M J Buckingham

Michael J Buckingham

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Proceedings of the Seventh European Conference on Underwater Acoustics, ECUA 2004
Delft, The Netherlands
5-8 July, 2004
RAPID ENVIRONMENTAL ASSESSMENT WITH AMBIENT NOISE
M. J. Buckingham
Michael J. Buckingham, Marine Physical Laboratory, Scripps Institution of Oceanography,
University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0238, USA
e-mail: mjb@mpl.ucsd.edu
At frequencies below 5 kHz, shallow rays from distant sources create a strong, near-
symmetrical horizontal lobe in the vertical directional density function of ambient noise in
shallow water. The angular half-width of the lobe is equal to the critical grazing angle, ac, of
the seabed. From a two-hydrophone measurement of the vertical coherence of the noise, the
critical angle may be estimated, from which the sound speed in the sediment may be
determined. Once the sediment sound speed is known, the remaining geo-acoustic parameters
may be obtained from the dispersion relations predicted by a recently developed theory of
wave propagation in saturated granular media. The combination of the noise measurement
and the new theory provides a basis for performing rapid, cost-effective geo-acoustic surveys
of shallow water sediments using more or less conventional sonobuoy technology.
1. INTRODUCTION
The spatial properties of ambient noise in shallow water are influenced by the local envir-
onment, in particular, by the proximity of the sea bed. Assuming a fast bottom, in which the
speed of sound is greater than that in the water column, there exists a critical grazing angle,
ac, for radiation incident from above. Typically, for a medium sand sediment in which the
speed of sound is in the vicinity of 1750 m/s, ac is around 30˚. A water-borne ray that is
incident on the bottom with a grazing angle greater than ac penetrates the interface and is
effectively removed from the channel after a few bounces. A shallower ray, propagating at
less than the critical grazing angle, undergoes total reflection, thus allowing it to survive
multiple bounces with little attenuation. Since this argument holds regardless of the nature of
the source, it follows that the field from distant noise sources consists predominantly of
shallow rays, with grazing angles less than the critical. Thus, the vertical directional density
function[1] of the noise in shallow water often exhibits a strong horizontal lobe with angular
half-width ac.
It follows that a measurement of the angular width of the horizontal noise lobe returns the
critical grazing angle, ac. Thus, the speed of sound, cp, in the sediment may be inferred, since
c
c
p
w
c
=
cosa
 , (1)
where cw, the speed of sound in the water column, is assumed known. The width of the noise
lobe could be determined by sweeping the narrow-band beam of a vertical line array of
hydrophones from the upward- to downward-looking vertical. Such an array, however, is
likely to be costly and not particularly convenient for rapid-deployment operations. An
attractive alternative is a pair of vertically aligned sensors, separated by about 1 m, from
which the vertical coherence function of the noise is obtained as a function of frequency[2].
The coherence function is the Fourier transform of the vertical directionality of the noise and
both therefore contain the same spatial information. In effect, aperture in the narrow-band
vertical array is traded for broader bandwidth in the vertical pair. A crude estimate of the
critical angle may be obtained from the first zero crossing of the real part of the coherence
function. Naturally, a more refined estimate is available from improved signal processing in
which the full coherence function is exploited.
Based on these ideas, a pair of elements mounted in a sonobuoy would appear to provide a
first step in a cost-effective system for assessing all the geo-acoustic parameters of the sea
bed. However, to advance from an estimate of just the sound speed in the sediment to the
values of the remaining geo-acoustic parameters, further information is required. From the
empirical relationships of Hamilton[3] and Richardson[4], it has been established that the
wave properties and the mechanical properties of sediments are correlated. Recently,
Buckingham[5] developed a grain-shearing (GS) theory of wave propagation in sediments,
which expresses these correlations through dispersion relationships for the compressional
wave and the shear wave. As described below, these new dispersion relationships provide the
necessary link between the noise measurement of the sound speed in the sea bed and all the
remaining geo-acoustic parameters of the sediment.
2. DISPERSION PAIRS
Marine sediments support a compressional wave and a shear wave. According to the GS
theory of wave propagation in sediments, the properties of these waves are governed by a
certain type of inter-granular sliding, which leads to dissipation and dispersion. The
dissipation occurs because wave energy is converted into heat, whilst the dispersion arises
because the grain-to-grain interactions make the material stiffer. The enhanced stiffness also
accounts for the predicted existence of a shear wave.
The wave properties of the compressional wave and the shear wave are expressed through
complex wave speeds, which may be split into real expressions for the two phase speeds and
the two attenuations. For the compressional wave, the phase speed, cp, is
c
c
c
j T
p
o
p s
o o
n
=
+
+È
Î
Í
˘
˚
˙
-
Re
( / )
( )
/
1
4 3
2
1 2
g g
r
w
(2a)
and the attenuation, ap, is
a
w g g
r
wp
o
p s
o o
n
c c
j T= - +
+È
Î
Í
˘
˚
˙
-
Im
( / )
( )
/
1
4 3
2
1 2
   . (2b)
The corresponding dispersion pair, cs and as, for the shear wave is
c
T
ns
s
o
n
=
Ê
Ë
ˆ
¯
g
r
w
p
( )
cos
/2
4
(3a)
and
a w
r
g
w
p
s
o
s
nT
n
= Ê
Ë
ˆ
¯
-( ) sin/2
4
    . (3b)
In these expressions, w is angular frequency, ro is bulk density, T = 1 s is a normalising
time, and co is Wood’s sound speed[6] for the equivalent suspension, which equals (ko/ro)
1/2,
where ko is the bulk modulus of the medium. Both ro and ko are given by familiar weighted
means [see Eqs. (6) and (7) below] of the corresponding values for the individual constituents
of the medium and, accordingly, both vary linearly with the porosity, N . The three
coefficients (gp, gs, n) are peculiar to the GS theory, characterising the intergranular sliding
process. The first two, gp and gs, are analogous, respectively, to the compressional and shear
moduli in elasticity theory. They depend on the mean grain diameter, ug, the depth in the
sediment, d, and porosity, N as follows:
g gp po
g
o go o
N u d
N u d
=
-
-
È
Î
Í
Í
˘
˚
˙
˙
( )
( )
/
1
1
1 3
(4a)
and
g gs so
g
o go o
N u d
N u d
=
-
-
È
Î
Í
Í
˘
˚
˙
˙
( )
( )
/
1
1
2 3
   , (4b)
where gpo and gso are compressional and shear coefficients, which are independent of the
macroscopic properties of the sediment. In Eqs. (4), the parameters with a subscript zero are
reference values, chosen for convenience (No = 0.37, ugo = 1000 mm, do = 0.3 m) and do not
represent additional unknowns. The remaining GS parameter, n, is a small positive constant
representing the non-linear mechanism of strain hardening, an effect that is postulated to
occur in the molecularly thin layer of pore fluid separating the grains[5].
The three GS parameters may be evaluated by inserting measurements of the comp-
ressional wave speed and attenuation, taken at a spot frequency, into Eqs. (2), and the shear
wave speed, also at a spot frequency, into Eq. (3a). Using this procedure with Richardson’s
38 kHz compressional-wave data from the SAX99 experiment in the northern Gulf of
Mexico, and Richardson’s 1 kHz shear-wave data from the Mediterranean, yields gpo = 3.90 x
108 Pa,  gso = 4.65 x 10
7 Pa and n = 0.085.
It should be noted that, although strain-hardening is a non-linear mechanism, it occurs in
the GS theory in connection with the stress induced by individual, microscopic sliding events.
A random, linear superposition of all such events yields the total stress. It follows that the
dispersion relations predicted by the GS theory, Eqs. (2) and (3), are strictly linear. These
expressions are also causal, satisfying the Kramers-Kronig relationships.
Another point to note is that the expressions in Eqs. (2b) and (3b) represent, respectively,
the intrinsic attenuation of the compressional and shear wave, that is to say, the loss due to
the conversion of wave energy into heat. In contrast, most measurements return the effective
attenuation, consisting of the intrinsic attenuation plus any additional loss due, for example,
to scattering from inhomogeneities such as shell fragments in the medium. It follows that
Eqs. (2b) and (3b) predict the lower bounds to the measured values of the two attenuations.
3. GRAIN SIZE AND POROSITY
The GS dispersion relations depend explicitly on the porosity, N, and the grain size, ug. If
values for both these variables were available from independent measurements on cores, for
example, it is recommended that they be used to evaluate the wave speeds and attenuations
from the GS dispersion relations. If, on the other hand, the only available variable were the
compressional wave speed, obtained from the vertical coherence of the ambient noise in the
water column, the remaining geo-acoustic variables may be derived from the GS dispersion
relations. To do this, however, it is necessary to identify a relationship between the porosity
and the grain size.
Such a relationship has been developed by Buckingham[7] on the basis of an argument
involving the random packing of rough spheres. The resultant expression for the porosity in
terms of the grain size is
N P
u
us
g
g
= -
+
+
Ï
Ì
Ô
ÓÔ
¸
˝
Ô
Ǫ̂
1
2
4
3
D
D
,   (5)
where Ps = 0.63 is the packing factor of a random “close” packing of uniform, smooth
spheres and D = 1 mm is a statistical measure of the departure from sphericity of the grains.
The expression in Eq. (5) is a smooth, monotonic decreasing function of the mean grain
diameter, ug, which follows the trend of the porosity versus grain size data. In the limits of
large and small grain size, Eq. (5) asymptotes, respectively, to Nmin = 1 - Ps = 0.37 and  Nmax
= 1 - (Ps/8) = 0.92, in good agreement with observations. At intermediate grain sizes,
especially those representative of silts and clays, the data show some spread, with Eq. (5)
representing reasonable average behaviour.
4. GEOACOUSTIC PARAMETERS FROM AMBIENT NOISE
With the aid of Eq. (5), the GS dispersion relations may be expressed as functions of either
the porosity or the grain size. This is illustrated in Fig. 1, which shows the compressional and
shear wave speeds  and  attenuations  plotted as  functions  of  the  porosity.  The sound speed
Fig. 1: Wave properties as functions of porosity, from the GS theory. a) Sound speed ratio,
b) sound attenuation, c) shear speed and d) shear attenuation.
ratio in Fig. 1a is the wave speed in the sediment normalised to the sound speed at the bottom
of the water column, a quantity which is relatively insensitive to changes in the temperature
of the environment.
Theoretical plots like those in Fig. 1 form the basis of the technique for determining the
geo-acoustic parameters of a sediment from measurements of the vertical coherence of
ambient noise in the water column. Suppose that the noise measurement returned a sediment
sound speed of 1650 m/s, corresponding to a sound speed ratio of 1.1 when the seawater
sound speed is 1500 m/s. According to Fig. 1a, the porosity of the sediment is 0.41, from
which the sound attenuation is 0.19 dB/m/kHz (Fig. 1b), the  shear speed is 96.2 m/s (Fig. 1c)
and the shear attenuation is 38 dB/m/kHz (Fig. 1d). In expressing the two (intrinsic)
attenuations in units of dB/m/kHz it is implicit that each varies linearly with the frequency,
which is consistent with the predictions of the GS theory.
0.2 0.4 0.6 0.8 1
0.95
1
1.05
1.1
1.15
1.2
1.25
porosity
so
un
d 
sp
ee
d 
ra
tio
0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
porosity
co
m
p.
 a
tte
n.
, d
B
/m
/k
H
z
0.2 0.4 0.6 0.8 1
0
50
100
150
200
porosity
sh
ea
r 
sp
ee
d,
 m
/s
0.2 0.4 0.6 0.8 1
0
50
100
150
200
250
porosity
sh
ea
r 
at
te
n.
, d
B
/m
/k
H
z
a) b)
c) d)
With the porosity now known, the grain size may be estimated from Eq. (5) and is found to
be ug ≈  88 mm, which is characteristic of a very fine sand. The bulk density of the sediment,
ro, may be calculated from the weighted mean
r r ro w gN N= + -( )1   ,                 (6)
where rw = 1005 kg/m
3 is the density of seawater and rg = 2730 kg/m
3 is the density of the
mineral grains. Eq. (6) yields ro = 2023 kg/m
3. Similarly, the bulk modulus of the medium,
ko, may be evaluated from the weighted mean
1 1
1
1
k k ko w g
N N= + -( )   , (7)
where kw = 2.37 x 10
9 Pa is the bulk modulus of seawater and kg = 3.36 x 10
10 Pa is the bulk
modulus of the mineral grains. Eq. (7) yields ko = 5.26 x 10
9 Pa. Finally, Wood’s sound speed
may be deduced, since co = (ko/ro)
1/2 = 1612 m/s. This is the value of the compressional wave
speed in the limit of low frequency, when the two-phase medium acts as a simple suspension
in which grain-to-grain interactions are negligible.
5. CONCLUDING REMARKS
A shallow-water technique has been discussed for estimating the geo-acoustic parameters
of a marine sediment from the critical angle of the seabed. The critical angle itself is obtained
from a broadband measurement of the vertical coherence of the ambient noise in the water
column. From the critical angle, the sound speed in the sediment is estimated, and all the
remaining geo-acoustic parameters are deduced using the dispersion relations for the
compressional wave and the shear wave, as predicted from a recently developed theory of
wave propagation in saturated porous media.
6. ACKNOWLEDGEMENTS
This work is supported by Dr. Ellen Livingston, Ocean Acoustics Code, the Office of
Naval Research, under grant number N00014-04-1-0063.
REFERENCES
[1] H. Cox, “Spatial correlation in arbitrary noise fields with application to ambient sea
noise,” J. Acoust. Soc. Am., 54, pp. 1289-1301, 1973.
[2] M. J. Buckingham and S. A. S. Jones, “A new shallow-ocean technique for determining
the critical angle of the seabed from the vertical directionality of the ambient noise in the
water column,” J. Acoust. Soc. Am., 81, pp. 938-946, 1987.
[3] E. L. Hamilton, “Acoustic properties of sediments,” in Acoustics and the Ocean Bottom,
Editors A. Lara-Saenz, C. Ranz Cuierra, and C. Carbo-Fité, Consejo Superior de
Investigacions Cientificas, Madrid, pp. 3-58, 1987.
[4] M. D. Richardson and K. B. Briggs, “Empirical predictions of seafloor properties based
on remotely measured sediment impedance,” In High-Frequency Ocean Acoustics,  La
Jolla, California, USA, Editors M. B. Porter, M. Siderius and W. A. Kuperman, in press,
2004.
[5] M. J. Buckingham, “Wave propagation, stress relaxation, and grain-to-grain shearing in
saturated, unconsolidated marine sediments,” J. Acoust. Soc. Am., 108, pp. 2796-2815,
2000.
[6] A. B. Wood, A Textbook of Sound, 3rd ed., G. Bell and Sons, London, pp. 610, 1964.
[7] M. J. Buckingham, “Theory of acoustic attenuation, dispersion, and pulse propagation in
unconsolidated granular materials including marine sediments,” J. Acoust. Soc. Am., 102,
pp. 2579-2596, 1997.


Rapid environmental assessment with ambient noise

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Rapid environmental assessment with ambient noise

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