EFFECTS OF ENVIRONMENTAL VARIABILITY
ON FOCUSED ACOUSTIC FIELDS
B. EDWARD MCDONALD, JOE LINGEVITCH AND MICHAEL COLLINS
US Naval Research Laboratory, Washington DC, USA
E-mail: [email protected]
A number of experiments have demonstrated the ability to produce tightly focused
acoustic fields in the ocean using time reversal mirrors. The resulting focal regions
may serve as acoustic probes or communications sites. We will review the effects of
various types of environmental variability on the quality of the focus. Among these are
bathymetric variation and water column inhomogeneities that may result from internal
waves, bubble clouds, fish schools, or water mass intrusions. We will give high resolution simulation results from the RAM code to illustrate and quantify the impact of
environmental variability upon the focal region.
1 Introduction
A number of laboratory >dH and ocean experiments [2–5] show that an acoustic field from
a point source after propagating through a complex environment can be time reversed and
back propagated, resulting in a tightly focused field near the source. The ability to produce
a well focused field results from reciprocity, which in turn depends on (1) the propagation
medium being nearly static between forward and back propagation; and (2) the medium
being relatively free of attenuation. In ocean experiments, the back propagation is carried
out with a vertical send-receive array (SRA) some distance from a probe source. If the
ocean were azimuthally symmetric about the SRA, the resulting focused field would be a
circular annulus centered on the SRA and passing through the probe source. The acoustic
field at different azimuths around the annulus might be used to probe the environment
and/or targets.
If the environment is not azimuthally symmetric about the SRA, then focal properties
will be azimuth dependent. This paper will examine some theoretical and numerical
modeling results relating the acoustic variability of the water column and bathymetry to
changes in the focal annulus.
2
Theory
Modeling studies for SACLANTCEN’s Focused Acoustic Fields 1999 (FAF99)
experiment >EH performed off the west coast of Italy in July 1999 implied that the primary
effect of azimuthally dependent bathymetry was a range shift of the focal spot. Some
simple closed form results for the focal range shift due to environmental variation may
be obtained by adiabatic mode theory and the WKB approximation applied to a pressurerelease shallow water waveguide >H. The frequency range used in FAF99 was 3–4 kHz.
377
N.G. Pace and F.B. Jensen (eds.), Impact of Littoral Environmental Variability on Acoustic Predictions and
Sonar Performance, 377-383.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
378
B.E. MCDONALD ET AL.
The results presented here are for the center frequency 3500 Hz. We take the acoustic far
field of a point source at radian frequency ω to be
p→
!
i
An ψn (z)e
n
"r
0
kn dx
(1)
where ψn is acoustic eigenmode number n, and mode amplitude An is only weakly
dependent on r. If we assume for a basic state a range independent isovelocity pressure
release waveguide (surface and bottom) with sound speed c and depth D we have
ω 2 # nπ $2
.
(2)
kn2 = 2 −
c
D
Focal properties may be determined from the acoustic intensity field,
"r
!
!
∗ i 0 (kn −km )dx
|p|2 =
|An ψn |2 + 2Re
An A∗m ψn ψm
e
(3)
n
n>m
subject to environmental perturbations. From Eq. (2) a depth perturbation
D → D + δD
(4)
implies
kn → kn + δkn ,
δD # nπ $2 .
δkn =
kn D D
(5)
The perturbed focal range is that which leaves the interference integral term in Eq. (3)
unchanged for mode pairs (n,m):
"r
(δkn − δkm )dx
δr = − 0
k − km
% r n
∂kn
∂δkn
(6)
→ −(
dx)/(
).
∂n
∂n
0
%
2 r
δDdx
#
D 0
The last expression is independent of mode number, so it gives the range shift of the
intensity of the total acoustic field within the range of validity for linear perturbation
theory applied to waveguides [8]. For the environments encountered in FAF99 the final
expression in Eq. (6) was found to be within approximately 20% of range shifts determined
in modeling studies [6] performed at high resolution with the RAM code [9].
The pressure release waveguide also leads to a simple expression for the change in
focal depth as a result of perturbations, as long as mode coupling may be neglected. Since
ψn ∝ sin(nπz/D), then as long as mode amplitudes are preserved in relative proportion,
the fractional change in focal depth zf is equal to the fractional change in D:
δzf = zf
δD
.
D
(7)
VARIABILITY EFFECTS ON FOCUSED FIELDS
379
Time reversal of a near-bottom probe source was carried out as part of FAF99 in order
to investigate the feasibility of using selectively placed bottom reverberation to probe
distant bathymetry [7]. During the modeling studies of [6] it was found that the focal
depth behaved in approximate agreement with Eq. (7) for weak changes in bathymetry
and for focal spots near the bottom. For mid-water column focii, however, it was noticed
that strong bathymetric variation dispersed the mid-water focus to such a degree that its
location was not well defined.
3
Simulation
3.1 Bathymetric Variation
Modeling studies are presented here to quantify the level of bathymeric variation at which
the approximations made in Eqs. (1–6) begin to fail. Numerical parameters for the modeling are as follows. The vertical grid spacing used in RAM was 5 cm, while the range step
was 2 m. We place a point source one meter above bottom of a 112 m deep ocean with a
summer sound speed profile appropriate to FAF99 [6] and bottom sound speed 1582 m/s.
RAM propagates the signal to an SRA (Fig. 1a) whose aperture is 15–93 m depth as in
FAF99. The range to the SRA is taken to be 5 km for the simulations presented here.
The complex acoustic field on the SRA is then conjugated and back propagated (Fig.
1b). One sees a strong focal spot near the bottom in Fig. 1b at the probe source range, as
verified by the bottom ensonification plot (expressed as transmission loss re 1 m) included
in the lower portion of Fig. 1b. Bottom ensonification is determined by resolving the
complex pressure field into up and down going waves, then taking the intensity of the
downgoing wave at the ocean bottom.
Table 1. Bathymetry for Back Propagation
Range(km)
0.0
3.0
4.5
6.5
7.4
10.0
Depth(m)
112
120
125
130
137
150
Back propagation from the SRA is next considered for an azimuth whose bathymetry
is different (Table 1) from that of the foward propagation. Results are shown in Fig. 2.
Examination of the bottom ensonification in Fig. 2 reveals that the focus at the bottom
has been shifted in range by approximately 1.3 km. The focal shift predicted by Eq. (6)
is 926 m, so the simple result of Eq. (6) is in error by approximately 32 percent for this
case. In order to determine how the accuracy of Eq. (6) degrades with increasingly strong
bathymetric variation, we performed a total of nine simulations similar to those of Figs.
1b and 2, with bathymetry determined by a scaling factor α:
D(r) = D1 (0) + α · [D1 (r) − D1 (0)]
(8)
where D1 (r) is taken from Table 1. The nine α values used in the simulations are (0,
.25, .5, .75, 1, 1.125, 1.25, 1.5, 2). Figure 1b is the range independent result for α = 0,
380
B.E. MCDONALD ET AL.
Back Propagation Range Independent SD 111 3500 Hz
Forward Propagation Range Independent SD 111 3500 Hz
60
70 TL (dB) 80
100 50
90
60
70 TL (dB) 80
90
100
100 Ensonification 50
(dB)
SRA Aperture
40
60
200 180 160 140 120 100 80
Depth (m)
SRA Aperture
20
0
50
rms bottom slope = 0.
0
1
2
3
4
5
6
7
8
9
10 0
1
Range (km)
(a)
2
3
4
5
6
7
8
9
10
Range (km)
(b)
Figure 1. (a): A probe source 1 m from the botton ensonifies the SRA. (b): Back propagation from
the SRA to the probe source. The lower curve is the bottom ensonification.
0
SSP for FAF99
Back Propagation Range Dependent SD 111 3500 Hz
60
70 TL (dB) 80
90
100
rms bottom slope = 4.302E-03
0
1
(a)
2
3
4
5
6
7
8
9
10
Range (km)
1500 1520 1540 1560 1580 1600
c (m/s)
100 Ensonification 50
(dB)
Depth (m)
200 180 160 140 120 100 80
60
40
SRA Aperture
60
90
150
120
Depth (m)
20
0
30
50
(b)
Figure 2. (a): Sound speed profile for Fig. 1. (b): Back propagation from the SRA along a different
azimuth from that of the probe source. The lower curve is the bottom ensonification.
and Fig. 2 is equivalent to α = 1. We performed seven other back propagations with α
values up to 2. Results are shown in Fig. 3 as a function of the root-mean-square slope
of the bathymetry between the SRA aperture and the 5 km range of the probe source.
The solid curve is the prediction from Eq. (6), the points marked × are taken from the
bottom ensonification curves as in Figs. 1b and 2, and the points marked ◦ are taken from
the visible focal maximum. For bathymetry slope less than 3×10−3 , Eq. (6) is within
about 15 percent of the simulation results for focal shift. At greater values of slope, the
simulation focal shifts increase at approximately twice the rate predicted by Eq. (6).
381
800
1200 1600 2000
Bathymetric Focal Shift
o
x
Focal Range 5km
__ Theory
x Bottom Ensonification
o
x
o Visible Maximum
o
x
o
x
x
o
x
o
x
o
x
o
400
0
Focal Shift(m)
VARIABILITY EFFECTS ON FOCUSED FIELDS
o
x
0
1.5
3
4.5
-3
6
7.5
r m s Bathymetry Slope x 10
Figure 3. Shift in focal range caused by varying bathymetry.
3.2 Water Column Variation
The dominant variability of focal properties in FAF99 was attributed to internal waves
which seemed to appear after wind events. One can get a feel for the internal wave
environment by converting the sound speed profile of Fig. 2(a) to a temperature profile
assuming salinty of 38 psu. The result is shown in Fig. 4(a) accompanied by the resulting
Brunt-Vaisala frequency profile in Fig. 4(b). Although the profiles were taken at times
not following wind events, the BV frequency in Fig. 4(b) is of high enough amplitude
to indicate a receptive environment for internal waves. Wind events would sharpen the
mixed layer and BV profile even further.
Internal wave modes obey the equation [10]
& 2
'
N (z)
−
1
q(z) = 0,
(9)
q## (z) + k2
Ω2
where the vertical mass flux is wρ(z) = q(z) exp(i(kx − Ωt)) and the (radian) BV
frequency is given by
N 2 (z) =
g dρ
ρ dz
(10)
with g the acceleration of gravity and z positive downward. In the limit of long horizontal
internal waves compared to the width of the thermocline, the modes approach the sharp
interface form with dispersion equation [10]
1
ρ2
Ω2 = gk ln
2
ρ1
(11)
where ρ2 and ρ1 are the densities below and above the thermocline.
In a range independent waveguide of depth D (Fig. 1) with a sharp interface at depth
z0 , internal wave modes have the form
q(z) =
(
sinh(kz)/ sinh(kz0 ),
sinh(k(D − z))/ sinh(k(D − z0 )),
z < z0
z > z0
(12)
382
B.E. MCDONALD ET AL.
BV Frequency
x
0
0
Temperature profile
x
x
x
xx
48
Depth (m)
x
96
72
96
x
72
48
x
Depth (m)
x
x
xx
24
24
x
120
120
x
10
14
18
22
26
30
0
T deg C
(a)
6
12
18
24
30
cph
(b)
Figure 4. (a): Termperature profile derived from SSP of Fig. 2a. (b): Brunt-Vaisala frequency in
cycles per hour.
Back Propagation with Internal Waves SD 111 3500 Hz
60
70
80
90
100
100 Ensonification 50
(dB)
200 180 160 140 120 100 80
Depth (m)
60
40
20
0
50
rms bottom slope = 0.
0
1
2
3
4
5
6
7
8
9
10
Range (km)
Figure 5. Dispersal of focus by internal waves added to the environment of Fig. 1b. Undisturbed
thermocline depth z0 is 24 m.
A set of three internal waves of the form (12) was introduced into the √
environment
√
used in Fig. 1b. The horizontal wavelengths were chosen as 100 m ×(1, 1.1, 1.2)
with irrational proportions so as to exclude artificial periodicities in the wave train. Amplitudes and phases were randomized and the wave train normalized to root-mean-square
thermocline displacement of 5 m. The result of back propagation through these internal
waves is shown in Fig. 5. One sees a negative shift in the focal range and a loss of
VARIABILITY EFFECTS ON FOCUSED FIELDS
383
approximately 6db in the peak relative to Fig. 1b. Due to the stochastic nature of internal
waves it is much more difficult to extract theoretical estimates for their effect than for
the case of bathymetric variation. Not surprisingly though, a vertical variability of order
5 m in the sound speed structure between foward and back propagation seriously disrupts
the focal structure of Fig. 1b.
4
Summary
We have shown that simple estimates from waveguide invariant theory applied to bathymetric perturbations of a focused acoustic field are reasonably accurate for mild bathymetric variation. A series of RAM simulations with increasingly steep bathymetry showed
(Fig. 3) a clear departure from theory at a root-mean-square bathymetric slope of 3×10−3
in a 112 m deep waveguide and focal range of 5 km. The presence of internal waves on a
sharp thermocline was found in our simulation to have a strong disruptive effect on focal
properties. This is in qualitative agreement with degradations in focal properties following
wind events in FAF99. We found in Fig. 5 that a 5 m rms fluctuation in the thermocline
had a more disruptive effect on the focus than did a 15 m change in bathymetry (Fig. 2b).
Acknowledgements
Work supported by NRL and the US Office of Naval Research.
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