A Bayesian method to estimate the depth and the range of phonating sperm whales using a single hydrophone

TL;DR: The author describes a Bayesian method to combine the information contained in those acoustic data plus visual observations that 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.

Abstract: 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 (direct path, sea surface, and seafloor reflections) 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 (using shallow hydrophones towed by research vessels) direct path transmissions and sea surface reflections of sperm whale phonations (clicks). 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.

Summary

Introduction

• Submitted on 7 Mar 2013 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not.
• 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. pp. 1519-1528. ISSN 0001-4966 Open Archive Toulouse Archive Ouverte is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.

A Bayesian method to estimate the depth and the range of phonating sperm whales using a single hydrophone

• Christophe Laplanchea 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 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.
• 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 can seldom clearly detect the seafloor echoes of clicks emitted by sperm whale using a hydrophone close to the sea surface towed by a research vessel.
• The Bayesian approach has already proven to be efficient to locate sound sources using TOADs Spiesberger, 2005 .

A. Trajectory model

• First, one would need a representation of the underwater trajectory of the whale using a mathematical model.
• Let t A be the time when the sperm whale flukes-up and starts diving, t B the time when the whale starts clicking, t C the time when the whale stops clicking, and t D the time when the whale resurfaces.
• One can then define the trajectory T using the depth, range, and heading of the whale.

B. Prior information

• This latter definition of T uses 3n+5 parameters, that is to say redundant information.
• Indeed, the coordinates of the last summit E n+1 can be calculated given the coordinates of the first summit E 1 and using the values of speeds and change of heading in the n segments.
• Sperm whales usually keep a constant vertical speed while descending to reach their prey and while ascending to reach the sea surface Miller et al., 2004a .
• Relative to the research vessel, between the fluking and the resurfacing points.
• Such uncertainties are modeled in the following section using random variables.

E. Probability distribution of the prior

• The author will give a more accurate description of the prior distributions appearing in Eq. 13 .
• By choosing such a prior, the author indicates that the underwater movement of the whale is not erratic, and that it is likely that the whale tends to swim in a given direction.
• The probability density functions have a maximum at the most likely prior value of the parameters, and its width illustrates the confidence the authors grant to this most likely prior value.
• One could choose for instance gamma probability distributions.
• For practical reasons, the author has chosen truncated normal distributions.

G. MCMC algorithm

• One can use a Markov Chain Monte Carlo MCMC algorithm to draw samples of the posterior.
• One can use the Metropolis-Hastings algorithm Robert and Casella, 2004 to draw such samples.
• Calculate the acceptance ratio of this sample to determine if the new sample Instead of drawing a whole new trajectory at each iteration i, draw the summits one by one.
• In practice, the second method requires a lower total amount of samples, as it requires less than I /C samples per subchain to lead to a correct estimate T̂. The author has used the subchain method.

H. Data set

• The author has run simulations using the free software SBPLASH implemented in MATLAB.
• The efficiency of the algorithm is illustrated using synthetic data.
• The depth, range, vertical speed, and horizontal speed of the whale when following this trajectory are given in Figs. 3–6, respectively.
• The initial value of the trajectory is in the author’s simulations the rectilinear, constant speed trajectory linking the prior locations of the points E 1 and E n+1 Figs.
• These parameters here are constants which are empirically chosen.

A. Convergence

• The algorithm generates trajectories with values close to the data after I0=100 iterations.
• The acceptance orithm left, circles of the depth z of the sperm whale throughout the dive.
• Draws of the first s=1 and last s=n+1 summits are more often accepted than the others 2 s n , given the different constraints and priors they are bound to.

B. Depth, range, and speeds

• The vertical and horizontal speeds of the whale are estimated using Eq. 19 .
• The estimate of the depth and the range of the whale are calculated using these values.
• Results of vz, vr, z, and r are plotted plus/minus twice their standard deviation.

IV. DISCUSSION AND CONCLUSION

• The algorithm correctly estimates the depth and the range of the whale throughout the dive Figs. 3 and 4 .
• The author is, however, confident regarding the choice of I1 for the given data set.
• The parameters used in the probability distributions of the priors have been fixed and empirically chosen.
• As stated regarding the standard deviation of the prior speed, an optimal value of could be estimated while sampling and estimating the trajectory parameters.
• Not requiring seafloor echoes, the method could be used to estimate the depth and the range of foraging sperm whales using a single hydrophone in any location no matter what the depth, the relief, or the constitution of the seafloor might be.

Figures

FIG. 5. True, initial, and estimated values of the vertical speed vz of

FIG. 6. True, initial, and estimated values of the horizontal speed vr of the

FIG. 4. True, initial, and estimated values of the range r of the sperm

FIG. 3. True value left, thick solid line and initial value used in the MCM Estimation of the depth of the sperm whale right, thick line plus/minus twi set presented in Fig. 7 and the ten 21 000-iteration Markov subchains. Par plus/minus twice their standard deviation right, thin bright dashed lines fo

FIG. 7. Surface echo delay of the K=1771 clicks emitted by the sperm whale during the dive depicted in Figs. 3–6. The solid line describes the data set M used in the tests. The circles represent the values of the trajectory used to initiate the algorithm.

FIG. 1. The whale dives at t= t A , starts clicking at t= t B , stops clicking at t= t C , and resurfaces at t= t D . The whale is at the depth z=z s at t= t s = t B + s−1 0 s 1, . . . ,n+1 , in this example n=14 . The vertical speed of the whale is constant and equal to vz s for t t s , t s+1 .

FIG. 2. The whale is at E s at t= t s . The hydrophone is at H. Ep s and Hp are the projections of E s and H on a horizontal plane. The whale moves in a constant heading and at a constant horizontal speed vr s for t t s , t s+1 along the segment S s to reach E s+1 at t= t s+1 . The angles b s and e s are defined as the angles between Ep s Ep s+1 and HpEp s , and Ep s Ep s+1 and HpEp s+1 , respectively. The change of heading from S s−1 to S s is s = e s−1 − b s . The horizontal range and the azimuth of the whale at t= t s are s s s

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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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makes it freely available over the web where possible.
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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
共s兲
, E
p
共s兲
, z
共s兲
, r
共s兲
, and
␪
共s兲
be the random variables corre-
sponding to the parameters E
共s兲
, E
p
共s兲
, z
共s兲
, r
共s兲
, and
␪
共s兲
共for

Frequently asked questions

Q1. What are the contributions in "A bayesian method to estimate the depth and the range of phonating sperm whales using a single hydrophone" ?

Laplanche et al. this paper proposed a Bayesian method to estimate the depth and the range of phonating sperm whales using a single hydrophone.