EFFECTS OF SOUND SPEED FLUCTUATIONS
DUE TO INTERNAL WAVES IN SHALLOW WATER
ON HORIZONTAL WAVENUMBER ESTIMATION
KYLE M. BECKER
Acoustics Group, The Pennsylvania State University, Applied Research Laboratory
P.O. Box 30, State College PA 16803, USA
E-mail: [email protected]
GEORGE V. FRISK
Dept. of Applied Ocean Physics and Engineering, Woods Hole Oceanographic Institution
MS #11, Woods Hole MA 02543, USA
E-mail: [email protected]
There is considerable current interest in the influence of water-column variability on
acoustic propagation and its effects on geoacoustic inversion. In general, the effect of
sound speed fluctuations due to internal waves in the water column is to promote the
coupling of energy between propagating acoustic modes. The effects of mode coupling
include fluctuations in individual modal amplitudes and arrival times along with time
spreading of the original pulses. In contrast to the broadband case, little research has
been conducted on the effects of internal waves on cw modal-based inversion methods. These techniques require estimates of the propagating modal eigenvalues for a
cw point source field as input data to the inversion algorithm. In much of the literature, wavenumber estimation is performed with the assumption that pressure is given
by an adiabatic mode sum. Changes in modal content as a function of range are then
attributed to local changes in the waveguide boundaries, specifically, the bottom. For a
shallow-water waveguide including internal waves, the adiabatic assumption is violated
and estimates of local wavenumber content is affected. This paper addresses the nature
of the these affects on the wavenumber estimation problem. In particular, numerical
studies of internal wave effects are conducted with respect to identification and bias
of individual modes along with the ability to resolve closely spaced eigenvalues. Preliminary results for a weak internal wave field show that mode coupling leads to an
enhancement of the wavenumber spectral estimates due to the energizing of weak modes
that were previously not excited.
1 Introduction
Sound propagation in range-independent shallow-water waveguides can be concisely represented by the exact Hankel transform relationship between complex pressure, measured
as a function of range, and the depth-dependent Green’s function, expressed in the horizontal wavenumber domain >dH. Further, from the modal interpretation of this relationship, a linearized inversion method can be developed relating modal eigenvalues, given
by discrete horizontal wavenumber values, to sediment geoacoustic properties >1H. A key
aspect of this approach is the accurate estimation of the individual modal eigenvalues
385
N.G. Pace and F.B. Jensen (eds.), Impact of Littoral Environmental Variability on Acoustic Predictions and
Sonar Performance, 385-392.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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K.M. BECKER AND G.V. FRISK
contributing to the propagating field. This problem becomes more complicated when
spatial dependence enters into the problem, through variation in sediment properties with
range [3], or variability in the water column due to sound speed fluctuations or changes
in bathymetry.
In the cases of changes in sediment properties or bathymetry that occur slowly over
range, horizontal wavenumber estimates can be obtained by assuming regions of local
range-independence in the waveguide and applying techniques analogous to short-time
Fourier transform methods [4]. However, although the effects of sound speed fluctuations
due to internal waves on modal propagation, as well as their corresponding effects on
broadband geoacoustic inversion, have been treated in the literature recently (see, for
example, [5] and [6]), their effects on horizontal wavenumber estimation have received
less attention. Two general assumptions are usually made for wavenumber analysis based
on the Hankel transform. The first assumption is the adiabatic mode approximation, which
states that no energy is transferred amongst the different modes and that the number of
propagating modes remains constant. The second assumption is that the effects of sound
speed fluctuations in the water column are minimized due to the averaging effect that
occurs when transforming data over sufficiently large apertures.
In the case of internal waves, it is well known that the adiabatic assumption does
not apply and that a complex competition between mode coupling and mode stripping
occurs [7]. In this paper, a study is presented addressing the wavenumber estimation
problem for a range-dependent acoustic field where mode coupling is known to occur. Synthetic complex pressure fields are generated using a range-dependent parabolic
equation (PE) model [8] for both a range-independent shallow-water waveguide, and a
waveguide with a deterministic internal wave (IW) field that introduces range-dependence
into the waveguide through perturbations to a background sound speed profile. Applying
short-window methods based on the Hankel transform to the resulting fields, modal content is plotted as a function of range and compared for the different models. In addition,
using a fully coupled mode code [9], reference values of modal amplitudes and horizontal
wavenumbers are determined as a function of range for both the background and internal wave models. Comparisons are made between estimated wavenumbers and reference
values for both acoustic fields. The influence of the internal wave field on wavenumber
estimates is examined in terms of the mean and variance of the estimates compared to
their reference values. A comparison of modal amplitudes between the different models
is also made yielding information about the detectability of particular modes.
2
Methods
For a cw point source, and assuming horizontal stratification, the equations governing
sound propagation in a shallow-water waveguide can be reduced to a Hankel transform relationship between the complex-pressure field p(r; z, zo ) and the depth-dependent
Green’s function g(kr ; z, zo ),
p(r; z, zo ) =
g(kr ; z, zo ) =
!
∞
!0 ∞
0
g(kr ; z, zo )J0 (kr r)kr dkr ,
(1)
p(r; z, zo )J0 (kr r)rdr.
IW EFFECTS ON WAVENUMBER ESTIMATION
387
In the above, pressure is measured as a function of range r, and the Green’s function is a
function of horizontal wavenumber kr and satisfies the inhomogeneous depth-dependent
wave equation, along with impedance boundary conditions at the surface (z = 0) and the
bottom (z = h) [10],
"
%
#
$
d
1 d
d2
2
2
+ ρ(z)
+ k (z) − kr g(kr ; z, zo ) = −2δ(z − zo ),
dz 2
dz ρ(z) dz
iZT (kr ) ∂g T (kr ; 0)
g T (kr ; 0) +
= 0,
ωρ(0)
∂z
iZB (kr ) ∂g B (kr ; h)
g B (kr ; h) +
= 0.
ωρ(h)
∂z
(2)
In Eq. (2), the source and receiver depths, zo and z, are treated as parameters in the
problem, ρ(z) is density, and k(z) = ω/c(z) is the total wavenumber for the angular frequency ω. The solution to the depth-dependent Eq. (2) is obtained by incorporating into
the total solution, the independent solutions gT (kr , z) and gB (kr , z) which satisfy impedance boundary conditions at the top and bottom, represented by ZT (kr ) and ZB (kr ),
respectively. The general form of the depth-dependent Green’s function satisfying both
boundary conditions can then be written,
g(kr ; z, zo ) = −2
gT (kr , z< )gB (kr ; z> )
,
W (zo )
0 ≤ z ≤ h,
(3)
where z< = min(z, zo ), z> = max(z, zo ), and W (zo ) is the Wronskian evaluated at the
source location and given by,
&
&
∂gB &&
∂gT &&
−
gB (zo ).
(4)
W (zo ) = gT (zo )
∂z &zo ∂z &zo
For a horizontally stratified ocean, and assuming the top surface satisfies a pressurerelease boundary condition, g(kr ; z, zo ) becomes a function of the impedance boundary
condition at the water-bottom interface alone. Through the impedance relationship, the
Green’s function is a complete characterization of the waveguide as needed for the solution
of Eq. (1). For shallow-water waveguides, the Green’s function is characterized by sharp
peaks, or resonances, that occur when the Wronskian in Eq. (3) goes to zero. These
peaks occur at distinct values of horizontal wavenumber that correspond to the modal
eigenvalues kn in a modal expansion of the field. The mode sum can be obtained by
inserting Eq. (3) into Eq. (1) and applying complex contour integration methods, to obtain
p(r, z) =
nmax
iπ '
(1)
φn (zo )φn (z)H0 (kn r) + I(r),
ρ(zo ) n=1
(5)
where φn are mode functions evaluated at the source and receiver depths, nmax is the
maximum number of propagating modes, and I(r) represents a continuum contribution
that can often be ignored for ranges greater than a few water depths from the source.
For analysis purposes, by applying the large argument asymptotic approximation to
the Bessel function, J0 (kr r) in Eq. (1), the Hankel transform operation can be replaced
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K.M. BECKER AND G.V. FRISK
with a Fourier Transform [10],
eiπ/4
g(kr ; z, zo ) ∼ √
2πkr
!
∞
−∞
√
p(r; z, zo ) re−ikr r dr, kr r % 1, kr > 0.
(6)
The asymptotic form allows the Green’s function
to be obtained through a Fast Fourier
√
Transform (FFT) of the product p(r; z, zo ) r. This form is particularly useful in that
various spectral estimation tools can be applied, including high-resolution methods, to
determine wavenumber content. Using these methods, horizontal wavenumbers corresponding to modal eigenvalues are determined from the peak locations in the Green’s
function estimate.
The relationship between the depth-dependent Green’s function and complex pressure
field described above is strictly valid for range-independent environments. For the rangedependent case, a convenient representation of the pressure field is given by the adiabatic
mode sum expressed as,
(r
√
nmax
i
k (r! )dr!
e 0 n
2πeiπ/4 '
φn (0, zo )φn (r, z) )(
.
(7)
p(r, z) ∼
r
ρ(0, zo ) n=1
k (r# )dr#
0
n
In this representation, mode functions and their corresponding eigenvalues adapt themselves to the local waveguide properties as they change in range. By performing the integral in (6) over a finite range interval given by −L/2 ≤ r ≤ L/2, where L is the aperture
length, a local estimate of the Green’s function can be obtained, where it is assumed that
the waveguide environment is approximately range-independent over L. For a truly adiabatic environment, moving the aperture along in range using short range steps gives a
picture of the modal content of the waveguide as it evolves with range [4]. The adiabatic
representation provides an intuitive way to understand how modal content evolves with
range, but is not a necessary requirement for extracting range-dependent modal information using the Hankel transform. For the synthetic pressure fields generated for this study,
the introduction of the internal wave field to the otherwise range-independent waveguide
causes energy in the propagating modes to be transferred amongst each other violating the
adiabatic assumption. However, estimates of the local depth-dependent Green’s function
can still be obtained by integrating the field over a finite aperture as described above.
3
Numerical examples and wavenumber estimation
The waveguide environment for the numerical studies was one of several designed as
a test bed for benchmarking range-dependent acoustic propagation models [11]. The
waveguide for this study was a fluid model with range-independent sediment properties
and a constant water/sediment interface depth of 200 m. Density in the water column
was 1 g/cm3 with no attenuation. The bottom had a compressional wave speed of 1700
m/s, a density of 1.5 g/cm3 , and an attenuation of 0.1 dB/λ. The background sound speed
profile was generally downward refracting with a slight duct occurring above 26 m depth
and was given by
c(z) =
1515 + 0.016z, z < 26 m
c(z) = co [1 + a(e−b + b − 1)], z ≥ 26 m,
(8)
389
IW EFFECTS ON WAVENUMBER ESTIMATION
a) TL vs Range (100 Hz)
b) 0−5 km (100 Hz)
−30
−30
background
internal wave
TL (dB)
−40
−40
−50
−50
−60
−60
−70
−70
−80
−80
−90
−90
−100
0
−100
0
5
10
15
20
1
TL (dB)
c) 10−15 km (100 Hz)
−50
−60
−60
−70
−70
−80
−80
−90
−90
11
12
13
14
3
4
5
d) 15−20 km (100 Hz)
−50
−100
10
2
15
−100
15
16
Range (km)
17
18
19
20
Range (km)
Figure 1. 100 Hz TL plots for internal wave study. 5 km sub-plots show departure of acoustic
field for internal waves from range-independent case.
where z is the depth in meters, co = 1490 m/s, a = 0.25, and b = (z − 200)/500 [12]. To
the background sound speed was added a perturbation as a function of depth and range
to represent the internal wave field. The sound speed perturbation, δc, was given by,
δc(z, r) = 4(z/B)e−z/B
5
'
cos(Ki r),
(9)
i=1
where Ki = 2π[2000 − 300(i − 1)]−1 m−1 and B = 25 m [12]. Using this model, the
maximum sound speed perturbation is about 7.5 m/s. Using the above models, PECan [8],
was used to generate the acoustic field at 100 Hz for both the background and perturbed
sound speed environments from 0 to 20 km on a 5 meter range grid. Transmission loss
data are shown in Fig. 1 for the full aperture and for several 5 km sub-apertures for a
source depth of 45 m and receiver depth of 60 m. The modal interference patterns in the
figure result from the constructive and destructive interference of different propagating
modes. The differences in interference patterns for the two fields are most evident at long
ranges and indicate the influence of the internal waves on the modal interactions. Using
the complex pressure fields, wavenumber estimates were obtained using a high resolution
sliding widow estimator based on an autoregressive spectral estimator [13]. The estimator is based on a parametric representation of the signal and requires the model order as
an input parameter. For the 20 km apertures of the synthetic data fields, wavenumber
estimates were made for 3000 m sliding window sub-apertures with a step size of 200
m and a model order of 1/3 the number of data points in each sub-aperture. Additionally, reference values of horizontal wavenumbers for the range-independent waveguide
environment with the background sound speed profile were determined using a normal
mode code [9]. The resulting wavenumber estimates with the reference results overlayed
are shown in Figs. 2 and 3. In the figures, modes identified using the Hankel transform
method correspond with the most energetic modes for the acoustic fields. Where there
is a match between an estimated wavenumber and a reference value, the reference value
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K.M. BECKER AND G.V. FRISK
Figure 2. Sliding window wavenumber estimates compared to reference results (dashed) plotted
for unperturbed case. The dark lines indicate the locations of the peaks of the Green’s function.
Figure 3. Sliding window wavenumber estimates compared to reference results (dashed) plotted
for IW case. The dark lines indicate the locations of the peaks of the Green’s function.
is plotted as a dashed line. The plots show that for the unperturbed case, 6 modes are
estimated that can be correlated with the reference results. For the IW case, 8 modes
are identified that correlate with the reference results. In both cases, the wavenumber
estimates are stable with range, which is especially evident for the higher-order modes.
This is an indication that the waveguide boundary conditions are range-independent. For
the low-order modes, there are small undulations about the background wavenumbers
that occur with range. However, there is no apparent wavenumber shift observed for the
modes obtained from the two fields. Together, the observations suggest that for the given
source/receiver geometry, certain modes are not excited or contain relatively little energy.
In order to verify this claim and gain a better understanding of the wavenumber estimates
shown, it is necessary to examine both modal amplitudes and horizontal wavenumber
content as a function of range.
391
IW EFFECTS ON WAVENUMBER ESTIMATION
mode 1
dB
40
20
0
0
5
10
dB
15
20
mode 3
20
5
10
15
20
mode 4
20
0
0
5
10
15
20
mode 5
40
0
5
10
20
15
20
mode 6
40
20
0
0
0
5
10
15
20
mode 7
40
dB
0
40
0
dB
Modal Amplitude (dB)
0
40
0
5
10
20
15
20
mode 8
40
20
0
0
0
5
10
15
20
mode 9
40
dB
mode 2
40
20
0
5
10
20
15
20
mode 10
40
20
0
0
0
5
10
Range (km)
15
20
0
5
10
Range (km)
15
20
Figure 4. Mode amplitude difference (internal wave - background) vs. range.
To examine mode amplitudes and wavenumbers, the COUPLE [9] range-dependent
normal mode code was used. Horizontal wavenumbers and modal amplitudes were output
at a receiver depth of 60 m for each range in the problem. The horizontal wavenumbers
corresponding to the propagating modes for the background problem are shown as the
straight lines in Figs. 2 and 3. For the IW case, discrete values of horizontal wavenumbers
were selected from the peak positions of the Green’s function at each range. Means and
variances over range were calculated for the individual eigenvalue estimates and compared
to the background, or reference, eigenvalues. The variances of the individual estimates
were small, consistent with the variance expected for the high-resolution estimator, as if it
were applied to a signal with additive noise and a high signal to noise ratio (SNR greater
than 45 dB) [13]. For the 8 modes identified in the IW case, the biases between the
estimates and the background values were less than 1 standard deviation for all modes.
This suggests that the mean values for the estimated horizontal wavenumbers could be
used as input data for a range-independent geoacoustic inversion algorithm.
Figure 4 shows the difference in individual modal amplitudes between the internal
wave field and background field as a function of range for the first 10 propagating modes.
The figure shows the influence of the internal wave field on the individual modes. While
it is seen that all of the modes are affected, modes 6 and 10 each have a large amount of
additional energy in them due to the internal wave field. This accounts for the enhanced
number of modes observed in the wavenumber estimates in figure 3, where the coupled
mode code verified that modes not observed were only weakly excited in either case.
Further, the smaller perturbations seen in the mode amplitudes can account for some of
the scintillations observed in the wavenumber estimates.
4
Conclusions
Internal waves greatly influence the propagation of sound in shallow water by redistributing energy amongst different propagating modes. This in turn has an effect on the
estimation of local modal content for use in linear inversion algorithms. In the case of the
weak internal wave field used as a model for this study, the following conclusions can be
392
K.M. BECKER AND G.V. FRISK
drawn. The effect of the internal wave field on local horizontal wavenumber estimates is
minimized by the averaging effects controlled by the size of the local aperture used in the
processing. A more complete study needs to be done to access an optimal aperture size
for a given environment. Finally, the presence of internal waves serves to excite modes
that otherwise would not be excited for a particular source receiver geometry and leads to
an enhancement of the wavenumber estimates. This enhancement yields additional data
which can be used in an inversion algorithm to determine seabed properties.
Acknowledgements
Richard B. Evans was extremely generous with his time in helping to get the coupled
mode code up and running. PECan data was generously provided by Gordon Ebbeson
of the Defense Research Establishment of the Atlantic (DREA), Canada. This work
was partially supported by an ONR Special Postdoctoral Fellowship Award in Ocean
Acoustics [Contract No. N00014-02-1-0334].
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