Modelling three-dimensional directivity of sound scattering by Antarctic krill : progress towards biomass estimation using multibeam sonar

Target strength (TS) estimation is a principal source of uncertainty in acoustic surveys of Antarctic krill (Euphausia superba). Although TS is strongly dependent on krill orientation, there is a paucity of information in this regard. This paper considers the potential for narrow-bandwidth, multibeam-echosounder (MBE) data to be used for estimating the orientations of krill beneath survey vessels. First, software was developed to predict MBE measurements of the directivity patterns of acoustic scattering from individual or aggregated krill in any orientation. Based on the distorted-wave, Born approximation model (DWBA), scattering intensities are predicted vs. MBE angles for specified distributions of krill orientations (pitch, roll, and yaw angles) and swarm densities. Results indicate that certain distributions of orientations, perhaps indicative of particular behaviour, should be apparent from the sonar data. The model results are compared with measurements on krill made using a 200-kHz MBE deployed from a small craft off Cape Shirreff, Livingston Island, Antarctica, in summer 2006. The stochastic DWBA model is then invoked to explain disparities between the model predictions and MBE measurements.


Introduction
High-frequency (!70 kHz) acoustic scattering from Antarctic krill (Euphausia superba) is highly directional.For example, the distorted-wave, Born-approximation model (DWBA;McGehee et al., 1998) and the Stochastic DWBA model (SDWBA; Demer and Conti, 2003a;Conti and Demer, 2006) predict that the target strength (TS) of krill at 120 kHz is 20 dB lower for a tilt angle u ¼ 158 off the dorsal incidence than at u ¼ 08.Consequently, the TS of krill at typical sonar frequencies can vary substantially, particularly for u , 308.
The orientations of in situ krill are variable and largely unknown.What is known about u for krill was derived from measurements of ex situ krill and inferences from data on in situ krill.For krill hovering in an aquarium, Kils (1981) and Endo (1993) estimated u ¼ 458 (s.d.¼ 308) and u ¼ 458 (s.d.¼ 208), respectively.For krill swimming in a flume, Kils (1981) measured u % 5-308, with the smaller angles corresponding to faster speeds.From acoustic observations, Chu et al. (1993) estimated u % 0-308 for swimming, encaged krill, whereas Demer and Conti (2005), Conti et al. (2005), and Conti and Demer (2006) estimated u % 11-158 (s.d. ¼ 4-58) for krill beneath a survey vessel.Lawson et al. (2006) estimated u % 108 (s.d.¼ 608; median u ¼ -0.58) visually for krill passing through a towed, video plankton recorder.Most of these observations indicate that krill orientate head-up (positive u) when at rest and are orientated nearly horizontal (u % 08) when swimming.Consequently, it must be assumed that krill in the wild can adopt any orientation, depending on their behaviour.
For acoustic surveys of krill conducted using a single-beam echosounder (SBE), u modulates the incidence angle and hence the measured TS.Measurements of volume-backscattering strength (S v ¼ TS þ 10 log N, where N is the number of scatterers per cubic metre) are therefore also modulated by u.Therefore, the generally unknown orientation of krill is a principal source of uncertainty in acoustic estimates of their density, abundance, and biomass.Better knowledge of krill orientations, particularly during acoustic observations, would constitute a major step towards reducing uncertainty in surveys with SBEs.Perhaps such information would also allow the use of multibeam echosounders (MBEs), that have swaths of 90-1808, for quantitative surveys of krill.At present, the multiple angles of incidence associated with the various beams make quantitative translation of echo intensity to numerical density essentially impossible.
Although some TS models of krill are parameterized for any incidence angle, only variations in TS vs. u have yet been considered (Martin Traykovski et al., 1998;McGehee et al., 1998;Demer and Conti, 2003a).Distributions of u have been estimated for krill by inverting TS models using broad-bandwidth (Martin Traykovski et al., 1998) and multifrequency measurements of S v (Chu et al., 1993;Conti et al., 2005;Demer and Conti, 2005;Conti and Demer, 2006).These methods are ineffective, however, if the scattering-directivity pattern (SDP) is complicated or if the acoustic beam is non-vertical.
Krill can adopt any orientation because of natural behaviour, water motion, or reactions to the presence of a survey vessel as can fish (Gerlotto and Fre ´on, 1992;Soria et al., 1996;Gerlotto et al., 2004).The combination of krill orientations and the range of beam directions from a wide-swath MBE, or a motionuncompensated SBE, means that krill could be insonified from any direction.Therefore, this study considers not only u, but also any orientation resulting from the pitch, roll, and yaw of the krill, combined with variations in the direction of the acoustic beam.Here, the TS values for any incidence angle are modelled using the DWBA (McGehee et al., 1998), resulting in complete, three-dimensional, frequency-specific, SDPs for krill, similar to those produced for fish by Jech and Horne (2002) and Towler et al. (2003).Using the krill SDPs, and a simulation model of the MBE developed by Cutter and Demer (2007), distributions of scattering intensity are predicted for many beam directions and various distributions of krill orientations and aggregation densities.These simulation results are compared with MBE data.Although the work focuses on krill, the findings here are also relevant to surveys of any pelagic species conducted with MBEs and SBEs.

Modelling krill shape
The DWBA was parameterized with a generic krill shape (Figure 1) comprising multiple, contiguous cylinders distributed along a curve (McGehee et al., 1998).Shape parameters include the cylinder radii a (m), the density contrast g, the sound-speed contrast h, and the locations of the centres of each cylinder r, in rectangular coordinates (Table 1).As in Demer andConti (2003b, 2005) and Conti and Demer (2006), the radii (a) were 40% larger than those of the starved krill modelled in McGehee  The density contrast was g ¼ 1.0357, and the sound-speed contrast h ¼ 1.0279.The number of cylinders, N bcyl , was 6, 10, 15, and 25 at frequencies 38, 70, 120, and 200 kHz, respectively.

Modelling SDPs
The DWBA was used to model krill SDPs for the commonly employed acoustic frequencies of 38, 70, 120, and 200 kHz.Following the reference frame and angle definitions in McGehee et al. (1998), the krill shapes were transformed, first by translating the central axis of the krill to the x-axis, then rotating the krill about that axis over 3608 at 18 intervals.For each roll-angle interval, the model predicted TS for 18 intervals over 3608 in the vertical plane.Therefore, TS data were modelled for all directions, and the results at each interval were combined to form the complete SDP.SDPs were similarly generated for 120 and 200 kHz using the SDWBA.TS values were estimated for ten realizations with random phases and s.d.(f) ¼ 0.7070 radians; see Demer and Conti (2003a, b) for details.

Modelling multibeam measurements
The 200-kHz SDPs calculated with both the DWBA and SDWBA were used in a simulation program (Cutter and Demer, 2007) to predict MBE measurements of scattering intensity from krill aggregations with specified densities (number per m 3 per beam) and orientation distributions, defined by the mean + s.d. for normal angle distributions and the minima and maxima for uniform distributions.Krill orientations were controlled by three angles.The pitch describes rotation about the lateral axis of the krill, or head-up, tail-down motion; the roll describes rotation about the head -tail axis; and the yaw describes rotation about the vertical axis (Figure 2).In each simulation, the number of krill was varied from 25 to 50 per m 3 .These densities were large enough to produce distributions with approximately normal pitch and uniform yaw.The same distributions of krill orientation were used for all beams.It was assumed that each krill was insonified on the beam axis, where the acoustic sensitivity is at a maximum.Outputs of the simulation included TS for each beam and each krill, and the mean and distribution of S v , normalized to one animal per unit volume for the krill insonified by each beam.Hence, the model results can be adjusted to predict S v for any number (N) of krill as the sum of the normalized S v and 10 log(N).Data were simulated for a subset (at intervals of 108) of beams spanning an athwartships swath of 1808 (+908 to each side of the vertical).

Multibeam data
A 200-kHz MBE (Kongsberg-Mesotech SM2000/SM20) was used to survey krill in shallow water (,200 m) off Cape Shirreff, Livingston Island, Antarctica (Cox et al., in press) from 2 to 9 February 2006.The MBE was deployed from a small (6 m long) inflatable boat (Mark V Zodiac).The MBE has 128 beams spanning a 1208 swath.The transmit power was 'medium', pulse duration was 825 ms, maximum range was 200 m, TVG was 20 log R þ 2 aR (where R is the range and a the absorption coefficient).The raw data were recorded and gain compensation was applied during post-processing.
The S v measurements from the MBE were calibrated (offset ¼ -71.2 dB) against measurements made concurrently with a calibrated 200-kHz SBE (Simrad ES60; see Cox et al., in press, for details).The S v from krill swarms were exported from Echoview (Myriax, Hobart, Australia) for calculations of the mean and s.d. of S v vs. beam direction as in Cutter and Demer (2007).These data were compared with patterns predicted by the simulation.Additionally, S v vs. range and bearing were exported for all swarms identified using Echoview's schooldetection routine in the multibeam module (Cox et al., in press).A random set of 100 pings from these data was examined for patterns of S v vs. beam direction.

Krill SDPs
SDPs based on the DWBA at 38, 70, 120, and 200 kHz were imaged three-dimensionally (Figure 3).The complexity and directivity increase with frequency.At 38 kHz, the SDP resembles a simple disc or compressed sphere, with a nearly circular main lobe (TS % -80 dB) in the plane defined by the lateral (y) and dorso-ventral (z) axes (Figure 3a).The SDPs at 70 and 120 kHz have a narrow main lobe and two (70 kHz; Figure 3b) to four (120 kHz; Figure 3c) side lobes, tilted in the y-z plane towards the head -tail axis.The SDP for 200 kHz has a very narrow main lobe and several narrow side lobes in close proximity to dorsal incidence, with peaks nearly equal to the main lobe (Figure 3d).Among these frequencies, krill TS was highest at 120 kHz with a peak value of 265.4 dB at dorsal incidence.
For comparison, the SDPs based on the SDWBA at 120 and 200 kHz are presented in Figure 4.The phase variations modelled in the SDWBA had little effect on the dorso-ventral scattering.However, the SDWBA predicted an increase in the TS observed from the head and tail directions and reduced directivity off the main lobes.These effects were most pronounced in the 120-kHz SDP, where the scattering was nearly uniform for all incidence angles away from the primary lobe, and include a more subtle, tilted secondary lobe.

Modelled multibeam measurements
The 200-kHz SDP from the DWBA was used to simulate measurements of S v vs. beam direction for an MBE with 1808 swath.The resulting S v vs. pitch is displayed in Figure 5.When krill are horizontal and aligned with the ship, the MBE beams intersect the main lobe of the SDP at all incidence angles, resulting in a nearly constant TS of approximately 270 dB.If pitch ¼ 58, the S v observed in the vertical beam changed by 25 dB; and by nearly 210 dB in the beams 308 to either side of the vertical.
To simulate MBE measurements of S v at 200 kHz from a krill aggregation with a pitch distribution N(118, 48), as described by Conti and Demer (2006), and a uniform distribution of yaw, DWBA-SDPs representing 25 krill were insonified by each beam.The resulting mean S v were 5 -10 dB higher in the beams +208 from the vertical compared with the outer beams.When the simulated pitch distribution was broader [e.g.N(118, 248)], the values of S v were nearly equivalent, with approximately equal variance vs. beam direction (Figure 6).

Multibeam survey data
Various patterns of S v vs. beam direction were apparent in the MBE data, but none was dominant.The observed patterns included nearly uniform, trending, alternating high and low, and occasionally an  absence of krill in the beams at and near vertical incidence (Figure 7).The S v vs. beam direction was variable, with S v ranging +2-10 dB from the mean value of each pattern (Figure 7).Trends and patterns of S v vs. beam direction were evident in some pings, but none could be clearly matched to the simulation results.For example, the simulation predicted that for a mean pitch ¼ 118, S v should be 5 dB lower for the vertical vs. outer beams when yaw ¼ 08, and higher in the near-vertical beams for non-zero, variable yaw.Lower S v in near-vertical beams was apparent in at least one case, but this could not be attributed uniquely to yaw.The result could also have been caused by non-uniform krill density that could occur naturally or a reaction to the vessel.

Discussion
When surveying krill with an SBE at the common frequencies of 70, 120, and 200 kHz, TS and S v are highly sensitive to small changes in pitch, but roll has negligible effect.However, when surveying with a high-frequency MBE, the pitch, roll, and yaw of krill are all important, because the SDPs are complex and the large swath insonifies the krill at many incidence angles.
For SBEs without motion compensation, the vessel's roll and pitch will result in non-vertical beams.The SDPs for krill at 70 -200 kHz are most variable within +458 of dorsal-aspect incidence.Within a narrow range of beam directions (+158), TS of krill can vary by 10 dB (at 200 kHz), depending on the krill orientations.If a krill yaws a mere 158 off the vessel-track direction, and the beam direction varies from 08 to 158 because of vessel motion (for the SBE), or for that portion of an MBE swath, the measured TS can range from 270.4 to 281.7 dB.Therefore, krill yaw is important even for surveys with SBEs, particularly at frequencies .120kHz.
The krill TS and the SDP depend on the animal size relative to the sound wavelength.The krill model used in this study was chosen to be consistent with the size of krill modelled in previous studies (McGehee et al., 1988;Demer and Conti, 2003b).Demer and Conti (2003b) measured krill length-frequency distributions from 20 to 50 mm, with an overall mean of 31.6 mm (s.d.¼ 6.6 mm).Larger krill would probably have slightly higher mean TS values and somewhat more complicated SDPs at lower frequencies (,200 kHz).Similarly, the SDP of smaller krill at 200 kHz would probably have a simpler geometry, a lower directivity, and a lower mean TS averaged over all angles.
For krill orientated with pitchN(118, 48) (Conti andDemer, 2006) andyawU(21498, 1398), the simulations based on the SDWBA at 200 kHz indicate an obvious pattern in the multibeam data (Figure 8).Specifically, measured scattering intensities should be nearly 8 dB higher for the near-vertical beams relative to the outer beams.This was not observed in the field data.
Many explanations are possible for the absence of the expected patterns in the MBE field data.For example, the pitch distribution might not be the N(118, 48) estimated by Conti and Demer (2006) for krill beneath a large vessel surveying at speed, or the krill might have been dispersed throughout each beam, as opposed to being only on the beam axes.That is, the MBE measurements were convolved with the beam patterns of each beam.A refinement of the simulation model should therefore account for beam-pattern effects.Alternatively, measurements with a fully calibrated MBE, particularly a split-beam instrument such as the Simrad ME70, could be deconvolved with the known beam patterns before comparison with the simulation results.Another hypothesis is that the krill changed their natural orientations or locations in vessel-avoidance reactions, as suggested by the swarms detected in the outer beams compared with those in the inner beams (Figure 7).
Non-uniform orientation distributions are difficult to estimate even for scatterers with high and known SDPs.Hence, the estimation of scatterers' orientation from narrowband MBE data requires restrictive conditions.However, concurrent multibeam and multifrequency split-beam measurement spanning 38-200 kHz could better constrain the problem and provide better estimates of scatterer' orientations.

Conclusions
If the orientation angles of the insonified krill are not known, the measured TS and S v include large uncertainties.A calibrated MBE is not prerequisite to detect the S v vs. beam-direction patterns associated with known consistent distributions of krill orientations.Therefore, krill orientations can be estimated from narrowband, multibeam data if all krill in an aggregation are orientated similarly across all beam directions, for example if krill are responding to a vessel in a predictable manner.However, the MBE must be calibrated to allow a quantitative comparison with a calibrated, split-beam SBE.The complexities of high-frequency SDPs make it difficult to estimate krill-pitch distributions from measurements with a single-frequency MBE.In fact, the MBE measurements of S v vs. beam direction in this study did not conform consistently to any of the simulation predictions.The simulations may be improved by accounting for additional factors, such as krill-length distributions and variations in their orientations within a swarm.In addition, the beam pattern, of the MBE should be taken into account.Either the simulated S v should be convolved with the expected beam patterns, or the measured S v should be deconvolved with the calibrated beam patterns.Future studies should include independent observations of krill orientation during acoustic surveys, perhaps using visual techniques. of the Southern Ocean, and the UK National Environmental Research Council, Royal Society, and the US Antarctic Marine Living Resources Program for co-sponsoring the data-collection portion of this investigation.We thank Jeff Condiotty of Simrad, USA, for the use of the SM2000.

Figure 1 .
Figure 1.Three views of the shape used to represent a generic krill for the DWBA and SDWBA models; (a) left side, (b) dorsal, and (c) head.Labels in (a) indicate the dorsal ('d') and ventral ('v') surfaces, and the head ('h') and tail ('t') of the krill.

Figure 2 .
Figure 2. Definitions of pitch, roll, and yaw used in the simulations.A horizontal krill with the dorsal side upwards has zero pitch.Positive pitch corresponds to head-up rotations.

Figure 3 .
Figure 3. SDPs representing the TS (dB) for any beam-incidence angle about a krill (E.superba); results from the DWBA model for (a) 38 kHz, (b) 70 kHz, (c) 120 kHz, (d) 200 kHz, and (e) the krill shape depicted within the 120 kHz SDP illustrating the incidence directions.

Figure 4 .
Figure 4. SDPs representing the TS (dB) for any beam-incidence angle about a krill (E.superba), resulting from the SDWBA model for (a) 120 kHz, and (b) 200 kHz.