ArticlePDF Available

Variability of phase and amplitude fronts due to horizontal refraction in shallow water

Authors:
  • University of Haifa Israel
Article

Variability of phase and amplitude fronts due to horizontal refraction in shallow water

Abstract and Figures

The variability of the interference pattern of a narrow-band sound signal in a shallow water waveguide in the horizontal plane in the presence of horizontal stratification, in particular due to linear internal waves, is studied. It is shown that lines of constant phase (a phase front) and lines of constant amplitude/envelope (an amplitude front) for each waveguide mode may have different directions in the spatial vicinity of the point of reception. The angle between them depends on the waveguide's parameters, the mode number, and the sound frequency. Theoretical estimates and data processing methodology for obtaining these angles from experimental data recorded by a horizontal line array are proposed. The behavior of the angles, which are obtained for two episodes from the Shallow Water 2006 (SW06) experiment, show agreement with the theory presented.
Content may be subject to copyright.
Variability of phase and amplitude fronts due to horizontal refraction in shallow water
Boris G. Katsnelson, Valery A. Grigorev, and James F. Lynch
Citation: The Journal of the Acoustical Society of America 143, 193 (2018);
View online: https://doi.org/10.1121/1.5020274
View Table of Contents: http://asa.scitation.org/toc/jas/143/1
Published by the Acoustical Society of America
Articles you may be interested in
A comparison between directly measured and inferred wave speeds from an acoustic propagation experiment in
Currituck Sound
The Journal of the Acoustical Society of America 143, 237 (2018); 10.1121/1.5021244
Sound Propagation through the Stochastic Ocean: Rebuttal to June 2017 JASA Book Review
The Journal of the Acoustical Society of America 143, 13 (2018); 10.1121/1.5019477
Acoustic inversion method for parameters of sediments based on adaptive predatory genetic algorithm
The Journal of the Acoustical Society of America 143, 141 (2018); 10.1121/1.5020272
Children's early bilingualism and musical training influence prosodic discrimination of sentences in an unknown
language
The Journal of the Acoustical Society of America 143, EL1 (2018); 10.1121/1.5019700
Striation-based source depth estimation with a vertical line array in the deep ocean
The Journal of the Acoustical Society of America 143, EL8 (2018); 10.1121/1.5020267
Pile driving acoustics made simple: Damped cylindrical spreading model
The Journal of the Acoustical Society of America 143, 310 (2018); 10.1121/1.5011158
Variability of phase and amplitude fronts due to horizontal
refraction in shallow water
Boris G. Katsnelson
a)
Leon Charney School of Marine Sciences, University of Haifa, Mount Carmel, Haifa 3498838, Israel
Valery A. Grigorev
Voronezh State University, Universitetskaya sq 1, Voronezh 394018, Russia
James F. Lynch
Woods Hole Oceanographic Institution, 98 Water Street, MS No.12, Woods Hole, Massachusetts 02543, USA
(Received 16 July 2017; revised 8 December 2017; accepted 13 December 2017; published online
16 January 2018)
The variability of the interference pattern of a narrow-band sound signal in a shallow water wave-
guide in the horizontal plane in the presence of horizontal stratification, in particular due to linear
internal waves, is studied. It is shown that lines of constant phase (a phase front) and lines of con-
stant amplitude/envelope (an amplitude front) for each waveguide mode may have different direc-
tions in the spatial vicinity of the point of reception. The angle between them depends on the
waveguide’s parameters, the mode number, and the sound frequency. Theoretical estimates and
data processing methodology for obtaining these angles from experimental data recorded by a hori-
zontal line array are proposed. The behavior of the angles, which are obtained for two episodes
from the Shallow Water 2006 (SW06) experiment, show agreement with the theory presented.
V
C2018 Acoustical Society of America.https://doi.org/10.1121/1.5020274
[JAC] Pages: 193–201
I. INTRODUCTION
A variability of the oceanic waveguide’s parameters in
the horizontal plane (due to bathymetry or the sound speed
profile) lead to the set of effects in sound propagation, which
are called “horizontal refraction,” or more generally “3D
effects.” These effects have been studied theoretically and
experimentally both in the deep ocean (Collins et al., 1995)
and in shallow water (Heaney and Murray, 2009;
Katsnel’son et al., 2007). As has been shown in numerous
past papers, there are a number of acoustical manifestations
of horizontal refraction due to coastal wedges. The first
observation and measurement of horizontal angle effects
was carried out in a coastal wedge by (Doolittle et al., 1988).
Subsequently, the following phenomena were studied: focus-
ing/defocusing of the sound field in the horizontal plane in
the presence of nonlinear internal waves (Katsnelson and
Pereselkov, 2000;Badiey et al., 2005); variations of the
interference pattern in the horizontal plane in the area of a
coastal wedge in (Deane and Buckingham, 1993;Katsnelson
et al., 2013), or in the area of the underwater canyon in
(Duda et al., 2011;Y.-T. Lin, 2013); multipath propagation
in the horizontal plane in an experiment in the Florida Strait
(Heaney and Murray, 2009); and many others. In the papers
(Bender et al., 2014;Ballard et al., 2012), the importance of
taking horizontal refraction into account for inverse prob-
lems was discussed as well.
The quantitative description of sound propagation in
shallow water in the presence of horizontal inhomogeneities
can be based upon the decomposition of the complex
amplitude of the sound field Pðr;zÞat some frequency xinto
waveguide modes (vertical modes) wnðz;rÞwhich depend
upon the horizontal coordinates r¼ðx;yÞ, as well as on ver-
tical parameters (zis the vertical coordinate directed down-
ward, and z¼0 corresponds to the surface). Modal
amplitudes Anðr;xÞin the sound field decomposition satisfy
a wave equation which can be solved either in the ray
approximation (horizontal rays and vertical modes)
(Weinberg and Burridge, 1974) or using the parabolic equa-
tion method (Collins et al., 1995;Colosi, 2016).
One of the key features of these phenomena is the fre-
quency dependence of the modal amplitude Anðr;xÞ(in par-
ticular, the amplitude and phase of the horizontal rays) and,
in turn, the interference pattern of the sound field in the hori-
zontal plane. In other words, after separating out the vertical
coordinate z, a two-dimensional (x-y plane) wave equation is
obtained, which describes wave propagation in a two-
dimensional dispersive medium.
Given this background, the goal of this paper is to study
the specific physical features of the interference pattern due
to its frequency dependence or, more specifically, the behav-
ior of the phase and amplitude fronts of a sound pulse propa-
gating in a shallow water waveguide. Such frequency
dependence in the presence of horizontal refraction plays
significant role in shallow water acoustics (Katsnelson and
Pereselkov, 2004;Dong et al., 2015).
II. PHASE AND AMPLITUDE FRONTS IN THE
INTERFERENCE PATTERN
Let us consider a narrow-band sound signal propagating
from a point source in shallow water with parameters (for
example, the sound speed profile and/or bathymetry)
a)
Electronic mail: bkatsnels@univ.haifa.ac.il
J. Acoust. Soc. Am. 143 (1), January 2018 V
C2018 Acoustical Society of America 1930001-4966/2018/143(1)/193/9/$30.00
depending on the horizontal (x-y) coordinates, and being
received by some receiving system. The complex sound
field, which depends on both time and the spatial point ðr;zÞ
has the form
Pðr;z;tÞ¼ðX
n
Anðr;xÞwnðz;r;xÞexp ðixtÞdx;
(1)
where wnðz;r;xÞare the waveguide modes, which depend
weakly on the horizontal coordinates as adiabatic modes and
smoothly upon the frequency (excluding the region near the
cutoff frequency). The corresponding modal eigenvalues qn
(propagation constants) depend on rand xas well:
qn¼qnðr;xÞ.
If the spectrum of the signal is narrow enough, i.e.,
x1<x<x2, where Dx¼x2x1x, then the eigen-
function, due to the smooth frequency dependence of the
mode shape, can be taken outside the integral (1) at some
central frequency x1<x0<x2,
Pðr;z;tÞ¼X
n
wnðz;r;x0ÞðAnðr;xÞexp ðixtÞdx
¼X
n
wnðz;r;x0ÞPnðr;tÞ;(2)
where we have introduced the modal amplitude, which
depends on the horizontal coordinates and time, as
Pnðr;tÞ¼ðAnðr;xÞexp ðixtÞdx:(3)
Neglecting mode coupling, Anðr;xÞsatisfies the two-
dimensional Helmholtz equation where the wave number is
qnðr;xÞ. Using standard techniques, it is possible to find the
spectral amplitude Anðr;xÞin the ray approximation
Anðr;xÞ¼anðr;xÞexp isnðr;xÞ½;(4)
where snðr;xÞis the eikonal in the horizontal plane, corre-
sponding to the mode n, which can be found from the two
dimensional eikonal equation
r?snðr;xÞ½
2¼q2
nðr;xÞ;(5)
where r?¼ð@=@x;@=@yÞis the gradient in the horizontal
plane.
Due to the frequency dependence of the eigenvalues on
the right side of Eq. (5), the horizontal rays have different
paths for different frequencies On the basis of ray theory, it is
possible to find horizontal rays joining the source and receiver
(the so-called eigenrays). The frequency dependence of the
ray path is the main peculiarity of the ray pattern in our given
case, in other words eigenrays corresponding to different fre-
quencies have different trajectories [Fig. 1(a)].
Next, we will analyze the sound field of a particular
mode with index number n; for brevity, we will omit show-
ing this index number explicitly in the following discussion.
If we introduce a vector q:(jqq), tangent to the horizon-
tal ray path, then at the locations of the source and receiver,
the vectors qðxÞ, corresponding to different frequencies,
will be different both in modulus and direction. In the Fig.
1(a), the vectors q1and q2are shown for frequencies x1and
x2at the location of the receiver.
It is seen that, for the sound field of a narrowband signal
in the neighborhood of the receiver, an interference pattern
is formed. For this pattern, a rather simple illustration can be
presented for the superposition of two horizontal rays
(locally plane waves). To illustrate the interference pattern
in the horizontal plane formed by two such rays with wave
vectors q1and q2, corresponding to two close frequencies
x1and x2[Fig. 1(a)], also with equal amplitudes for sim-
plicity, coming into the receiver, we write the sound field as
Pðr;tÞ¼Aexp iðq1rx1tþu1Þ½
þAexp iðq2rx2tþu2Þ½
¼Aðpr tÞexp iðqr xtþuÞ½;(6)
where u1and u2are some phases, q¼ðq2þq1Þ=2,
x¼ðx2þx1Þ=2, p¼Dq=Dx,u¼ðu2þu1Þ=2, Dq¼q2
q1,Dx¼x2x1,Du¼u2u1, and Aðpr tÞ¼2A
cos ½ðDxðpr tÞþDuÞ=2.
So, in the vicinity of the receiver the sound field is mod-
ulated in space and time, with carrier frequency xand com-
plex amplitude Aðpr tÞ. The scales of this modulation in
the horizontal plane, i.e., the interference pattern similar to
Fig. 1(b), depends upon the parameters of the waveguide.
Let us now introduce a phase front (lines of constant phase)
qr xt¼constant, an amplitude front (line of constant
amplitude) pr t¼constant, and normals to these lines,
i.e., the vectors q(the direction of propagation of the phase
front) and p(the direction of propagation of the amplitude
front).
If the spectrum of the source is contained in the band
x1<x0<x2, then the wave vectors of the horizontal rays
coming into the receiver will be located in the sector between
FIG. 1. (a) Two horizontal rays for dif-
ferent frequencies coming into a
receiver. (b) Directions of the ampli-
tude and phase fronts (wave vectors p
and q). (c) Spatial Fourier transform in
the horizontal plane.
194 J. Acoust. Soc. Am. 143 (1), January 2018 Katsnelson et al.
q1and q2. We can get an expression for the received signal
by integration of Eq. (3) assuming a narrow spectrum where
we can use a linear decomposition of the wave vector: q0
¼qðx0Þ¼qðxÞþpðxÞðx0xÞ¼qþpðx0xÞ,where
xis the carrier frequency, for example x¼ðx2þx1Þ=2,
pðxÞ¼dq=dx. So the received signals have the form
Pðr;tÞ¼ðx2
x1
Aðr;x0Þexp iðq0rx0tþu0Þ

dx0
¼Aðpr tÞexp iðqr xtÞ½;(7)
where the amplitude is Aðpr tÞ¼Ðx2
x1Aðr;x0Þexp
½iðx0xÞðpr tÞþiu0dx0. In particular, assuming that
within this band the spectral amplitude is constant, i.e.,
Aðr;x0Þ¼A, we get
Apr t
ðÞ
¼2ADxsin Dxpr t
ðÞ

Dxpr t
ðÞ
:(8)
The time duration of the envelope of this signal is 2p=Dx,
and the spatial scale of the envelope in the direction of prop-
agation is 2p=jDqj. The interference pattern of the signal
in the horizontal plane has rather complex character when
the lines of constant amplitude have a different direction
than the lines of constant phase. Waves of this type are
called inhomogeneous (Born and Wolf, 1968), and are simi-
lar to those presented in Fig. 1(b). [A similar interference
pattern has been observed for a wedge model by Katsnelson
et al. (2013).]
We note that if the spectral components in Eqs. (6) and
(7) have different amplitudes, then the result of their summa-
tion has a more complex form than in the right side of Eqs.
(7) or (8), and in that case we will define the amplitude front
as a line (or some area in the horizontal plane) of maximal
amplitude.
III. THEORETICAL ESTIMATES OF AMPLITUDE AND
PHASE FRONT VARIATIONS
Let us consider theoretical and numerical estimations of
the possible angles for the amplitude front (AF) and phase
front (PF) directions within the framework of a simplified
model for typical shallow water conditions.
As an example, consider sound propagation in the
Shallow Water 2006 experiment (Newhall et al., 2007)in
the presence of linear internal waves with small amplitude,
having some anisotropy, and propagating mainly toward the
coast. This situation has been seen to take place before and
after the passage of a train of intense nonlinear internal
waves [see, for example, Sabinin and Serebryani (2005)].
Let the source and receiver be placed at a distance Drel-
ative to each other along the raxis (Fig. 2), which is approxi-
mately parallel to the coastal line. For SW06, D2104
m. The unperturbed sound speed profile cðzÞis characterized
by a rather sharp thermocline, with thickness much less than
the water depth, that is htH(for SW06 ht=H0:1). The
displacement of the isodensity surface from its equilibrium
shape (plane) fðr;zÞcharacterizes the internal waves and the
corresponding variation of the sound speed profile cðr;zÞ
¼cðzÞþdcðr;zÞ. The correction to the sound speed profile
can be written (Flatte, 1979)
dcr;z
ðÞ
cz
ðÞ ¼QN2z
ðÞ
fr;z
ðÞ
;(9)
where NðzÞis the Brunt-Vaisala frequency (buoyancy fre-
quency), and the parameter Qdepends on the water column
properties. In shallow water Q2:4s
2
/m.
For linear internal waves, the gradient of the displace-
ment in the horizontal plane c¼r
?fðr;zÞhas the value
jcj0:01 and the perturbation of the sound speed in the area
of the thermocline due to the displacement is jdcj1m/s.
The eigenvalue of the nth acoustic mode qðr;xÞin the
presence of the internal waves can be found using perturba-
tion theory
qðr;xÞ¼q0ðxÞþdqðr;xÞ;(10)
where the first order correction dqðr;xÞhas the form
(Katsnelson et al., 2012)
dqr;x
ðÞ
¼Qk2
q0f0r
ðÞ
ðH
0
w0z
ðÞ
hi
2
N2z
ðÞ
Uz
ðÞ
dz;(11)
where fðr;zÞ¼f0ðrÞUðzÞ,f0ðrÞis the displacement of the
point corresponding to maximal amplitude, UðzÞis the first
mode of the internal gravity wave normalized to its maximal
value maxUðzÞ¼1, w0ðzÞand q0are the eigenfunction and
the eigenvalue of the mode nin the absence of an internal
wave (mode number omitted), and k¼x=~
c, where ~
cis
some average representative value of the sound speed in
water.
dqðr;xÞ=q0ðxÞis the first order correction to the refrac-
tive index in the horizontal plane, determining the trajectory
of the ray paths from Eq. (5), and depending on mode num-
ber and frequency.
Let us next suppose that the displacement fdepends on
a coordinate s, transverse to the acoustic track (Fig. 2).
Further, in the area of the ray paths (which have small hori-
zontal angles in our problem) this displacement depends
only linearly on s:fðr;zÞ¼cs, in the sense of a Taylor
expansion. In this case, the correction to the refraction index
in horizontal plane has the form
FIG. 2. Eigenrays joining source S and receiver R for two frequencies. We
define positive angles in the counterclockwise direction. IW denotes the direc-
tion of motion of an internal wave. Estimates based upon references
(Katsnelson et al.,2012;Katsnelson et al., 2013;Weinberg and Burridge, 1974)
give: jujjaj12;jDbj14;jDuj0:1for Dx10 Hz.
J. Acoust. Soc. Am. 143 (1), January 2018 Katsnelson et al. 195
dqr;x
ðÞ
q0x
ðÞ ex
ðÞ
s;(12)
e¼cQk2
q0
ðÞ
2ðH
0
w0z
ðÞ
hi
2
N2z
ðÞ
Uz
ðÞ
dz:(13)
In some cases, in particular for SW06, it is possible to
approximate that the function under integral (13) is concen-
trated within the thermocline of thickness ht. Rough estima-
tion of the integral in Eq. (13) gives N2
0ht=H, where N0is the
value of the Brunt-Vaisala frequency inside the thermocline
layer (its maximal value). The Brunt-Vaisala frequency can
be expressed via the vertical gradient of temperature T:N2ðzÞ
¼ðg=qÞðdq=dzÞ¼bTgðdT=dzÞ, where g10 m/s
2
,qis
the density of water, and bT1:3104. 1/K is the water
thermal compressibility coefficient. The variation of salinity
in our situation is negligible.
So, in the area of the thermocline, where the temperature
gradient can be 1 K/m, the maximal value of the squared
Brunt-Vaisala frequency is estimated as: N2
0ð11:5Þ
103Hz
2
. Neglecting the frequency dependence, which is
small for a narrow-band signal, jejjcjQN2
0ht=Hð2:54Þ
106v
1
.
All the angles we deal with are defined in Fig. 2.
Positive values of angles are in the counterclockwise direc-
tion. It is possible to estimate the horizontal angle uof an
arriving ray relative to the acoustic track (Fig. 2) using a
well known expression for the ray cycle distance D
(Katsnelson et al., 2012), assuming a linear dependence of
the refractive index on the coordinate sonly Eq. (12),
D¼2 tan u
e2u
e:(14)
Thus, as a result
uDe
2:(15)
For the distance 20 km, juj12. The sign of this angle
is determined by the gradient of the soundspeed along the s
axis. So, in Fig. 2we have u<0 for @v=@s>0 (if e>0).
Let us next find the angle between horizontal eigenrays
corresponding to different frequencies for the same vertical
mode. Vectors qðxÞand qðxþDxÞcome to the location of
the receiver at different angles uðxÞand uðxþDxÞ
¼uðxÞþDu. The angle Ducan be estimated as
Du¼D
2
de
dxDx:(16)
To estimate the value de=dx, we use the expression
(Katsnelson et al., 2012) for the connection between the
phase and group velocities of a waveguide mode, which in
simplified form is
q
x
@q
@x1
~
c2:(17)
For the Pekeris model, Eq. (17) is an exact expression where
in the right hand side one sees the sound speed in the water
layer. In our more general case, the right side is an average
value of the sound speed over the depth and Eq. (17) is an
estimate. Next, using Eqs. (12) and (17) we see
de
dx¼ d
dx
1
q
@q
@s

¼1
q2
@q
@x
@q
@s1
q
@
@s
@q
@x

2xe
q2~
c22e
x:(18)
Thus we get for Du, from Eq. (16) corresponding to the fre-
quency interval Dx,
Du¼De
xDx¼2u
xDx:(19)
It follows from Eq. (19) that for horizontal eigenrays which
are close to straight lines, the arriving and outgoing angles
decrease with increasing frequency.
For a distance of 20 km, a main pulse frequency f¼x=
ð2pÞ¼300Hz, and a frequency band Df¼Dx=ð2pÞ¼10Hz,
we estimate that jDuj0:1, and we also can write Du=Df
0:01 deg/Hz.
Next, let us estimate the angle Dbbetween the ampli-
tude and phase fronts (Fig. 2), or equivalently between the
vectors p¼Dq=Dxand q, using the triangle formed by the
vectors qand qþDqfor frequencies xand xþDx. In this
case, taking into account the smallness of /,
Dbq
DqDu2uq
Dq
Dx
x:(20)
The factor containing ðq=DqÞðDx=xÞis close to the ratio of
the modal group velocity to the modal phase velocity. If we
estimate the group velocity to phase velocity ratio roughly as
1, then Db2u, which leads to the estimate of the angle
between the phase and amplitude fronts as jDbj14.
The angle a(Fig. 2), determining the direction of the
amplitude front (vector p) relative to the direction of the
acoustic track, is
a¼uþDb;(21)
or roughly au. It is interesting to note that the normal
vector to the amplitude front is directed to the other side, rel-
ative to the acoustic track, than the normal vector to the
phase front. We remark that this is the result of the rather
idealized model used for our estimates; in a more realistic
situation the ray pattern, and in turn the connection between
angles, can be much more complex.
We should also note that the above-mentioned esti-
mates, which display increasing effects of horizontal refrac-
tion with decreasing frequency, are obtained assuming a
linear dependence of the displacement fand are valid for ray
paths in this assumed region.
IV. A METHOD TO FIND THE ANGLE BETWEEN THE
AMPLITUDE AND PHASE FRONTS FROM
EXPERIMENTAL DATA, AND ANALYSIS OF SW06
RESULTS
Let us now consider how to find the values of the afore-
mentioned angles on the basis of experimental acoustical
196 J. Acoust. Soc. Am. 143 (1), January 2018 Katsnelson et al.
data. Information about the details of the interference pattern
in the horizontal plane (for example, the directions of the
normals to the amplitude front and the phase front) can be
obtained using a spatial two-dimensional Fourier transform
(FT). For fixed time and depth, for example near the bottom,
the amplitude of the sound field is Pðr;H;t0Þ¼Pðx;yÞ(sup-
pressing depth Hand time t
0
) and the FT has the form
Gðqx;qyÞ¼ððPðx;yÞexp iðqxxþqyyÞ

dxdy
:
(22)
A typical spatial FT in the plane ðqx;qyÞfor the sound
field formed by close horizontal rays is shown in Fig. 1(c),
where the parameters of the problem (vectors qand p) and
the corresponding angles are denoted.
The spatial spectrum produced by Eq. (22) is a result of
integrating the sound field in the horizontal plane, but by
using dispersion relationships for the waveguide modes it is
possible to get Gðqx;qyÞdirectly from experimental data
measured by an L-shaped array.
Let us denote direction of the horizontal line array
(HLA) as the xaxis, which means that the set of observa-
tional data can be represented as the function Pðx;tÞ. As the
first step, we construct the space-time spectrum of the
received signal depending on the component qxand the fre-
quency x,
Gðqx;xÞ¼ððPðx;tÞexp iðqxxxtÞ½dxdt
:(23)
Let us next use the dispersion relation giving the con-
nection between the components of the vector q,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q2
xþq2
y
q¼qðxÞ;(24)
where qðxÞ, as a function of frequency, can be obtained from
the Sturm-Liouville eigenvalue problem for the vertical cross-
section at the receiver position. From Eq. (24), we can express
frequency as a function of the components ðqx;qyÞand after
substitution into Eq. (23), it is possible to get Gðqx;qyÞand in
turn to estimate all the aforementioned angles.
We next apply this methodology to data processing and
analysis of the SW06 results.
As an example, consider two episodes from the Shallow
Water 2006 experiment. The layout of the experiment is
shown in Fig. 3. NRL300 is the Naval Research Laboratory
source, fixed near the bottom. It radiated linear frequency
modulated (LFM) signals: the duration of each one was
2.048 s, the frequency band was 270330 Hz, and the time
period of radiation was 4.096 s. The receiver was an L-
shaped array: the horizontal part of it (the horizontal line
array, or HLA) was of length L¼465 m and contained 32
hydrophones spaced by 15 m. Its direction is the xaxis, as
seen in Fig. 3, with L=2xL=2. The vertical part of
the array (the vertical line array, or VLA) contained 16
hydrophones, and its horizontal coordinates are given by (L/
2, 0). The angles in the horizontal plane determining the ori-
entation of the HLA relative to the acoustic track are shown
in Fig. 3(a):b1¼25:75,b0¼26:04,b2¼26:34. The
depth of the sea along the acoustic track was 80 m.
A strong characteristic of the experimental region is the
existence of nonlinear internal waves (NIW) with amplitudes
525 m, propagating approximately toward the coast (perpen-
dicular to the acoustic track). These arise at the M2 tidal period
(roughly two times per day) and form the dominant anisotropy
of the internal wave (IW) field. In this paper, two transmission
time intervals from 19 of August 2006 are considered:
10:01–10:08 Greenwich mean time (GMT) (Fig. 4)and
08:30–08:38 GMT. Variations of the sound speed profile are
determined by three thermistor strings, denoted SW45, SW54,
and SW32, deployed approximately along the acoustic track.
Let us first consider the time period 10:01–10:08 GMT.
In Fig. 5(a), thermistor records for this period are shown,
with the instruments placed in the lower part of the thermo-
cline. As we can see, vertical oscillations of the thermocline
layer in area of the source and receiver are seen with ampli-
tude 1 m; in the middle part of acoustic track, their ampli-
tude is 3 m. In Fig. 3(b), the amplitude of one of the pulses
jPðx;tÞj received by the HLA is shown after matched filter-
ing. (“Matched filtering” means that the spectrum of the
received signals is multiplied by the complex conjugate
spectrum, which leads to the compression of the received
LFM pulse from 2 to 0.03 s.) As a result, the signal-to-
noise-ratio (SNR) is increased by about 70 times, and this
makes it possible to separate modes using arrival times. In
Fig. 3(b) we can see two dark lines corresponding to modes
3 and 4. The shape of the intensity distribution on the VLA
FIG. 3. (a) Scheme of the SW06 exper-
iment. (b) Amplitude of one pulse,
received by the HLA after matched
field compression, showing arrival
time dependence.
J. Acoust. Soc. Am. 143 (1), January 2018 Katsnelson et al. 197
can be used to determine the mode number: for mode 3,
there are 3 maxima, and for mode 4, there are 4 maxima.
We note that the maximal compression for the 60 Hz
band, which is 1=60 0:02 s, was 1.5 times less than that
observed in the experiment, probably due to different phase
shifts for different frequency components provided by intra-
modal dispersion, and due to different phase shifts along dif-
ferent horizontal rays corresponding to different frequencies.
FIG. 4. Data processing. (a) Space-time spectrum of mode 3; the amplitude is shown in Fig. 3(b). (b) Filtering of spectral maxima for frequencies in the band
290 65 Hz. (c) Calculation of the spatial spectrum in coordinates ðqx;qyÞusing the space-time spectrum.
FIG. 5. Episode 10:01–10:08 GMT. (a) Thermistor records (in degrees Centigrade) on SW45, SW32, and SW54. These were located at: the NRL300, in the
middle of acoustic track and at the receiver (VLA). (b) Temporal variations of the amplitude front and the phase front. The three dotted lines correspond to
angles b1,b2, and b0, giving the directions from the source to the ends and middle point of HLA [Fig. 3(a)]. Values of angles for amplitude and phase fronts
are denoted by numbers for the frequencies shown in the figure.
198 J. Acoust. Soc. Am. 143 (1), January 2018 Katsnelson et al.
Let us consider the following analysis for mode 3. In Fig.
4(a), the space-time spectrum Gðqx;xÞis shown, calculated
using Eq. (23) for mode 3 and for one of received pulses
Pðx;tÞ. The next step is to get the spatial spectrum, similar to
Fig. 1(c), using Eq. (24), and to find the direction of the vector
pin coordinates ðqx;qyÞusing a linear interpolation of the
points corresponding to maxjGðqx;qyÞj.InFig.4(b),themax-
ima of the spectrum Gðqx;xÞare shown, obtained for fre-
quencies in the band 290 65 Hz, which correspond to the
neighborhood of the main maximum at 290Hz. Using the val-
ues of xand qxfor each maximum, we find qyfrom Eq. (24),
where the dispersion law qðxÞand in turn xðqÞwere obtained
from the SturmLiouville problem using experimental data
about the waveguide parameters at the location of the
receiver. So, the positions of the maxima are found at the
coordinates ðqx;qyÞ, and also the straight line of maxima for
the spatial spectrum Gðqx;qyÞ[Fig. 4(c)]. Angles bam and
a¼b0bam, characterizing the angle of the amplitude front
with the HLA and the acoustic track (the angle between pand
the axes xand r), were found from the slope of the line of
maxima relative to the axis qx. Using the slope of the lines
from the origin to the points corresponding to frequencies
285, 290, and 295 Hz, we have found the angles bph and
u¼b0bph, which determine the angles between the phase
front and the HLA and the acoustic track (angle between q
and axes xand r) for the given frequencies [Fig. 4(c)]. A fre-
quency band of 10 Hz (290 65 Hz) was chosen for calcula-
tions to insure the condition of linearity of the variation of the
wave vector in this interval. Geometrically, this means that
the vectors q¼ðqx;qyÞin this frequency interval are placed
in a straight line.
For the time interval 10:01–10:08 GMT, 98 pulses were
processed. The results are shown in Fig. 5(b), where the tem-
poral dependence of the angles characterizing the amplitude
front and the phase front are presented. On the vertical axis
on the left side, the values of bph and bam are placed, and on
the right side uand a. These angles are shown for frequen-
cies 285 and 295 Hz.
Analysis of the angles obtained above leads to a good
correspondence with the theoretical estimates (Fig. 2): the
average value of the angle between the directions of the nor-
mal to the phase fronts for frequencies 285 and 295 Hz is seen
to be jDuj0:1in accordance with the estimate from Eq.
(19); the average modulus of the angle between the directions
of the amplitude front and the phase front for frequency
290 Hz is jDbjbph bam j2:9[Eq. (20)]; the average
value of angle between the direction of the normal to the
phase front and the acoustic track at frequency 290Hz is
juj0:63[Eq. (15)]; the average value of the angle between
the direction of the normal to the amplitude front and the
acoustic track is jaj2:7. We note that the fluctuations of
the steering angle in SW06 that were reported in the paper by
(Duda et al., 2012) for signals at 200 Hz, using the coherence
function of the horizontal line array in the presence of nonlin-
ear internal waves, were of the same order.
Next, let us consider the following. At 10:05 GMT, the
angle of the amplitude front aand two angles of the phase
front uat the frequencies 285 and 295Hz are equal to each
other. In accordance with the simplified theory, this can only
take place for straight-line trajectories of the horizontal eigen-
rays where bam ¼bph ¼b0¼26:04. In the experiment, at
10:05 GMT we have bam bph 26:8,thatisgreaterthan
b0almost by 1. This can be explained either by a systematic
error in the measurement of angles, or by a more complex
shape of the real IW than was supposed in the simplified the-
ory. In the latter case, in spite of real horizontal refraction, the
directions of the amplitude front and the phase front are the
same in the neighborhood of the receiver. The systematic
error can be produced by the deviations from a straight-line
shape of HLA, as was discussed in (Duda et al., 2012).
Next, we note that, in accordance with the theory pre-
sented above, if direction of horizontal gradient of the sound
speed profile is changed, then the signs of angles uand aare
changed also. This leads to the following inequalities for
angles bph1 and bph2, corresponding to 285 and 295 Hz, and
angle bam:bam <bph2 <bph1 if the gradient is directed
toward positive values of s, and bam >bph2 >bph1, in the
case of the opposite gradient direction.
In the experiment at time 10:05 GMT, a change of the
sign of the angles takes place that can be interpreted as a
change of the direction of gradient of the sound speed or a
change of the thermocline slope (e.g., the forward face or
back face of an internal wave).
Let us consider the variations of angles bam and bph
together with the thermistor records. It is seen from Fig. 5(a)
that, on average in the time interval 10:01–10:04 GMT, the
forward front of the moving IW is passing through the acous-
tic track (SW45 shows an approximately constant level of
thermocline displacement, whereas SW32 and SW54 show an
increase in displacement). In this case, we should have deflec-
tion of the eigenrays toward the area s>0, and so the
inequality bam <bph2 <bph1 takes place (Fig. 2). In the inter-
val 10:04–10:08 GMT, the character of the IW motion is
changed; specifically, the thermistor string SW32 shows
mainly a decrease of the thermocline’s displacement, whereas
SW45 and SW54 show an approximately constant level of
displacement. Here, it is probable that the back edge of the
IW is passing through the acoustic track, and the values of the
angles correspond to: bam >bph2 >bph1 [Fig. 5(b)].
Similar work was done for another time interval,
08:30–08:38 GMT, where 110 pulses were processed.
Thermistor records are shown in Fig. 6(a). The variations of
the angles determining the directions of the amplitude front
and the phase front are shown in the Fig. 6(b).
It is seen that the properties of the angle variability have
the same character as the first example. In particular, at
08:37 GMT, the directions of the amplitude front and the
phase front are the same. Average values of the angles in
this period are jDuj0:07,jDbj2:1,juj0:67,
jaj2:7, in accordance with theoretical estimations.
Next, let us examine the angles bph and bam for the first
and second periods for frequencies 290 and 305 Hz, taking
into account our confidence/error interval. In the first case:
bph ¼26:760:2,bam ¼2663; in the second case:
bph ¼26:760:1,bam ¼28:861:5. These confidence/
error intervals correspond to the mean square deviations. As
we can see, the mean square fluctuations of the amplitude
front’s angle are bigger than those of the phase front by
J. Acoust. Soc. Am. 143 (1), January 2018 Katsnelson et al. 199
approximately by 15 times. In other words, the phase front is
essentially more stable with respect to variations of the
waveguide parameters in the horizontal plane than the ampli-
tude front. This is a standard and expected result.
Finally, we should note that, in our theory, we neglect the
coupling between vertical modes of different number which is
determined by the ratio of matrix element Vmn Ðwm
ð@wn=@rÞdz to the difference jqmqnj. In our case this ratio
jVnm=jqmqnj103for the closest pair of modes: 3–2
and 3–4. Thus, neglecting mode coupling is reasonable.
V. CONCLUSIONS
Let us present our conclusions as a simple list.
(1) It has been shown that the frequency dependence of tra-
jectories of horizontal rays (or of the modal amplitude
distribution in horizontal plane) leads to a specific inter-
ference pattern in the horizontal plane in the area of
reception of the sound pulse. This pattern is interpreted
as a combination of two structures: phase fronts (lines of
constant phase) and amplitude fronts (lines of constant/
maximal amplitude or envelopes). These lines have
different directions and different spatial scales of vari-
ability. We note that a similar interference pattern takes
place in nonlinear and nano-optics for the propagation of
two frequency beams in a crystal (Bakunov et al., 2012).
(2) On the basis of analytical estimations, it is shown that in a
typical situation, if horizontal refraction is due to internal
waves of small amplitude (1 m), then the angle of revo-
lution of the phase front (direction of the horizontal ray)
is about u12for an acoustical track of typical
length 20 km. The angle Dubetween horizontal rays, cor-
responding to different frequencies, depends on the differ-
ence between the frequencies. Using the estimate
Du=Dx0:01 deg/Hz, we get for a narrow-band pulse
with bandwidth 10 Hz that this angle is approximately
Du0:1We note that our estimates should be modified
in the presence of nonlinear internal waves (Duda et al.,
2012) where experimental measurement of these angles is
more difficult.
(3) The angle bdetermining the direction of the normal to
the amplitude front can be essentially greater than the
angle determining the normal to the phase front. It
implies a more remarkable manifestation of horizontal
FIG. 6. Results of data processing for the period 08:30–08:38 GMT.
200 J. Acoust. Soc. Am. 143 (1), January 2018 Katsnelson et al.
refraction than one would suppose using the trajectory of
the horizontal rays. The variation of the direction of the
amplitude front (or its sensitivity in the direction to
waveguide parameter variations) is essentially higher
than that for the phase front.
ACKNOWLEDGMENT
The authors are grateful to Professor O. Godin for
helpful discussions. This work was supported by the Israel
Science Foundation, Grant No. 565/15, and the Ministry of
Education and Sciences of the Russian Federation, Grant
No. 14.Z50.31.0037.
Badiey, M., Katsnelson, B. G., Lynch, J. F., Pereselkov, S., and Siegman,
W. L. (2005). “Measurement and modeling of three-dimensional sound
intensity variations due to shallow-water internal waves,” J. Acoust. Soc.
Am. 117(2), 613–625.
Bakunov, M. I., Tsarev, M. V., and Mashkovich, E. A. (2012). “Terahertz
difference-frequency generation by tilted amplitude front excitation,” Opt.
Exp. 20(27), 28573–28585.
Ballard, M., Lin, Y.-T., and Lynch J. (2012). “Horizontal refraction of prop-
agating sound due to seafloor scours over a range-dependent layered bot-
tom on the New Jersey shelf,” J. Acoust. Soc. Am. 131(4), 2587–2598.
Bender, C. M., Ballard, M. S., and Wilson, P. S. (2014). “The effects of
environmental variability and spatial sampling on the three-dimensional
inversion problem,” J. Acoust. Soc. Am. 135(6), 3295–3304.
Born, M., and Wolf, E. (1968). Principles of Optics (Pergamon, Oxford).
Collins, M. D., McDonald, B. E., Heaney, K. D., and Kuperman W. A.
(1995). “Three-dimensional effects in global acoustics,” J. Acoust. Soc.
Am. 97, 1567–1575.
Colosi, J. (2016). Sound Propagation through Stochastic Ocean (Cambridge
University Press, London), 424 pp.
Deane, G. B., and Buckingham, M. J. (1993). “An analysis of the three-
dimensional sound field in a penetrable wedge with a stratified fluid or
elastic basement,” J. Acoust. Soc. Am. 93, 1319–1328.
Dong, H., Badiey, M., and Chapman, R. (2015). “Matched mode geoacous-
tic inversion of broadband signals in shallow water,” J. Acoust. Soc. Am.
137, 2390.
Doolittle, R., Tolstoy, A., and Buckingham, M. J. (1988). “Experimental
confirmation of horizontal refraction from a point source in a wedge-
shaped ocean,” J. Acoust. Soc. Am. 83(6), 2117–2125.
Duda, T., Collis, J., Lin, Y-T., Newhall, A., Lynch, J., and DeFerrari, H.
(2012). “Horizontal coherence of low-frequency fixed-path sound in a con-
tinental shelf region with internal-wave activity,” J. Acoust. Soc. Am.
131(2), 1782–1797.
Duda, T. F., Lin, Y.-T., Zhang, W. G., Cornuelle, B. D., Lermusiaux, P. F. J.
(2011). “Computational studies of 3D Ocean sound fields in areas of com-
plex seafloor topography and active ocean dynamics,” edited by C.-F.
Chen et al.,inProceedings of the 10th International Conference on
Theoretical and Computational Acoustics, NTU, Taiwan, 12 pp.
Flatte, S. M., ed. (1979). Sound Transmission Through a Fluctuating Ocean
(Cambridge University Press, London), 299 pp.
Heaney, K. D., and Murray, J. J. (2009). “Measurements of three-
dimensional propagation in a continental shelf environment,” J. Acoust.
Soc. Am. 125(3), 1394–1402.
Katsnelson, B., Malykhin, A., and Tckhoidze, A. (2013). “Propagation of
wideband signals in shallow water in the presence of meso-scale horizon-
tal stratification,” Acoust. Australia 41(1), 45–83.
Katsnelson, B., and Pereselkov, S. (2000). “Low-frequency horizontal
acoustic refraction caused by internal wave solitons in a shallow sea,”
Acoust. Phys. 46(6), 684–691.
Katsnelson, B., and Pereselkov, S. (2004). “Space–frequency dependence of
the horizontal structure of a sound field in the presence of intense internal
waves,” Acoust. Phys. 50(2), 169–176 (2000).
Katsnelson, B., Petnikov, V., and Lynch, J. (2012). Fundamentals of
Shallow Water Acoustics (Springer, New York), 540 pp.
Katsnel’son, B. G., Badiey, M., and Lynch, J. (2007). “Horizontal refraction
of sound in a shallow water and its experimental observations,” Acoust.
Phys. 53(3), 313–325.
Lin Y.-T. (2013). “A higher-order tangent linear parabolic-equation solution
of three-dimensional sound propagation,” J. Acoust. Soc. Am. 134(2),
EL251–EL257.
Newhall, A. E., Duda, T. F., Keith von der Heydt, Irish J. D., Kemp J. N.,
Lerner S. A., Liberatore S. P., Ying-Tsong Lin, Lynch J. F., Maffei A. R.,
Morozov, A. K., Shmelev, A. A., Sellers, C. J., and Witzell, W. E. (2007).
“Acoustic and oceanographic observations and configuration information
for the WHOI moorings from the SW06 experiment,” Woods Hole
Oceanog. Inst. Tech. Report.
Sabinin, K. D., and Serebryani, A. N. (2005). “Intense short-period internal
waves in the ocean,” J. Mar. Res. 63(1), 227–261.
Weinberg, H., and Burridge, R. (1974). “Horizontal ray theory for ocean
acoustics,” J. Acoust. Soc. Am. 55(1), 63–79.
J. Acoust. Soc. Am. 143 (1), January 2018 Katsnelson et al. 201
Thesis
Full-text available
Modelling of sound propagation and analyzing of sound fields are important parts in underwater acoustics. With the frequency band of sonar being extended towards low frequency, it is necessary to consider bottom elasticity in sound propagation modelling, because low-frequency sound waves can penetrate deep into the elastic bottom, and then transmit back to the water column after reflection. Meanwhile, with the operating distance of sonar growing, it is important to take account of the horizontal refraction effects, which are due to azimuthal variation of the environment with respect to the source. In this thesis, modelling of sound propagation in wedge-like oceans and analyzing of horizontal refraction effects on vector fields are studied. First, a parabolic-equation (PE) model applicable to wedge-like oceans with elastic bottom is established. Second, with an attempt to verify the PE model, the source images method, which is a benchmark model for the problem of sound propagation in wedge-like oceans, is modified, and the acoustic fields calculated by the source images method and the PE model are compared. Third, the influences of horizontal refraction on vector-field characteristics are analyzed. More details for the studies are listed as following. (1) ⭐Establishment of the PE model for wedge-like oceans with elastic bottom// A parabolic-equation (PE) model for sound propagation in wedge-like oceans with elastic bottom is established. First, the elastic parabolic equations (PEs) are derived from the elastic motion equations. Second, the mapping approach is used to handle the sloping fluid-elastic interface. Third, by using Fourier transform, the three-dimensional (3D) PEs are simplified to two-dimensional (2D) PEs, which are much easier to solve. Finally, the 3D sound fields are obtained through inverse Fourier transform. The accuracy of the model is tested in a horizontal waveguide and a wedge-like waveguide, and azimuthal limitation of the model is analyzed. According to the theory of Padé approximation, the limitation azimuthal angle can reach between 70 and 80 degrees when the order of Padé approximation is 8, and the limitation azimuthal angle increases with the decreasing of the marching step. An analysis of the sound fields in wedges and in transitional areas between continental slope and continental shelf reveals that, the larger the azimuth in the considered waveguides, the stronger the horizontal refraction effects. In addition, the sound fields in the transitional area indicate that, the horizontal refraction effects in the sloping region affect the sound fields in farther region, though in the farther region the bottom is horizontal. At last, the sound fields calculated by the PE model are compared to the results measured in a tank experiment, where a plastic plate is placed at the tank bottom to play the role of an elastic sea bottom. It is shown that in the case of horizontal bottom, the simulated results and the measured results agree well with each other; while in the case of a sloping bottom, the simulated results and the measured results deviate with each other, and further study is required to know the exact reason. (2)⭐ Study on the problem of branch selection for the reflection coefficients in source images method// The source images method is regarded as a benchmark model for 3D sound fields calculation in wedge-like oceans. An important issue in the source images method is how to choose the correct branch of the reflection coefficients. In principle, the criterion for branch selection is the infinite radiation condition. Nevertheless, for portions of the plane-wave components emitted from the source, the correct branch of the reflection coefficient is not obvious when one tries to judge it according to the infinite radiation condition. P. S. Petrov published a code for the source images method, in which a detailed branch selection rule is given, but a strict proof for this branch selection rule is not found in published literature. In this thesis, we give a systematic discussion of the branch selection problem. In the case of lossless bottom, by combining the infinite radiation condition with a series of numerical examples, a branch selection rule of the reflection coefficients can be concluded: the normal component of the refracted wavenumber should be in the same quadrant with the normal component of the incident wavenumber. The physical meaning of the above branch selection rule is that the propagation direction and the decaying direction of the refracted wave normal component should be the same with those of the incident wave normal component. The branch selection rule in Petrov’s code elects the same branches as the above branch selection rule when the bottom is lossless, thus Petrov’s code is applicable to the lossless-bottom case. In the case of lossy bottom, it is found that for portions of the plane-wave components, the branch of reflection coefficient can not be judged by the branch selection rule above. Nevertheless, if choosing the branch with the rule of keeping the propagation direction of the refracted wave normal component the same with that of the incident wave normal component, the corresponding acoustic fields produced by the source images method are still with high accuracy. The 3D sound fields in the ASA wedge with lossy bottom is calculated by the source images method with the above branch choice rule, and the results are compared to the results produced by other 3D sound propagation models as well as the results produced by Petrov’s code. The comparisons reveal a high accuracy of the source images method based on the branch selection rule in this thesis. Meanwhile, it is shown that the results based on the branch selection rule in this thesis are smoother, i.e., with less oscillation, than those calculated by Petrov’s code. Finally, the solutions of sound fields in a series of wedge-like oceans, which differs only in bottom shear-wave speed, are calculated, and these results can be used to verify the accuracy of other 3D sound propagation models. (3)⭐ Analysis of the effects of horizontal refraction on acoustic vector-field characteristics// The effects of horizontal refraction on acoustic vector-field characteristics are studied. Based on the basic difference between a 3D sound propagation problem and a cylindrically symmetric 2D sound propagation problem, i.e., whether the environmental parameters vary along the azimuth direction with respect to the source, two effects of horizontal refraction on acoustic vector-field characteristics are deduced, i.e., the elliptical polarization of the horizontal particle displacement and the horizontal deflection of the acoustic energy flux. First, the measured data for particle displacements in a sea trial is analyzed. The results show the horizontal displacement at the receiver point was always elliptically polarized regardless of the change of source position, implying in realistic oceans the propagation of sound waves is usually accompanied with the effects of horizontal refraction due to the horizontal inhomogeneity of the media and the horizontal variation of the bathymetry. Second, the horizontal deflection angle of the acoustic energy flux, which is equal to the deviation between the direction of arrival (DOA) measured by a single vector hydrophone and the true bearing of the source, in coastal wedges with a 2.86º sloping bottom are simulated using the source images method. The results show that the horizontal deflection angle of acoustic energy flux can be larger than 10 degrees in some regions in the considered wedges, indicating that horizontal refraction of sound waves can lead to non-negligible error in bearing estimation. Keywords: elastic sea bottom; wedge-like ocean; sound propagation; horizontal refraction; acoustic vector fields
Article
Full-text available
This paper discusses some effects of the interaction between internal waves and low frequency currents. Experimental data that have been obtained in various regions of the world ocean are presented. The following features are evident in the field measurements: the frequency of the predominant short-period internal waves exceeds the frequency of the sharp narrowing of the waveguide in the осеап: waves with а propagation component in the opposite advection tо the current dominate; and the mоге the deviation is from the counter direction, the smaller become the wave amplitudes and wavelengths.
Book
The ocean is opaque to electromagnetic radiation and transparent to low frequency sound, so acoustical methodologies are an important tool for sensing the undersea world. Stochastic sound-speed fluctuations in the ocean, such as those caused by internal waves, result in a progressive randomisation of acoustic signals as they traverse the ocean environment. This signal randomisation imposes a limit to the effectiveness of ocean acoustic remote sensing, navigation and communication. Sound Propagation through the Stochastic Ocean provides a comprehensive treatment of developments in the field of statistical ocean acoustics over the last 35 years. This will be of fundamental interest to oceanographers, marine biologists, geophysicists, engineers, applied mathematicians, and physicists. Key discoveries in topics such as internal waves, ray chaos, Feynman path integrals, and mode transport theory are addressed with illustrations from ocean observations. The topics are presented at an approachable level for advanced students and seasoned researchers alike. Offers a comprehensive treatment of wave propagation theory, observations, and relevant ocean dynamics, particularly internal waves Presents simple conceptual models as well as state of the art models, making it relevant to a broad range of readers from students to researchers Provides an exposition of acoustic fluctuations in a broad variety of ocean environments showing key differences in applying acoustic fluctuation theories.
Chapter
In this chapter, we will consider some simple models of the shallow water waveguide. Such models allow us to obtain and understand the main features of SW sound propagation quickly. Such simple models can also be perturbed to take into account more realistic properties of the environment, thus giving them far more power than one might think at first.
Article
A matched mode geoacoustic inversion is presented for broadband acoustic signals recorded in experiments at a shallow water region of the New England Bight. First the modal behavior of the waveguide in the presence of 3D effects due to the water column and bottom bathymetry is examined. Spatial measurements of the temperature and salinity profiles and bathymetry provide data for calculating 3D propagation of the broadband acoustic wave field. Frequency shift of the broadband signal along one of the source-receiver paths in the experiment is attributed to internal waves propagating in the water column. Geoacoustic model parameters of the seabed along a source-receiver track with constant bathymetry are estimated by matched mode processing through matching the phase of the mode signals recorded at a Vertical Line Array. A warping transform was applied to identify and isolate modes, and the filtered modes were inverse transformed back to the time domain to reconstruct the mode signals. The inversion results are compared with the results from other inversion methods to assess the performance of the inversion scheme for estimating reliable values of the geoacoustic parameters in the experiment region.
Article
The overall goal of this work is to quantify the effects of environmental variability and spatial sampling on the accuracy and uncertainty of estimates of the three-dimensional ocean sound-speed field. In this work, ocean sound speed estimates are obtained with acoustic data measured by a sparse autonomous observing system using a perturbative inversion scheme [Rajan, Lynch, and Frisk, J. Acoust. Soc. Am. 82, 998-1017 (1987)]. The vertical and horizontal resolution of the solution depends on the bandwidth of acoustic data and on the quantity of sources and receivers, respectively. Thus, for a simple, range-independent ocean sound speed profile, a single source-receiver pair is sufficient to estimate the water-column sound-speed field. On the other hand, an environment with significant variability may not be fully characterized by a large number of sources and receivers, resulting in uncertainty in the solution. This work explores the interrelated effects of environmental variability and spatial sampling on the accuracy and uncertainty of the inversion solution though a set of case studies. Synthetic data representative of the ocean variability on the New Jersey shelf are used.
Article
Low‐frequency ocean acoustics problems are solved in three dimensions over the entire globe. Adiabatic mode solutions are obtained at 1 Hz using the parabolic equation method by marching through both the oceans and the continents from the location of the source to the antipode. The examples indicate that azimuthal coupling can be important for global‐scale problems. One of the examples illustrates the broadening of a shadow behind the Hawaiian Islands by horizontal refraction. The other examples involve sources at the locations of the sources used in the Perth–Bermuda and Heard Island experiments.
Article
Horizontal stratification in shelf zone of the ocean is provided by existence of coastal wedge, temperature fronts, nonlinear internal waves, slopes, and canyons, where typical scales are up to tenths of kilometer in range and up to tenths minutes in time, for some perturbations spatial scales are essentially different in different directions in horizontal plane. In this case, there is remarkable horizontal refraction in sound propagation and frequency dependence of horizontal ray trajectories. It means that Fourier components of wideband signal propagate along different paths joining source and receiver in the horizontal plane. Distribution of spectral components in horizontal plane has crescent-like shape and restricted by rays, corresponding to boundary frequencies in spectrum. Propagating wide-band signal has additional spectral distortion as a result of different phase shift for spectral components, propagating along different paths. Also there is difference in directions of wave vectors for difference spectral components (tangents to horizontal rays), leading to phenomena similar to spatial dispersion: different directions of phase and group velocities, compression and decompression of pulses, additional time delay of signal, etc. Mentioned phenomena are considered for models of coastal wedge and temperature fronts. Analytical estimations are presented, as well as results of numerical modeling.
Article
A higher-order square-root operator splitting algorithm is employed to derive a tangent linear solution for the three-dimensional parabolic wave equation due to small variations of the sound speed in the medium. The solution shown in this paper unifies other solutions obtained from less accurate approximations. Examples of three-dimensional acoustic ducts are presented to demonstrate the accuracy of the solution. Future work on the applications of associated adjoint models for acoustic inversions is proposed and discussed.