PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Estimation of bottom parameters using measurement and analysis
of sound intensity fluctuations in the presence of moving nonlinear
internal waves in shallow water
Boris KATSNELSON1; Valery GRIGOREV2; Yanyu JIANG1, Yun REN3
1

University of Haifa, Israel

2

Voronezh University, Russia

3

Institute of Acoustics, China

ABSTRACT
Methodology of estimation of bottom parameters is proposed using data processing and analysis of soundintensity fluctuations due to mode coupling in the presence of moving nonlinear internal waves (NIW) inshallow water channel. As an examples experiments Shallow Water 2006 (Atlantic shelf of USA) [1] andASIAEX 2001 (South China sea) [2] are considered. In these experiments NIW generated in area of the
shelf break were moving approximately along acoustic track with the length about a few tens of km. Spectraof intensity fluctuations of the sound intensity of pulses during a few hours on vertical line array (VLA) due
to mode coupling in both experiments have specific peaks corresponding to the difference between wave
numbers of coupling pairs of modes. Analysis of fluctuation spectra includes both temporal dependence offluctuations on separate hydrophones and temporal dependence of intensity summarized over VLA. It isshown that geoacoustic inversion in given case can be achieved using comparison of experimental spectral
and temporal dependencies and results of theoretical modeling, containing bottom characteristics (soundspeed density and attenuation) as a fitting parameters.
Keywords: Shallow water sound propagation, Nonlinear internal waves

1. INTRODUCTION
Nonlinear internal waves (NIWs) exist in many different areas of the ocean, often in the form of
trains consisting of up to 10-12 separate “solitons” of large amplitude. They are found particular in thecontinental shelf zone (shallow water), where they play an important role in various oceanographic,
biological and chemical processes. The influence of NIWs on sound propagation makes the study of theinteraction of acoustical signals and NIWs seems rather interesting, and can allow us to consider acousticalmonitoring of NIWs. A useful manifestation of the influence of NIWs on sound propagation are thefluctuations of the sound field observed at the receiving system. We will consider array receivers here, forexample vertical and/or horizontal line arrays (VLAs and/or HLAs). The properties of the fluctuations
depend on the angle α between the direction of the sound propagation and the direction of the NIW front [1].
In particular, if the angle α is not very small (roughly >10 degrees), then the mechanism of the fluctuations ismode coupling.
We would note that fluctuations of the sound amplitude and intensity caused by moving NIWs, asmeasured at a fixed depth, takes place even in the absence of attenuation. The spectral properties of such
fluctuations were studied in [3,4], both in the ray approximation and using modal analysis. In these papers, it
was assumed that (see Fig.1): 1) the NIWs are entirely located between the source and receiver, which
excludes the adiabatic mechanism of fluctuations taking place due to variations of the sound speed profile atthe source and/or receiver and 2) the angle  between the wave front of the NIW and the acoustic track is
not too small, so that we can to first order neglect horizontal refraction.
In this scenario, mode coupling (using the modal picture for low frequency sound) or scattering ofrays (using the ray picture for high frequency sound) are main mechanisms of the fluctuations. Both theories
give a description and explanation of the temporal fluctuations of the sound signals at the receiver. The
existence of a so called “predominant frequency” providing a maximum amplitude in the spectrum is themain feature of interest.
1

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bkatsnels@univ.haifa.ac.il

In the present work, modal theory is applied for the description of the propagation of 300±30 Hz
signals in the presence of NIWs in the SW06 experiment, and in ASIAEX experiment, where we haverather detailed information about the oceanographic situation, including NIWs. This allows us to carry out
detailed analysis and theoretical modeling adequate to the experiment, and comparison of the observational
data with theory.

Fig.1

2. THEORY OF INTENSITY FLUCTUATIONS
Let’s consider a waveguide with a water column of constant depth H and an unperturbed soundspeed profile c0 z  , corresponding to the averaged experimental data, with density   1 g/cm3.
Our bottom model is a liquid homogeneous half-space with sound speed cb  1570 m/s anddensity  b  1.8 g/cm3. The attenuation coefficient is unspecified so far. As we will see, one of main
results of this paper will be an estimation of the sediment induced bottom loss using a comparison of theexperimental data with the results of our modeling. This “optimal estimate” attenuation coefficient will beseen to be    0.3 dB/. This value is something like an intermediate value in the interval 0.1–1 dB/,where minimal value was used in [Ballard et al, 2012] for the same region, and the maximal value iscalculated using the empirical formula of Hamilton [Hamilton, 1972], obtained with the same value ofporosity, 0.49. Other bottom parameters correspond to data from the authors of [Yardim et al, 2012; Ballard
et al, 2012].
Let’s suppose that NIW train of the length L moves along r-axis at the speed v. A moving NIW
train leads to the perturbation of the sound speed profile c0 z  . For a description of the perturbed soundspeed profile, the following expression is used: c z , x  0, x  L,
c~x, z    0
c r1  x, z , T1 , 0  x  L(1)where x  r  rT , rT  r1  v r T  T1  , T is time, and v r is the projection of the NIW velocity on ther axis during the time of observation T . We can remark that, if T  T1 , r  r1 then x  0 . If during
the interval T the motion of the NIW takes place toward the source at velocity v , then v r  v .
Note that in Eq. (1) the NIW train is taken to be limited in the horizontal direction to L  13
km (seven separate ISWs); it is moving with the same shape and velocity v toward the source. At an
arbitrary moment of time T , the coordinates of the starting and final points of the NIW are: r  rT andr  rT  L . Out of the NIW area, the sound speed profile is unperturbed. i.e. c0 z  .
Let s be the spectrum of radiated signal. In the frozen time approximation, neglecting
horizontal refraction (argumentation of this statement is presented in the sec. VI), the spectrum of the signalP r , z , T  at an arbitrary point r, z  can be found from the Helmholtz equation and the corresponding
boundary conditions:5409

1      2
 r 2,
 r   2  k  P (r , z , T )   s  z  z0 2r
 r r  r  z

1 P
1 P
P z 0  0, P z  H   P z  H  ,, z z  H  b z z  H 



(2)



k02 n 2   , 0  z  H

c
2
2
2
where k  , k0  , n ,   x, z   c c~  c0  ,2
c 0 z ckb , z  H2k b  cb1 1  i  54.6  ,  is Dirac delta-function, and z 0  z1 H H 1 is the effective depth of the
NRL300 source. If z in Eq. (2) is the depth of hydrophone at the distance r  R , then z should be
calculated as z  z 2 H H 2 .
In area 0  r  rT there are no NIWs and   0 . The solution to Eq. (2) in this case does notdepend on T and has the form:P r , z   
m

C m0
qm r

 m z expi m r  ,(3)0

where C m are the modal amplitudes at the source

C m0 

expi 4 s m z 0  ,8

(4)

 m z  and  m  qm  i m 2 are the eigenfunctions and complex eigenvalues (due to attenuation) of the
Sturm-Liouville problem:

 d2
2 2
2
 2  k0 n   m  m z   0,
 dz

H
d m
g  2 H  m 0  0,  m H   g m
 0,  m2 z dz  m m 2  1,
dz z  H
2m1
0
where g m  im1(5)

k b2   2m , and m1   b  .

In the region of the NIW rT  r  rT  L , which corresponds to 0  x  L . Taking into
account the smallness of  , the solution of Eq. (2) has a form similar to Eq. (3):C m r , T  m z expiqm r  .
(6)
qm r
The modal amplitudes Cm r , T  depend on r, T in the exp-function and instead of  m there is now qm .
P r , z , T   mAfter the substitution of Eq. (12) into Eq. (2), and performing standard evaluations, specifically: using the22
forward scattering approximation by neglecting by d C m dr , ignoring all terms of the second order ofsmallness, and using Eq. (11) and the orthogonality of the eigenfunctions, we get the usual system of firstorder differential equations describing the mode coupling:dC m  m C m  i Vmn C n exp iqmn r  ,
dr
2
n

(7)

which, after the replacement of variable r by x becomesdBm  m Bm  i Vmn Bn exp iqmn x  ,dx
2nwhere Bm x, T   C m expiqm rT  , qmn  qm  qn , and

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(8)

H

k02
Vmn ( x, T )  x, z  m z  n z dz .
2 qm qn 0
Remark, that analysis of different mode coupling equations was done, for example, in [Godin,1998].
A rough estimation for the mode couping coefficient for experimental conditions is

Vmn ( x, T ) ~ k0  / 2 ~ 5 104  105 m-1, much less than the the minimal qmn ~ 5 103 m-1.
The boundary condition for equation system (8) on the left boundary at r  rT has the form
C m rT , T   C m0 exp  m rT 2  . Correspondingly at x  0 for the equation system (8) it is:

Bm 0, T   C m0 expi m rT  .

(9)The “initial” condition” for the moving perturbation includes a phase shift depending on the time T.
Let’s find a solution to Eq. (14), satisfying boundary condition in Eq. (9) using the matrix form:dB
 WB ,dx(10)

where B is column Bm , W is matrix with elements Wmn    m  mn 2  iVmn exp ixqmn  , and mn is the Kronecker delta . Let Sx  be fundamental matrix of the system (16):dS
 WS, S0  I ,
dx

(11)

where I is the unit matrix. Then the solution to Eq. (10) for x  L is:

BL, T   SLB0, T  .

(12)

The corresponding equation for modal amplitudes C m has the form:

C m rT  L, T    S mn LC n0 exp  n rT 2 exp iqmn rT  .

(13)

n

In the region r  rT  L , up to the VLA receiving array at r  R , we have   0 and(14)
Cm r , T   Cm rT  L, T  exp  m r  rT  L  2 .
After the substitution of Eq. (14) into Eq. (12) at r  R , and denoting P z , T   P R, z , T  ,we get the sound field at the VLAP z , T    Pmn exp iqmn v r T  ,

(15)

m,n

 m z 

expi m R  i mn v r T2   m L 2   mn v r T  ,
(16)
qm R
where  mn   m   n ,  mn   m   n  2 , and T2  T1  r1 v r  06:26:52 is the moment of time
when the moving point x  0 coincides with the position of the source. We remark that in absence of
NIWs, S mn L   mn exp  m L 2  and the solution of Eq. (16) at r  R turns into the unperturbed
solution Eq. (9). We also note that Eq. (21) is expressed through v r (the projection of the NIW velocity
Pmn  С n0 S mn Lon the r axis), which is comfortable for the analysis of experiments when the NIW moves toward the source(in this case v r  v ) or toward the receiver (here v r  v ).
Let’s denote  mn  vqmn and take into account that in our experiment v r  v . Then, in
accordance with Eq. (4), the intensity of the sound field isI  z , T   2c 

1

 P P expi 
*
mn kl

mn

  kl T  .

(17)m , n , k ,lExpression (17) allows us to calculate the fluctuations of intensity due to NIW motion and comparethem with the experimental data. As a final step, we take into account the bottom loss in the model of the5411waveguide.

3. APPLICATION OF THEORY TO ASIAEX AND SW06 EXPERIMENTS
In the ASIAEX experiment we have the following situation (Fig.2) . NIW with the length L moves
along acoustic track with the velocity v , where the left end of NIW at the time moment T (measured in
UTC scale) is determined by the coordinate rT  r0  vT , (we consider temporal interval
T0  8 : 30 UTC  T  T1  11 : 30 UTC ) , the corresponding positons of NIW are shown in the Fig.2,
and r0  3 km.

Fig. 2 Schematic of the acoustic propagation path (S1-R) and three positions of NIW. Sound speed profiles
for two areas are shown in the right panel.

Figure 3. Time period 8:30-11:30 on May 7. a)-sound intensity as a function of time and depth;

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b) –

intensity for the depth 53.25 m; c) - spectrum of sound intensity for different depth; d) – spectrum for thedepth of 53.25. All plots are normalized by maximal value equal to1
In the Fig.3 results of calculation of spectra (17) for experiment ASIAEX. Values of numbers of interacting
modes are shown in the Fig.3.
So in temporal dependence of the depth distribution of intensity we clearly see periodical character offluctuations. (Figs.3a).In the corresponding plot of spectrum at fixed depth the biggest peak (predominating
frequency) has the value 1.3 cph which is close to theoretical values provided by coupling of two different
pairs of modes (2,3) and 3,4) having rather close values of the beating scales  , and probably two
different peaks cannot be resolved. Some displacement of theoretical values for frequencies can be
explained by some error in speed of NIW or in waveguide parameters. For other peaks, also there are
candidates of mode numbers providing the corresponding frequencies. Very low frequency fluctuations(less than 0.5 cph, provided by tide) are excluded.
We can estimate predominating frequency of fluctuations using ray approximation for given waveguede
using idea of predominating frequency of fluctuations. Let’s take linear approximation for the sound5
speed profile: c( z )  c0 (1  z ) where for our case (depth H  250 m) c0  1535 m/s,   8 10-1m . Cycle of ray, having maximal length, corresponding to modes with minimal attenuation coefficients(numbers 2,3,4) can be calculated using expression D ~ 2 tan  /  , where  is grazing angle,
corresponding to ray, touching the surface (having maximal length of the ray cycle). After simple
evaluations, we have D ~ 2 2 H /  ~ 5 km. The corresponding predominating frequency is about ~ v / D ~ 1.25 cph, in a good correspondence with the modal theory and experiment
Let’s consider the Shallow Water 2006 experiment. Here, the Naval Research Laboratory“NRL300” is a source of linear frequency modulated (LFM) signals with a duration 2.048 sec each, afrequency band of 300±30 Hz, and a period of repetition of 4.096 sec. Pulses were radiated in sequences of
110 separate pulses, and the duration of each series is 7.5 min every half an hour. For our present analysis,
the time interval of radiation 06:00:00–06:07:30 GMT from August 14 2006 is taken. The depth of thesource is z1=72 m, and the sea depth is H1=82.5 m. VLA denotes the vertical line array, as previously noted.
We analyzed the signal receptions for nine hydrophones within the depth range z2=17.25–77.25 m, with aspacing distance between them of 7.5 m. The depth of the sea at the array site is H2=79 m. The distance
between the NRL300 source and the VLA is R=20181 m.

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Fig.4 Spectrum of intensity fluctuations and numbers of interacting modes.

Fig.5. Experimental and model dependencies of relative total intensity fluctuations J T  for parameters
denoted in the Figures. (a) – without loss, (b) with bottom attenuation. v  3.2 m/s.

Spectra of fluctuations both using experimental data and theoretical modeling are shown in the Fig.4. In thelower panel mode numbers, providing mode coupling and in turn sound intensity fluctuations are shown as
well.
In order to explain the observed fluctuations of the total intensity (Fig.5) , let’s introduce bottom loss. As iswell known, the existence of attenuation in the sediment leads to modal attenuation, i.e. depending explicitly
on mode number. Moreover, the attenuation coefficient usually increases with mode number, at least
beginning with some number.
The value    0.3 dB/ corresponds to the best agreement between the measured and calculated
amplitudes of the fluctuations, and also in the positions of the main peaks (~15 and ~25 cph) in theexperimental and calculated spectra.

CONCLUSION
Significant fluctuations take place in the presence of NIWs inside the acoustic track, and rather small
fluctuations are seen in the absence of NIW
The theory of ode coupling by moving NIWs explains the properties of fluctuations measured by ahydrophone at a fixed depth (i.e. the positions of the peaks in the spectrum and their relative amplitudes).
Mode coupling in the presence of bottom loss can lead to fluctuations of total intensity, due to thefluctuations of modal amplitudes having different attenuation coefficients. The positions of the peaks in the5414

spectrum of fluctuations of total intensity are the same as for the spectrum of fluctuations at fixed depth, but
the amplitudes are different, depending on the attenuation coefficient. Comparison of the calculated spectrumof total intensity with experimental data can provide an estimate of bottom attenuation.

ACKNOWLEDGEMENTS
Work was supported by ISF fund. Grant 565/15

REFERENCES
1. Badiey M., Katsnelson B., Lynch J., Pereselkov S. Frequency dependence and intensity fluctuations
due to shallow water internal waves. J. Acoust. Soc. Am V.122(2) p.747-760, (2007)
2. J.F. Lynch, S.R. Ramp, C.-S. Chiu, T.Y. Tang, Y.J. Yang, and J.A. Simmen, “Research Highlightsfrom the Asian Seas International Acoustics Experiment in the South China Sea,” IEEE J. OceanicEng. 29, 1067-1074 (2004).
3. B.Katsnelson, V.Grigorev, J.Lynch. Intensity fluctuations of mid-frequency sound signals, passing
through moving nonlinear internal waves. JASA EL, 2008, vol.124, N3, EL78-84
4. V. A. Grigorev, B. G. Katsnelson, and J. F. Lynch. Energy fluctuations of high frequency soundsignals in a shallow water in the presence of nonlinear internal waves. Acoustical physics, 2013,
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