Validation of the stochastic distorted-wave Born approximation model with broad bandwidth total target strength measurements of Antarctic krill

Total-scattering cross-sections ( r t ) of Antarctic krill ( Euphausia superba ) were measured over a broad bandwidth (36–202kHz) using a new technique based on acoustical reverberation in a cavity. From 18 February to 9 March 2002, mean total target strengths ð TTS ¼ 10 log ð r t = 4 p ÞÞ , were measured from groups of 57–1169 krill (average standard length ¼ 31.6mm; standard deviation ¼ 6.6mm) at the Cape Shirreﬀ ﬁeld station, Livingston Island, Antarctica, and aboard RV ‘‘Yuzhmorgeologiya’’. Chirp pulses were transmitted sequentially by an omni-directional emitter into one of three glass carboys containing groups of krill swimming in 9.3, 19.3, or 45.9 liters of seawater (0.6 (cid:1) C (cid:2) temperature (cid:2) 4.0 (cid:1) C). Between each pulse the krill moved within the ﬁxed-boundary tank and the modulated reverberations were sensed bi-statically with three omni-directional receivers. At each center frequency (f c ), the coherent energy in 200-pulse ensembles identiﬁed sound scattered by the tank. The incoherent energy described total sound scattering from the krill. Thus, the TTS at each f c was extracted from a correlation analysis of energy reverberated in the tank. Measurement bias was determined to be (cid:3) 0.4dB from an experiment using metal sphere reference targets, and the precision was estimated as (cid:3) 0.8dB from the variability in the krill TTS (f c ) measurements. The empirical estimates of mean r t corroborated a krill-scattering modelbased onthedistorted-wave Bornapproximation(DWBA),enhanced bytheauthorsto account for the stochastic nature of sound scattering (SDWBA), integrated over all scattering angles and averaged over all incident angles (SDWBA TTS ). The SDWBA, solved for target strength of Antarctic krill, may be the best predictor of backscatter for this important species andmayalsoprovidebackscatteringspectraforimprovingtheiracousticidentiﬁcation.Theseadvancesmayhelptoreduceuncertaintyinkrill-biomassestimationusingmulti-frequencyechosounderdataandecho-integrationmethods.


Introduction
The United States of America's Antarctic Marine Living Resources Program (AMLR) uses multi-frequency echosounders and echo-integration to map the dispersion of Antarctic krill (Euphausia superba) over large areas and to estimate their abundance (Hewitt and Demer, 2000). The bias and precision of the survey results depend primarily on the uncertainties in identifying acoustical backscatter from krill and in estimating the mean backscattering crosssection (r), or target strength ðTS ¼ 10 logðrÞÞ of krill (Demer, in press).
Model estimates of TS are based either on empirical data or on the physics of sound scattering. For Antarctic krill, Greene et al. (1991) proposed a linear model of TS versus total length (L), based on measurements of various crustacean zooplankton (Wiebe et al., 1990) and corroborated at frequency f ¼ 120 kHz for krill over a small range of L Hewitt and Demer, 1991). The implications of using the Greene et al. model were explored  and the model was adopted as an international standard for estimating krill biomass (CCAMLR, 1991).
Taking the physics-based approach, McGehee et al. (1998) used the distorted-wave Born approximation (DWBA; Morse and Ingard, 1968) to model the TS of krill versus f, and animal body-mass density (q), sound speed (c), L, shape (s), and angle of orientation relative to the incident wave (U). The krill s is modeled as a string of cylinders having varying diameters and positions along a curve, and TS is estimated from a coherent summation of backscatter from these elements. With TS(U) measurements of tethered, live krill in a tank, they validated the DWBA model near broadside incidence (ca. 75 U 105 ) at f ¼ 120 kHz. However, there was poor agreement between the DWBA and their TS measurements for U away from the main lobe. Demer and Conti (2003) developed a stochastic version of the DWBA (SDWBA) that models the effects of animal flexure, shape complexities, and noise with a random-phase term for each of the cylinders. The resulting degree of incoherence explains the discrepancies between McGehee et al.'s empirical data and theory over all angles of incidence (or krill orientation).
It is now desirable to gather empirical corroboration of the new SDWBA model over a broad bandwidth. Unfortunately, the constraints of conventional techniques for making free-field measurements of krill TS with a broad bandwidth and known distributions of animal sizes and orientations make this a formidable challenge (see Demer et al., 1999;Ona, 1999). Recently, however, a new technique has been developed (De Rosny and Roux, 2001) for conveniently and accurately (Demer et al., 2003) making broad-bandwidth measurements of total scattering cross-sections (r t ), or total target strength ðTTS ¼ 10 logðr t =4pÞÞ of one or more scatterers moving in an echoic cavity with static boundaries.
Contrary to the free-field requirement of conventional TS-measurement techniques, the new method extracts measurements of TTS from quotients of the incoherent and coherent energies in ensembles of reverberation time-series. Also intriguing is that absolute measurements of sound scatter can be made without the usual system calibration-the system parameters cancel in the quotient-and the animals' orientations and positions within the acoustical beam are inconsequential because the reverberant sound field is homogeneous. By employing this novel method, the aims of our investigation are: (1) to make broad-bandwidth TTS measurements of swimming krill; (2) to use these measurements to validate the SDWBA model over a broad bandwidth (the SDWBA model can be evaluated for TS (SDWBA TS ) and TTS (SDWBA TTS )); and (3) thus provide both an improved tool for predicting krill TS, and a broad-bandwidth spectrum for acoustically identifying krill.

Methods
Empirical TTS of Antarctic krill From 18 to 24 February 2002, TTS measurements of Antarctic krill were made at AMLR's Cape Shirreff field station on Livingston Island, Antarctica. Details of the general processing steps are outlined in De Rosny and Roux (2001) and Demer et al. (2003). Descriptions of the equipment and procedures specific to these measurements follow.
The krill were captured near the South Shetland Island archipelago using a 2-m Isaacs-Kidd midwater trawl (IKMT), deployed from RV ''Yuzhmorgeologiya''. They were kept alive and transferred ashore via Zodiac in 20 liter buckets of seawater. For each experiment, a glass carboy (9.3 AE 2%, 19.3 AE 1.6%, or 45.9 AE 0.9% liter) was filled completely with ambient seawater at temperatures ranging from 0.6 to 4.0 C. Groups of 57-1169 krill were then added and the top was closed with a rubber stopper containing an emitter, three receivers, and a thermocouple ( Figure 1). Displacing small amounts of water in the process, the resultant cavity had no air-water interface. This setup and procedure provided an echoic cavity with fixed boundaries, a requisite while conducting the experiments on a moving ship. From 26 February to 9 March, the TTS measurements were continued aboard AMLR's chartered RV ''Yuzhmorgeologiya''.
At center frequencies (f c ) ranging from 36 to 202 kHz, frequency-modulated pulses (0.4 V p-p ; 500 ls) with a 2 kHz bandwidth were generated (Hewlett Packard 33120A arbitrary-waveform generator), amplified 20 dB (Krohn-Hite 7500 power amplifier), and transmitted twice-persecond using an omni-directional, broad-bandwidth emitter and received bi-statically with three omni-directional broad-bandwidth receivers. During sequential pulses ðk ¼ 1--200Þ, the animals moved within the fixed-boundary tank and t ¼ 0 to 10, 20, or 32 ms of the modulated reverberation (h k (t)) were digitized at 410 kHz using a 12-bit analog-to-digital converter (National Instruments Daqpad 6070E). The lengths of the recorded time-series depended on the signal-to-noise ratio (SNR) and thus mainly on the carboy volume and the number of krill therein (see Table 1). To reduce noise, the h k (t) were match-filtered by cross-correlation with the transmitted signal. The coherent energy in 200-pulse ensembles identified sound scattered from the echoic tank. Because the positions of the animals were uncorrelated from ping-to-ping, the incoherent energy described sound scattering from the krill. The ratio of uncorrelated ðhh k ðtÞh kþ1 ðtÞiÞ and correlated ðhh k ðtÞ 2 iÞ energies decayed exponentially: The exponential decay of S(t) was estimated for each 200pulse ensemble by separately low-pass filtering the numerator and denominator in the linear domain ðf cutoff ¼ 500 HzÞ, and fitting a least-squares slope ðd lnðSðtÞÞ=dtÞ, while requiring 2 t 9 ms for the 9.3 and 19.3 liter cavities, 3 t 13 ms for the 45.9 liter cavity, and lnðSðtÞÞ ¼ 0 at t ¼ 0. Knowing the volume of the cavity (v), the number of krill (n), and the sound speed in seawater (c), an estimate of r t was made for each group of krill and f c : Thus, TTS(f ) measurements were made of 12 aggregations of 57-1169 swimming krill (Table 1) from the reverberation sensed at three receiver locations, and during one or more 3-h runs. Following the measurements of each aggregation, L was measured to the nearest millimeter (from the anterior tip of the rostrum and the posterior end of the uropods, excluding their terminal setae) before preserving in sample jars with ethanol. Demer et al. (2003) used precision metal spheres to demonstrate that this TTS-measurement technique is remarkably accurate (AE0.4 dB) and precise (AE0.7 dB). How-ever, v should not be too large compared to the total volume of the scatterers; the reflectivity of the boundaries must be high and the reverberation time-series must be long enough to precisely estimate d lnðSðtÞÞ=dt. Also, a large number of modes must be excited in the cavity to obtain a homogeneous sound field. As a guideline limit, v ! 100k 3 ðk ¼ c=f Þ, or v ! 7 liter at 36 kHz (De Rosny and Roux, 2001). Thus, the working bandwidth in these experiments was limited by the reflectivity of the glass carboys and the frequency responses of the emitter and receivers.

Theoretical TTS of Antarctic krill
To predict the empirical estimates of r t , the SDWBA model was integrated over all scattering angles and averaged over all incidence angles (SDWBA TTS ). The computation is detailed in Appendix A. Parameters include the generic krill shape , c ¼ 1455 m s ÿ1 , the nondimensional sound-speed and density contrasts (h ¼ 1:0279 and g ¼ 1:0357, respectively) from Foote (1990) and Foote et al. (1990), and random phase chosen from a normal distribution ðu ¼ N½0 ; 40:5 Þ from Demer and Conti (2003). The generic krill shape, derived for a krill with L ¼ 38:35 mm, was proportionately scaled to represent the smaller krill in these experiments (average L ¼ 31:6 mm). Thus, SDWBA TTS was evaluated from 36 to 202 kHz.

Results
The mean TTS of Antarctic krill was measured acoustically over a broad bandwidth (36-202 kHz), on land and at sea, from 18 February to 9 March 2002 ( Figure 2). By matchfiltering the reverberation time-series to reduce noise (e.g. from the ship and the electronics), the TTS measurements made aboard RV ''Yuzhmorgeologiya'' were comparable to those made at Cape Shirreff. In general, the TTS measurements increased monotonically versus f with a gradual reduction in slope. In 9 of 12 runs, the TTS(f c ) from about 90 to 202 kHz showed remarkable agreement with the SDWBA TTS calculated for the mean L in each aggregation. The TTS measurements from two of the aggregations, 117 and 326 krill, respectively, were, in fact, nearly identical matches to the SDWBA TTS over the entire measurement bandwidth.
Anomalous increases in TTS below about 150 kHz occurred in the measurements with 86 and 173 krill ( Figure  2). This is true, but to a lesser extent, for the measurements with 176 krill. This characteristic is probably an artifact of residual aeration in the carboy at the beginnings of the runs. As f c was scanned from 36 to 202 kHz over about 3 h, the uncorrelated bubble scatter diminished over time and did not bias the results at higher frequencies.
The TTS measurement precision (s.d.) is estimated as AE20% (AE0.8 dB) from three recordings per krill aggrega-tion. Judging from Demer et al. (2003), the systematic error of these krill TTS measurements could be estimated as AE0.4 dB. However, because krill are weaker scatterers, and the cavity volumes were smaller than for the standardsphere measurements, some additional measurement bias might be expected, especially at the lower frequencies.
The krill length-frequencies were variable between aggregations and varied from quasi-uniform to normal distributions ( Figure 3). The overall distribution was negatively skewed with lengths ranging between approximately 20 and 50 mm (mean L ¼ 31:6 mm; s:d: ¼ 6:6 mm).
A mean TTS spectrum was obtained by averaging the results of the individual runs, excluding the measurement from the 86 and 173 krill (Figure 4). Despite the omission of the two anomalously noisy data sets, the slope of TTS(f c ) is questionably flat below about 60 kHz. This feature, and decreasing SNR below about 80 kHz as indicated by virtually identical reverberation time-series for all the pings within an ensemble, suggests that the measurements at frequencies f c ¼ 38 to 58 kHz are unreliable and those between 60 and about 80 kHz may have a small positive bias due to noise. The 1-to 2-dB spike in the TTS measurements from 196 to 200 kHz was observed in all the runs and may be an artifact of the cavity geometry and hydrophone placement.
The mean krill TTS were compared to the SDWBA TTS calculated with the L probability density function for all the measured krill (Figure 3). Considering mean TTS from f c ¼ 60 to 202 kHz, the measurements ranged from about ÿ84.6 to ÿ71.0 dB, and their AE1 s.d. lines encompass the SDWBA TTS predictions (Figure 4). Over the same range of f, the predicted TTS ranges from about ÿ85.6 to ÿ72.0 dB. The two curves match to within a fraction of 1 dB from 60 to about 130 kHz, and within 1 dB at higher frequencies.

Discussion
The currently accepted model for krill TS (Greene et al., 1991), which depends linearly only on log(L), was developed empirically for f ¼ 420 kHz, and is scaled to different f assuming a frequency-squared relationship.   However, the krill TTS measurements in this study do not indicate a strong linear relationship between scattering intensity and animal length (see Figure 5 and note the frequency-dependent slopes, y-intercepts, and low R 2 values). Also, for a given L, krill shapes, volumes, and material properties can be dramatically variable. Moreover, krill orientation has a dominant effect on TS variability (see Demer and Martin, 1995), and both gender and maturation state affect U. For example, mature female krill with swollen cephalothoraxes are thought to have larger tilt angles than males (Endo, 1993). The DWBA model  accounts for all these parameters (i.e. TS(f, c, g, h, s, L, U)). Lavery et al. (2002) developed another version of the DWBA that replaces the generic krill shape  with highresolution, three-dimensional measurements of the animal's exterior boundary. However, in view of the uncertainties associated with the other model parameters, animal flexure and motion, and the animal-to-animal variability in s, this degree of model complexity may provide little practical improvement, especially at lower frequencies. Demer and Conti (2003) accounted for the stochastic nature of sound scattering in a model that provides probabilities of krill TS versus all angles of orientation (SDWBA). The SDWBA has now been corroborated with TTS over a broad range of frequencies and can easily accommodate distributions of animal density and sound speed contrasts, shape, size, and orientations, assuming they can be characterized. For very high frequencies it can also be extended to three-dimensional shapes (viz. Lavery et al., 2002).
To predict r or TS for Antarctic krill, the expected value of SDWBA TS was estimated as a function of f c and L ( Figure 6). As in Demer and Conti (2003). SDWBA TS was averaged over 100 realizations computed with interelement phase variability ðu ¼ N½0 ; 40:5 Þ, and over the Gaussian krill-orientation distribution from Kils (1981); ðU ¼ N½45:3 ; 30:4 Þ. Without U data on free-swimming krill, the latter assumes that the relatively stationary tank populations measured by Kils exhibited the same U distributions as potentially more active wild populations. Multifrequency (e.g. Chu et al., 1993) andbroad-bandwidth techniques (e.g. Martin Traykovski et al., 1998) have been suggested for measuring krill-orientation distributions in situ, but to date no such data have been published.
In addition to predicting TS(f) for krill, the SDWBA TS can be used for acoustical classification of these important marine organisms. Backscatter at two and three frequencies has been used to classify acoustic backscatter from krill and other sources (e.g. Madureira et al., 1993a, b;Brierley et al., 1998;Demer et al., 1999), but accurate scattering models are required for successful implementations (Greenlaw and Johnson, 1983). Backscatter at three or more frequencies, or over a broad bandwidth, provides even more information for successful taxa classification (e.g. Simmonds et al., 1996;Zakharia et al., 1996). Thus, the SDWBA TS spectra ( Figure 6) may improve the accuracy of multi-frequency and broad-bandwidth, echo-trace classification techniques.

Conclusions
A new method for measuring numbers of mobile targets in an echoic cavity from reverberation time-series (De Rosny and Roux, 2001) has been adapted here for accurately and precisely measuring the total sound scatter of swimming krill over a broad bandwidth. With this method, absolute measurements of r t of live Euphausia superba were obtained conveniently without the usual system calibration. Moreover, the animals' orientations and positions during the measurements were inconsequential. These TTS(f c ) data corroborate the new SDWBA krill-scattering model (Demer and Conti, 2003) by matching the theoretical predictions to better than about 1 dB from f c ¼60 to 202 kHz. In this bandwidth, the minor differences in spectrum shapes can be explained by reduced SNR in the lower-frequency measurements. The offset between the TTS measurements and theory (<1 dB) can be explained very easily by measurement error and uncertainties in g, h, and s. Thus, the SDWBA TS may provide the best estimate of TS(f ) for Euphausia superba, it should aid in the acoustical identification of krill, and it offers reduced measurement error in krill-biomass estimates. Figure 4. The average TTS of 10 aggregations of Euphausia superba totaling 57-1169 animals. TTS data from 36 to 60 kHz had low SNRs (gray circles); those above 60 kHz are considered accurate to about 0.4 dB (black circles). The AE1 s.d. error bands (dashed lines) encompass the SDWBA TTS predictions (solid gray), computed with g ¼ 1:0357, h ¼ 1:0279, generic s, and the overall krill length-frequency distribution (see Figure 3). Because the random-phase term caused variations in SDWBA TTS of less than 0.1 dB, expected values of TTS were effectively computed at each f using only a single random realization of phase. Error bounds on the prognosticator are thus negligible.
2002 zooplankton team (Nancy Gong, Emma Bredesen, Shelly Peters, Lorena Linacre-Rojas, Mike Force, Adam Jenkins, Valerie Loeb, and Rob Rowley) for providing us with live animals from the net catches. Rob Rowley was especially helpful in designing and constructing an equipment rack for transporting the electronics to and from the island. Finally, thanks to the team at Cape Shirreff (Iris Figure 5. TTS measurements versus mean krill total lengths. For L ranging from 26.6 to 39.6 mm, TTS at 38, 70, 120, and 200 kHz trend upward. At each frequency, there is high variability about the lines fitted to the data and the R 2 values are low. The low slope at 38 kHz is probably an artifact of a low SNR. The slopes are high and similar at 70 and 120 kHz, but decrease at 200 kHz when the wavelength is much smaller than the animal dimensions. Fitting the data to TTS ¼ mlog(L) þ b gives similar results: m ¼ 24. 251, 47.054, 48.214, and 29.231; b ¼ ÿ123.551, ÿ154.351, ÿ150.516, and ÿ116.286; and R 2 ¼ 0.149, 0.560, 0.778, and 0.565 respectively.