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IEEE SIGNAL PROCESSING MAGAZINE [120] NOVEMBER 2013 1053-5888/13/$31.00©2013IEEE
Digital Objec t Identifier 10.110 9/MSP .2013 .226 7651
Date of publica tion: 15 October 2013
T ime-frequency (T-F) analysis of signals propagated
in dispersive environments or systems is a challeng-
ing problem. When considering dispersive wave-
guides, propagation can be described by modal
theory. Propagated signals are usually multicompo-
nent, and the group delay of each mode (i.e., each component)
is nonlinear and varies with the mode number. Consequently,
existing T-F representations (TFRs) covariant to group delay
shifts (GDSs) are not naturally adapted to this context. To over-
come this issue, one solution is to approximate the propagation
using simple models for which the dispersion properties do not
vary with the mode number. If the chosen model is both simple
and robust to uncertainties about the waveguide, it can be used
to define adapted TFRs, such as the power-class with a suitable
power coefficient. This article focuses on a context where this
[
Julien Bonnel, Grégoire Le Touzé, Barbara Nicolas, and Jérôme I. Mars
]
[
Power class utilization with
waveguide-invariant approximation
]
Physics-Based
Time-Frequency
Representations for
Underwater Acoustics
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© istockphoto.com/–m–i–s–h–a–
IEEE SIGNAL PROCESSING MAGAZINE [121] NOVEMBER 2013
methodology can be applied: low-frequency acoustic propaga-
tion in shallow water. In this case, the global oceanic dispersion
can be summarized using a single scalar b called the waveguide
invariant. This parameter can be used to approximate the group
delay of each mode with a power law. Consequently, it is possi-
ble to use power-class TFRs with a b-based power coefficient.
Their practical use is demonstrated on two experimental data
sets: a man-made implosion used for underwater geoacoustic
inversion, and a right-whale impulsive vocalization that can be
used to localize the animal.
INTRODUCTION
Many physical or technical situations involve the propagation of a
wave through a dispersive medium, e.g., ultrasonic waves in
bones [1], electromagnetic whistlers in the magnetosphere [2],
and ground-penetrating radar [3]. When considering broadband
propagation, each frequency travels with its own speed, and a
propagating nonstationary signal is distorted during its propaga-
tion. Dispersion usually tends to complicate the received signal,
particularly in complex propagation environments that are domi-
nated by multiple modal components, and this often limits the
ability to directly recover information from the signal. On
the other hand, the dispersion effects convey information about
the propagation medium and the source/receiver configuration. If
properly characterized using suitable signal processing methods,
dispersion can be used as the basis of inversion algorithms
(source localization and/or environmental estimation) [4]–[6].
When considering a transient source signal in a single receiver
configuration, the received signal is nonstationary, and T-F analy-
sis is the most common tool for dispersion analysis.
The application field considered in this article is underwater
acoustics, although the proposed methodology can be extended
to other dispersive situations. In particular, the article examines
the practical application of power class TFRs for underwater
acoustics, especially the low-frequency impulsive sources in
shallow water. In this context, the propagation is described by
modal theory. The oceanic environment acts as a dispersive
waveguide, and the received signal contains several nonlinear
frequency-modulated components called propagating modes.
T-F analysis of the received signal and modal filtering are
important issues for the underwater acoustic community. Using
a single receiver, this allows for source localization and environ-
mental estimation [4]–[6]. Dispersion analysis in the T-F
domain has been studied by the signal processing community.
Several approaches have been proposed, mainly for underwater
acoustics [6]–[11], but also for other fields of application [1],
[12]. The majority of these methods are physics-based and inject
a priori environmental knowledge into the T-F processing. For
such methods, the key is to make the best use of the available a
priori information, while at the same time being insensitive to
misknowledge in this a priori information.
Many TFRs have been proposed by the signal processing
community [13], but none of them can be used successfully in
every situation. All the TFRs have advantages and drawbacks,
and are adapted to analyze signals with particular properties and
particular T-F content. The TFRs are thus often classified based
on the properties they satisfy. Critical properties for T-F analysis
are the covariance properties. A TFR is said to satisfy a covari-
ance property if this TFR preserves certain T-F changes in the
signal. It is also said that the TFR is covariant to that property.
As an example, the Cohen’s class is covariant to constant time-
shifts and frequency-shifts, while the affine class is covariant to
constant time shifts and to T-F scaling [13]. However, because of
dispersion, none of these covariance properties are adapted to
our underwater acoustics context. There is the requirement for
covariance properties that take into account the non linear
group delay of the signal considered. Such a covariance prop-
erty is called the GDS covariance, and the corresponding TFR
class was introduced by Papandreou-Suppappola et al. [14], [15].
This article is centered on a subclass of the GDS covariant
class: the power class with an associated power time-shift covari-
ance property [16]–[18]. The power class is thus adapted to disper-
sive signals with power-law group delays. For physical and
underwater acoustic applications, the power class covariance
should allow the inclusion of any available a priori information
about the environment. To do that, the article establishes a link
between the power class TFRs and the waveguide invariant b [19],
[20]. The waveguide invariant is a scalar that summarizes the dis-
persion in the waveguide and allows the approximation of the
phase of each mode as a frequency power function. This connec-
tion allows power class TFRs to be used to analyze the propagation
of impulsive sounds in dispersive multimodal shallow water wave-
guides. This is demonstrated on both simulated and experimental
marine data sets that correspond to both active and passive under-
water acoustics (geoacoustic inversion and bioacoustics).
TIME-FREQUENCY REPRESENTATIONS COVARIANT
TO GROUP DELAY SHIFTS
The group delay shifT covariance
As stated in the introduction, the constant time shift covariance
property of many TFRs is not adapted to the context of disper-
sive propagation. One requires a TFR covariant to dispersive
time shifts corresponding to the nonlinear group delay function
of the medium.
Considering a signal
()
Xf in the frequency domain, the out-
put
()
Yf of an allpass dispersive system is given by ()Yf =
() .Xfe(/ )jff20
rp- The group delay information is given by
() (/ ).fdf
dff
0
xp
= (1)
It is proportional to the derivative of the one-to-one phase func-
tion (/ ),ff
0
p where f0
02 is a normalization frequency. To sat-
isfy the GDS covariance, a TFR T should preserve the
frequency-dependent time shift ().fx In a more general frame-
work, the GDS operator of a signal
()
Xf is ()DXf
() =
a
p
() ,Xfe(/ )jff20
rap- with a a real number. For a given phase func-
tion ,p the parameter a quantifies the amount of dispersion on
the signal ().DXf
()
a
p A TFR T is GDS covariant if the TFR of
the output
()
DXf
()
a
p corresponds to the TFR of the input
()
Xf
IEEE SIGNAL PROCESSING MAGAZINE [122] NOVEMBER 2013
shifted in time by the change in group delay
()
fax introduced
by the GDS operator
(, )( ()
,)
.TtfT
tf
f
DXX
() ax=-
a
p (2)
Successful use of the GDS covariant TFRs requires that the
group delay of the TFR in (2) matches the group delay of the
signal studied. Indeed, significant distortion can occur if this is
not the case [21]. Moreover, the GDS covariance follows directly
from the constant time-shift covariance according to the uni-
tary equivalence principle [22], and thus requires knowledge
about the time origin. These are two strong constraints for
practical/experimental T-F analysis. Both of these will be taken
into account in the context of underwater acoustics for
unknown low-frequency impulsive sources in shallow water.
The power class
A particular part of the GDS covariant class is the power class
[16]–[18]. It is adapted to the case of the power-law covariance
property, for which the phase and group delay functions are
power functions defined as
() () () ,
() () ,
ffsgnf f
ff
ff
f
00
1
pp
xx l
==
==
ll
l
l-
(3)
(4)
where l is the power parameter, f0 is a fixed reference fre-
quency, and
()
sgnf provides the sign of the frequency .f The
GDS covariance in (2) thus becomes the power time-shift cova-
riance. The lth power class TFRs satisfy two specific covariance
properties. As an extension of the affine class, the first covari-
ance property is covariance to scale change
(, )(
,)
,TtfTat a
f
() ()
CXX
a
=
ll (5)
where Ca is the scaling operator that is defined as
() (/ )/
||
.CXf Xf
aa
a= The second covariance property is cova-
riance to power time shifts that matches the signal group delay
(, )( (),)
,,
TtfTtff
Tt ff
ff
() ()
()
DXX
X00
1
ax
al
=-
=-
lll
ll-
a
l
cm
(6)
where Da
l is the power time shift operator given by
() () .DXf Xfe(/ )jff20
=
a
lrap
-l Once again, the parameter a is a real
number that expresses the amount of dispersion in the signal
under study. It will be of particular interest in the section
“Waveguide Invariant-Based Time-Frequency Representations.”
Hlawatsch et al. have shown [18] that any member of the
th
l power class can be expressed as
(, )(/, /)
() () ,
Ttffffffe
Xf Xfdf df
1
()
*
Xjtf
f
f
f
f
12 2
1212
12
#
C=
3
3
3
3
lrl
pp
--
-
ll
eeoo
>
G
##
(7)
where C is a two-dimensional kernel that uniquely character-
izes the TFR. Note that the affine class [13] is a particular case
of the power class for .1l=
The power class can also be obtained using unitary warping
[18], [22]. The power-warping operator is
() (/ )(/)
|||/ |()|/
|,
WXf df
dff Xff
ff
Xf sgnf ff
1
1010
02
1
00
1
pp
l
=
=
lll
l
ll
--
-
^
`
h
j
(8)
where
()
f
1
pl
- is the inverse function of ().fpl Any member of
the power class T
()
l can be computed as an affine TFR T
()
A
(i.e., a TFR of the affine class) of the warped signal (),WXf
l fol-
lowed by a nonlinear T-F coordinate transform
(, )()
,(/),
|/ |,()| /| .
TtfT ff
tfff
Tff
tfsgn fff
() ()
()
W
W
XX
A
X
A
000
01
00
xp
l
=
=
l
ll
l
l
-
l
l
c
c
m
m
(9)
As an example, the power Wigner distribution corresponds to
the case where T
()
A is the Wigner distribution [13] and the pow-
ergram corresponds to the case where T
()
A is the scalogram
[13]. Note that as T
()
A is a member of the affine class, it is a qua-
dratic TFR. Both T
()
A and T
()
l are thus influenced by cross-
terms in the case of multicomponent signals. The cross-term
geometry is detailed in [17].
Power class definition is based on a single power group delay
().fxl Consequently, the power class cannot handle multicom-
ponent signals with different dispersion power laws. However,
power class definition is independent of the parameter .a Con-
sequently, power class is a suitable method to analyze multi-
component signals with different amounts of dispersion a (as
long as the phase function pl is the same for every component).
In the modal propagation context, the received signal contains
several components called modes. Successful use of the power
class requires that 1) each mode has a power group delay; 2) for
a given signal, the power parameter l is the same for every
mode; and 3) ideally, the value of l can be determined using
physical considerations. In the following, we will demonstrate
that it is possible to fulfill these three expectations.
ACOUSTIC PROPAGATION IN SHALLOW WATER
Modal propagaTion
When considering low-frequency acoustic waves in shallow
water, propagation can be described by normal mode theory
[20]. Considering an impulsive source, the received signal
()
Yf
after propagation in the acoustic waveguide over a range r con-
sists in N dispersive waves called modes
() () .Yf Afe()
m
m
N
jf
1
2m
=rz
=
-
/ (10)
The phase function of each mode ()
()
fr
kf
mm
z= depends on
the source/receiver range r and on the waveguide properties
through the horizontal wavenumber ().
kf
m The modal ampli-
tude
()Af
m depends both on the waveguide properties and on
the source/receiver configuration (range and depth). Because of
dispersion, each mode can be associated to a nonlinear group
IEEE SIGNAL PROCESSING MAGAZINE [123] NOVEMBER 2013
delay () (/
)(
).ddf
ff
mm
xz
= If the range and the waveguide
properties are known, the modal phase
()
f
m
z can be computed,
and it is possible to define GDS covariant TFRs that are perfectly
adapted to a given mode .m
When considering an ideal waveguide that consists of an iso-
velocity water column between a pressure-release upper bound-
ary and a rigid bottom, the modal phase can be expressed as [20]
() , ,fc
rff
ff
m
id
wmm
222z=- (11)
where cw is the water sound speed, and fm is the cutoff fre-
quency of mode m that depends linearly on the mode number
.m Equivalently, a time-domain received signal
() ()yt Bte
()
m
N
m
jt
1
2m
=r{
=
-
/
obtained as the inverse Fourier
transform of (10) can be considered. In an ideal waveguide, the
time-domain modal phase is [11], [20]
() (/ ),
/.
tftrctrc
m
id mw w
22
2{=- (12)
Note that from (12), the modal phase depends linearly on .fm It is
thus possible to decompose
()
t
m
id
{ into a mode-dependent
amount of dispersion fm and a mode-independent dispersive
phase (/
).
trcw
22
- If the waveguide properties and range are
known, it is thus possible to define covariant TFRs that are
adapted to the whole signal (i.e., to every mode at once) in an ideal
waveguide. Various studies have used the ideal waveguide (11) and
(12) to perform dispersion-based warping [9], [11]. Le Touzé et al.
also proposed modal-based TFRs using a similar model [10].
Except for this ideal simplistic waveguide, however, the dis-
persion characteristics evolve nonlinearly with mode number,
and it is impossible to define TFRs adapted to the whole
received signal. Although it would be possible to define GDS
covariant TFRs adapted to each mode, this solution does not
appear practical for experimental data, especially if the environ-
ment is not known. The next section presents the waveguide
invariant approximation that allows this issue to be overcome.
The waveguide invarianT
In the guided modal propagation context, interference structures
naturally appear in the
rf
- domain (where r is the source/
receiver range and f is the frequency). These interferences follow
a striation pattern that can be summarized by a single scalar b
called the waveguide invariant [19]. The waveguide invariant has
been widely used in underwater acoustics for various applications,
including passive source localization [23], [24], geoacoustic inver-
sion [25], [26], and active sonar [27], [28].
Classical uses of the waveguide invariant require range diver-
sity (i.e., several source/receiver ranges), which can be obtained
using a horizontal line array or the combination of a single
receiver and a moving source. By its nature, the waveguide invari-
ant b describes physical interferences between modes and sum-
marizes the dispersive behavior of the whole waveguide. However,
in 1999, D’Spain and Kuperman [29] provided an approximation
for single-mode dispersion using the waveguide invariant (see [29,
Appendix A]). In particular, they derived the frequency dependence
of the phase slowness ()
/,
Sf
kf
2
p
mmr= which was extended to
wavenumbers by Gao et al. [30]. Recalling that () (),fr
kf
mm
z=
one obtains the modal phase
()
[]
,fc
r
ff
mm
1
-
zc
-
bb
- (13)
where c is the average sound speed in the water column, and m
c
is a scalar that depends only on the mode number .m When con-
sidering a single receiver and an impulsive source, Gao et al. [30]
showed that it is possible to use (13) as the basis of a “dedispersion
transform.” This transform computes
() ,Ys YA
dedisp s
12
= by
projecting the received signal
()
Yf on the atoms
()
[]
.expAf jc
rfsf2
s
1
r=- -b
-
`
j
(14)
In the following, we will generalize the work of Gao and show
that (13) can be used as the basis of b-based power class TFRs and
warpings. Before this, note that (13) is based on several approxi-
mations that are inherent to the waveguide-invariant approxima-
tion. In particular, the waveguide invariant is actually invariant
only within a group of closely spaced modes and within a fre-
quency interval in which it is possible to define a functional rela-
tionship between group and phase speeds, which does not depend
on mode number [19], [20]. This relationship usually holds true
for low-frequency and long-range propagation in range-
independent shallow water, where the acoustic field consists of a
few modes that are weakly affected by refraction in the water col-
umn [20]. If a range-dependent environment is considered, it is
also required that the spatial variations of the waveguide are suffi-
ciently weak, so that an adiabatic modal approximation is valid
[20], [29]. Note that in classical shallow-water environments, it
can be demonstrated that .1-b In particular, 1b= in the ideal
waveguide presented in the section “Modal Propagation” [31]. The
link between the waveguide invariant and the ideal waveguide, i.e.,
the link between (11) and (13), will be presented in the section
“Comparison with Physics-Based Time-Frequency Representa-
tions Based on the Ideal Waveguide Model.”
WAVEGUIDE INVARIANT-BASED TIME-FREQUENCY
REPRESENTATIONS
power class and waveguide invarianT
According to (13), the modal phase can be decomposed into
■ a nondispersive component (/)rc f that corresponds to a
plane wave with constant group delay /
rc
t0=
■ a dispersive component (/ )rc f(/ )
m1
c-b- with group delay
(/)( /) .rc f1(/)
m
11
bc
b
--
As ,1-b the dispersive component corresponds to a disper-
sive wave with high frequencies arriving before low frequencies,
with the limit that infinite frequency arrives first at time .
tt
0
=
One can consider only the dispersive component by making t0 the
new time axis origin. This operation is not a practical problem, as
t0 corresponds to the time when energy first arrives at the receiver.
The modal phase thus becomes
() .fc
rf
mm
1
zc=-
bb
-
u (15)
IEEE SIGNAL PROCESSING MAGAZINE [124] NOVEMBER 2013
Equation (15) shows that the power class with power parameter
/1
lb
=- is adapted to every mode. One can thus define the
b-power phase and group delay
(16)
() ()|| ,
() .
fsgn ff
fff
f1
/1
00
1
p
xb
=
=-
bb
b
b
b
-
-
+
(17)
(In the following, the chosen notations are not consistent with
classical power class notations; however, no ambiguity arises and
these concise notations will be used throughout.) The correspond-
ing b-warping operates as
()
|/ |
|| () .WXf
ff
Xfsgnf f
f
02
100
b
=
bb
b
+
-
cm
(18)
The b-power warping linearizes the modal phase of each mode,
while the b-power TFRs are adapted to the whole received signal.
Note that inverse b-warping is given by .
WW
/
11
=
bb
-
To normalize dispersion to mode 1, i.e., to have ,1
1
a= one
must define
,
fr
c
m
m
0
1
1
c
ac
c
=-
=
b-
cm
(19)
(20)
where f0 is the reference frequency, and m
a is the amount of
dispersion of mode .m As a reminder, the concept of the
amount of dispersion was introduced in (2). Physically, it is a
good thing that the amount of dispersion depends on the
mode number .m In terms of signal processing, the modal
expansion of the acoustic field corresponds to the power signal
expansion over a finite number of power impulses [16], [18]
(each power impulse being a physical mode). As a result, the
inverse Fourier transform of the frequency-warped signal
()
WXf
b shows the modes as a discrete set of impulsions [18]
[see the section “Waveguide Invariant-Based Time-Frequency
Representations” and Figure 1(b)]. As a consequence, we will
call the inverse Fourier transform of the warped frequency
domain the modal domain. In the modal domain, mode m lies
at the position
/.
f
m0
a
coMpuTaTion and inTerpreTaTion
According to (9), any member of the b-power class can be
computed as an affine TFR of the b-power warping signal fol-
lowed by a nonlinear T-F coordinate transform. Until the end
of the article, we focus on the case of the b-powergram, for
which the original affine TFR is the scalogram. This choice
minimizes the cross-terms between the different components
of the signals, i.e., between the modes. This is an important
feature for practical underwater acoustics applications with
more than two propagating modes, as it prevents overlap
between modes and cross terms. In the following, we present
the equations for the b-powergram with power class formal-
ism and show that an equivalent formulation can be obtained
using a physics-based approach.
POWER CLASS FORMALISM
The powergram is obtained when the considered affine TFR T
()
A
is the scalogram. The scalogram (, )Stf
X of the signal
()
Xf can
be computed as the squared magnitude of the wavelet transform
(, )|
,|
,StfXH,
Xt
f2
12
= (21)
where H,
tf
is the time-shifted and scaled version of the wavelet
:H .
))
(|(/)| (( /)
ff
Hf fHffe
,/
tf jtf
012 02
=r
ll
lNote that /
ff
0 is the
scale coefficient. Next, the scalogram of the b-warped signal
(, )
St
f
WX
b can be obtained using unitary equivalence [22]:
(, )| ,||,
|.WW
StfXHXH
,/
,WXt
ft
f
212
12
12
==
bb
b (22)
Finally, the b-powergram (, )Stf
()
X
b is obtained from (9) by the
nonlinear T-F coordinate transform:
/)f/)f
)
(, )()
,(/)
(||
(/ )
(.
StfS ff
tfff
f
fXf f
f
Hf ff
f
ed
f
()
/
*() (
W
XX
jf
tf
000
012
0
1
00
02
2
0
xp
b
p
p
=
=
3
3
b
bb
b
b
b
brxp
-
-
-
+
-
b
bb
ll
ll
l
c
c
m
m
#
(23)
PHYSICAL APPROACH
It is also possible to obtained a b-based scalogram (, )
St
f
()phy
X
with a physical approach that does not use power class for-
malism or unitary equivalence. The change in formalism
does not provide a better TFR, but it helps to understand the
equations and to physically interpret the results. To compute
(, ),
St
f
()phy
X the wavelet must be adapted to the studied signal.
This can be done using a wavelet family for which the wave-
let phase corresponds to the modal phase. As the modes have
the phase () (/ ),
ff
f
mm0
zap=b one can choose the wavelet
family
{}
G,mf with
)/)f
)(( ,Gf Gf e
,(
mf f
jf
2m0
=ra p-b
ll l (24)
where )(Gf
fl is a window function that is concentrated
around frequency .f As an example, )(Gf
fl can be a scaled
version of the wavelet G: .
))
(|(/)| (( /)
ff
Gf fG
ff
/
f012 0
=
ll
This
choice allows the easy computing of (, )
St
f
()phy
X by modifying
an existing scalogram code. Whatever the choice of )
(,
Gf
fl
the family
{}
G,mf leads to a mode-frequency wavelet
transform
(,), ,SfXG
() ,
mod
X
mm
f
12
a= (25)
where the square magnitude is similar to ,SWX
b apart from a
compressional factor /f10 in the modal dimension. Conse-
quently, mode m lies at position m
a in the modal dimension
associated to ,S
()
mod
X while it lies at position /f
m0
a in the modal
dimension associated to .SWX
b
The final step to obtain a b-based scalogram (, )
St
f
()phy
X
using the physical approach is to go from the mode-frequency
domain to the T-F domain. This is achieved by substituting m
a
with the time t that corresponds to the mode m group delay
() /().tf
tf
mm
+ax
ax==
bb
(26)
IEEE SIGNAL PROCESSING MAGAZINE [125] NOVEMBER 2013
The following b-based T-F scalogram is thus obtained:
/)f
))
(, )| (/ (),)|
(( .
StfStff
Xf Gf
ed
f
() ()
*() (
phymod
XX
fjf
tf
2
20
x=
=
3
3
b
rxp
-
-
bb
2
ll l
l
# (27)
Note that in (27), the phase in the integral is equal to the
phase in the integral for powergram computation [see
(23)]. Moreover, Gf can be chosen so that the wavelet
transform
S()
mod is invertible (it must thus respect the clo-
sure condition). Note also that Gf can be chosen so that
.
SS
()
()
phy
=
b The corresponding Gf is obtained by compar-
ing (27) and (23).
siMulaTion exaMple
The proposed methodology is applied to realistic simulated data
that mimics shallow water conditions during the summer. The
configuration considered consists of a 100-m-deep water col-
umn over a layered bottom. The sound speed profile in the
water column is as follows:
■ constant sound speed 1,525 m/s from the surface z0=
m to the depth z45= m
■ linear variation 1,525 m/s "1,495 m/s from z45= m to
z55= m
■ constant sound speed 1,495 m/s from z55= m to the
bottom z100= m.
The seabed is modeled as a fluid (a classical assumption in
underwater acoustics). It consists of a homogeneous sediment
layer on top of a homogeneous semi-infinite basement. The
seabed parameters are
■ sediment: width 10 m, sound speed 1,800 m/s, density 1.8
■ basement: sound speed 2,000 m/s, density 2.
The source/receiver range is r10= km. Both the source and
the receiver lie on the bottom, so that the modes are (nearly)
equally excited. The frequency-domain impulse response of the
waveguide was computed using the ORCA modal code [32]. The
source is assumed to be perfectly impulsive. The received signal is
thus the simulated impulse response of the waveguide. No noise
has been added to the simulation, so that the results are more
easily understood. The robustness of the method against noise
will be demonstrated on real data in the next section.
The scalogram of the simulated signals is presented in Fig-
ure 1(a) with mode numbers labeled in white. Figure 1(a)
shows seven modes with relatively good T-F resolution. High-
order modes are closer to each other than low-order modes,
and they suffer from T-F interference. Mode 1, which has the
lowest frequency content, suffers from poor time resolution.
Figure 1(b) shows the scalogram of the inverse Fourier trans-
form of the b-warped signal. As explained in the section
“Waveguide Invariant-Based Time-Frequency Representa-
tions,” warped modes are impulsive signals. Note that warped
modes are not perfect impulses, which shows that the wave-
guide invariant, and particularly (13), is an approximation.
However, waveguide invariant approximation is accurate
enough such that the T-F representation of the warped modes
shows them well separated. The b-warping is thus a useful
tool for modal filtering, which is an important issue for the
underwater-acoustics community. Indeed, filtered modes serve
as input for matched mode processing [33] and other inver-
sion schemes [5], [34]. Figure 1(c) shows the b-powergram.
The seven modes are resolved with good resolution. Note that
both power warping and the powergram were computed using
..
12b= This value was chosen empirically to match the T-F
curvature of the modes.
As stated in the section “The Waveguide Invariant,” the wave-
guide-invariant value is usually 1-b in shallow water. However,
b is known to change according to the environmental conditions
[35]. For example, b can be linked to seabed properties in a
100
Frequency (Hz)
90
80
70
60
50
40
30
20
10
0 0.5 1
Time (s)
1.5
100
Frequency (Hz)
90
80
70
60
50
40
30
20
10
0 0.5 1
Time (s)
1.5
60
Warped Frequency (Arbitrary Unit)
55
50
45
40
35
30
25
20
15 02 4
Modal Dimension (Arbitrary Unit)
(a) (c)(b)
6
1
2
2
2
3
3
3
4
4
4
5
5
5
6
1
1
7
6
6
7
7
[FIG1] The simulated results. (a) A scalogram of the simulated signal. (b) A scalogram of the inverse Fourier transform of the b-warped
signal. (c) b-powergram. The power warping and powergram were computed using
..
12b= The scalograms and powergram were
computed using Morlet wavelets.
IEEE SIGNAL PROCESSING MAGAZINE [126] NOVEMBER 2013
Pekeris waveguide [36]. Depending on the available a priori
knowledge of the environmental context, the b value can be cho-
sen empirically or defined accurately using the knowledge of the
environment. In either case, it is important that the correspond-
ing TFR is robust to a relative mismatch in the b value. The
robustness of the b-powergram is considered in Figure 2. It pres-
ents powergrams that were computed for b= 0.5, 1, 1.5, and 3.
All of these powergrams show a good T-F resolution for each
mode, which demonstrates that the method is robust to mis-
match in the b value. However, Figure 2(a) and (d), which corre-
sponds to b=0.5 and 3, respectively, shows relatively poor T-F
resolution for high-order modes. Indeed, these modes are the
modes that interact most with the seabed, which explains why
they are more sensitive to an incorrect b value (and thus to an
incorrect environmental model).
coMparison wiTh physics-based
TiMe-frequency represenTaTions
based on The ideal waveguide Model
The b-power class provides a general and theoretical framework
to analyze the signals received in a dispersive waveguide. In this
section, this is compared with other physics-based solutions that
allow for dispersion T-F analysis [7], [8], [10]. Physics-based TFRs
inject a priori knowledge about the waveguide properties to mod-
ify the T-F tiling. This knowledge is usually based on a simple
waveguide model [7], [10], although it can be adaptively refined
using optimization [8]. The most common model for dispersion-
based signal processing is the ideal waveguide presented in the
section “Modal Propagation” [7], [9], [11]. It is thus interesting to
compare it with our b-based model.
The modal phase in the ideal waveguide is given by (11), while it
is given by (13) under the waveguide-invariant model. If the fre-
quency f is well above the cutoff frequency ,fm the ideal model and
the waveguide-invariant model with 1b= are both equivalent to a
nondispersive wave that would travel at the water sound speed
() ~()~ .
ff
c
rf
m
id
fmfw
1
zz
""33
b= (28)
This is consistent with modal propagation. In shallow water, high
frequencies correspond to low grazing angles (the energy propa-
gates nearly horizontally), while low frequencies correspond to
high grazing angles (the energy propagates nearly vertically).
Consequently, high frequencies are not strongly influenced by the
waveguide properties. In the limit of infinite frequency, the waves
travel horizontally and the waveguide does not influence propaga-
tion anymore (there is no dispersion). On the other hand, low fre-
quencies interact strongly with the bottom. It is thus necessary to
model the seabed correctly to study low-frequency dispersion,
which is the superiority of the waveguide-invariant model.
Indeed, the ideal waveguide model has a perfectly rigid bottom
100
Frequency (Hz)
80
60
40
20
00.5 1
Time (s)
1.5
100
Frequency (Hz)
80
60
40
20
00.5 1
Time (s)
1.5
(a) (b)
(c) (d)
100
Frequency (Hz)
80
60
40
20
00.5 1
Time (s)
1.5
100
Frequency (Hz)
80
60
40
20
00.5 1
Time (s)
1.5
[FIG2] The simulated b-powergram dependence on the value of .b (a)
..
05b= (b) .1b= (c)
..
15b= (d) .3b= The powergrams
were computed using Morlet wavelets.
IEEE SIGNAL PROCESSING MAGAZINE [127] NOVEMBER 2013
that does not model seabed propagation (i.e., no energy pene-
trates the seabed) while the waveguide invariant conveys informa-
tion about a more realistic seabed [25], [ 36]. Consequently, the
b-based TFRs can be used to perform accurate dispersion analy-
sis in complex waveguides. This is particularly suitable when sea-
bed interactions are important and can be modeled using a
suitable b value.
Waveguide invariant approximation, however, does not cor-
rectly model frequency dispersion between the cutoff frequency
and the Airy phase. For a given mode, the Airy phase is the fre-
quency at which the group speed is minimum, and thus it corre-
sponds to the last energy arrival in the T-F domain. The energy
that corresponds to frequencies between the cutoff frequency and
the Airy phase was called ground wave by Pekeris [37]. It propa-
gates in the seabed with high group speed and high attenuation.
Its T-F content is a slow upsweep going from cutoff frequency to
Airy phase, creating a T-F turnaround at the Airy phase. As an
example, mode 3 cutoff frequency and Airy phase can be seen in
Figure 1(a) at 28 Hz and 31 Hz, respectively. As ground waves are
highly attenuated, they are usually masked by noise in experi-
mental marine data (see Figures 3 and 4). However, ground waves
can be encountered when considering powerful explosions [4].
Analysis of the T-F turnaround associated with ground waves can-
not be achieved for all modes at once. This would require the
GDS covariant TFR based on accurate single mode dispersion.
EXPERIMENTAL DATA APPLICATION
This section demonstrates the practical use of the b power class
on experimental data. Two different underwater acoustic con-
texts are considered: active geoacoustic inversion and passive
study of marine mammal vocalization. In both cases, the con-
sidered source is impulsive and the environment is a complex
waveguide (nonisovelocity sound speed profile in the water col-
umn, layered seabed, and weak range dependence).
lighTbulb daTa: shallow waTer 2006 experiMenT
First, we consider an experimental signal that was recorded on
the continental shelf of New Jersey (United States) during the
Shallow Water 2006 Experiment [38]. The considered source is
a lightbulb implosion, and the corresponding acoustic field was
recorded by the Marine Physical Laboratory vertical line array (a
single hydrophone is used for this study). In this active context,
the T-F dispersion of the signal can be used to infer information
about the seabed properties (geoacoustic inversion). More
details about the environment and this specific signal can be
found in [5].
The source/receiver range, r7- km, is relatively short for
such a low-frequency inversion configuration. The scalogram of
the received signal is presented in Figure 3(a). The modes are
labeled in white for easier understanding. It can be seen that they
are close to each other and interfere together. Classical scalo-
grams cannot be used to infer information about the modal dis-
persion. The b-powergram
(.
)125b= is presented in Figure
3(b). It shows better resolution than the classical scalogram, and
it can be used to infer dispersion information. The value
.125b= has been chosen empirically to match the T-F curva-
ture of the modes.
righT-whale vocalizaTion: fundy bay daTa
Next, we consider an experimental signal that was recorded in
Fundy Bay (Canada) as part of the 2003 Workshop on Detec-
tion, Localization, and Classification of Marine Mammals
Using Passive Acoustics [39]. The source considered is a right-
whale impulsive vocalization that was recorded on a single
bottom-moored hydrophone (labeled H during the experi-
ment). In this passive context, the T-F dispersion of the signal
can be used to estimate the range between the whale and the
receiver. More information about the environment can be
found in [39], and about this specific signal in [40].
(a)
0.05 0.1 0.15
Time (s)
1.2
(b)
0.05 0.1 0.15
Time (s)
0
240
220
200
180
160
Frequency (Hz)
140
120
100
80
60
40
240
220
200
180
160
Frequency (Hz)
140
120
100
80
60
40
11
2
2
3
3
4
4
[FIG3] SW06 experimental results. (a) Scalogram of the received signal. (b) b-powergram of the received signal, computed using
.125b=. The scalogram and the powergram were computed using Morlet wavelets.
IEEE SIGNAL PROCESSING MAGAZINE [128] NOVEMBER 2013
In this passive context, the source/receiver range is not known.
It has been estimated in various studies, and it should be r8-
km. The scalogram of the received signal is shown in Figure 4(a).
It is nearly impossible to distinguish the modal content of the sig-
nal, and the modes are labeled in white to facilitate understanding.
Note that because of a low-speed sediment layer, mode 1 is proba-
bly trapped in the seabed and so does not appear in the T-F
domain. The b-powergram
(.)08
b= is shown in Figure 3(b),
and it has relatively good modal resolution. The b-scalogram is
thus a valuable tool to analyze experimental low-frequency data,
even when they are recorded in a passive context and in a compli-
cated environment. Note that the value .
08
b= was chosen
empirically to match the T-F curvature of the modes.
CONCLUSIONS
The power-class TFRs, or, more generally, the GDS covariant
TFRs, are suitable TFRs to analyze signals with dispersive group
delay. However, their use is limited to monocomponent signals,
or to multicomponent signals for which the dispersive proper-
ties do not vary from one component to the other. Successful
use of the power-class TFRs to analyze signals propagated in a
dispersive medium requires that the dispersion can be modeled
using a power law. Another desirable property is that the power
coefficient l can be defined using robust physical a priori infor-
mation about the medium.
This article focuses on dispersive propagation in an oceanic
waveguide; especially the low-frequency impulsive sounds propa-
gated in shallow water. This context is particularly challenging as
propagation is described by modal theory. Indeed, the signals con-
sidered are multicomponent, and each component (called a
mode) has a nonlinear group delay that varies from mode to
mode. However, it is possible to approximate the waveguide dis-
persion with a power law ,f/1 b- where b is the waveguide
invariant. The waveguide invariant is a scalar widely used in
underwater acoustics that characterizes propagation in shallow
water. It is possible to define a class of TFRs that are adapted to the
modal propagation context. This class benefits from the formalism
of the power class (with power coefficient
/)
,1
lb
=- while at the
same time allowing the inclusion of robust physical a priori infor-
mation through the waveguide invariant .b Note that single-
receiver modal filtering using b-warping can be obtained as a
by-product of the power class methodology.
This method is illustrated both on realistic simulation data
and experimental data. Two marine data sets that correspond to
different underwater acoustic contexts are considered: active
geoacoustic inversion and passive bioacoustics. In these difficult
practical situations, the proposed method allows us to resolve
several modes in the T-F domain, which was impossible with
classical TFRs. The b-power class is thus a promising method
for analysis of underwater acoustic single-receiver data.
AUTHORS
Julien Bonnel (julien.bonnel@ensta-bretagne.fr) received the
Ph.D. degree in signal processing from Grenoble-Institut
National Polytechnique (INP), France, in 2010. Since 2010, he
has been an assistant professor at Lab-STICC (UMR 6285), Ecole
Nationale Supérieure de Techniques Avancées Bretagne in
Brest, France. His research in signal processing and underwater
acoustics include time-frequency analysis, source detection/
localization, geoacoustic inversion, acoustical tomography, pas-
sive acoustic monitoring, and bioacoustics. He is a Member of
the IEEE and the Acoustical Society of America.
Grégoire Le Touzé (gregoire.le-touze@gipsa-lab.grenoble-
inp.fr) received the master’s degree in signal processing from
Grenoble-INP, France, in 2004 and the Ph.D. degree in 2007.
From 2008 to 2010, he was a postdoctoral researcher at the
[FIG4] The Fundy Bay experimental results. (a) Scalogram of the received signal. (b) b-powergram of the received signal, computed
using
..
08b= The scalograms were computed using Morlet wavelets.
(a)
0.2 0.4 0.6
Time (s)
0.8
(b)
0.200 0.4 0.6
Time (s)
0.8
100
90
80
70
60
Frequency (Hz)
50
40
30
20
10
100
90
80
70
60
Frequency (Hz)
50
40
30
20
10
2
3
4
2
3
4
IEEE SIGNAL PROCESSING MAGAZINE [129] NOVEMBER 2013
Laboratoire de Mécanique et d’Acoustique, Marseille, France,
and from 2010 to 2012 at the GIPSA-Lab, Grenoble, France. His
research interests include seismic and acoustic signal process-
ing, wavefield separation methods, time-frequency and chirplet
anaysis, and array processing.
Barbara Nicolas (barbara.nicolas@gipsa-lab.grenoble-inp.fr)
received the M.S. and Ph.D. degrees in signal processing from
Grenoble-INP, France, respectively, in 2001 and 2004. In 2005,
she held a postdoctoral position at the French Atomic Energy
Commission (C.E.A.) in medical imaging. Since 2006, she has
been a CNRS researcher at the GIPSA-Lab in the Image Signal
Department, Grenoble, France. In 2008, she was a visiting sci-
entist at SipLab in Faro, Portugal. Her research in signal pro-
cessing and underwater acoustics include TFRs, array
processing, acoustic tomography, underwater source detection
and localization, and geoacoustic inversion.
Jérôme I. Mars (jerome.mars@gipsa-lab.grenoble-inp.fr)
received the Ph.D. degree in signal processing (1988) from the
Institut National Polytechnique of Grenoble. From 1989 to 1992,
he was a postdoctoral researcher at the Centre des Phénomènes
Aléatoires et Geophysiques, Grenoble. From 1992 to 1995, he was
a visiting lecturer and scientist at the Materials Sciences and Min-
eral Engineering Department at the University of California,
Berkeley. He is currently a professor of signal processing in the
Image-Signal Department at GIPSA-Lab (UMR 5216), Grenoble
Institute of Technology. He is the head of the Image-Signal
Department. His research interests include seismic and acoustic
signal processing, source separation methods, time-frequency,
and time-scale characterization. He is a Member of the IEEE and
the European Association of Geoscientists and Engineers.
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[SP]
