, Princeton) 927 pp.

    Stanton T.K., Chu D., Wiebe P.H. (1998) Sound scattering by several zooplankton groups. II: Scattering models. Journal of the Acoustical Society of America 103:236–253.[CrossRef][Web of Science][Medline]

    Wiebe P.H., Greene C.H., Stanton T.K., Burczynski J. (1990) Sound scattering by live zooplankton and micronekton: empirical studies with a dual-beam acoustical system. Journal of the Acoustical Society of America 88:52346–2360.[CrossRef][Web of Science]


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