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061215-092
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 . 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 acoustics, sediment attenuation, nonlinear attenuation
I.INTRODUCTION
S
ediment 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 compressional wave attenuation in the upper layer of sediment [1]. Careful examination of experimental data, summarized in [2], 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, , in the near-water sediment layer. However, the simplified theory of Biot [3,4] predicts a quadratic (nonlinear) frequency dependence, . 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 calculations to explain the less than quadratic frequency dependence, , 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
where is the reference frequency, is the frequency exponent, and is the intrinsic attenuation profile in dB/m at the reference frequency. Note that the reference frequency is chosen to be 1 kHz and thus for frequencies less than 1 kHz a lower value results in greater attenuation.
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,
where is range in meters, is the EAC, and the third term is cylindrical spreading. The EACs are slopes of the least squares fit of the range- and depth-averaged transmission loss with cylindrical spreading removed.
The environment chosen for this analysis is the New Jersey Continental Shelf, using a simplified model representation developed from past experiments [5]. Comparisons between measured and calculated transmission loss for frequencies between 400 Hz and 1 kHz, have indicated a frequency power of 1.8 0.2 and the attenuation at the water-sediment interface to be 0.35 0.02 dB/m [6]. The measured bathymetry was almost range-independent at 73 m, the sub-bottom fairly uniform, and was approximated by a 3-layer sediment model. The top layer is 5 m thick with compressional sound speed 1560 m/s, density 1.86 g/cm3, and attenuation determined by the nonlinear sound speed dependence. The second layer is 5 m thick with compressional sound speed of 1610 m/s, density 1.96 g/cm3, and attenuation 0.7677 dB/. The bottom layer has compressional sound speed 1740 m/s, density 2.09 g/cm3, and attenuation 1.511 dB/.
III.Examples
Numerical calculations using parabolic equation codes [7,8], were performed with a nonlinear frequency dependent attenuation in the first layer of sediment. Calculations produced 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 = 1000 Hz, and unless noted otherwise, the frequency exponent . The intrinsic attenuation is = 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 frequency dependence of compressional wave attenuation? The gradient sound speed profile in the first 5 m of sediment is determined by
,
where is the depth (in meters) below the ocean surface, and are the depths of the ocean bottom and first sediment-sediment interface, is the sediment sound speed, and the parameter determines the gradient.
Figure 1 shows the EAC between 100 and 1000 Hz, for a gradient profile () and two constant sound speed profiles with and . The gradient profile (, dashed-dotted line) and constant profile (, dashed line) both indicate greater attenuation than predicted by the constant profile (, solid line). It can be seen that the gradients result in increased effective attenuation. The slope of the gradient profile result differs from the constant profile results and this difference increases with frequency. The differences between the constant profile cases, and , decrease with increasing frequency.
The second case, B, considers that elastic effects of the bottom are an alternative physical mechanism causing attenuation of compressional acoustic waves. The question asked is whether shear waves and their attenuation can explain the observed frequency dependence? The geoacoustic profiles are the same as that used in the previous case, but with a shear wave speed profile of m/s. Shear wave attenuation is assumed frequency dependent in the first 5 m of sediment with a surficial attenuation value of 3.5 dB/m. The shear attenuation is assumed 2.0 dB/ below 78.0 m.
Figure 2 shows the results (presented in A) for constant profiles (, dashed line and , solid line) with the elastic constant profile (, dashed-dotted line). An increased value of the effective attenuation coefficient is observed. The differences between these results reduce as the frequency approaches 1 kHz. The result for the elastic geoacoustic profile behaves quantitatively the same as the fluid sediment geoacoustic profile result for , with increased attenuation and differences that decrease with frequency.
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, , yet experimental evidence is consistent with . 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 as a result of the gradient profiles is consistent with an increased 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 result.
Elasticity can result in greater effective attenuation than is expected from a frequency exponent of . These effects are near-inversely proportional to frequency and are in general agreement with expected values. Also, differences between the fluid and elastic bottom cases for 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.
References
[1]I. Rozenfeld, W. M. Carey, P. G. Cable, and W. M. Siegmann, “Modeling and analysis of sound transmission in the Strait of Korea,” IEE J. Oceanic Eng., vol. 26, pp. 809–820, 2001.
[2]J. D. Holmes, S. Dediu, and W. M. Carey, “Low-frequency attenuation of sound in marine sediments,” Proc. Oceans 2005, Brest, FR, June 2005.
[3]M. Biot, “Theory of propagation of elastic waves in a fluid saturated porous solid. I. Low frequency range,” J. Acoust. Soc. Am.., vol. 28, pp. 168–178, 1956.
[4]A. D. Pierce, W. M. Carey, and W. M. Zampoli, “Nonlinear frequency dependent attenuation in sandy sediments,” IEEE J. Oceanic Eng., vol. 23, pp. 439–447, 2005.
[5]R. B. Evans and W. M. Carey, “Frequency dependence of sediment attenuation in two low-frquency shallow-water acoustic experimental data sets,” IEE J. Oceanic Eng., vol. 23, pp. 439–447, 1998.
[6]S. Dediu, W. M. Carey, and W. L. Siegmann, “Statistical analysis of sound transmission obtained on the New Jersey Continental Slope,” Proc. Oceans 2006, Boston, USA, Sept. 2006.
[7]M. D. Collins, “User’s Guide for RAM Versions 1.0 and 1.0p,” ftp://ftp.css.nrl.navy.mil/pub/ram/RAM, 1995.
[8]M. D. Collins, “A Higher-Order Parabolic Equation for Wave Propagation in an Ocean Overlying an Elastic Bottom,” J. Acoust. Soc. Am., 86, 1459–1464, 1989.
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: 617-555-5555; e-mail: jcollis@ bu.edu).