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A Bayesian method to estimate the depth and the range

of phonating sperm whales using a single hydrophone

Christophe Laplanche

To cite this version:

Christophe Laplanche. A Bayesian method to estimate the depth and the range of phonating sperm

whales using a single hydrophone. Journal of the Acoustical Society of America, Acoustical Society

of America, 2007, Vol. 121, pp. 1519-1528. �10.1121/1.2436644�. �hal-00797709�

To link to this article: DOI: 10.1121/1.2436644

http://dx.doi.org/10.1121/1.2436644

This is an author-deposited version published in: http://oatao.univ-toulouse.fr/

Eprints ID: 5607

To cite this version: Laplanche, Christophe A Bayesian method to estimate the

depth and the range of phonating sperm whales using a single hydrophone.

(2007) The Journal of the Acoustical Society of America (JASA), 9ol. 121 (n°3).

pp. 1519-1528. ISSN 0001-4966

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A Bayesian method to estimate the depth and the range of

phonating sperm whales using a single hydrophone

Christophe Laplanche

a兲

Laboratoire Images, Signaux et Systèmes Intelligents, Groupe Ingénierie des Signaux Neuro-Sensoriels,

Université Paris 12, Créteil, France

Some bioacousticians have used a single hydrophone to calculate the depth/range of phonating

diving animals. The standard one-hydrophone localization method uses multipath transmissions

共direct path, sea surface, and seaﬂoor reﬂections兲 of the animal phonations as a substitute for a

vertical hydrophone array. The standard method requires three multipath transmissions per

phonation. Bioacousticians who study foraging sperm whales usually do not have the required

amount of multipath transmissions. However, they usually detect accurately 共using shallow

hydrophones towed by research vessels兲 direct path transmissions and sea surface reﬂections of

sperm whale phonations 共clicks兲. Sperm whales emit a few thousand clicks per foraging dive,

therefore researchers have this number of direct path transmissions and this number of sea surface

reﬂections per dive. The author describes a Bayesian method to combine the information contained

in those acoustic data plus visual observations. The author’s tests using synthetic data show that the

accurate estimation of the depth/range of sperm whales is possible using a single hydrophone and

without using any seaﬂoor reﬂections. This method could be used to study the behavior of sperm

whales using a single hydrophone in any location no matter what the depth, the relief, or the

constitution of the seaﬂoor might be.

关DOI: 10.1121/1.2436644兴

I. INTRODUCTION

Sperm whales undertake long foraging dives to catch

their prey. They breathe at the sea surface, ﬂuke-up and swim

downwards to reach their prey, hunt at depth, and reascend

back to the sea surface 共Miller et al., 2004a兲. During forag-

ing dives, sperm whales emit echolocation clicks 共Backus

and Schevill, 1966兲. They emit echolocation clicks at a tre-

mendous source level 共Møhl et al., 2003, 2000兲 and in series

共Whitehead and Weilgart, 1990兲.

Since sperm whales emit long series of clicks of high

source level, passive acoustics is an effective tool to study

the foraging behavior of these animals. Researchers have de-

veloped and used different passive acoustic localization tech-

niques. These techniques require synchronous recordings

made on tridimensional 共Watkins and Schevill, 1972兲, bidi-

mensional 共Thode, 2004兲, or unidimensional 共Villadsgaard et

al., 2007兲 arrays of hydrophones.

To locate the sound source using an array of n hydro-

phones, one would need to isolate one signal emitted by the

source and to measure the n times of arrival 共TOA兲 of this

signal on the n hydrophones of the array. The differences

between TOAs 共TOAD兲 are calculated. A TOAD provides

information on the location of the source: The source is on a

sheet of a two-sheeted hyperboloid of geometry given by the

TOAD itself and by the location of the hydrophones used to

calculate the TOAD. By repeating this localization process

using different TOADs, one can, using the required amount

of hydrophones 共Spiesberger, 2001兲, geometrically or ana-

lytically compute the intersection of the hyperboloid sheets

and therefore be able to more accurately identify the location

of the source. A unidimensional array requires at least three

hydrophones 共Villadsgaard et al., 2007兲. In this case the hy-

perboloid sheets deﬁned by the TOADs intersect into a

circle. The plane containing this circle is perpendicular to the

line of the array and the center of the circle sits on this line.

Therefore, if the unidimensional array is vertical, then the

circle is horizontal, and its depth and radius give the depth

and the horizontal range of the sound source.

One noteworthy passive acoustic localization technique

requires a single hydrophone 关see for instance Thode et al.

共2002兲 or Laplanche et al. 共2005兲兴. Signals emitted by sound

sources may reﬂect on the sea surface and the seaﬂoor while

propagating to the hydrophones. The detection on a hydro-

phone of the echoes from the surface/seaﬂoor serves as a

substitute of an unidimensional vertical hydrophone array

共Urick, 1983兲. By measuring the TOADs of the echoes 共later

referred to as echo delays兲 relative to the nonreﬂected trans-

mitted signal, one can calculate the depth and the range of

the sound source. If the source is a phonating sperm whale,

one can theoretically, by repeating the localization process

on every click emitted by the whale during a whole dive, plot

the values of the depth and the range of the whale during this

dive.

Unfortunately, sperm whales usually forage above con-

tinental slopes or abyssal plains, i.e., areas of either deep or

high relief seaﬂoor. One can seldom clearly detect the seaf-

loor

echoes

of clicks emitted by sperm whale using a hydro-

phone 共close to the sea surface兲 towed by a research vessel.

Usually one can only detect seaﬂoor echoes at the beginning

a兲

Electronic mail: laplanche@gmail.com

of the whale’s dive, while the whale both swims and clicks

downwards 共Thode et al., 2002兲. In low sea state conditions,

using a hydrophone close to the sea surface, one can how-

ever clearly detect surface echoes during the whole dive 关see

for instance Thode 共2004兲 or Laplanche et al. 共2005兲兴.

Nevertheless, the measurement of a single echo delay

共e.g., a surface echo delay兲 is not enough to calculate the

depth and the range of a phonating sperm whale, since, as

aforementioned, by using a single TOAD one can only know

that the whale is on a hyperboloid sheet. Is it not possible to

more accurately identify the location of the whale using a

single hydrophone but not using seaﬂoor echoes?

Actually one can still estimate the depth and the range of

the whale under such constraints, and the aim of this work is

to demonstrate the feasibility of this process. Every single

surface echo delay contains information regarding the loca-

tion of the whale. By combining the pieces of information

contained in the set of the surface echo delays of the clicks

emitted by the whale during a dive, one should be able to

give a more accurate description of the location of the whale

during this dive. The efﬁcient combining of these pieces of

information can be achieved in a Bayesian frame. The Baye-

sian approach has already proven to be efﬁcient to locate

sound sources using TOADs 共Spiesberger, 2005兲.

The author proposes a Bayesian passive acoustic tech-

nique to estimate the depth and the range of foraging sperm

whales using a single hydrophone and without using any

seaﬂoor echoes. To be applied, this technique requires a set

of values of the sea surface echo delay of clicks emitted by

the whale during the whole dive. It also requires the mea-

surement, by a visual observer, of the approximate location

共i.e., range and azimuth relative to the research vessel兲 of the

whale when beginning and ending the foraging dive.

II. MATERIAL AND METHODS

A. Trajectory model

First, one would need a representation of the underwater

trajectory of the whale using a mathematical model. Let t

共A兲

be the time when the sperm whale ﬂukes-up and starts div-

ing, t

共B兲

the time when the whale starts clicking, t

共C兲

the time

when the whale stops clicking, and t

共D兲

the time when the

whale resurfaces. The author will decompose the trajectory

of the whale for t 苸 关t

共B兲

,t

共C兲

兴 into n苸N

*

pieces of equal

duration

0

=共t

共C兲

−t

共B兲

兲/ n 共Fig. 1兲. Let t

共s兲

=t

共B兲

+共s−1兲

0

be

the time when the whale is at the beginning 共s苸 兵1,... ,n其兲

or at the end 共s 苸 兵2,...,n+1其兲 of such trajectory pieces.

Let E

共s兲

be the location in the terrestrial reference frame,

let z

共s兲

be the depth, let r

共s兲

be the horizontal range, and let

共s兲

be the azimuth of the whale at time t

共s兲

共Fig. 2兲. Let E

p

共s兲

and H

p

be the projections of E

共s兲

and H 共H is the location of

the hydrophone兲 on a horizontal plane. Let

␥

b

共s兲

and

␥

e

共s兲

be

the angles 共E

p

共s兲

E

p

共s+1兲

ជ

,H

p

E

p

共s兲

ជ

兲 and 共E

p

共s兲

E

p

共s+1兲

ជ

,H

p

E

p

共s+1兲

ជ

兲.

The value of n is chosen high enough to be able to

assume that the whale moves at constant speed and

constant heading for t 苸 关t

共s兲

,t

共s+1兲

兴共s苸 兵1, ... ,n其兲. Let S

共s兲

=关E

共s兲

E

共s+1兲

兴 be the segment deﬁning the location of the

whale for t苸 关t

共s兲

,t

共s+1兲

兴. Let

v

z

共s兲

苸 R and

v

r

共s兲

苸 R

+

be the

vertical and horizontal speeds of the whale along the seg-

ment S

共s兲

. Each segment S

共s兲

is entirely deﬁned by the coor-

dinates of the points E

共s兲

and E

共s+1兲

. The trajectory of the

whale for t苸 关t

共B兲

,t

共C兲

兴 is labeled T. It itself is entirely de-

ﬁned by the location of the summits E

共s兲

共s 苸 兵1, ...,n +1其兲,

that is to say T ⬅共E

共1兲

, ...,E

共n+1兲

兲. This deﬁnition of T re-

quires, by using the coordinates in the terrestrial reference

frame of the n+ 1 summits E

共s兲

, a set of 3n +3 parameters.

One can then deﬁne the trajectory T using the depth,

range, and heading of the whale. Each segment S

共s兲

can be

recursively deﬁned by writing

S

共1兲

⬅共z

共1兲

,r

共1兲

,

共1兲

,z

共2兲

,r

共2兲

,

␥

b

共1兲

兲,

S

共s兲

⬅共z

共s+1兲

,r

共s+1兲

,

␥

b

共s兲

兩S

共s−1兲

兲 for s 苸 兵2, ... ,n其共1兲

and the trajectory T is entirely deﬁned by the set of 3n +3

parameters

T ⬅共z

共1兲

,r

共1兲

,

共1兲

,z

共2兲

,r

共2兲

,

␥

b

共1兲

, ... ,z

共n+1兲

,r

共n+1兲

,

␥

b

共n兲

兲. 共2兲

This leads to the following deﬁnition of T, which is

required by the algorithm described later. Let ⌬

␥

共s兲

=

␥

b

共s兲

−

␥

e

共s−1兲

be the change of heading of the whale at time t

共s兲

.

Each segment S

共s兲

is recursively deﬁned by writing

FIG. 1. The whale dives at t = t

共A兲

, starts clicking at t= t

共B兲

, stops clicking at

t=t

共C兲

, and resurfaces at t=t

共D兲

. The whale is at the depth z =z

共s兲

at t=t

共s兲

=t

共B兲

+共s −1兲

0

共s苸 兵1, ... ,n+1其, in this example n =14兲. The vertical speed

of the whale is constant and equal to

v

z

共s兲

for t 苸 关t

共s兲

,t

共s+1兲

兴.

FIG. 2. The whale is at E

共s兲

at t= t

共s兲

. The hydrophone is at H. E

p

共s兲

and H

p

are

the projections of E

共s兲

and H on a horizontal plane. The whale moves in a

constant heading and at a constant horizontal speed

v

r

共s兲

for t苸 关t

共s兲

,t

共s+1兲

兴

along the segment S

共s兲

to reach E

共s+1兲

at t = t

共s+1兲

. The angles

␥

b

共s兲

and

␥

e

共s兲

are

deﬁned as the angles between 共E

p

共s兲

E

p

共s+1兲

兲 and 共H

p

E

p

共s兲

兲, and 共E

p

共s兲

E

p

共s+1兲

兲 and

共H

p

E

p

共s+1兲

兲, respectively. The change of heading from S

共s−1兲

to S

共s兲

is ⌬

␥

共s兲

=

␥

e

共s−1兲

−

␥

b

共s兲

. The horizontal range and the azimuth of the whale at t=t

共s兲

are

r

共s兲

=H

p

E

p

共s兲

and

共s兲

, respectively.

S

共1兲

⬅共z

共1兲

,r

共1兲

,

共1兲

,

v

z

共1兲

,

v

r

共1兲

,

␥

b

共1兲

兲,

S

共s兲

⬅共

v

z

共s兲

,

v

r

共s兲

,⌬

␥

共s兲

兩S

共s−1兲

兲 for s 苸 兵2, ... ,n其共3兲

given the coordinates of the ﬁrst summit 关E

共1兲

, deﬁned by the

triplet 共z

共1兲

,r

共1兲

,

共1兲

兲兴 and the whale vertical speed, horizon-

tal speed, and change of heading in the n segments. Alterna-

tively, given the coordinates of the last summit 关E

共n+1兲

, de-

ﬁned by the triplet 共z

共n+1兲

,r

共n+1兲

,

共n+1兲

兲兴 and the whale

vertical speed, horizontal speed, and change of heading in

the n segments, each segment S

共s兲

is recursively deﬁned by

writing

S

共s兲

⬅共

v

z

共s兲

,

v

r

共s兲

,⌬

␥

共s+1兲

兩S

共s+1兲

兲 for s 苸 兵1, ... ,n −1其

S

共n兲

⬅共

v

z

共n兲

,

v

r

共n兲

,

␥

e

共n兲

,z

共n+1兲

,r

共n+1兲

,

共n+1兲

兲. 共4兲

Using Eq. 共3兲, the trajectory T is also entirely deﬁned by the

set of 3n +3 parameters

T ⬅共z

共1兲

,r

共1兲

,

共1兲

,

v

z

共1兲

,

v

r

共1兲

,

␥

b

共1兲

,

v

z

共2兲

,

v

r

共2兲

,⌬

␥

共2兲

, ... ,

v

z

共n兲

,

v

r

共n兲

,⌬

r

共n兲

兲, 共5兲

or alternatively, using Eq. 共4兲, by the set of 3n+3 parameters

T ⬅共

v

z

共1兲

,

v

r

共1兲

,⌬

␥

共2兲

, ... ,

v

z

共n−1兲

,

v

r

共n−1兲

,⌬

␥

共n兲

,

v

z

共n兲

,

v

r

共n兲

,

␥

e

共n兲

,z

共n+1兲

,r

共n+1兲

,

共n+1兲

兲. 共6兲

B. Prior information

By choosing

共1兲

=0,

共n+1兲

represents the change in azi-

muth of the whale, relative to the research vessel, between

the points E

共1兲

and E

共n+1兲

. One can combine the deﬁnitions of

T given in Eqs. 共5兲 and 共6兲 by writing

T ⬅共z

共1兲

,r

共1兲

,

v

z

共1兲

,

v

r

共1兲

,

␥

b

共1兲

,

v

z

共2兲

,

v

r

共2兲

,

⌬

␥

共2兲

, ... ,

v

z

共n兲

,

v

r

共n兲

,

⌬

␥

共n兲

,z

共n+1兲

,r

共n+1兲

,

共n+1兲

兲. 共7兲

This latter definition of T uses 3n+5 parameters, that is to

say redundant information. Indeed, the coordinates of the last

summit E

共n+1兲

can be calculated given the coordinates of the

first summit E

共1兲

and using the values of speeds and change

of heading in the n segments. The consequences of such

redundancy during the estimation process will be discussed

later. The aim of combining Eqs. 共5兲 and 共6兲 into Eq. 共7兲 is to

gather in a single definition of T information on the fluking

and resurfacing points of the whale.

Let z

共A兲

=0 m and r

共A兲

be the depth and the horizontal

range of the whale at the time t=t

共A兲

. Let

v

z

共A兲

and

v

r

共A兲

be the

average vertical and horizontal speeds of the whale for t

苸 关t

共A兲

,t

共B兲

兴. The depth and the range of the whale at t =t

共1兲

are then z

共1兲

=

v

z

共A兲

共t

共B兲

−t

共A兲

兲 and r

共1兲

. Let z

共D兲

=0 m and r

共D兲

be

the depth and the horizontal range of the whale at the time

t= t

共D兲

. Let

v

z

共D兲

and

v

r

共D兲

be the average vertical and horizon-

tal speeds of the whale for t 苸 关t

共C兲

,t

共D兲

兴. The depth and the

range of the whale at t =t

共n+1兲

are then z

共n+1兲

=

v

z

共D兲

共t

共C兲

−t

共D兲

兲

and r

共n+1兲

.

Sperm whales initiate their foraging dives by ﬂuking-up

and diving vertically 共as observed from the sea surface at the

very beginning of the dive 兲. Sperm whales usually keep a

constant vertical speed while descending to reach their prey

and while ascending to reach the sea surface 共Miller et al.,

2004a兲. There are exceptions however: Sperm whales may,

for instance, horizontally translate during the ascent likely

due to the presence of conspeciﬁcs 共Miller et al., 2004b兲.

Assuming that the main objective of the whale while de-

scending is to reach bathypelagic prey and that the main

objective of the whale while ascending is to reach oxygen at

the sea surface, the whale would swim vertically for t

苸 关t

共A兲

,t

共B兲

兴 and t苸 关t

共C兲

,t

共D兲

兴. This leads to

v

r

共A兲

⯝0ms

−1

,

v

r

共D兲

⯝0ms

−1

, r

共1兲

⯝r

共A兲

, and r

共n+1兲

⯝r

共D兲

. In this case

共n+1兲

represents the change in azimuth of the whale, relative to the

research vessel, between the ﬂuking and the resurfacing

points. A visual observer can measure the parameters z

共1兲

,

r

共1兲

, z

共n+1兲

, r

共n+1兲

, and

共n+1兲

from the research vessel. The

visual measurement process is inaccurate and thus results in

uncertainties on z

共1兲

, r

共1兲

, z

共n+1兲

, r

共n+1兲

, and

共n+1兲

. The as-

sumption of verticalness may also be inaccurate resulting in

additional uncertainties on these variables. Such uncertain-

ties are modeled in the following section using random vari-

ables.

C. Likelihood

The algorithm which will be described later is used to

estimate the depth and the range of the whale during a dive,

and requires the set of values of the surface echo delay of the

clicks that the whale has emitted during this dive. Let K

e

be

the number of echolocation clicks that the whale has emitted

for t 苸 关t

共B兲

,t

共C兲

兴. The author assumes that K艋K

e

clicks are

correctly detected 共both the direct path and the surface echo兲.

The observer then makes K consistent measurements of the

surface echo delay at the time t 苸 兵t

1

, ...,t

K

其共t

1

=t

共B兲

and t

K

=t

共C兲

兲, labeled

M = 共

共t

1

兲, ... ,

共t

K

兲兲. 共8兲

The trajectory model T is close to the true trajectory that

the whale follows. Let f共T , t

k

兲 be the value at the time t

k

共k

苸 兵1,...,K其兲 of the surface echo delay if the whale were on

the trajectory T. Such value, however, due to the inaccuracy

of the measurement and modeling processes, is not exactly

equal to

共t

k

兲. The difference

⑀

共T , t

k

兲 between the data

共t

k

兲

and the model f共T,t

k

兲 is deﬁned as

共t

k

兲 = f共T,t

k

兲 +

⑀

共T,t

k

兲 for k 苸 兵1, ... ,K其. 共9兲

The author assumes that the errors due to such inaccu-

racies are centered 共the mean of the error is equal to zero兲,

independent 关the error made on

共t

i

兲 is independent with the

error made on

共t

j

兲, i⫽ j兴, and of equal variance. In that case,

one can model the above-described inaccuracies by interpret-

ing them as an additive white Gaussian noise. Let

⑀

be a

centered, white Gaussian noise of standard deviation

,

⑀

⬃N共0,

兲.

The noise on the data is modeled using the random vari-

able

⑀

. One can model the ﬂuctuations in the values of the

parameters previously deﬁned using random variables. Let

E

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, E

p

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, z

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, r

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, and

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be the random variables corre-

sponding to the parameters E

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, E

p

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, z

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, r

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, and

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共for