TOMOCRAPHIC APPROACH TO ACOUSTIC NOISE MAPPING IN SHALLOW SEA
Vinayak Dutt and G.V. Anand
Department of Electrical Communication Lngineering
Indian Institute of Science
Bangalore 560 012, INDIA
ABSTRACT
This paper presents a method for the acoustic
noise mapping of a shallow ocean using a tomographic
approach. The output of a planar array of hydrophones
in an ocean is expressed as a weighted line integral of
the acoustic field. A set of measurements of the array
output for different steering directions and several
array locations can be inverted using algebraic
reconstruction technique to reconstruct the acoustic
field
.
where
D
(ev)
m '
=
COSY- k cos0 )I
.sin
. . . .[(2N+1)
. . . . . . . . .(d/2)
. . . . . .(f! __________________
sin [ (d/2)
(em cosy - k cos0
is the directivity function of the array, k
the wave number; and
1
=
(4)
w/c is
INTRODUCTION
The possibility of using a tomographic technique
for noise mapping of the ocean was first suggested by
Rockmore [ I ] . One important applicatiori of noise
mapping is in ocean surveillance to detect and localise
discrete acoustic sources which would stand out as
peaks in the noise map. Rockmore has formulated the
noise mapping problem for an unbounded medium. In this
paper we have extended the formulation to shallow ocean
by taking into account the multipath effects due to
reflections from the top and the bottbm surfaces. For
this purpose, it is necessary to replace linear arrays
by planar arrays. The acoustic field due to a source
can be expressed as a sum of normal modes. Each
vertical aoluum of a planar array is used as a mode
filter to extract the signal due to a single normal
mode. Each vertical coluum is designed to select the
same (say, the mth mode).
The mode-filtering
eliminates the multipath effect, and the planar array
output is seen to be a family of fan-beam projections
of the depth-averaged noise intensity distribution when
the array is steered in different directibns.
Projections from several arrays can be inverted by
tomographic methods to reconstruct the map of depthaveraged noise intensity. We have used an algebraic
reconstruction technique [ 21 for performing the
tomographic inversion. Computer simulation results are
presented
for two different noise
intensity
distributions over a region of finite support.
FORMllLATION OF THE PROBLEN
The normal-mode representation for acoustic field
at a receiver located at (xo,yo, zo) due to a source
of amplitude A(x,y,z) located (x,y,z) is given by [ 3 ] ,
* exp(- dmr - j w t )
(1 1
where w is the angular frequency of the source,.qm is
the attenuation coefficient of the mth mode, M 1 s the
total number of propagating modes, E, is the horizontal
component of the wave number of the #th mode, 9 (z) is
the eigen function for the mth mode, Hto(Fmr)"is the
Hankel function of the first kind and
2
2 1/2
r = [(x-x,) + (y-yo) 1
is the bearing of the source.
In order to avoid the multi-mode f i e l d at the
array, we replace each element of the horizontal array
by a vertical column of (2M-1) hydrophones. Each
column is designed as a null-steering mode filter [&I
to select the first mode. A normal mode is a standing
wave pattern formed by the interference of two quasiplane waves travelling at angles +@, with respect to
the horizontal, where,
8, = cos-1 (tm/k),
m
=
I,
..., M.
(6)
Mode filtering is achieved by steering nulls in the
directions of arrival of all unwanted modes. To reject
the mth mode, a 3-elemental vertical array with the
weighting coefficient sequence
W, = { I , -2 cos [(k2 -Im2) 1/2A 9 1 }
(7)
is required, where A is the inter-element spacing of
the vertical array. All modes higher thar the first
can be rejected using a (2M-1) element vertical array
with weighting coefficient sequence,
W
=
w2
* w2
...
it
wM
where * denotes the convolution. It is important to
note that the weighting coefficients are independent of
the source bearing angle and the array depth. The
weighting coefficients are also real and symmetric,
which is an added advantage during implementation.
After the mode filtering, only the contribution of
the first mode remains. Hence, the total field due to
source distribution throughout the volume can be
expressed in the integral form as,
5
FD
p(8)
=TI
j
0
9
h
rdr rd-f]dz A(xo + r cosy, yo + r sin1 ,z)
0
0
D,(e,Y) exp(-Q',r
- jwt)
(9)
where h is the ocean depth. Separating the integral
with respect to z as
Hence, the output of a horizontal line array of (2Nt1)
elements with inter-element spacing as d, steered in
the direction 8 , centred at (xo, yo, zo) , is given by
29.3.1
CH2766 - 4/89/0000 - 0581 0 1989 IEEE
581
we get,
m
p(8)
=nj
rdr
n
7
Then,
dy F(xo
f
+ r cosy.
r cosy, yo + r sin7 )
J'
0
~Dl(e,Y)\yl(zo)
H t o ( F l r ) exp(-vlr - jwt)
1 p(e~\~l
I ( e ) = E[
exp(-2q1r) r dr;
(11)
The power output of the array is
(12)
where E is the expectation operator.
Assuming that the noise distribution
is
uncorrelated and has slow variations along the depth,
representing the depth-averaged noise intensity as
j'
+
j
r sinY.)
J
= 1,
. .., NP
(21)
For the reconstruction of the noise intensity map,
the scanned area is divided into a uniform grid, with
the reconstruction being performed to estimate the
noise intensity at these grid points (see Fig. 1 ) . The
value of u 2 at other points is computed by
interpolation of the values at the grid points. Now
approximating the integral in the equation (21) by a
summation,
M.
IJB(xjk,yjk)
exp(-2y1 rjk) r.
ar;
Jk
(22)
j = 1 , ..., N
P
where M, is the number of steps in the jth summation,
J
and Q2(xjk, yjk) is the estimate of the noise intensity
S, =
k=l
and using the far-field approximation for the Hankel
function,
equation (12) can be written as
zn
at (xjk, y ) computed by interpolation of the grid
jk
point noise intensity values, gi (i = 1 , . .., Ng , where
oo
N
g
Now if the inner integral of equation (15)
represented as,
is
=
number of grid points),
2(xjk, yjk)
Q
N
=
Lg d.. gi; j=1,
i=l lJk
60
k=l,
..., NP'
..., Mj *
*
(23)
and d.. is the contribution of gi to the estimate of
0
*
exp(-2q1r)r dr,
(16)
1Jk
c 2 at (x.
Jk
S. =
J
Defining q = cosy and P=(k/qm) cos 8 ,
j=l,
we can write,
Let
where
G(x) = sin2 [ ((2N+1)/2) d.x. ]/sin2[d.x/2]
then,
N
S. = x g a.. gi;
J
i=l 1 J
(20)
(22) can be written as,
N
M.
zg
[ LJ exp(-2q1 rjk) rjk dijkArl.gi;
i=l k=l
(18)
is the directivity function. Hence, from equation (19)
it can be seen that I ( 0 ) is a convolution integral. So
S(y) can be obtained from I(0) by deconvolution as G ( 4 )
is a known function. Equation (16)indicates that S(Y)
is a family of fan-beam projections of the function
r2(x,y), the vertex of the fan being at (x , y,).
Several families of projectioons can be obtajned by
deploying arrays at several locations. From these
families of projections, the depth-averaged sound
intensity distribution r'(x,y) can be reconstructed
using the algebraic reconstruction technique [ 21.
, yjk). So equation
..., NP
j=l,
(24)
..., NP
(26)
Equation (26)is a linear set of equations in g. which
can be recursively solved by the
afgebraic
reconstruction technique (ART), where the ART iteration
is given by [2]:
J
J
SIMULATION RESULTS
Let the deconvolved mode power S(y) obtained for
various receiver locations be denoted by S. where
subscript j denotes the deconvolved power for Jbearing
and receiver at (x , yj). Let the total number of
j
such measurements be N
P'
5
Computer simulations were carried out for a
Pekeris model of the ocean with the parameters, depth
as 5Om, speed of sound in the ocean and the bottom
medium as 1500 m/s and 2000 m/s respectively, ratio of
density of bottom medium to that of ocean water as 1.2
and 5 , = 0.2426 radlm.
29.3.2
582
The area scanned was a square L+OOOrn x 4OOOm with
the coordinates of its corners as (O,O), (0,4000),
(4000,4000)and (4OOO,O). The line array had 21
elements with inter-element spacing 2m. The four
arrays were located at (0,1000), (0,2000)
, (0,3000)and
(1000,0),
centred at a depth of 1%.
60 scans were
taken from each array.
In these simulations, actually there was no
deconvolution done as a delta function approximation
was made for the array directivity function during the
reconstruction.
Figure 2 gives the results for the two source
distributions taken f o r the simulations. The actual
depth-averaged noise intensity is shown in Figs. 2(a)
and 2(b) respectively. The intensities in the map are
scaled linearly in the range 0 to 9.
The result for noise intensity distribution given
by Fig. 2(a) is not as good as that for the
distribution given by Fig. 2 ( b ) . This is so because
all the arrays are on one side of the scanned area, and
the direction of the gradient of the noise intensity
distribution is along the scan directions f o r this
array configuration,'which results in inaccuracies in
the reconstruction.
CONCLUSIOE
The present work shows that the tomographic
approach for noise intensity mapping seems to give
fairly good results even though no deconvolution of the
array output was done and a small number of arrays was
used.
Presently work is being carried out on
deconvolution processing to see the improvement in the
reconstruction by this extra computational effort.
jEFERENCES
Rockmore, 'A Tomographic Approach to
Wtiarray Ocean Surveillance', IEEE Journal of
Ocean Engg., Vol. OG17, pp 83-89, April 1982.
[ I ] A.J.
Signal Processing,
[2] Simon Haykin, Ed.. , Arra
Prentice Hall 1985,C h . d p 408-419.
[ 3 ] L. Biekhovskikh and Yu. Lysanov, Fundamentals of
Ocean Acoustics, Springer-Verlag 1982, Ch. 6, pp
120-122.
-
Fig. 1. Illustration of the noise intensity projection
integral py a summation over a square
reconstruction region.
[ 4 ] H.N. Chouhan and G.V. Anand, ' A New Technique of
Acoustic Mode Filtering in Shallow Sea' , Submitted
to the Journal of Acoustical Society of America for
publication.
29.3.3
583
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( 4000.0)
Fig. 2. Simulation results. ( a ) and (b) give t h e a c t u a l
depth avera ed noise i n t e n s i t y map used for
simulations f c ) and ( d ) give t h e reconstruction
using t h e tomographic approach for ( a ) and (b)
respectively. (X i n p a r t ( c ) and ( d ) give t h e
positions of t h e a r r a y s ) .
29.3.4
584
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