Christophe Laplanche
email: laplanche@univ-paris12.fr

Olivier Adam

Maciej Lopatka

Jean-François Motsch

Male sperm whale acoustic behavior observed from multipaths at a single hydrophone

Sperm whales generate transient sounds ͑clicks͒ when foraging. These clicks have been described as echolocation sounds, a result of having measured the source level and the directionality of these signals and having extrapolated results from biosonar tests made on some small odontocetes. The authors propose a passive acoustic technique requiring only one hydrophone to investigate the acoustic behavior of free-ranging sperm whales. They estimate whale pitch angles from the multipath distribution of click energy. They emphasize the close bond between the sperm whale's physical and acoustic activity, leading to the hypothesis that sperm whales might, like some small odontocetes, control click level and rhythm. An echolocation model estimating the range of the sperm whale's targets from the interclick interval is computed and tested during different stages of the whale's dive. Such a hypothesis on the echolocation process would indicate that sperm whales echolocate their prey layer when initiating their dives and follow a methodic technique when foraging.

I. INTRODUCTION

The scientific community is compelled to research sperm whales ͑Physeter macrocephalus͒ for a variety of reasons; some researchers may strive to assess the risks associated with anthropogenic noise ͑Gisiner, 1998͒, while others conduct experiments to protect the whales from excessive boat traffic ͑Delory et al., 2002͒. Another motivation is the quest for new knowledge regarding the foraging behavior of sperm whales, since this aspect of their behavior is still the most puzzling ͑Fristrup and [START_REF] Fristrup | How do sperm whales catch squids?[END_REF][START_REF] Jaquet | How spatial and temporal scales influence understanding of sperm whale distribution: A review[END_REF] Underwater visual observations of free-ranging, foraging sperm whales have never been carried through convincingly ͑Mccarey and Rubin, 1998͒, and the information available on the sperm whale diet is limited to stomach or fecal samplings. This is due to the fact that sperm whales mostly feed on mid sea and deep sea squid and fish, and at such depths research is difficult to conduct ͑Santos et al., 2001;[START_REF] Smith | The diet of Galapagos sperm whales ͑physeter macrocephalus͒ as indicated by fecal sample analysis[END_REF]. However, many experiments have been conducted to address the whales' foraging behavior, using tagging techniques, visual observations, and passive acoustics.

Tagging sperm whales with depth-meters has led to an understanding of their diving behavior. Sperm whales undertake series of 30-90-min four-stage foraging cycles, composed of a breathing break at the sea surface, a vertical descent to the hunting depth, the hunt itself, and a vertical resurfacing ͑Watkins et al., 1993͒. Visual observations of sperm whales breathing at the sea surface provide some understanding of their social behavior. For instance, the males tend to travel solitarily, whereas the females and their calves travel in groups ͑Arnbom and [START_REF] Arnbom | Observations on the composition and behavior of groups of female sperm whales near the Galapagos Islands[END_REF] The use of passive acoustics enables scientists to better understand the sperm whale acoustic activity. Sperm whales emit a multitude of transient sounds ͑clicks͒ when diving. The function of the clicks may be to communicate ͑Watkins and [START_REF] Watkins | Sperm whale codas[END_REF]Weilgart, 1990͒ or echolocate ͑Jaquet et al., 2001;[START_REF] Mohl | Sperm whale clicks: Directionality and source level revisited[END_REF] The directionality and source level of the clicks are measured to determine their use in communication or echolocation [START_REF] Madsen | Male sperm whale ͑physeter macrocephalus͒ acoustics in a high-latitude habitat: implications for echolocation and communication[END_REF] Echolocating sperm whales emit usual click sequences ͑0.2 Ͻ ICIϽ 2 s, interclick intervals͒, interrupted by faster click trains ͑creaks, ICIϽ 0.2 s͒ and then silences ͑Mullins et al., 1988͒. Sperm whales emit echolocation clicks mostly during the descent and while hunting within the prey layer ͑Madsen et al., 2002b͒. Sperm whales within social groups also send stereotyped ICI structure click sequences, called codas ͑Weilgart, 1990͒. Male sperm whales may make some other rare vocalizations, called slow clicks [START_REF] Jaquet | Vocal behavior of male sperm whales: Why do they click?[END_REF][START_REF] Madsen | Male sperm whale ͑physeter macrocephalus͒ acoustics in a high-latitude habitat: implications for echolocation and communication[END_REF]. The use of passive acoustics is not only limited to the study of the click rythmics, but it also leads to an understanding of the click production mechanism ͑Mohl et al., 2003;[START_REF] Thode | Depth-dependant acoustic features of diving sperm whales ͑physeter macrocephalus͒ in the Gulf of Mexico[END_REF] to the reconstruction of sperm whale diving trajectories ͑Wahlberg, 2002͒.

New tagging techniques, using depth-meters, hydrophones, accelerometers, and magnetometers, have enabled scientists to gain knowledge of sperm whale behavior. These tags enable scientists to record sperm whale movements together with their acoustic activity ͑Johnson and Tyack, 2003; [START_REF] Zimmer | Combining data from a multisensor tag and passive sonar to determine the diving behavior of a sperm whale ͑physeter macrocephalus͒[END_REF] Fluking has been measured while sperm whales swim down and resurface ͑Miller et al., 2004a͒. Sperm whales maneuvering is measured during the hunting process ͑Miller et al., 2004b͒. Such tagging techniques, like passive acoustics, have also lead to the description of the click production mechanism and to the reconstruction of diving trajectories, but with increased levels of precision ͑Zimmer et al., 2005͒.

While the aforementioned tagging techniques provide insight into sperm whale behavior, the technique itself can disrupt or alter the natural tendencies of the tagged mammals. Also the tagging procedure is time consuming and difficult to implement ͑Madsen et al., 2002a͒. Because of this, the authors bring forward a passive acoustic technique that will uncover new behavioral information without such shortcomings. Passive acoustics may give results close to those attained using depth-meter/hydrophone tags ͑Madsen et al., 2002b͒. The depth ͑and range͒ of sound sources can be estimated by using a single hydrophone. This method has been successfully carried out on sperm whales ͑Thode et al., 2002͒ anddolphins ͑Aubauer et al., 2000͒. Nevertheless, the accuracy on the depth and range measurements is not enough for estimating subtle foraging sperm whale movements.

The authors estimate subtle sperm whale movements by measuring pitch angle variations using a single towed hydrophone. The comparison of these variations to the acoustic activity then leads to a hypothesis on the sperm whale echolocation system: sperm whales, like some other toothed whale species, would control click level and ICI when echolocating on targets at different ranges. The authors then compute an echolocation model estimating sperm whale target ranges from the ICI. They apply this echolocation model during the sperm whale descent and hunt, and assess its consistency. They estimate the water mass geometry analyzed by the whale's sonar during both stages, leading to the hypothesis of a dive-scale sperm whale foraging strategy.

II. MATERIAL AND METHODS

A. Single hydrophone localization

The authors need to assess the vertical distribution of click level to estimate sperm whale pitch angle. To this end, they compare the click levels emitted along different rays of multipath propagating sperm whale clicks. The authors estimate the depth and range of phonating sperm whales, before reconstructing the multipath ray geometry, to measure the attenuation of the click signals when transmitting along these rays.

The sea surface and the seafloor reflect and diffuse propagating acoustic waves ͑Clay and Medwin, 1977͒. The hydrophones collect multiple delayed, dephased, and attenuated versions of the signals emitted by sound sources ͑Fig. 1͒. The use of a single hydrophone, and the detection of n surface/seafloor echoes, serves as a substitute of a large virtual vertical ͑n +1͒-hydrophone array ͑one real, n virtual͒ ͑Aubauer et al., 2000͒. A sound source can be localized in depth and range from this array, by measuring the time of arrival differences of the source signal between the virtual hydrophones, then by measuring the time of arrival differences between the echoes ͑Fig. 2͒. One needs to detect two echoes of a click to estimate the depth and range of a phonating sperm whale. The detection of an additional echo leads to an estimation of the depth of the recording hydrophone ͑Thode et al., 2002͒.

B. Accuracy

The depth, range, and propagation loss are estimated using a spherical propagation model, by assuming that the speed of sound is constant along the water column. The validity of this convenient hypothesis is assessed by comparing this model with a ray-tracing model. The vertical variations of the speed of sound are estimated from temperature and salinity values given by the MERCATOR oceanic model ͑MERCATOR, 2004͒. The authors have compared the propagation times of sperm whale clicks by ray tracing, in a me-FIG. 1. Signals received from a single emitted sperm whale click. Here, assuming z h Ͻ z s Ͻ z bz h ͑Fig. 2͒, the direct path click signal ͑1͒ comes first, followed by the surface echo ͑2͒, the seafloor echo ͑3͒, and the-twice reflected seafloor/surface echo ͑4͒. Click signals reflected by the seafloor are more attenuated when propagating ͑by reflecting and by transmitting on a longer distance͒ than click signals transmitted via direct path. The amplitude of the former may anyway be greater than the amplitude of the latter, suggesting click directionality. FIG. 2. Multipath ray geometry. The depths of the sperm whale, the hydrophone, and the seafloor are respectively z s , z h , and z b . The sperm whale is at the range r s from the hydrophone. The direct path ray, surface reflected ray, and seafloor reflected ray form the algebraic angles ␣ 1 ͑␣ 1 Ͼ 0 ⇔ z s Ͼ z h ͒, ␣ 2 ͑␣ 2 Ͼ 0͒, and ␣ 3 ͑␣ 3 Ͻ 0͒ to the horizontal. The detection of two surface/ seafloor echoes serves as a substitute of a large virtual vertical hydrophone array of length 2z b . The whale depth and range define the geometry of the multipath rays ͑length and inclination͒. that in the context of the author's experiments, the assumption of a spherical propagation model would be enough. This assumption would be valid in some sensible settings, with a hydrophone and a source not too close to the sea surface, and a source at close enough range, so as to limit the interference and refraction phenomena.

The accuracy of the localization process when using the spherical propagation model is evaluated using a random variable error estimator ͑Laplanche et al., 2004͒ modeling with Gaussian random variables the nonlinear error estimation method presented in Wahlberg ͑2004͒. The source depth and range are calculated from the seafloor depth, the hydrophone depth, the surface/seafloor echo delays, and the mean speed of sound ͑Thode et al., 2002͒. A large number of samples ͑100 000͒ of these parameters are drawn around a mean ͑the measured value͒ with a standard deviation ͑half the accuracy of the measurement͒. The accuracies ͑twice the standard deviations͒ of the source depth and range are estimated from the depth and range samples calculated from the parameter samples. The linearity of the localization process is evaluated by estimating the kurtosis excess of the depth and range samples.

C. Click level

Acoustic waves are attenuated when propagating in the sea water and when reflecting on the sea water interfaces. Still using the spherical propagation model, the attenuation of an acoustic wave when traveling through a distance r of sea water is close to 20 log 10 ͑r͒ + ar ͑Aubauer et al., 2000͒. The absorption coefficient a depends on the frequency of the transmitting wave and on chemical properties of the sea water ͑such as pH, salinity, and temperature͒. In this particular case, it is close to a =10 -3 dB/ m using the Thorp model. The authors consider sperm whale click central frequency at 9 kHz, given the limited band of the recording system, as compared to the 15-kHz sperm whale click central frequency ͑Madsen et al., 2002a͒.

The attenuation of an acoustic wave when reflecting to the seafloor is estimated using the Hall-Watson model. It is contingent upon the incidence angle of the wave on the seafloor, the frequency of the wave, and the porosity ͑0 Ͻ p Ͻ 1͒ of the seafloor.

Using the sonar equation, the measurement of the input levels, and the estimation of the propagation attenuation of direct path transmitted sperm whale click signals and seafloor reflected click signals will then lead to an evaluation of the apparent source levels of the clicks in these ray directions ͑in dB re: 1 V/Pa͒. The comparison of the apparent source levels in these ray directions will then lead to an estimation of the vertical distribution of the click beam energy.

D. Sperm whale pitch angle variations

Let us consider a sperm whale at a depth z s and a range r s from a hydrophone, itself at a depth z h , in a basin of depth z b ͑Fig. 2͒. The direct path ray ͑1͒, the surface echo ray ͑2͒, and the seafloor echo ray ͑3͒ form the algebraic angles to the horizontal ␣ 1 = atan͓͑z sz h ͒ / r s ͔, ␣ 2 = atan͓͑z s + z h ͒ / r s ͔, and ␣ 3 = atan͓͑z s + z h -2z b ͒ / r s ͔. Let p 1 , p 2 , and p 3 be sperm whale click level sent in these three directions. The ratios ͑p i / p j ͒ dB = 20 log 10 ͑p i / p j ͒ would lead to an estimation of the vertical click level distribution.

Sperm whales may alter the click level while diving, as discussed below. There would be no consequences of click level variations on the values of the ratios ͑p i / p j ͒ dB , if the sperm whale click frequency content were constant. But sperm whales, like smaller odontocetes, alter jointly click frequency and level ͑Madsen et al., 2002a͒. Such a frequency emphasis would lead, after transmission, to an alteration of the measurement of the ratios ͑p i / p j ͒ dB , and a resulting bias in the estimation of the sperm whale pitch angle. The authors, given the limited band of their recording system, assume these effects to be negligible. Directionality variations, as directly induced by a modification of the shape of the click beam, would also alter the whale pitch angle estimation. However, such variations do not take place, as discussed below. The variations of the ratio ͑p i / p j ͒ dB would then be representative of a change in the direction the sperm whale click beam.

The surface ray is directed upwards ͑␣ 2 Ͼ 0͒, and the seafloor ray is directed downwards ͑␣ 3 Ͻ 0͒. The upwards/ downwards direction of the direct ray ͑Fig. 2͒ depends on the depth of the source ͑␣ 1 Ͼ 0 ⇔ z s Ͼ z h ͒. For short cable surface hydrophones ͑z h Ӷ z b ͒, the direct ray would be directed upwards, as sperm whales usually start clicking a few minutes after beginning a dive, stop clicking a few minutes before surfacing, and are usually silent at the sea surface level ͑Whitehead and Weilgart, 1990͒. For short cable hydrophones mounted on the seafloor ͑z bz h Ӷ z b ͒, the direct ray would be directed downwards. For long cable hydrophones, a preliminary source localization is necessary to conclude the direction of the ray. One can demonstrate that the aperture angles defined by the direct/surface rays and the seafloor ray are greater than / 4 for a usual source range/depth setting,

␣ 1 -␣ 3 ജ / 4 for r s ഛ 2͑z b -z h ͒ and z s Ͼ z h , ␣ 2 -␣ 3 ജ / 4 for r s ഛ 2z b and z s Ͼ z h .
The ratio ͑p 2 / p 3 ͒ dB ͓or ͑p 1 / p 3 ͒ dB , assuming a short cable surface hydrophone͔ would then assess the trend in the vertical direction of the click energy. Assuming a vertical symmetry of the sperm whale click acoustic beam ͑Zimmer et al., 2005͒, then p 2 / p 3 = 0 dB for clicks emitted in the direction ͑␣ 2 + ␣ 3 ͒ /2, p 2 / p 3 Ͼ 0 dB for clicks emitted in directions ␣ Ͼ ͑␣ 2 + ␣ 3 ͒ / 2, and p 2 / p 3 Ͻ 0 dB for clicks emitted in directions ␣ Ͻ ͑␣ 2 + ␣ 3 ͒ / 2. For ͉␣ 2 ͉Ӎ͉␣ 3 ͉, the sign of ͑p 2 / p 3 ͒ dB gives the vertical direction of the click beam. Assuming that sperm whales need to move their head to change the direction of the click beam, the variations of ͑p 2 / p 3 ͒ dB would then indicate changes in sperm whale vertical movement. The authors define the whale pitch angle ␤ as the angle formed by the whale's dorso-rostral axis and the horizontal.

E. Sperm whale pitch angle

The use of a sperm whale click beam pattern ͑Zimmer et al., 2005͒ theoretically leads to a quantification of the values of sperm whale pitch angles. Using a whale heading to the direction ␥ ͑defined as the projection of the off-axis angle on the horizontal plane, Fig. 3͒ with the pitch angle ␤, a trigonometry calculation leads to the value of the off-axis angle ␦ of the hydrophone to the whale click beam:

cos ␦ = cos ␣ 1 cos ␤ cos ␥ + sin ␣ 1 sin ␤.

͑1͒

And by defining ͔ -/2, /2͓ such that tan = cos ␥ / tan ␣ 1 , the whale pitch angle can be calculated from the off-axis angle, by selecting ␤ ͓-/2, /2͔ from

␤ = asinͩ cos ␦ cos sin ␣ 1 ͪ-or ͑2͒ ␤ = -asinͩ cos ␦ cos sin ␣ 1 ͪ-.
Using the sperm whale click beam pattern and the estimation of the click gain, the estimation of the click beam level emitted towards the hydrophone leads to the estimation of the ray off-axis angle ␦, which then leads to an estimation of the whale pitch angle ␤. The sperm whale heading ␥ is estimated from the three-dimensional whale trajectory, as described below.

F. Sperm whale trajectory reconstruction

The authors estimate small scale sperm whale pitch movements from the vertical distribution of click level. Large scale movements can be estimated from the sperm whale depth and range estimations, which are found by measuring click echo delays. Large scale movements will give an estimation of the sperm whale heading ␥, as required in the sperm whale pitch angle quantification process. Large scale movement will also lead to the rough three-dimensional reconstruction of sperm whale trajectories.

The authors deconstruct the sperm whale trajectory into segments, while assuming a uniform sperm whale movement ͑straight and constant speed͒ on each segment. Let us consider a whale at depth z 1 , range r 1 , and time t 1 heading straight to reach the depth z 2 , range r 2 , and time t 2 at the constant speed v 12 ͑Fig. 3͒. The sperm whale depth and range at a time t 1 ഛ t ഛ t 2 are

z s = z 1 + v z ͑t -t 1 ͒, ͑3͒ r s 2 = r min 2 + v r 2 ͑t -t min ͒ 2 ,
while the vertical and horizontal speeds of the whale are

v z = ͑z 2 -z 1 ͒/͑t 2 -t 1 ͒, ͑4͒ v r = ͱ r 1 2 -r min 2 /͑t 1 -t min ͒,
where r min is the horizontal distance of the hydrophone to the line containing the trajectory segment, and t min is the time when the whale would reach the range r min ͑Fig. 3͒. Interpolating the whale depth variations during ͓t 1 , t 2 ͔ with a straight line leads to an estimation of depths z 1 , z 2 and vertical speed v z . Interpolating the squared whale range variations r s2 with a parabola leads to an estimation of the ranges r 1 , r 2 , r min and the horizontal speed v r . Such interpolations lead to an estimation of the three-dimensional geometry of the trajectory segment ͑a symmetry and a rotation around the hydrophone vertical axis apart͒, the whale speed v 12 = ͱ v z

G. Recording system

The data were recorded during two missions in the Mediterranean Sea, in the Main Channel of the Strait of Gibralter in March 2003, and in the abyssal plain offshore Toulon in August 2004, using preamplified, omnidirectional, ceramic Vinci-Technologies HC2000 hydrophones, with an output sensitivity of, -155 dB re: 1 V/Pa, and a frequency response flat in the frequency window ͓80 Hz, 13 kHz͔. The signals were sampled at 44.1 kHz, PCM coded with 16 bits using a laptop with a Digigram VXpocket 440 soundcard, and recorded using the multitrack software Sek'd Samplitude. In this experiment, the signals have been highpassed at 3 kHz with a digital filter, so as to enhance the S/N ratio in the click detection process, given the relative frequency bands of sperm whale clicks and background noise. Solitary male sperm whales were recorded, from the CIRCé ship Elsa ͑10 m͒ during the 2003 mission, and from three Breach sail boats Brin d'alu ͑10 m͒, Saoufe ͑12 m͒, and Zino ͑14 m͒ during the 2004 mission.

Hydrophones were at 30-m depth in 2003 and 70 m in 2004. The sea bottom was at 900 m in 2003 and 2000 m in 2004. The depth of the seafloor was measured using marine charts and is defined as the depth below the hydrophones. The depth of the hydrophones z h was estimated using the method presented in Thode et al. ͑2002͒. The seafloor is assumed to be flat at a known depth and of known porosity.

The assumption of a flat seafloor is applicable in the author's examples but not in all other cases. When sperm whales are recorded in front of the continental slope, clicks reflect on the cliff into many echoes, making difficult their identification. The clear detection of seafloor echoes is a good test for substantiating the flatness of the seafloor. In both recording locations cited above, the seafloor was flat enough to identify FIG. 3. Sperm whale trajectory. The sperm whale trajectory is deconstructed into segments ͑here a single segment, projected into the horizontal plane͒, the whale moving straight at constant speed in each segment. In this example, the whale moves along the segment from the range r 1 at t 1 to the range r 2 at t 2 . The range of the whale while crossing the segment is r s = ͱ r min 2 + v r 2 ͑t -t min ͒ 2 , and its heading is defined by the angle ␥. The range r min is the horizontal distance of the segment to the hydrophone. t min is the time the whale passes ͑or would pass͒ at the range r min .

the detected echoes. Short-scale relief variations scatter seafloor echoes and decrease the accuracy of the measurement of their delays. The consequence of this lack of accuracy is estimated using the random variable error estimation method described above.

H. Delay/level measurements

The data is recorded from isolated male sperm whales. The authors study the acoustic behavior of isolated male sperm whale individuals and do not deal with group strategies. The methods ͑click detection͒ and results would apply within this framework. Nevertheless, hypotheses on the acoustic behavior of isolated sperm whales could also help in understanding the behavior of each individual within a group.

Surface echoes are always detected jointly with direct path click signals. Twice reflected seafloor/surface echoes are also always detected with once-reflected seafloor echoes ͑Fig. 1͒. Seafloor echoes are well detected during the first 10 min of dives, when sperm whales vertically swim down and echolocate towards the seafloor ͑Thode et al., 2002͒. The seafloor echo detection during the hunting stage depends on the seafloor properties ͑depth and relief͒ and the whale range. The seafloor echoes were more easily detected during the 2003 mission, by recording sperm whales from a longer range in a more shallow bassin. Surface and seafloor echo delays are measured by cross-correlating the echoes with the direct path signal. The localization and click level measurement processes have been automated by estimating the a priori probability density functions of the surface/seafloor delays as described in Laplanche et al. ͑2005͒.

III. RESULTS

A. Acoustic behavior

The variations of the surface echo delay during a complete dive of a single sperm whale ͑recorded during the 2004 mission͒ are plotted in Fig. 4

B. Click directionality and level

The estimated direct path and seafloor reflected click levels ͑p 1 and p 3 , respectively͒ and ICI variations during search/terminal phases are plotted in Fig. 6. Here, the whale is located at a depth z s = 600± 30 m ͑kurtosis excess 3 ϫ 10 -3 ͒, and a range r s = 3700± 250 m ͑kurtosis excess 2 ϫ 10 -3 ͒, from a hydrophone at a depth z h =32±2 m, in a basin of depth z b = 880± 30 m and mean speed of sound c 0 = 1512± 10 m / s, measuring surface echo delays ±0.1 ms and seafloor echo delays ±0.5 ms. The direct ray, the surface ray, and the seafloor ray then create respectively ␣ 1 = 8.5± 0.6, ␣ 2 = 9.5± 0.6, and ␣ 3 = -17± 1.3 ͑kurtosis excess 5 ϫ 10 -2 ͒ degree angles with the horizontal ͑Fig. 2͒. The 0 dB value of the level ratio ͑p 1 / p 3 ͒ dB would then correspond to a click emitted slightly downwards, forming a ␣ = ͑␣ 1 + ␣ 3 ͒ /2 = -4.2°± 0.6°angle with the horizontal.

The sperm whale click beam pattern is extremely narrow ͓͑p 1 / p 3 ͒ dB = -30 at t = 780 s, Fig. 6͔. Sperm whale clicks are also very powerful ͑Fig. 7͒, reaching 210 dB re: 1 Pa ͑pp͒ ͑limiting to the band ͓3 kHz, 13 kHz͔͒. These two results are consistent with previous findings, i.e., the sperm whale click generator can also serve as an echolocation system ͑Mohl et al., 2003͒. FIG. 4. Surface echo delay variations during a complete dive. A whale ͑2004 data set͒ emitted 2380 usual clicks and 31 creaks ͑vertical lines͒ during a 45-min dive. Two thousand clicks are detected using a threshold/crosscorrelation algorithm, leading to 2000 measurements of the surface echo delay ͑dots͒. The algorithm wild points are also plotted ͑200, crosses͒. The whale emitted the first creak at t = 10 min, at the depth z b = 400 m, with a 57-ms surface echo delay. The whale repeats 31 search/terminal phases, in this example ending with creaks.

C. Sperm whale movements

In this study, only relative ͑and not absolute͒ values of sperm whale pitch angles are estimated, by using the ratio ͑p 1 / p 3 ͒ dB . The complexity of the sperm whale click beam pattern as formed by various components of different levels and frequency contents ͑Zimmer et al. 2005͒, the lack of knowledge in the variations amongst sperm whale individuals of this click beam pattern, the lack of knowledge in the click level variations, and the use by the authors of a limited bandwith recording system kept the authors from conducting a reliable pitch angle quantification process ͓using Eq. ͑2͔͒.

Sperm whale pitch angle variations are plotted during a descent ͑Fig. 5͒ and during two successive search/terminal phases ͑Fig. 6͒. The respective acoustic activity is also plotted. The sperm whale pitch movements go together with the rhythmic variations while descending ͑Fig. 5͒. The sperm whale pitch movements while hunting are periodic and in sync with the rhythmic variations ͑Fig. 6͒. In the next section, the authors propose a hypothesis on the sperm whale biosonar, so as to clarify this correlation between the sperm whale acoustical and physical activity.

IV. DISCUSSION

A. Echolocation model

The recorded sperm whales emitted bursts of usual clicks each lasting 5 -15 s during the search phases, followed by clicks at a higher rate ͑creaks͒ and/or silence periods during the terminal phases ͑Fig. 6͒. As shown in Fig. 7, during the search phase, clicks from a burst are first emitted every ICI max second. The interclick interval then progressively decreases to ICI min , before gradually increasing to a new maximum ICI max Ј . These variations of the interclick in- terval are correlated to those of the levels p 1 and p 3 of the direct path/seafloor reflected click signals. During the search phases, 20-dB level peaks are simultaneously detected in both direct path and seafloor path directions. Such peaks were only detected when recording from directions close to the whale axis, as discussed below. Click level maximums p max emitted towards these paths are detected when the interclick interval of a 5 -15-s burst reaches its maximum ICI max . And click level minimums p min are detected when the interclick interval reaches its minimum ICI min ͑Fig. 7͒.

Four hypotheses could explain such apparent click level variations: vertical movement of the sperm whale head ͑pitch͒, horizontal movement ͑yaw͒, click directionality control, and click level control. The first hypothesis is rejected given the pitch angle variations plotted in Fig. 6. The second hypothesis is rejected, as maximums/minimums of click levels are always in sync with maximums/minimums of ICI ͑this also applies to the first hypothesis͒. The whale would have to systematically direct its click beam towards the hydrophone when clicking at a slow rate, and turn its head when clicking faster, which is highly unlikely. The third hypothesis would suggest that sperm whales would emit highly directional clicks when clicking fast, and widen the click beam when clicking slower. The fourth hypothesis would suggest that sperm whales would send less powerful clicks when clicking fast, and increase the click level when clicking slower. Homogenous off-axis angle variations of a measured sperm whale click beam pattern ͑Zimmer et al. 2005͒ would suggest that sperm whales would not change the click directionality when diving. The variations in click level given in Fig. 7 would rather suggest click level control. These click level variations are in sync with the ICI variations, suggesting click level and ICI adjustment to the target range, as found on some smaller odontocetes, as discussed below. Yet some sperm whale tag recordings show no correlation between click level and ICI variations ͑Madsen et al., 2002a͒. Click level variations as plotted in Fig. 7 were not detected when making off-axis recordings ͑especially in 2004, when recording from a shorter range͒. The authors then suggest that these level variations would only appear, using Mohl et al. ͑2003͒ notation, on the forward echolocation pulse ͑P 1 ͒. The initial backward pulse ͑P 0 ͒ level would not be altered by this process. Off-axis recordings, from a distant towed hydrophone or a tag attached to the whale's back, would then not detect the click level variations as plotted in Fig. 7. Still, what would explain P 0 20-dB level variations detected from the tags ͑Madsen et al., 2002a͒? Would click level adjustment to target range take place while transmitting within the nasal complex, and a different click level adjustment ͑to target kind?͒ take place when emitting the initial pulse at the phonic lips? Understanding click level variations may be a key point in learning the workings of the sperm whale's biosonar. The hypothesis on click level ͑and ICI͒ control to target range would lead to sensible results, as discussed below, and provide some clues regarding the understanding of the sperm whale's biosonar.

Results from biosonar tests carried out on some smaller odontocetes suggest click level control ͑Au and Herzing, 2003; [START_REF] Au | Echolocation signals of dusky dolphins ͑lagenorhynchus obscurus͒ in Kaikoura, New Zealand[END_REF][START_REF] Rasmussen | Source levels of clicks from freeranging white-beaked dolphins ͑lagenorhynchus albirostris gray 1846͒ recorded in Icelandic waters[END_REF] with target range. Dolphins would adjust the click level by 6 dB for every halved or doubled target range. ICI control has also been observed on bottlenose dolphins ͑Au, 1993͒. Such an acoustic behavior does not seem to be characteristic of all ondontocete species ͑Madsen et al., 2005͒. The sperm whale's and dolphin's biosonar share many similarities ͑Madsen et al., 2002a͒, and, given measurements illustrated in Fig. 7, it would be interesting to assess the consistency of the hypothesis of click level and ICI control to target range on sperm whales.

The author's echolocation model speculates that sperm whales control click level and ICI to target range as bottlenose dolphins do. The click level variations presented in Fig. 7 would then result in adjustments by 6 dB for every halved or doubled target range. The ICI would directly depend on the whale's target range r, ICI= 2r / c 0 + t proc ͑Au, 1993͒. The constant processing delay t proc would be the delay between the reception of the echo of a click and the production of the next click.

B. Processing delay

Sperm whales would adjust the level of clicks to compensate for the one-way transmission loss between the whale and its target. If a click emitted with a 0-dB gain aims at a target at a range r, a click emitted with a gain G ͑dB͒ would then aim at a target at a range 10 G/20 r, assuming only spherical spreading loss. Then, by labeling r max and r min the range of the targets aimed using clicks at the rates 1 / ICI max and 1 / ICI min , one arrives at r max =10 G/20 r min with G = 20 dB, leading to r max Ӎ ␣r min with ␣ = 10. ͑5͒

Assuming a processing time t proc of the click echoes independent of the range of the target, then The echolocation model is now calibrated by estimating t proc = 0.45 s. Its consistency is then evaluated during sperm whale descent and hunting.

C. Scan during the descent

Sperm whales emit echolocation clicks during most of the descent. What do they echolocate? Some authors have suggested that sperm whales echolocate to the sea floor. ICI variations are indeed correlated to the whale depth variations and consistent with a vertical scan of the sea bottom ͑Thode et al., 2002͒. The authors have in this case used the echolocation model ICI= 2r / c 0 , close to the one presented here but with a null processing delay. The seafloor scanning hypothesis, however, would not seem to be correct. Different authors have shown that the ICI of click sequences emitted during the descent may sometimes be much shorter than the two-way travel time to the sea bottom ͑Zimmer et al., 2005͒. These authors have suggested that sperm whales rather scan their prey layer while descending. ICI variations during the descent are indeed correlated to whale pitch angle variations ͑Zimmer et al., 2003͒.

The authors apply the echolocation model ICI= 2r / c 0 + 0.45 s to the descent. Pitch variations correlated to ICI variations are also noticed using the method presented by the authors ͑Fig. 5͒. The authors assess the consistency of the aforementioned hypothesis of prey layer scanning while descending. In the example of Fig. 4, the prey layer is at 400 m, which is equal to the depth of the sperm whale when emitting the first creak. Using the echolocation model, the estimated ICI during the descent, when using targets vertically below the whale inside the prey layer, is plotted in Fig. 5. Measured and estimated ICIs are equal at t = 200, 280, 400, and 570 s. The measured ICI is greater than the estimation between these instants; it increases for t = 200→ 250 s, t = 280→ 350 s, and t = 400→ 480 s; and decreases for t = 250→ 280 s, t = 350→ 400 s, and t = 480→ 570 s. These variations are correlated to those of the estimated pitch angle ͑Fig. 5͒. This would suggest, as mentioned in Zimmer et al. ͑2003͒, that the sperm whale indeed echolocated vertically at the prey layer ͑at t = 200, 280, 400 s͒, and echolocated at a horizontal prey layer strip by a vertical movement of its body while descending. The sperm whale pitch angle and the prey layer strip geometry can be quantified using the echolocation model. The sperm whale would start echolocating vertically, and make three scans of the prey layer, with pitch angles varying to 90°→ 60°→ 90°for t = 200→ 250→ 280 s, 90°→ 30°→ 50°for t = 280→ 350→ 400 s, and 50°→ 20°for t = 400→ 570 s, before entering the prey layer and hunting. Such pitch estimations are consistent with those found using digital tags ͑Zimmer et al., 2003͒. The prey strip is estimated as 600 m long ͑from the whale pitch angle and distance to the prey layer͒ and 80 m deep ͑from the whale pitch angle and its ICI standard deviation͒. The results presented in this section would suggest that sperm whales would indeed echolocate at their prey layer while descending, and follow an acoustic behavior which would be close to the aforementioned echolocation model.

D. Whale heading

The authors then estimate the whale trajectory from the interpolation of the whale range variations between t 1 = 16 min and t 2 = 37 min ͑Fig. 8͒. Such variations of the whale range could be explained assuming a straight trajectory at a constant speed, as described above. Based on this hypothesis, the whale would move, in the example of Fig. 8, in a straight line at v r = 0.8 m / s from the range r 1 = 480 m to the range r 2 = 650 m, passing at t min = 25 min at the minimum range r min = 200 m. The whale would then swim at a constant heading within the prey layer. Such behavior is consistent with surface visual observations, describing a general constant heading movement of male sperm whales based on the location of the surface breathing points ͑Watkins et al., 1999͒. The authors' measurements would support the hypothesis of a sperm whale broken-line trajectory, whereas trajectory reconstructions using tagged accelerometers would support more chaotic underwater behavior ͑Zimmer et al., 2005͒. The straightforward movement pointed out by the authors is a large scale movement and does not prevent short scale chaotic movements when closing on prey ͑Miller et al., 2004a͒.

E. Scan during the hunt

Sperm whales would move straightforward while hunting within the prey layer. The whale's physical activity while crossing the prey layer is estimated from the pitch angle variations, as described below. Its acoustic activity and the range of the whale's targets are estimated from the ICI using the echolocation model. The conjunction of all these bits of information would lead to the authors' hypothesis asserting that sperm whales would methodically scan a cone-shaped water mass searching for prey with usual clicking at depth. Sperm whales would then attack while silencing or creaking. This methodic scan is a hypothesis, and a consequence of the assessed assumption that sperm whales control ICI when searching for their prey.

The sperm whale's physical activity is estimated from the pitch angle variations ͑Fig. 6͒. In this example, the sperm whale would start usual clicking aiming horizontally ͑p 1 / p 3 =0 dB͒. It would then progressively point downwards ͑p 1 / p 3 Ͻ 0 dB, decreasing͒, before gradually ͑but more quickly͒ coming back to a horizontal line ͑p 1 / p 3 Ͻ 0 dB, in-creasing͒. During the terminal phase, the sperm whale would start with a horizontal click beam and move upwards, increasing its click rate or silencing, before aiming back horizontally so as to start a new search/terminal phase. The low S/N ratio during the terminal phases makes difficult the assessment of the acoustical/physical activity of the whale. Nevertheless, the fast click activity and silence would suggest feeding ͑Miller et al., 2004a͒.

From the value of the delay t proc , the authors estimate the ranges r min and r max during the search phases. At the beginning of the plotted search phases ͑ICI max = 1 s and ICI min = 0.5 s͒, the sperm whale would aim at targets located at r min Ӎ 42 m and r max Ӎ 420 m. At the end of the search phase ͑ICI max = 0.8 s and ICI min = 0.45 s͒, it would aim at targets located at r min Ӎ 30 m and r max Ӎ 300 m.

Given the echolocation and the progressive variations of the ICI during a single 5 -15-s click burst, decreasing from ICI max to ICI min before reincreasing to a maximum ICI max Ј ͑Fig. 7͒, the authors suggest that the whale would echolocate between the ranges r max and r min ͑ICI decreasing from ICI max to ICI min ͒ and then back to a maximum range r max Ј ͑ICI in- creasing from ICI min to ICI max Ј ͒ every 5 -15-s burst of clicks.

The authors suggest that during the search phase sperm whales would methodically scan a cone-shaped mass of water searching for prey ͑Fig. 9͒. Each scan would last 5 -15 s and would analyze the water twice between the upper ranges r max , the lower range r min = r max / ␣, and the upper range r max Ј . This scan would suggest that each sperm whale click is generated to aim in a specific direction at a specific range. Sperm whales would move physically to change the click beam direction, and control level and ICI to change the click target range. Similar results have been found during the sperm whale descent studied above.

There are some variations in the geometric features of the scanned water mass: variations within a dive, between dives, and from one sperm whale to another. These geometric features would depend on the number of scans, the ICI minimum and maximum values, the duration of the search phase, and the whale pitch angle variations. For instance, the cones scanned by the whale tracked in 2004 would be shorter ͑lower ICI͒ and thinner ͑only three to four 5 -15-s click bursts͒ than the cone scanned by the whale tracked in 2003, whose ICI sequence is plotted in Fig. 7. This result suggests that sperm whales would, if methodically scanning the prey layer when hunting, adapt their scanning depending on existing prey.

V. CONCLUSIONS

The authors have estimated sperm whale pitch angle relative ͑not absolute͒ values by studying multipaths at a single hydrophone. Such an estimation enables the authors to analyze sperm whale physical activity during different dive stages: descent and hunt. In both cases, the whale's physical activity correlates to the whale's acoustic activity.

The authors then point out a correlation between click level variations and ICI. The hypothesis explaining such a correlation would be click level control: sperm whales would click slowly at a high source level and faster at a lower source level. Inspired by results found for small odontocete species, the authors propose an echolocation model, click focusing. This echolocation model specifies that sperm whales would aim at closer targets while emitting clicks with lower source level at higher rhythmic rate, and aim at farther

  . The whale emitted 2380 usual clicks and 31 creaks. The additional detection of seafloor echoes at the beginning of the dive leads to the plotting of the sperm whale depth and range variations which indicate a vertical dive, as illustrated by Madsen et al. ͑2002a͒ and Thode et al. ͑2002͒. The ICI variations during the descent are plotted in Fig. 5. After vertically diving to 400 m, the whale repeats an acoustic pattern lasting between 30 and 180 s, composed of a block of usual clicks, hereafter referred to as search phase, ending with a creak or a pause, hereafter referred to as terminal phase. Such a notation, used by different authors regarding Blainville's beaked whales ͑Madsen et al., 2005͒, is discussed below. Some of these search/terminal phases are plotted in Fig. 6. Usual click sequences during search phases ͑as described below͒ are not always such stereotypes, but similar patterns were found in 2003 and 2004. Creaks are not always emitted during the terminal phases ͑Madsen et al., 2002b͒. If no creaks were detected in 2003, every search phase ended with a creak in 2004. And silences are also sometimes skipped during the terminal phases ͑Mullins et al., 1988͒.

FIG. 8 .

 8 FIG. 8. Sperm whale trajectory reconstruction. Sperm whale range variations ͑dots͒ and parabola interpolation ͑solid line͒ between t 1 = 16 min and t 2 = 37 min. The whale would move in a straight line at v r = 0.8 m / s from the range r 1 = 480 m to the range r 2 = 650 m, passing at t min = 25 min at the minimum range r min =200 m.

  dium of a varying speed of sound, and in a medium of a constant speed of sound. The curvature of the rays has negligible effects on the propagation times ͑Laplanche et al., 2004͒. An adjustment of the single hydrophone localization technique ͑Thode et al., 2002͒, taking into account speed of sound variations ͑Laplanche et al., 2004͒, would indicate

  FIG.5. ICI and pitch angle variations during the descent. ICI during the first 10 min of a dive ͑dots, at the top͒. Using the echolocation model, the estimated ICI during the descent, when using targets vertically below the whale inside the prey layer, is also plotted ͑solid line͒. The pitch angle variations are plotted ͓ratio ͑p 1 / p 3 ͒ dB , raw values and moving average, at the bot-tom͔. The bissecting angle ͓for which ͑p 1 / p 3 ͒ dB =0͔ between the direct ray and the bottom ray varies from -39°→ -27°to the horizontal for t =200→ 600 s. The whale pitch angle is then greater than 30°during the descent ͓as ͑p 1 / p 3 ͒ dB Ͻ 0͔. The ICI variations are correlated to the pitch angle variations. Pitch angle variations suggest a three-phase pitch movement ͑͑t =200→ 250 → 280 s, t = 280→ 350→ 400 s, t = 400→ 480→ 570 s͒. The ICI plotting ends ͓͑580, 600͔ s͒ with the first search phase inside the prey layer. FIG. 6. Sperm whale pitch angle and ICI during two successive search/terminal phases. Relative sperm whale pitch angle ͑p 1 / p 3 ͒ dB = 20 log 10 ͑p 1 / p

3 ͒ ͑solid line͒ and moving average ͑dashed line, at the top͒. It is positive for clicks emitted upwards, and negative for click emitted downwards. ICI ͑at the bottom͒.

  FIG. 7. Sperm whale click level and ICI during a search phase. ͑Top͒ Click source level p 1 emitted ͑upwards͒ towards the direct path ͑solid line͒, and click source level p 3 emitted ͑downwards͒ towards the seafloor reflected path ͑dashed line͒. ͑Bottom͒ ICI. Peaks of source level in p 1 and p 3 are detected when the sperm whale clicks at the lowest rate ͑ICI= ICI max ͒, and minimum values of level are detected when the sperm whale clicks at the highest rate ͑ICI= ICI min ͒. The difference of click level within a click burst may reach 20 dB.From the ICI values given for the first plotted 5 -15-s click burst ͑ICI max = 1 s and ICI min = 0.5 s͒ ͑Fig. 7͒, one can find the processing delay t proc = 0.45 s. This value would be equal to the minimum ICI corresponding to the shortest target echolocation range ͑r =0͒. This value ICI= 0.45 s would set the limit of validity of the sperm whale echolocation model described above. The same 0.45-s value has been found in a statistical analysis, setting the limit between usual clicks and creaks ͑Zimmer et al., 2005͒. Sperm whale acoustic activity then splits into two distinct modes, usual clicking ͑ICIജ 0.45 s͒ and creaking ͑ICI Ͻ 0.45 s͒, in view of the temporal pattern of the click sequences ͑search phase/terminal phase͒, the click rate production ͑Zimmer et al., 2005͒, and the echolocation model limit. The delay t proc has been defined as the delay between the reception of the echo of a click and the production of the next click. It may be interpreted as the sum of the processing delay of the first click target echo, and the delay required to produce the next click. The distinction between usual clicks and creaks mentioned above would then suggest that sperm whales would either process usual click and creak echoes differently, or produce usual clicks and creak clicks differently.

	ICI min =	2r min c 0	+ t proc and ICI max =	2r max c 0	+ t proc , ͑6͒
	leading to
	t proc =	␣ICI min -ICI max ␣ -1	.	͑7͒

+ v r 2 , and the whale heading ␥ when crossing this segment ͑Fig.

3͒.

ACKNOWLEDGMENTS

Thanks to Christophe Guinet of the CEBC-CNRS ͑Centre d'Etude Biologique de Chizé, France͒ for initiating this project. Recordings during the 2003 mission could not have been made without the help of Renaud De Stephanis from CIRCé ͑Conservation Information and Research on Cetaceans, Spain͒ and Xavier Demoulin from MAREE ͑Lorient, France͒. Thanks to the Association Breach ͑http:// www.breach-asso.org, France͒ for providing boats, skillful skippers, and ecovolunteers during the 2004 mission, and to the Association Keto⌺ ͑France͒ for undertaking the airborne survey. The MERCATOR Project ͑France͒ helped in providing the oceanographic data required for this project. This study was sponsored by the Association Dirac ͑France͒. Many thanks to the two anonymous reviewers for their contribution in improving the clarity and the rigorousness of this paper. Also thanks to L. Adam, M. Adam, and M. Campbell for refining the English. Many thanks to F. Rondepierre, L. Mortier ͑LODYC, Laboratoire d'Océanographie Dynamique et de Climatologie, France͒, N. Pérez Gimeno ͑Laboratorio de Acústica y Vibraciones de la Universidad de Cádiz, Spain͒, V. Toumazou ͑MERCATOR͒, and G. Vinay ͑MERCATOR͒.

targets emitting clicks with higher source level and slower rhythmic rate. Based on this hypothesis, the authors compute an echolocation model estimating the sperm whale target range from the ICI.

The authors find a click processing delay of 0.45 s, setting the limit of validity of the sperm whale echolocation model. This value is equal to what has been found by different sources to be the limit between usual clicks and creaks. The authors apply this echolocation model during the descent. The coherency of the model would suggest that sperm whales would indeed, as suggested by different authors, echolocate at the prey layer while vertically descending. Estimated sperm whale pitch angle variations based on this hypothesis are consistent with those found when using tags.

The authors then apply the echolocation model during the hunt at the prey depth. Periodic sperm whale target range variations and pitch movements suggest that sperm whales would scan the water before attacking when foraging. The authors estimate the scanned water mass dimension from the echolocation model.

Finally, the animal range interpolation suggests that sperm whales move at constant heading and speed while hunting. The synthesis of all the results leads the authors to suggest that sperm whales would follow a dive-scale strategy when foraging. Sperm whales would preanalyze their prey layer while descending, and cross this layer by repeating a methodic scanning/catching technique, before resurfacing.


Christophe Laplanche

Olivier Adam

Maciej Lopatka

Jean-François Motsch

Measuring the off-axis angle and the rotational movements of phonating sperm whales using a single hydrophone

The common use of the bent-horn model of the sperm whale sound generator describes sperm whale clicks as the pulse series ͕p0, p1, p2, p3, ...͖. Clicks, however, deviate from this standard when recorded using off-axis hydrophones. The existence of additional pulses within the ͕p0, p1, p2, p3, ...͖ series can be explained still using the bent-horn model. Multiple reflections on the whale's frontal and distal sacs of the p0 pulse lead to additional sets of pulses detectable using a farfield, off-axis hydrophone. The travel times of some of these additional pulses depend on the whale's orientation. The authors propose a method to estimate the off-axis angle of sperm whale clicks. They also propose a method to determine the nature of the movement ͑if it is pitch, yaw, or roll͒ of phonating sperm whales. The application of both methods requires the measurement of the travel time differences between pulses composing a sperm whale click. They lead, using a simple apparatus consisting of a single hydrophone at an unknown depth, to new measurements of the underwater movements of sperm whales. Using these methods shows that sperm whales would methodically scan seawater while searching for prey, by making periodic pitch and yaw movements in sync with their acoustic activity. ͓DOI: 10.1121/1.2184987͔

I. INTRODUCTION

Sperm whales ͑Physeter macrocephalus͒ mostly feed on meso-and bathypelagic squid and fish of various sizes and physical constitution ͑Kawakami, 1980; Whitehead, 2003͒. A 15-ton adult sperm whale would need to catch around half a ton of pelagic food in a day, assuming a daily consumption of 3% of its body weight ͑Lockyer, 1981͒. Sperm whales spend most of their time underwater and undertake long, deep dives to catch such an amount of prey ͑Miller et al., 2004;[START_REF] Watkins | Sperm whale surface activity from tracking by radio and satellite tags[END_REF]. However, the low-light, highpressure conditions reigning at these great depths make difficult the visual observation of foraging sperm whales. Consequently, one does not know precisely the technique͑s͒ sperm whales would use to successfully seek and catch their food.

Nevertheless, scientists have developed many methods of studying the underwater behavior of deep-diving marine mammals ͑e.g., sperm whales͒, notably electronic tagging and passive acoustics. By using such methods, scientists have been collecting over the last decades many clues regarding the foraging behavior of sperm whales.

Sperm whales have a sound generator located in their snout ͑Madsen et al., 2002͒. The snout represents one quarter to one third the length of the whale ͑Clarke, 1978b; [START_REF] Nishiwaki | Change of form in the sperm whale accompanied with growth[END_REF] the whale uses it primarily for biosonar purposes ͑Jaquet et [START_REF] Jaquet | Vocal behavior of male sperm whales: Why do they click?[END_REF][START_REF] Møhl | Sperm whale clicks: Directionality and source level revisited[END_REF][START_REF] Whitehead | Click rates from sperm whales[END_REF]. Sperm whales are indeed the most acoustically active toothed whales. They emit a series of transient sounds ͑clicks͒ when undertaking foraging dives ͑Backus and Schevill, 1966͒. Such clicks would be of a duration, frequency content, source level, and directionality suited to echolocate their prey [START_REF] Møhl | The monopulsed nature of sperm whale clicks[END_REF] Sperm whale dives are composed of a vertical descent, a hunt in a prey layer at depth, and a reascent ͑Miller et al., 2004͒. Sperm whales emit a long series of clicks at a slow rate ͑usual clicks, 0.45ഛ ICIϽ 2 s, ICI denoting the interclick interval͒ during the descent. At depth, sperm whales emit usual clicks followed by clicks of increasing rate ͑creaks, ICIϽ 0.45 s͒ and/or silences ͑Mullins et [START_REF] Mullins | Behavior and vocalizations of two single sperm whales, Physeter macrocephalus, off nova scotia[END_REF][START_REF] Zimmer | Threedimensional beam pattern of regular sperm whale clicks confirms benthorn hypothesis[END_REF] It has been advanced that sperm whales emit usual clicks when searching for prey ͑denoted searching phase͒ and creaks when closing in on prey ͑denoted approach and terminal phases͒. They repeat this searching/approach/terminal pattern at depth ͑Laplanche et [START_REF] Laplanche | Etude par acoustique passive du comportement de chasse du grand cachalot[END_REF]Madsen et al., 2005͒. According to Miller et al. ͑2004͒, sperm whales would keep crossing the prey layer during the searching phase and actively swim during the approach/terminal phases.

The sperm whale movements and precise use of their biosonar during these searching/approach/terminal phases are not perfectly understood yet. Laplanche et al. ͑2005͒ have proposed a model describing the sperm whale vertical movements and biosonar use during the searching phase. This model is, however, only partially correct and the authors offer some clarifications below. They propose a passive acoustic technique, requiring a single hydrophone, to determine the nature of the movements of diving sperm whales ͑i.e., if it is a pitch, yaw, or roll movement͒. They also present a new technique to estimate the off-axis angle of sperm whale clicks. The application of both techniques is grounded on a close description of the waveform of sperm a͒ Electronic mail: laplanche@univ-paris12.fr whale clicks, as predicted by the standard model of the sperm whale sound generator. The authors show that sperm whales synchronize their movements with their clicking activity. The authors then propose a more accurate model describing how sperm whales move and use their biosonar while searching for prey.

II. METHODS

A. The leaky bent-horn model

Today's hypothesis is that sperm whales generate their clicks with their snout ͑Norris and Harvey, 1972͒ according to the bent-horn model. This model, proposed by Møhl et al. ͑2003͒ and later confirmed by [START_REF] Zimmer | Threedimensional beam pattern of regular sperm whale clicks confirms benthorn hypothesis[END_REF], successfully predicts the function and location of the various organs composing the sperm whale's snout. The sperm whale's snout is composed of the nasal circuitry ͑left and right nares͒, a connective organ ͑phonic lips͒, two air sacs ͑distal and frontal sacs͒, two spermaceti compartments ͑spermaceti case and junk͒, muscles, and blubber, as described by Clarke ͑1978b͒ and illustrated in Fig. 1. According to the bent-horn model, and in the notation of Møhl et al. ͑2003͒, clicks are generated by the phonic lips ͑leading to the p0 pulse͒, roughly directed backward by the distal sac, focused by the spermaceti case, and reflected by the frontal sac. The frontal reflected pulse ͑labeled p1/2͒ is channeled and projected forward by the spermaceti junk, leading to the outgoing p1 pulse.

The use of this model also predicts that the sperm whale click waveform is multipulsed, i.e., composed of a series of pulses, ͕p0, p1, p2, ...͖ ͑Fig. 2, top͒. Indeed, the frontal reflection p1 / 2 travels back in the spermaceti case and reflects on the distal sac ͑leading to a p1Ј pulse͒. The distal reflection p1Ј follows an acoustic path inside the snout of the whale similar to the one followed by p0, leading to a frontal reflection p3 / 2, the junk pulse p2, and a distal reflection p2Ј, the distal reflection p2Ј itself leading to the triplet ͕p5/2, p3, p3Ј͖, and so on. The standard use of the bent-horn model predicts that the initial p0 and the junk ͕p1, p2, p3, ...͖ pulses exit from the sperm whale's snout, explaining the observed multipulse structure of sperm whale clicks. The bent-horn model also successfully predicts the measured variations of the apparent source level of the ͕p0, p1͖ pulses with the orientation of the hydrophone relative to the whale ͑Møhl et [START_REF] Whitehead | Sperm Whales: Social Evolution in the Ocean ͑The University of[END_REF][START_REF] Zimmer | Threedimensional beam pattern of regular sperm whale clicks confirms benthorn hypothesis[END_REF] Using this model ͑or its Norris and Harvey ancestor͒, some researchers have described the ͕p0, p1, p2, ...͖ series as regular, the time of arrival difference between successive pulses ͕pi , pi +1͖ ͑i ജ 0͒ being a function of the whale's length ͑Goold, 1996; [START_REF] Gordon | Evaluation of a method for determining the length of sperm whales ͑Physeter catodon͒ from their vocalizations[END_REF][START_REF] Rhinelander | Measuring sperm whales from their clicks: Stability of interpulse intervals and validation that they indicate whale length[END_REF]. However, the irregularity of the multipulse structure ͑i.e., variations of such time of arrival differences as well as the existence of additional pulses within the ͕p0, p1, p2, ...͖ frame͒ has been underlined by different re-FIG. 1. The initial pulse p0 is generated by the phonic lips, transmits through the spermaceti case, and reflects on the frontal sac into the p1 / 2 pulse. The p1 / 2 pulse transmits ͑1͒ through the junk and leads to the echolocation pulse p1, ͑2͒ back through the spermaceti case, reflects on the distal sac, and leads to the distal pulse p1Ј, and ͑3͒ through the whale's body to the receiver. The piЈ pulses ͑i ജ 1͒ then recursively lead to the p͑2i +1͒ /2, p͑i +1͒, and p͑i +1͒Ј pulses. The standard use of the bent-horn model stipulates that only the initial and the junk pulses ͕p0, p1, p2, p3, ...͖ exit from the sperm whale's snout. The leaky bent-horn model stipulates that ͑1͒ the above mentioned frontal ͕p1/2, p3/2, p5/2, ...͖ and distal ͕p1Ј , p2Ј , p3Ј , ...͖ reflections also leak, and that ͑2͒ ͕p0, p1, p2, p3, ...͖, ͕p1/2, p3/2, p5/2, ...͖, and ͕p1Ј , p2Ј , p3Ј , ...͖ form distinct sets of pulses that can be separated using an off-axis hydrophone in the farfield. [START_REF] Zimmer | Off-axis effects on the multipulse structure of sperm whale usual clicks with implications for sound production[END_REF] demonstrated that the frontal p1 / 2 pulse indeed leaks from the sperm whale's snout and that the p1 pulse exits at the flat anterior surface of its junk. Zimmer et al. ͑2005a͒ also demonstrated that the travel time differences of the p1/2 and p1 pulses to the p0 pulse depend on the orientation of the whale to the hydrophone. The authors of the present work do not reillustrate the soundness of such statements. They supplement the description of the waveform of a sperm whale click proposed by [START_REF] Zimmer | Off-axis effects on the multipulse structure of sperm whale usual clicks with implications for sound production[END_REF], asserting that it is also composed of the distal reflection p1Ј and the additional sets of pulses ͕p3/2, p5/2, ...͖, ͕p2, p3...͖, and ͕p2Ј , p3Ј , ...͖, as described above. Such a description is discussed below.

B. Definition of pulse delays and angles

The delay of the py pulse on the px pulse is noted xy , simplified into y if px = p0. The delays ͕ 1 Ј , 1 Ј 2 Ј , 2 Ј 3 Ј , ...͖ are all equal to the travel time required by a pulse to travel twice between the distal and frontal sacs through the spermaceti case. Such a travel time is still denoted IPI by the authors, being the interpulse interval used in the literature regarding the sperm whale length estimation process mentioned above.

The delays ͕ 1/2 , 3/2 , ...͖ of the frontal pulses ͕p1/2, p3/2, ...͖ to the initial p0 pulse, and the delays ͕ 1 Ј 1 , 2 Ј 2 , ...͖ of the junk pulses ͕p1, p2, p3, ...͖ to the distal pulses ͕p1Ј , p2Ј , p3Ј , ...͖ depend on the orientation of the whale to the hydrophone, as proven by Zimmer et al.

͑2005a͒ regarding 1/2 and 1 Ј 1 . Using a hydrophone at a fixed location in the farfield, ͕ 1/2 , 3/2 , ...͖ and ͕ 1 Ј 1 , 2 Ј 2 , ...͖, delays will change while the sperm whale moves underwater. Our aim in this work is to determine the nature of the whale's movements using the measurement of these delays. As described below, in the recordings used to illustrate this work, most of the time only pulses from the set ͕p0, p1/2, p1, p1Ј͖ are clearly detected. In the study of the whale's movements that follows, the authors have considered such pulses and used the delays ͕ 1/2 , IPI, 1 Ј 1 ͖ only.

The authors consider in the following the reference frame of the whale. They define the pitch, yaw, and roll movements of the whale as rotations on its left-right, updown, and dorsorostral axes. As a difference, Zimmer et al. ͑2005b͒ and Laplanche et al. ͑2005͒ have both considered the sperm whale movements in the terrestrial reference frame. The off-axis angle of the whale ͑labeled ␦ ͓0,͔, and estimated when the whale is emitting a click͒ is defined as the angle separating the whale dorso-rostral axis to the line joining the whale to the hydrophone ͑Fig. 3͒.

C. Estimation of the off-axis angle of a click

The off-axis angle of a whale represents its orientation to a reference point ͑e.g., the hydrophone͒. This angle varies with the whale's underwater movements, and, inversely, the variations of the off-axis angle of a whale provide information on the whale's underwater movements. The knowledge of this angle is critical in the following study, and the authors propose a simple method to carry out the estimation of the off-axis angle of a sperm whale click. The following demonstration applies for off-axis angles verifying ␦ ͓ /2,͔ ͑Fig. 3͒. The demonstration is slightly different if ␦ ͓0, /2͔, but leads to the same result, i.e., Eq.

͑5͒.

The leaky bent-horn model predicts, as illustrated in Fig. 3, that the p0 pulse follows the path ͓DKH͔, and the p1/2 pulse follows the path ͓DFH͔. The delay 1/2 of the pulse p1 / 2 on the pulse p0 is then equal to ͑ XY denoting the travel time of a pulse on the path ͓XY͔͒

1/2 = ͓ DF + FH ͔ -͓ DK + KH ͔. ͑1͒
The p0 and p1 / 2 pulses travel through the whale's body and through seawater to reach the hydrophone H. The off-axis angle ␦ = ͑FD ជ , DK ជ ͒ can be found from cos ␦ Ӎ -DK / FD. One then assumes, given the dimension of the whale's body, that the curvature of the rays due to refraction in the whale's body lead to negligible variations on the delay DK . Then, assuming that the p0 pulse travels along the path ͓DK͔ in a straight line at the mean speed of sound c sw = c DK , and along the path ͓DF͔ at the mean speed of sound c s , then ͪ.

͑6͒

D. Travel time difference between p1 and p1Ј

The leaky bent-horn model also predicts, as illustrated in Fig. 4, that the p1 pulse follows the path ͓DFJH͔ and the p1Ј pulse follows the path ͓DFDH͔ to reach the hydrophone H. The algebraic delay 1 Ј 1 of the echolocation pulse p1 on the distal pulse p1Ј also depends on the orientation of the whale to the hydrophone. As illustrated in Fig. 4 and demonstrated by Zimmer et al. ͑2005a͒,

1 Ј 1 ͓-DJ/c w ,DJ/c w ͔, ͑7͒
where J is at the center of the flat anterior surface of the junk. By writing the length DJ and the length L so of the spermaceti organ as functions of the whale's full length ͑noted L͒, DJ = ␣͑L͒L, and L so = ␤͑L͒L, approximating the length of the spermaceti organ as the length DF, and using DF = c s IPI/ 2, then

DJ c w = ␣͑L͒ ␤͑L͒ c s c w IPI 2 . ͑8͒
The authors then assume that the height to length ratio of the whale's spermaceti organ is equal to the height ͑L hs ͒ to The delay 1 Ј 1 indicates how much above or below the whale is situated to the hydrophone. It is positive if the whale swims its back toward the hydrophone and is negative if the whale swims its belly toward the hydrophone. It is equal to zero if the hydrophone is in the lateral plane of the whale.

E. Pitch, yaw, and roll movements

The off-axis angle indicates where the hydrophone is situated in relation to the whale. It is calculated from Eq. ͑5͒ using the measurement of the delay 1/2 . Assuming that the line ͑DF͒ is close to the dorsorostral axis of the whale, the delay 1/2 , and, consequently, the off-axis angle are constant while the whale makes a roll movement. On the other hand, 1 Ј 1 oscillates during such a movement around its mean value 0 and between the extrema given in Eq. ͑7͒. Variations of 1 Ј 1 with 1/2 being constant would then indicate a pure roll movement.

However, 1/2 and ␦ are expected to vary during a pitch, a yaw, or a combination of pitch and yaw movements. Assuming ͑JD͒ Ќ ͑DF͒, the delay 1 Ј 1 is constant during a yaw movement, whereas 1/2 oscillates around its mean value, IPI/2. Variations of 1/2 with 1 Ј 1 being constant would then indicate a pure yaw movement.

Simultaneous variations of 1/2 and 1 Ј 1 indicate either a combination of roll and yaw movements, or a pitch movement, or a combination of roll, yaw, and pitch movements. Whereas synchronized variations of 1/2 and 1 Ј 1 would be the result of a pure pitch movement ͑Fig. 4͒. Such variations are similar ͑both increasing or both decreasing͒ if the hydrophone is located in front of and below the whale or behind and above the whale. Such variations are opposite ͑one increasing, one decreasing͒ if the hydrophone is located in front of and above the whale or behind and below the whale. By studying the variations of the delays 1/2 and 1 Ј 1 of a click sequence, one can qualitatively determine the nature of the whale's movements while it emits this click sequence.

III. RESULTS

A. Multipulse structure

A diagram illustrating the variations in the multipulse structure of the clicks emitted by a sperm whale during a single dive is provided in The p0 pulse is clearly detected through the dive, given its low directionality ͑Zimmer et al., 2005b͒. The p1, p1/2, and p1Ј pulses are also well detected. As predicted by the leaky bent-horn model, the p1/2 and p1 pulses are best detected if the hydrophone is in front of the whale ͑␦ ͓0, /2͔ Û 1/2 ͓IPI/ 2 , IPI͔͒, the p1Ј pulse is best detected if the hydrophone is behind the whale ͑␦ ͓ /2,͔͒. The time of arrival of the p1 and p1 / 2 pulses fits with the model, as shown later using Eqs. ͑6͒ and ͑7͒, and clearly demonstrated by [START_REF] Zimmer | Off-axis effects on the multipulse structure of sperm whale usual clicks with implications for sound production[END_REF]. The p1Ј pulse is distinct with the p1 pulse, as unequivocally illustrated in the first sequence of Fig. 5. As predicted by the bent-horn model, the p2Ј pulse is noticeable only if p1Ј is detected ͑e.g., in the first sequence of Fig. 5͒. The p3 / 2 pulse is also detectable ͑e.g., in the third sequence of Fig. 5͒. Its travel time does not fit, however, with the model, as discussed later.

B. Pulse delays and off-axis angle

The travel time differences between the ͕p0, p1Ј , p2Ј͖ pulses are constant, as predicted by the model. The interpulse interval of the whale that clicks, plotted in Fig. 2 and Fig. 5, is IPI= 5.8 ms.

The application of Eqs. ͑6͒ and ͑7͒ requires the measurement of the whale's length L, the speed of sound c s in the spermaceti case, and the speed of sound c w in sea water at the recording location. The whale is of length L = 12.8 m, using the empirical formula proposed by Rhinelander and Dawson ͑2004͒ with IPI= 5.8 ms. The authors choose in the following a speed of sound c s = 1370 m s -1 in the spermaceti organ, using the results presented by Flewellen and Morris ͑1978͒ for a temperature of 33 °C and a pressure of 1 atm. Such a choice is discussed below. The speed of sound in seawater at the recording location was c w Ӎ 1500 m s -1 .

When applying Eq. ͑6͒, clicks are on axis and the hydrophone is in front of the whale ͑␦ ͓0, /4͔͒ if 0.82 IPI ഛ 1/2 ഛ IPIÛ 4.8ഛ 1/2 ഛ 5.8 ms and behind the whale ͑␦ ͓3 /4,͔͒ if 0 ഛ 1/2 ഛ 0.18 IPIÛ 0 ഛ 1/2 ഛ 1 ms. Most of the clicks of the sequences of Fig. 5 are off axis ͑␦ ͓ /4,3 /4͔, 1ഛ 1/2 ഛ 4.8 ms͒. Some clicks are almost on axis, for instance, click #750, which is / 6 off axis ͑ 1/2 = 5.5 ms͒, using ͑5͒ with c sw = c s .

Using Eqs. ͑8͒ and ͑9͒, the delay of p1 on p1Ј should verify ͉ 1 Ј 1 ͉ ഛ 0.18 IPIӍ 1.05 ms. Cranford ͑1999͒ suggested that the distance between the blow hole and the eye is a good estimate for the length of the spermaceti organ. Linear regressions of the results provided by Nishiwaki et al. ͑1963͒ then lead to a tip of snout to center eye distance close to ͓15+ 26L /15͔L / 100 m, a tip of snout to blowhole distance close to ͓3.5+ 4.5L /15͔L / 100 m, hence a blowhole to center eye distance ␤͑L͒L Ӎ͓11.5+ 21.5L /15͔L / 100 m. This leads to the estimate of the distance between the phonic lips and the location of point J on the anterior surface of the junk DJ = ␣͑L͒L = 0.12L = 1.54 m. Zimmer et al. ͑2005a͒ has found DJ = 1.2 m for a sperm whale of length L =12 m. Given the similar lengths of the whales observed in both works, one should expect ␣͑L͒ = 0.10 for the 12.8 m whale observed in the present work. This discrepancy comes from the fact that the height to length ratio of the whale's spermaceti organ is actually inferior to the height to length ratio of the whale's skull. This should be expected, in view of the relative importance of the maxillonasalis muscles in the constitution of the whale's snout ͑Clarke, 1978b͒. One should correct previous statements by writing ␣͑L͒ / ␤͑L͒ = ͓␥͑L͒ / ͑L͔͒ / with = 1.2. In that case, the delay of p1 on p1Ј should verify ͉ 1 Ј 1 ͉ ഛ 0.15 IPIӍ 0.9 ms.

C. Movements

In the example presented in Fig. 5, the whale vertically dives to 400 m from click #1 to click #420 ͑Laplanche et al., 2005͒ ͑only clicks #1 to #200 of this descent are represented in Fig. 5͒. During this descent, both 1/2 and 1 Ј 1 delays oscillate, as the consequence of a physical movement of the whale. The whale barely rolls during the descent, the delay 1 Ј 1 being always positive. The whale creates a slow pitch movement, resulting in the synchronous, slow variations of 1/2 and 1 Ј 1 . Variations of 1 Ј 1 during this pitch movement are in sync with the vertical movement described in Laplanche et al. ͑2005͒. Maxima of the delay 1 Ј 1 ͑at t ͕282, 415, 560͖ s, clicks #͕125, 245, 380͖͒ are detected during the inflections of the vertical movement of the whale ͓at t ͕280, 400, 570͖ s, Fig. 5 of Laplanche et al. ͑2005͔͒. The slow pitch movement found using the present method and the vertical movement described in Laplanche et al. ͑2005͒ must represent the same slow, vertical, pitch movement. The whale also creates a repetitive and faster yaw movement ͑10-click period variations of 1/2 with 1 Ј 1 con-stant͒ during the descent.

Sperm whales also move actively when emitting usual clicks at depth. The variations of 1/2 and 1 Ј 1 being constant during the click sequence #670→ #730 would result of a periodic yaw movement. Synchronous variations of 1/2 and 1 Ј 1 during the click sequence #730→ #770 would result of a periodic pitch movement. The former movement is in sync with the click rhythm, as illustrated in Fig. 6. The authors suggest that the whale methodically scans the content of expanses of water during the descent and at depth, as discussed later.

IV. DISCUSSION

A. Multipulse structure

The description of the multipulse structure proposed by the authors and Zimmer et al. ͑2005a͒ using the bent-horn model correctly predicts the existence, the time of arrival differences, and the relative levels of the pulses ͕p0, p1/2, p1, p1Ј , p2Ј͖. The model also predicts the existence of the p3 / 2 pulse. The travel time difference of p3/2 to p0 is, however, shorter than expected. During the click sequence #1000→ #1025, one can measure 3/2 Ӎ 8.1 ms ͑Fig. 5͒, though one should expect 3/2 = IPI+ 1/2 Ӎ 9.4 ms. Whereas the delays 1/2 and 3/2 slightly change during this click sequence, the travel time difference 3/2 -1/2 of p3/2 to p1 / 2 remains constant. This suggest that the p3 / 2 pulse is indeed the result of a multipath propagation inside the head of the whale and not the reflection by objects surrounding the whale. The travel time difference 3/2 is, however, 1.3 ms lower than predicted by the model. According to the benthorn model, the p0 pulse travels through the spermaceti case and reflects on the frontal sac ͑p1 / 2 pulse͒. The p1 / 2 pulse leaks into the spermaceti junk ͑leading to the p1 pulse͒ and case ͑leading to the p1Ј pulse͒. This statement has lead the authors to speculate that the p1 / 2 pulse might also leak in sea water. The distal reflection p1Ј, when clearly detected, is followed by an IPI-delayed pulse ͑the p2Ј pulse͒. This has lead to the assumption that p1Ј might follow an acoustic path inside the whale's snout similar to the one followed by p0. The p1Ј pulse would then reflect on the frontal sac, and, as p0 did, lead to a frontal reflection leaking in seawater: the p3 / 2 pulse. The discrepancy underlined above regarding the value of the delay 3/2 might be the consequence of the inexactitude of this model. The authors did not though manage to explain the reason of such discrepancy.

The p1 / 2 pulse has been defined as the reflection of p0 on the frontal sac. The p1 / 2 pulse can, however, be detected using a hydrophone placed behind the whale ͑see, for instance, the fourth click sequence of Fig. 5͒. In such cases, the transmission of the p1 / 2 pulse would reach and cross the frontal sac, before leaking in the sea water through the whale's thorax. The p1 / 2 pulse has been described by Zimmer et al. ͑2005a͒ as arriving as scattered subpulses. Zimmer et al. ͑2005a͒ have suggested that this could be the result of multiple reflections on the frontal sac. The p1 / 2 pulse is also, regarding the dataset used through this work, composed of subpulses when recorded in front of the whale ͑Fig. 6͒. It is, however, not time spread when detected from behind the whale. This would rather suggest that the p0 pulse reaches the frontal sac at a single point. This does not forbid multipath propagation of p0 within the spermaceti case, as it is expected to occur as a result of the heterogeneous temperature and chemical composition of the case ͑Flewellen and Morris, 1978͒. Leakage of the p0 pulse through the frontal sac might reflect on the whale's lungs. Such reflections could result, using a hydrophone in front of the whale, of p1/2 arriving as scattered subpulses. One should also expect, assuming multiple reflections on the frontal sac resulting from multipath propagation within the spermaceti case, p1/2 to form a single broad spread pulse. Subpulses are, however, clearly identifiable ͑Fig. 6͒, bearing out the hypothesis describing p1 / 2 as the result of multiple reflections on distinct surfaces. Such a hypothesis, however, does not take into account lung collapsing with pressure at depth, and does not explain why the sperm whale's frontal sac is large and parabolic. Such size and geometry might though indicate a double function of the frontal sac ͑e.g., an acoustic receptor, as suggested by Kozak et al.͒.

B. Off-axis angle

The off-axis angle is estimated using Eq. ͑6͒. The authors have assumed that ͑1͒ the speed of sound along the path ͓DF͔ is constant through the dive and of known value c s , ͑2͒ the path ͓DF͔ is rectilinear and parallel to the click axis, ͑3͒ the speed c sw along the path ͓DK͔ ͑if ␦ ͓ /2,͔͒ or ͓FK͔ ͑if ␦ ͓0, /2͔͒ is equal to the speed of sound in seawater c w . The speed of sound in the core of the spermaceti case is expected to change through the dive, as a consequence of variations in temperature and pressure, c s ͑T , P͒ ͑Goold, 1996͒. Such variations could be the result of depthdependent variations of the thermodynamical properties of the sea water or the result of whale-controlled changes within the snout ͑thermoregulation or the use of the maxillonasalis muscles͒. The sperm whale is expected to control the temperature of its spermaceti case if it uses it to efficiently focus sounds ͑Flewellen and Morris, 1978͒, though it does not seem to have the required extensive vascularization system ͑Cranford, 1999͒. Anyone knows the precise variations of the temperature of the spermaceti case and junk through the dive. Gordon ͑1991͒ have considered c s = 1350 m s -1 , Goold ͑1996͒ c s = 1430 m s -1 . The author's choice of c s = 1370 m s -1 is here arbitrary, and has been chosen as the speed of sound in spermaceti at a temperature of 33 °C ͑Clarke, 1978a͒ and at the sea surface level.

In light of such spatial and temporal variations c s ͑x , y , z , t͒ within the spermaceti case, the path ͓DF͔ is likely not rectilinear. The speed of sound c s used in Eq. ͑6͒ is not the mean speed of sound c sc in the core of the spermaceti case along the curved path ͓DF͔, but the speed required for an acoustic wave to travel rectilinearly between the points D and F with the travel time IPI/2. Assuming that the p0 pulse reflects into p1 / 2 at the center of the frontal sac, the line ͑FD͒ is inclined relatively to the axis of the rostrum. The latter is, however, close to the axis of the junk. The speed of sound c sw along the path ͓DK͔ or ͓FK͔ is actually a function of the off-axis angle, c sw ͑␦͒. The authors have assumed c sw ͑␦͒ = c s if ͉␦ -/2͉ ജ / 4 and c sw ͑␦͒ = c w if /4ഛ ␦ ഛ 3 / 4. The knowledge of the spatial and temporal variations of the temperature within the spermaceti organ could resolve all the above approximations, leading to a precise measurement of ͑1͒ the speed of sound c s , ͑2͒ the location of the reflection point F, ͑3͒ the empirical law c sw ͑␦͒.

The off-axis angle estimator proposed by the authors in Eq. ͑6͒ is then not accurate. Nevertheless, it is a beginning, and this work is worth being pursued. The applications of an off-axis angle estimator are various. One of them could allow to reconstruct the beam pattern of sperm whale clicks using a single hydrophone. The simultaneous measurements from different locations of the off-axis angle of a sperm whale could also be used to find the three-dimensional orientation ͑in the terrestrial reference frame͒ of this whale.

C. Rotational movements

Laplanche et al. ͑2005͒ have presented a model describing how sperm whales might move and echolocate while searching for prey. This model is partially correct. It is grounded on the hypothesis that sperm whales control click level and rhythm during echolocation. Such a hypothesis has been made in light of the apparent synchronization of click level and ICI ͑for instance, in Fig. 7 of this reference͒. Nevertheless, such synchronization could result when the whale synchronizes its click rhythm to an oscillatory movement, and when the observer's position is less than ideal. Laplanche et al. ͑2005͒ rejected such a hypothesis, thinking that an ICI-movement synchronization would lead to an ICIlevel synchronization only for on-axis recordings. Actually, the lateral asymmetry in the acoustic behavior of sperm whales while searching for prey described later could explain why click level and rhythm can so often be in apparent sync.

As illustrated regarding the click sequence #670 → #730 ͑Fig. 6͒, the sperm whale clicking activity can be in sync with the fast, periodic movement it creates ͑in this case a yaw movement͒. In this example, the hydrophone is in front of the whale ͑␦ ഛ /2͒ and the maxima of the ICI correspond to the maxima of the off-axis angle. A similar relationship ͑i.e., ICI max Û ␦ max ͒ can be observed using a hydrophone behind the whale ͑␦ ജ /2͒.

The authors think of three models describing the fast, periodic, rotational movement of the whale in sync with its ICI ͑Fig. 7͒. Such a rotational movement is assumed to be planar in the following. The first model stipulates that the whale echolocates at slow rate ͑ICI max ͒ in front of it and faster ͑ICI min ͒ on its sides. The second model stipulates that the whale echolocates at high rate ͑ICI min ͒ in front of it and slower ͑ICI max ͒ on its sides. The third model stipulates that the whale echolocates at slow rate ͑ICI max ͒ on one side and faster ͑ICI min ͒ on the other.

One can show that during a periodic, planar, rotational movement, the off-axis angle reaches its extreme, either when the hydrophone is in the plane containing the whale dorsorostral axis and the rotational axis or during the inflections of the movement. Whereas the synchronization of the ICI and click level may appear approximate ͑e.g., during the click sequence #730→ #770 of Fig. 6͒, the synchronization of the ICI and the off-axis angle is unequivocal ͑e.g., during the click sequence #670→ #730͒. The authors, whereas they should have not so easily rejected the hypothesis that the ICI-apparent level synchronization being the result of the hydrophone being in a less than ideal location ͓Laplanche et al. ͑2005͔͒, here they reject the hypothesis that the ICI-␦ synchronization mentioned above is apparent only. They assume in the following that the extreme of the off-axis angle ͑as those illustrated in Fig. 6͒ are the consequence of inflections in the rotational movement of the whale.

One can then show that only the third model could lead to an ICI max Û ␦ max synchronization using a hydrophone located either in front of or behind the whale. Such an important result, an asymmetry in the use of the sperm whale's biosonar, need to be borne out by analyzing more data before being considered as representative of the sperm whale acoustic behavior. It could be, for instance, the result of the asymmetry in the anatomy of the sperm whale sould generator, or part of a metabolic saving foraging strategy ͑Laplanche, 2005͒.

Such an asymmetry in the sperm whale acoustic behavior also explains the often observed synchronization of the ICI and apparent click level ͑AL͒. As illustrated in Fig. 7, the relationship ICI max Û AL max is detectable using a hydrophone located in a large portion of space ͓on side ͑C͔͒. The ICI-␦-AL appear to be decorrelated when using a hydrophone in front of the whale ͓in sector ͑B͔͒.

The technique presented in Laplanche et al. ͑2005͒ has been used to measure the vertical movements of diving sperm whales. It does not detect horizontal movements, hence the inexactitude of the model previously presented by the authors. However, the existence of horizontal movements does not alter the measurement of vertical movements, and the technique used by Laplanche et al. ͑2005͒ remains valid. The description of the vertical movements of sperm whales while descending and foraging at depth is then to be supplemented with the description of their horizontal movements.

The sperm whale, regarding the dive chosen as an example, makes a slow vertical pitch movement and a faster yaw movement with no rolling during the descent. Data presented by [START_REF] Zimmer | Off-axis effects on the multipulse structure of sperm whale usual clicks with implications for sound production[END_REF] would rather suggest that sperm whales create a fast pitch movement during the descent as a result of fluking. The fast movement, has, however, not been detected by Laplanche et al. ͑2005͒, suggesting that it is horizontal. The authors' present results suggest that, in the example illustrating this work, the whale would analyze the content of a horizontal water expanse during this descent, by making a slow, vertical, pitch movement and create a faster, horizontal, yaw movement.

At depth, regarding the sequence #670→ #730 ͑Fig. 6͒, the whale makes a slow vertical pitch movement, barely rolls, and creates a faster yaw movement. The latter movement would be horizontal, for reasons similar to those mentioned above, though this statement is highly speculative, because Laplanche et al. ͑2005͒ have used a different dataset to show that sperm whales would not create fast, vertical movements during the searching phases. The whale would, in this case, analyze the content of a vertical water expanse. Concerning the sequence #730→ #770 ͑Fig. 6͒, the fast movement would be a pitch movement.

The authors have not analyzed enough data to conclude on how sperm whales precisely move when searching for prey. The method itself is qualitative, as a result of making 2 measurements-1/2 and 1 Ј 1 -when trying to estimate three angles-pitch, yaw, and roll. The authors could not use it to precisely determine the geometry of the water masses analyzed by sperm whales when foraging. Nevertheless, the method presented is promising, and the first results demon-FIG. 7. Three models describing the fast, periodic movement of the whale in sync with its ICI. Only the third model could lead to the synchronization ICI max Û ␦ max using a hydrophone located either in front of or behind the whale ͓it is located in ͑a͔͒. The click apparent level and the ICI are in sync ͑ICI max Û AL max ͒ if the hydrophone is located in ͑c͒. There is no apparent synchronization between the ICI, the off-axis angle, and the click AL if the hydrophone is in sector ͑b͒.

strate that sperm whales actively search for their prey, by creating a fast, periodic movement in sync with their acoustic activity.

V. SUMMARY AND CONCLUSIONS

The traditional view of the sperm whale click multipulse structure describes a click as a regularly spaced series of pulses ͕p0, p1, p2, p3, ...͖. Nevertheless, additional pulses are often detected within this structure. Their presence has been explained using the standard bent-horn model of the sperm whale sound generator. Frontal reflections ͕p1/2, p3/2, p5/2, ...͖ and distal reflections ͕p1Ј , p2Ј , p3Ј , ...͖ would also leak from the sperm whale snout, leading to a more complex sperm whale click multipulse structure when recorded by a farfield, off-axis hydrophone.

The delays 1/2 ͑of p1/2 on p0͒ and 1 Ј 1 ͑of p1 on p1Ј͒ are variable and depend on the whale's orientation to the hydrophone. The off-axis angle of the whale can be calculated using the measurements of the standard IPI and the delay 1/2 . The nature of the movements of the whale ͑if it is pitch, yaw, or roll͒ can be measured by analyzing the variations of the delays 1/2 and 1 Ј 1 . Variations of 1/2 alone ͑with 1 Ј 1 constant͒ indicate pure yaw movements, variations of 1 Ј 1 alone ͑with 1/2 constant͒ indicate pure roll movements, and synchronous variations of 1/2 and 1 Ј 1 indicate pure pitch movements. The analysis of the waveform of usual click sequences then leads to some insight on the sperm whale movements when foraging. Sperm whales would barely roll when descending and when searching for prey at depth. In the example illustrating this work, the authors have shown that the whale made a slow vertical pitch movement and at the same time created a faster, periodic yaw movement when descending. When searching for prey at depth, the sperm whale made a slow pitch movement and created a faster pitch or yaw movement in sync with the clicking activity. The authors have suggested that sperm whales would, at depth, make an asymmetric scan of the surrounding water. The method presented is easily implemented, is repeatable, and could lead to new insights on the foraging behavior of sperm whales or other toothed whales using a very simple apparatus consisting of a single hydrophone.

FIG. 2 .

 2 FIG.2. Both clicks were emitted by the same sperm whale. Sperm whale clicks are often described as a series of regularly spaced pulses ͕p0, p1, p2, . . .͖ ͑top͒. Clicks recorded in the farfield rarely fit to this model and are composed of a more complicate set of pulses ͕p0, p1/2, p1, . . .͖ ͑bottom͒.

FIG. 3 .

 3 FIG.3. The off-axis angle of the whale is defined as the angle between the whale dorsorostral axis and the line joining the whale to the hydrophone H. The point D is at the center of the phonic lips ͑point of emission of the p0 pulse͒. F is the point of frontal reflection p0 → p1/2. K is the orthogonal projection of F on the line ͑DH͒.

  FIG.4. 1/2 ഛ IPI/ 2 if the hydrophone is behind the whale, 1/2 ജ IPI/ 2 if it is in front of the whale. 1 Ј 1 ഛ 0 if the hydrophone is below the whale, 1 Ј 1 ജ 0 if it is above the whale. The sign of the variations of 1/2 and 1 Ј 1 during a pitch movement depend on the location of the hydrophone to the whale.

Fig. 5 .

 5 Four sequences of 200 successive clicks are plotted, using the 2004 dataset presented in Laplanche et al. ͑2005͒. Only usual clicks ͑with ICIϾ 0.45 s͒ are considered.

FIG. 5 .

 5 FIG. 5. The normalized, apparent level of 4 ϫ 200 usual clicks emitted by a sperm whale during a single dive. Sea surface reflections are here visible from clicks #0 → #40. A sperm whale click is composed of distal ͕p0, p1Ј , p2Ј , . . .͖, frontal ͕p1/2, p3/2, . . .͖, and junk ͕p1, p2, . . .͖ pulses.

FIG. 6 .

 6 FIG. 6. The whale creates a fast yaw movement during the click sequence #670→ #730 and a fast pitch movement during the click sequence #730 → #770. The ICI is in sync with the movement during the first sequence and in sync with the click apparent level during the second sequence.

  searchers ͓see[START_REF] Bahl | Towards identification of sperm whales from their vocalizations[END_REF][START_REF] Goold | Signal processing techniques for acoustic measurement of sperm whale body lengths[END_REF][START_REF] Møhl | The monopulsed nature of sperm whale clicks[END_REF] for instance͔ without being elucidated.The standard multipulse series ͕p0, p1, p2, ...͖ does not actually fully represent the waveform of sperm whale clicks. Clicks, when recorded off axis, deviate from this standard ͑Fig. 2, bottom͒. The authors propose a new and more complete description of the multipulse structure of sperm whale clicks, however, maintaining the standard bent-horn model, and describe what one could call the leaky bent-horn model.

	This description clarifies the discrepancies between the mul-
	tipulse structure of clicks predicted by the standard model
	͕͑p0, p1, p2, ...͖͒ and what is actually observed in practice.
	Different researchers have reached similar conclusions ͑Zim-
	mer et al., 2005a͒. The authors of the present work then do
	not reillustrate the soundness of the new model, as it has
	been convincingly done by Zimmer et al. ͑2005a͒. However,
	they briefly redefine it as a basis for the application described
	below. They also supplement the description of the sperm
	whale click multipulse structure proposed by Zimmer et al.
	͑2005a͒, as it is not complete. For the sake of clarity, the
	authors of the present work use a notation similar to the
	notation that Zimmer et al. ͑2005a͒ introduced.

  Refraction takes place during this transmission ͑Flewellen and[START_REF] Flewellen | Sound velocity measurements on samples from the spermaceti organ of the sperm whale ͑Physeter cat-odon͒[END_REF][START_REF] Urick | Principles of Underwater Sound ͑McGraw-Hill[END_REF] Refraction in seawater leads to negligible variations of the quantity FH -KH , assuming a hydrophone in the farfield, the path ͓KH͔ and ͓FH͔ being close to each other. In this case,

FH -KH Ӎ 0. ͑2͒

In view of the previous statement IPI= 1 Ј , then DF = IPI/ 2 leads to 1/2 Ӎ IPI 2 -DK . ͑3͒

  ͓0, /2͔ ͑K being in that case the orthogonal projection of D on the line ͑FH͒ and choosing c sw = c FK as the mean speed of sound on the path ͓FK͔͒. The mean speed of sound c sw is a function of the off-axis angle, c sw ͑␦͒. The authors choose c sw = c s for onaxis angles ͑␦ ͓0, /4͔ or ␦ ͓3 /4,͔͒ and c sw = c w

	DK FD	Ӎ -	c sw DK c s IPI/2	.	͑4͒	␦ Ӎ arccosͩ	c w c s	2 1/2 -IPI IPI
	Equations ͑3͒ and ͑4͒ lead to the following estimator of the
	off-axis angle of a click, as a function of the standard inter-
	pulse interval IPI and the delay 1/2
	␦ Ӎ arccosͩ	c sw c s	2 1/2 -IPI IPI	ͪ.
	͑naming c w the speed of sound in seawater͒ for off-axis
	angles ͑␦ ͓ / 4,3 /4͔͒. Such an approximation is dis-
	cussed below.
	Equation ͑5͒ cannot be properly used to estimate the

cos ␦ Ӎ -͑5͒ This formula also applies for ␦ whale's off-axis angle if it is on-axis ͑d ␦ / d 1/2 → + ϱ if 1/2 → 0 or 1/2 → IPI͒. A click with a normalized delay ͓1 + ͑c s / c w ͒ / ͱ 2͔ /2ഛ 1/2 / IPIഛ 1 is on axis and the hydrophone is in front of the whale ͑␦ ͓0, /4͔͒. A click with a normalized delay 0 ഛ 1/2 / IPIഛ ͓1-͑c s / c w ͒ / ͱ 2͔ / 2 is on axis and the hydrophone is behind the whale ͑␦ ͓3 /4,͔͒. A click with a normalized delay ͓1-͑c s / c w ͒ / ͱ 2͔ /2ഛ 1/2 / IPIϽ ͓1 + ͑c s / c w ͒ / ͱ 2͔ / 2 is off axis ͑␦ ͓ /4,3 /4͔͒, and its off- axis angle can be estimated using

ACKNOWLEDGMENTS

Thanks to L. Conteville ͑Université Paris XII, France͒ for helping in analyzing the data. Recordings could not have been made without the help of the Association Breach ͑http:// www.breach-asso.org, France͒, which provided boats, skippers, and ecovolunteers. Thanks to the Association Keto⌺ ͑France͒ for undertaking the airborne survey. Many thanks to S. Wiener for reviewing an earlier version of the manuscript and to L. Adam, M. Adam, and C. Bédard for refining the English. The authors also thank M. Macé for his help on the bibliography and W. Zimmer for his useful and encouraging comments on the accepted version of the manuscript.