The existence of acoustic velocity dispersion and frequency dependence of attenuation in marine sediments is investigated using *in situ* measurements from a wideband acoustic probe system during the Shallow Water 2006 experiment. Direct-path pulse propagation measurements show evidence of velocity dispersion within the $10\u201380kHz$ frequency band at two silty-sand sites on the New Jersey Shelf. The measured attenuation in dB/m shows linear frequency dependency within the $10\u201380kHz$ frequency band. The measured velocity dispersion and attenuation curves are in good agreement with those predicted by an extended Biot theory [Yamamoto and Turgut, J. Acoust. Soc. Am. 83, 1744–1751 (1988)] for sediments with a distribution of pore sizes.

## I. Introduction

Compressional wave velocity and attenuation are two of the most important geoaocustic parameters that control sound propagation in shallow water. The existence of velocity dispersion and nonlinear frequency dependence of attenuation within the seabed might drastically change the predictions of propagation models that commonly use constant velocity and linear frequency dependence of attenuation. Several measurement techniques have been used to measure the compressional wave velocity and attenuation at different frequency bands in different types of marine sediments. Strong velocity dispersion in well-sorted beach sediments in the $1\u201330kHz$ frequency band has been observed by using cross-probe measurements.^{1} Critical-angle measurements of reflection coefficient^{2} and shotgun/sediment-probe measurements^{3} also indicated strong velocity dispersion in granular marine sediments. A lesser degree of velocity dispersion within the $25\u2013100kHz$ frequency region has been observed in medium-sand sediments in the Gulf of Mexico by using a sediment probe system.^{4} More recent time-of-flight measurements by using buried source and receivers in the same area showed strong velocity dispersion within the $1\u20135kHz$ region.^{5} A reflection measurement technique^{6} showed almost negligible velocity dispersion within the $0.1\u201310kHz$ frequency region in silty-sand sediments of Malta Plateau and New Jersey Shelf. An earlier version of the wideband acoustic probe system^{7} showed weak velocity dispersion in silty sediments and no velocity dispersion in muddy sediments within the $20\u2013100kHz$ frequency band. In this paper, wideband $(10\u201380kHz)$ measurements of velocity dispersion and attenuation in silty sand are reported.

In Sec. II, a brief description of measurements and methods for measuring *in situ* velocity dispersion and attenuation are given. Then, measured velocity dispersion and attenuation are compared with those calculated from an extended Biot model.^{8} In Sec. III, the extended Biot model is briefly described and sound speed and attenuation predictions are presented for sediments having different distribution of pore sizes. Finally, in Sec. IV, summary and conclusions are given.

## II. Experiment

As a part of sediment charcterization effort during the SW06 experiment, a wideband acoustic probe system was deployed at two different sites on the New Jersey Shelf. At Site-1 (39.0235N, 73.0348W), additional high-frequency scattering and propagation measurements were conducted. At Site-2 (39.00145N, 73.1202W), independent measurements of sound speed, density, porosity, and grain size were available from a $3-m$-long sediment core. The acoustic probe system includes a self-contained data acquisition unit, four acoustic probes (one source and three receivers), and about $400kg$ of lead weights. The self-contained data acquisition unit is programmed on board the research vessel based on a planned pulse transmission schedule. The system can be operated at water depths up to $2000m$ with up to $2m$ penetration into the sediment. Each probe is $20mm$ in diameter and has an adjustable length. The acoustic transducers are cylindrical rings with $OD=15mm$, $ID=12mm$, and $height=20mm$. Loading effects might be significantly different when the acoustic transducer is placed in hard (sandy) sediments than those in the water column. These differences might introduce measurement errors if the waveforms measured by a single receiver in the water and sediment. In our measurements, the wideband pulse signals received at two buried receivers are used for measuring velocity and attenuation as a function of frequency. Spectral ratio method^{1} is used to calculate and phase velocity and attenuation from the measured pulse spectra. The phase delay between two receivers is calculated as

where $S1(\omega )$ and $S2(\omega )$ are spectra of the pulses received at distances $d1$ and $d2$, respectively. When the distance between two receivers is larger then the acoustic wavelengths, the above equation provides a wrapped phase delay in the range $[\u2212\pi \pi ]$. To remove the phase wrapping, a reference velocity $c0$ is used to align the received pings by applying a constant phase shift to $S2(\omega )$ as $S2\u2032(\omega )=S2(\omega )exp[\u2212i\omega (d2\u2212d1)\u2215c0]$. In Eq. (5), the spectrum $S2(\omega )$ is replaced by $S2\u2032(\omega )$ and a new phase delay $\Delta \varphi \u2032$ is obtained. Then, the phase speed is calculated using the new phase delay as

The spectral ratio of two pulses received by two receivers is also used for measuring attenuation as a function of frequency. Spectral ratios are calculated both in the water column and sediment so that geometrical spreading and different receiver sensitivities can be taken into account automatically. The attenuation in dB/m is calculated from the spectral ratio as

where $Q1(\omega )$ and $Q2(\omega )$ are spectra of the pulses transmitted in water and received at distances $d1$ and $d2$, respectively. Three Gaussian-windowed LFM pulses ($5\u201325kHz$, $20\u201350kHz$, and $40\u2013100kHz$) are transmitted by the source transducer, and measured pulse spectra $S1(\omega )$ and $S2(\omega )$ are used in Eqs. (1)–(3) to calculate phase velocity and attenuation. Figure 1 shows the measured phase velocity and attenuation at Site-1 and Site-2. Note that compressional wave velocity gradually increases for each frequency band from $10to80kHz$. The velocity dispersion is slightly stronger at the Site-1 especially within $30\u201340kHz$ frequency band. The gaps in the measured data are due to the discarded low signal to noise ratio regions at both sides of the Gaussian-shaped spectra. An extended Biot theory, described in the next section, was used to calculate velocity and attenuation curves for silty sand with a log-normal pore-size distribution. The sediment physical properties, used in the calculations, are inferred from the core data and given in Table I. The calculated phase velocity and attenuation curves are also plotted in Fig. 1 with pore-size standard deviations of $\sigma =0.0$, $\sigma =1.25$, and $\sigma =2.0$. The Biot theory with the uniform pore size assumption $(\sigma =0.0)$ predicts a dispersion region below $10kHz$ and there is no significant dispersion within the $10\u201380kHz$ frequency band. The extended Biot theory predictions agree with the measurement results when silty-sand sediments, with pore-size standard deviations between $\sigma =1.25$ and $\sigma =2.0$, are considered. In Fig. 1(b), measured and calculated attenuation coefficients are compared. The measured attenuation values are in agreement with the theoretical predictions when sediments with distributed pore sizes are considered. Figure 1(c) shows velocity and porosity profiles near Site-2, obtained from the sediment core analysis. At the probe depths $(0.5m)$, the sound velocities measured by a $500kHz$ acoustic core logger are slightly higher than those measured by the *in situ* acoustic probe system. Grain size analysis at the probe depth $(0.5m)$ showed a mixture of gravel (11%), sand (70%), silt (12%), and clay (7%) at Site-2. Assuming a correlation between grain size and pore size distributions, this confirms our acoustic prediction of nonuniform $(\sigma \u22600)$ pore size distribution. The frequency dependency of the attenuation in sediments with various pore-size distributions will be elaborated in the next section.

## III. Extended Biot theory predictions

Marine sediments can be regarded as a fluid-filled porous medium, and acoustic wave interaction with the bottom can be physically modeled by Biot’s theory.^{9} An extension of the Biot theory has been developed^{8} for marine sediments with statistically distributed pore sizes and successfully validated for air-filled porous granular materials.^{10} In the Biot theory, acoustic wave attenuation is primarily attributed to the viscous losses due to relative motion between the pore fluid and the skeletal frame. The viscous interaction is assumed to take place according to Darcy’s law of fluid flow through porous media with the modification that the viscosity is made dependent on the frequency of the elastic wave. A frequency dependent viscosity correction factor, $F$, is defined by comparing the ratio of the total friction to the average fluid velocity in the oscillatory flow and steady laminar flow regimes. It is also assumed that the variation of viscous friction with frequency in a porous material follows the same laws as that in a single pore represented by a two-dimensional duct or a capillary tube of uniform cross section. Yamamoto and Turgut^{8} have suggested a new viscous correction factor by calculating total viscous resistance and average seepage velocity for a macroscopic unit element with nonuniform pore size distribution. The new viscous correction factor is calculated as

where $\kappa =r(\omega \rho f\u2215\eta )1\u22152$ is a nondimensional parameter, $\omega $ is the angular frequency, $ks$ is the permeability, $\beta $ is the porosity, $r$ is the pore radius, $e(r)$ is the pore radius distribution function, $\eta $ is the dynamic viscosity, $\rho f$ is the pore fluid density, and

in which ber and bei are the real and imaginary parts of Kelvin function and ber’ and bei’ are their derivatives.

In Fig. 2, pore size distributions of New Jersey Shelf sediments, measured by mercury injection technique, are shown. The pore-size distributions in Figs. 2(a) and 2(b) are approximated as log-normal [or $\varphi $-normal, $\varphi =\u2212log2r$ ($r$ in mm))] radius distributions with the standard deviation $\sigma $ and mean $\varphi \xaf$ [$\sigma =1.5$, $\varphi \xaf=5.5$ in Fig. 2(a), and $\sigma =2.2$, $\varphi \xaf=7.8$ in Fig. 2(b)]. In Fig. 3, the measured and calculated phase velocity and attenuation curves are plotted in logarithmic scale to identify the velocity dispersion regions and to examine the frequency dependency of attenuation. In Figs. 3(a) and 3(b), it can be seen that, in sediments with nonuniform pore size distribution, the velocity dispersion and nonlinear attenuation regions due to viscous friction, are widened and shifted toward higher frequencies. Again, theoretical predictions by using $\sigma =1.25$, and $\sigma =2.0$ agree with the measured velocity and attenuation much better than that of using $\sigma =0.0$. In Figs. 3(c) and 3(b), theoretical predictions of frequency dependency of attenuation are shown for $\sigma =0.0$, $\sigma =1.25$, and $\sigma =2.0$. Using the definition $\alpha =kfn$ ($k$ is a constant and $f$ is the frequency in kHz), the power exponents of $n=0.6$, $n=0.79$, and $n=1.0$ are found by line fitting within the $10\u2013100kHz$ band. Similarly, within the $0.1\u20131kHz$ frequency band, power exponent values of $n=1.74$, $n=1.45$, and $n=1.35$ are found for $\sigma =0.0$, $\sigma =1.25$, and $\sigma =2.0$, respectively. Sediments with sand, silt, and clay mixture are represented by higher values of $\sigma $, while sediments with more uniform grain sizes are represented by smaller value of $\sigma $, indicating more uniform pore sizes. Within the $0.1\u20131kHz$ frequency band, the estimated power exponents are in good agreement with that of several low-frequency measurements, recently reported by Holmes *et al.*^{11}

## IV. Summary and conclusions

*In situ* measurements of sound speed and attenuation were performed by using a wideband acoustic probe system, deployed at two silty-sand sites on the New Jersey Shelf. The spectral ratios of broadband pulses, transmitted between the $0.5-m$-deep probes, showed phase velocity dispersion within the $10\u201380kHz$ frequency band. The frequency region of observed velocity dispersion seems to be higher than that of previous measurements, performed in well-sorted beach sand and granular marine sediments.^{1,3,5} As compared to the original Biot theory predictions, measured phase velocity dispersion results agree better with those predicted by the extended Biot theory with nonuniform pore size distribution. The observed linear frequency dependency of the attenuation also agrees with the theoretical predictions when sediments with nonuniform pore size distributions are considered.

## Acknowledgments

This work was supported by the Office of the Naval Research. The authors thank the crew of the R/V Knorr for the excellent support during the 2006 Shallow Water Experiment. They also thank Chief Scientist Dr. D. J. Tang for the scientific support.