
061215-092
1
A Physical Explanation For Less Than Quadratic
Recorded Attenuation Values
Jon M. Collis, William M. Carey, and Allan D. Pierce

Abstract— A review of experimental evidence shows that the
low frequency attenuation of compressional acoustic waves in
sandy marine sediments obeys a simple power law consistent with
a
frequency
dependent
attenuation
proportional
to
f n , n  1.8 .
This observation is in general agreement with
Biot’s model (1956) for frequencies less than 1 kHz. Recent
theoretical work [A. D. Pierce et al., Proc. Oceans 2005, Brest, FR.
(2005)] shows that in this lower frequency range, a quadratic
dependence (n=2) is to be expected and is decreased by
predictable modal propagation characteristics in shallow water.
Physical properties and depth-dependent characteristics of the
sediment are used to explain the apparent n=1.8 power law
dependence of attenuation. Calculations performed using
parabolic equation codes with simplified Biot geoacoustic profiles
are compared to experimental results.
Index Terms— ocean
nonlinear attenuation
acoustics,
sediment
attenuation,
I. INTRODUCTION
S
calculations to explain the less than quadratic frequency
dependence, n  1.8 , of compressional attenuation and the
importance of the simplified geoacoustic profile. To quantify the
attenuation properties of shallow-water waveguides, various
physical mechanisms are considered as possible influences on

propagation.
In particular, the influence of sound speed and
attenuation gradients and effects due to elasticity are discussed.
II. BACKGROUND
The focus of this discussion is on the consequences of
frequency dependence of sediment volume attenuation in the
uppermost sediment layer. Nonlinear frequency dependent
attenuation is expected in the water saturated portion of the
sediment, especially the first 5 m [5]. The analysis presented
herein is based on previous experimental results [1,2] and
comparisons follow the procedure developed in [5]. Parabolic
equation propagation codes have been modified to apply a
nonlinear frequency dependent attenuation in the first layer of
sediment, determined by
 f n
 (z, f )   (z, f 0 )  ,
 f 0 
where f 0 is the reference frequency, n is the frequency
exponent, and  (z, f 0 ) is the intrinsic attenuation profile in
attenuation has an important effect in many
underwater acoustics problems, in particular those of shallow
waters. Results have shown that the accurate calculation of
the acoustic field in a shallow-water waveguide with sandy
sediment bottoms requires a nonlinear frequency-dependent
dB/m at the reference frequency. Note that the reference
compressional wave attenuation in the upper layer of sediment
frequency
 is chosen to be 1 kHz and thus for frequencies less
[1]. Careful examination of experimental data, summarized in [2],
than
1
kHz a lower n value results
 in greater attenuation.
show that the accurate calculation of shallow-water sound

transmission between 100 Hz and 1 kHz in a waveguide with
sandy-silty bottom requires a nonlinear frequency-dependent
attenuation, n  1.8  0.2 , in the near-water sediment layer.

However, the simplified theory of Biot [3,4] predicts a quadratic
(nonlinear) frequency dependence, n  2 . Though both theory
and experiment show nonlinear frequency dependence, there is a

difference
between theory and experiment.
Intrinsic sediment attenuations can be estimated from careful

sound transmission measurements
and numerical methods.
These numerical methods usually employ simplified geoacoustic
representations of sediment characteristics such as a three-layer
sediment profile. This paper discusses parabolic equation
EDIMENT
Manuscript received March 30, 2007. This work was supported by the
U.S. Office of Naval Research, in particular an Ocean Acoustics Postdoctoral
Fellowship Award to the first author.
J. M. Collis, W. M. Carey, and A. D. Pierce, are with the Aerospace and
Mechanical Engineering Department, Boston University, Boston, MA 02215
USA (Corresponding Author: Jon Collis, phone: 508-540-8665; fax: 617-5555555; e-mail: jcollis@ bu.edu).
Fig. 1. Effective attenuation as a function of frequency for constant versus
gradient profiles. Note that a frequency dependent attenuation of n=1.8 results
in greater attenuation for frequencies less than 1 kHz.
061215-092
2
Fig. 2. Effective attenuation coefficient versus frequency with constant
profiles. Fluid and elastic parabolic equation models are compared for different
frequency exponent n .

A metric for the influence of nonlinear frequency dependence
of sediment attenuation on the range degradation of
transmission
loss is the effective attenuation coefficient (EAC),

defined as the slope of the linear least square fit to range- and
depth-averaged transmission loss. For each comparison the
calculated depth-averaged transmission loss data are first range-
averaged over the length of the propagation track using a moving
window. Then the window-averaged calculated transmission

losses are fit using the expression,



depth-averaged transmission loss, and used measured
geophysical properties, water sound-speed profiles, and
bathymetry, to a maximum range of 26 km, with a source depth
of 36 m, for 25 m range steps. Finally, the depth-averaged
transmission loss was range averaged over the interval 3-21 km
using a 1 km moving window. A sub-bottom attenuating layer
was used to avoid artificial bottom reflections.
Frequency-dependent attenuation was applied in the upper 5
m of sediment, with a reference frequency f 0 = 1000 Hz, and
unless noted otherwise, the frequency exponent n  2 . The
intrinsic attenuation is  (z, f 0 ) = 0.35 dB/m. Comparisons are
of the EAC, calculated at 10 frequencies, 100*i, i=1..10 Hz.
The first case, A, considersthe influence of sedimentary
sound speed and attenuation gradients. 
Could the surface sound
speed gradient resulting from changes in porosity explain the
n  1.8 
frequency dependence of compressional wave
attenuation? The gradient sound speed profile in the first 5 m of
sediment is determined by
c(z)  c(z0 )  (c(z1 )  c(z0 ))
1 e(zz0 )
,
1 e(z1 z0 )
where z is the depth (in meters) below the ocean surface, z 0
and z1 are the depths of the ocean bottom and first sedimentsediment interface, c is the sediment sound speed, and the
parameter  determines the gradient.
 Hz, for a
Figure 1 shows the EAC between 100 and 1000
 gradient profile (  1.8,n  2.0 ) and two constant sound
 with n  1.8 and n  2.0 . The gradient profile
speed profiles
TL   eff r  b  10log( r),
( n  2.0 , dashed-dotted line) and constant profile ( n  1.8 ,
dashed line) both indicate greater attenuation than predicted by
where r is range in meters,  eff is the EAC, and the third term
the constant
 profile ( n  2.0 , solid line). It can be seen that the
is cylindrical spreading. The EACs are slopes of the least squares gradients
result in increased

effective attenuation. The slope of
fit of 
the range- and depth-averaged transmission loss
 with the gradient profile result differs from 
the constant profile
cylindrical spreading removed.
results and this difference increases with frequency. The
The environment
 chosen for this analysis is the New Jersey differences
 between the constant profile cases, n  1.8 and
Continental Shelf, using a simplified model representation
n  2.0 , decrease with increasing frequency.
developed from past experiments [5]. Comparisons between
The second case, B, considers that elastic effects of the
measured and calculated transmission loss for frequencies bottom are an alternative physical mechanism causing
between 400 Hz and 1 kHz, have indicated a frequency power of attenuation of compressional acoustic waves.
 The question asked
1.8  0.2 and the attenuation at the water-sediment interface
 to is whether shear waves and their attenuation can explain the
be 0.35  0.02 dB/m [6]. The measured bathymetry was almost observed frequency dependence? The geoacoustic profiles are
range-independent at 73 m, the sub-bottom fairly uniform, and the same as that used in the previous case, but with a shear wave
was approximated by a 3-layer sediment model. The top layer is speed profile of c  300 m/s. Shear wave attenuation is
s
5 m thick with compressional sound speed 1560 m/s, density
assumed frequency dependent in the first 5 m of sediment with a
3
1.86 g/cm , and attenuation determined by the nonlinear sound
surficial attenuation value of 3.5 dB/m. The shear attenuation is
speed dependence. The second layer is 5 m thick with
compressional sound speed of 1610 m/s, density 1.96 g/cm3, assumed 2.0 dB/  below 78.0 m.
Figure
 2 shows the results (presented in A) for constant
and attenuation 0.7677 dB/  . The bottom layer has
profiles ( n  1.8 , dashed line and n  2.0 , solid line) with the
compressional sound speed 1740 m/s, density 2.09 g/cm 3, and
elastic constant profile ( n  2.0 , dashed-dotted line). An
attenuation 1.511 dB/  .
increased
 value of the effective attenuation coefficient is
observed. The differences between these results reduce as the
III. EXAMPLES

 frequency approaches 1 kHz. The result for the elastic
Numerical calculations using parabolic equation codes [7,8], geoacoustic profile
 behaves quantitatively the same as the fluid
were performed
with a nonlinear frequency dependent sediment geoacoustic profile result for n  1.8 , with increased

attenuation in the first layer of sediment. Calculations produced attenuation and differences that decrease with frequency.

061215-092
3
Other possible variations have been considered, though are
not shown here. Changing the surficial shear attenuation to 0.66
dB/m or changing the shear attenuation at 78.0 m to 0.2 dB/ 
has little or no effect on EAC values. Assuming that the shear
attenuation is not frequency dependent, yields slightly smaller
EAC values, but does not change the trend of the plot.
Assuming a constant sound speed of 1520 m/s
in the water,
instead of downward refracting, causes EAC values to fall
dramatically. Range-dependent bathymetry can have an effect
similar to surface gradients, though this is highly dependent on
the nature of the bathymetry. Different layer approximations of
the sediment environment have had little effect on EAC line plot
trends.
IV. CONCLUSION

Theory predicts a quadratic frequency-dependent sediment
attenuation, n  2.0 , yet experimental evidence is consistent
with n  1.8 . Sediment sound speed gradients in the first 5 m
of sediment can cause an increase in effective attenuation that is
proportional to frequency. While the higher attenuation observed
a result of the gradient profiles is consistent with an increased
as
attenuation expected with less than quadratic frequency
dependence, EAC value differences grow with frequency. The
variation of effective attenuation results with frequency was
found to not agree with the expected n  1.8 result.
Elasticity can result in greater effective attenuation than is
expected from a frequency exponent of n  2.0 . These effects
are near-inversely proportional to frequency and are in general
 expected values. Also, differences
agreement with n  1.8
between the fluid and elastic bottom cases for n  2 decrease
with increasing frequency. 


ACKNOWLEDGMENT
The authors would like to acknowledge the contribution of
Prof. William Siegmann for his discussions and insights into
shallow-water sound propagation. Discussions with Simona
Dediu are acknowledged with much appreciation. This work was
supported by the U.S. Office of Naval Research, in particular an
Ocean Acoustics Postdoctoral Fellowship Award to the first
author.
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[1]
[2]
[3]
[4]
[5]
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[6]
[7]
[8]
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