Nonlinear frequency-dependent attenuation in sandy sediments Open Access

The accurate calculation of sound propagation in shallow-water waveguide has been shown to depend on the use of geoacoustic profiles and the correct frequency dependence of depth-dependent attenuation profiles. Sonar engineers require a correct estimate of sound transmission loss and time spread over a wide band of frequencies. Calculations with sediment attenuation with a linear frequency dependence were found to yield inconsistent results with careful sound transmission loss and time spread measurements. However, the oceanic community has been slow in accepting the necessity for such a nonlinear dependence. Here we adopt a geophysical hypothesis that those areas primarily formed by deposition of sediments and sea level changes will have similar properties. The coast of the Eastern Gulf of Mexico, the East Coast of the United States between Cape Hatteras and Long Island, sections of the Asian Coast, the North Indian Ocean, the Strait of Korea, the East China and Yellow Seas, to mention a few, have sandy depositional layers that are classified as fast or critical-angle bottoms formed by the raising and lowering of the sea. In order to compare results between experiments consistent semantics need be used. Here the American Geophysical Union Classification system characterization of sediments composed of sand, silt, clay by particle size is employed. This characterization is shown on the diagram of Shepard¹ where such terms as sand, silt-sand, and clayey are consistent with the lower left-hand portion of this sediment triangle. The eastern coastal margins and seas examined here are between 40 and $200m$ deep and have consistent bottom sediment layers formed by deposition of sands and silt.

We exclude many other coastal areas where volcanism has resulted in a thin sediment layer over layers of hard substrata. Another exception are areas determined by the depositional fans of major rivers composed of fine silts and the resulting slow bottoms, i.e., the sound speed is less than that in the overlying water. These fans are a consequence of river outflow and fine dendritic materials are constantly being added to the rivers by land runoff and human activity.

This paper summarizes experimental evidence for the exponent power, $n$⁠, of the frequency dependence of the attenuation (see summary in Table I) by the following.

The paper states that the accurate calculation of the transmission for narrowband and broadband sound in shallow-water waveguides, with a fast, sandy-silty bottom, requires nonlinear frequency dependence for attenuation in the upper sediment layer for frequencies less than a few kilohertz.

There are four techniques for in situ measurement of the frequency dependence of attenuation in ocean sediments currently found in the literature.

(1) Modal techniques incorporate a vertical hydrophone array and use dispersion or wave number analysis to separate modes (e.g., see Ref. 2). Comparison of measured mode properties to theoretical properties, assuming some bottom attenuation frequency dependence, reveals the best match to data.

(2) Transmission loss (TL) techniques use TL versus range measurements at multiple frequencies with narrowband sources or broadband with impulsive sources. The estimation of frequency dependence of the attenuation is accomplished by using geoacoustic profiles that are based on geophysical measurements and forward propagation calculations. A comparison between calculated and measured TL, along with a metric to determine best fit (e.g., see Refs. 3–5), provides the exponent of the frequency dependence and a confidence interval. The constraints imposed on this process are independently measured bottom properties and, consequently, a reduction in the number of free-ranging parameters.

(3) Inversion schemes are also used and encompass many techniques, such as generic algorithms or perturbative inversions. These methods use a multi-parameter space and global search techniques, such as simulated annealing (e.g., see Ref. 6). The lack of constraints on key parameters and knowledge of the underlying probability distributions often leads to uniqueness problems and inconsistent results with the actual bottom. These techniques are numerically intensive, have only rarely been tried over a broad frequency band, and are the subject of current research.

(4) Finally, measurement of the angle-dependent reflection coefficient successfully used at higher frequencies may also be possible with narrow beam sources and line arrays at the lower frequencies. This method often makes use of the fact that the subcritical reflection loss depends on the sediment attenuation for fast bottoms; in the case of a slow bottom, the reflection coefficient at the angle of intromission depends on the sediment attenuation (e.g., see Ref. 7).

Nonlinear frequency-dependent sediment attenuation is not a new finding in shallow-water waveguides with fast, sandy bottoms. In 1973, Ingenito² observed a nonlinear dependence between 400 and $750Hz$⁠. Nine years later, for two sites with medium to coarse sand, Beebe³ found factors between 1.57 and 1.83. In the mid-1980s, Rogers et al.⁶ summarized several previous experiments^8,9 in the Yellow Sea and observed that the frequency-dependent attenuation factor ranged between 1.6 and 1.9. In 1993, Tappert¹⁰ found in his analysis of an experiment on the West Coast of Florida that $(n=2)$ was the best-fit parameter consistent with the Biot porous medium model. Tatersal and Chizhik (1992–1993)^11,12 used the Biot theory to compute sediment parameters for a fluid bottom and an elastic bottom; calculations with a fast-field program and a multipath expansion code were found to agree with measured transmission loss over the $100Hz$ to $8kHz$ range when the Biot bottom was used with a frequency-dependent $(n=2)$ attenuation coefficient. Additionally, Rozenfeld¹³ found a frequency-dependent power of $(n=1.8)$ was required to describe both narrowband and broadband TL measurements in the complex Strait of Korea. Results from experiments in the East and South China Seas conducted by the Office of Naval Research in 2003 and reported by Peng et al.¹⁴ and Zhou et al.⁹ showed dependencies between 1.6 and 1.63, while the results of Knobles et al.¹⁵ were inconclusive.

Evans and Carey⁴ showed that the accurate calculation of the shallow water sound transmission in a waveguide with sandy-silty bottom required a nonlinear frequency-dependent attenuation, with an exponent of $(n=1.5)$⁠, relative to a reference frequency of $50Hz$ over the interval $50–1000Hz$⁠. Dediu, Carey, and Siegmann⁵ reexamined this analysis using $1kHz$ as the reference frequency with the attenuation constant in the range specified by Hamilton et al.¹⁶ and a refined sound speed profile determined from the experimental measurements. Extensive comparisons between measured and calculated transmission loss for frequencies between $400Hz$ and $1kHz$ yield a frequency power of $(n=1.85±0.15)$ and a reference-attenuation

Holmes et al.^17–19 conducted the Nantucket Sound Experiment to quantify the acoustic properties of the bottom sediment between 220.5 and $1228Hz$ and to determine the frequency-dependent attenuation characteristics. The attenuation at $1kHz$ was assumed to be consistent with Hamilton’s value at that frequency. Iterative comparison of the measured to calculated transmission loss while adjusting the frequency-dependent attenuation revealed that the best fit of attenuation over all frequencies was

Table I provides a chronological summary of previous work on the frequency dependence of attenuation, including the bottom type, the type of experiment, and the estimated critical angle of the bottom as a basis of comparison. From the results in Table I, it is clear that there is substantial evidence that in the frequency range of $100Hz–1kHz$⁠, sediment attenuation follows a nonlinear frequency dependence with site dependent exponent $1.6<n<2.0$ for sandy-silty bottoms (see Refs. 1 and 16 for definition of sandy silt). These results compare with a summary by Zhou and Zhang²⁰ drawn from a larger and less restrictive group of experimental results, that is to say the inclusion of experiments with a greater uncertainty with respect to knowledge of the bottom sediment, yielded $n=1.84$⁠, $α(fo)=0.34$⁠, where the reference frequency was not specified.

The accurate calculation of propagation quantities in shallow-water range-dependent waveguides with a sandy bottom boundary requires specification of the sub-bottom sound speed, density, and attenuation profiles. Normally these are based on cores, grab samples, and sub-bottom profiling. The detailed structure is usually simplified to a three- to four-layer geoacoustic model. A key parameter in this model is the frequency-dependent attenuation profile in the near surface sediment layer. The experimental evidence summarized here shows that the near-water sediment layer has a nonlinear frequency dependence. The Biot theory^21–24 predicts an $(n=2)$ frequency dependence. A consequence of the modal nature of the shallow-water waveguide is that the measured dependence should, in most instances, be less than an exponent of $(n=2)$⁠, resulting from the depth dependence of the modal functions and the geoacoustic properties of the sediment.²⁴ Furthermore it is known that sound speed profiles and gradients can have an important influence on the attenuation of sound; however, range-dependent calculations that use sound speed profiles in the water and geoacoustic profiles in the sediment account for these effects. For such calculations to be accurate, the use of the sub-bottom nonlinear frequency dependence is required but a dependence of $(n=1.8)$ is found to provide the best fit. The $(n=1.8)$ dependence has not been explained.

The purpose of this paper was to provide a summary of experimental evidence for a nonlinear frequency-dependent attenuation estimated from sound transmission measurements in shallow-water areas with sandy sediments. This nonlinear dependence is

We conclude that nonlinear frequency dependence in oceanic sandy sediments is well established by the published experimental evidence.

We should also mention that early sedimentary measurements date back to those made by Wood and Weston²⁵ in 1964. However, their measurements considered a finer-grained low-density sediment (mud at porosity of 0.76) in Emsworth Harbor, England, and a wide frequency range of $4–50kHz$⁠. They demonstrated a linear dependence on frequency. In 1982, Beebe, McDaniel, and Rubano³ determined that a linear dependence was required to provide an adequate fit to data for a mud sediment. A recent study,²⁶ performed south of New England in the middle Atlantic bight, for a silt-clay sediment, used a multi-parameter inversion scheme and concluded that the measurements of the attenuation values were in agreement with the previous work.²⁸ The physical reason for linear frequency dependence would be a decrease in the relative particle velocity of the liquid and sediment particles, as discussed by Stoll¹⁶, pp. 1–15].