Feature-based maximum entropy for geophysical properties of the seabed^a)

As part of a larger experiment in May 2022, whose purpose was to characterize the seabed structure on the New England shelf and slope, acoustic recordings were made on a 16-element vertical line array (VLA) during the passage of a towed broadband source. The source tow track and position of the VLA were on the 200 m contour, which is the approximate position of the shelf break. An examination of the time-frequency spectrograms revealed features in the form of peaks and nulls in the received intensity time-frequency spectrograms at the closest position of approach (CPA) time. The densities of the peaks and nulls generally increased at the shallower receiver depths. It was ascertained in a pre-analysis of the data that the details of the frequency-dependent acoustic intensity were sensitive to the properties of the seabed, source-receiver range, and depth of the receiver.

The properties of the sound speed profile (SSP) and depth of the source (120 m) permit a statistical inversion of seabed properties using a ray analysis. A finite sediment layer over a basement halfspace are represented by the visco-elastic viscous grain shearing (VGS) model (Buckingham, 2000, 2007). The VGS model parameter space includes marine sediment properties, such as average porosity, grain bulk modulus, average grain density, and grain size, and several additional parameters that characterize the dynamics of a grain-to-grain contact. The model output is the frequency-dependent sound speed, attenuation, and bulk density, which then allow the complex reflected component of the field to be computed. This paper explores the information content on the geophysical properties of the sediment from frequency features which arise from the coherent combination of the direct and reflected ray components.

The remainder of this paper is organized as follows. Section II discusses the acoustic measurements and Sec. III presents a ray methodology for computing the acoustic field. Section IV presents the VGS model parameterization and prior assumptions. Section V presents the statistical inversion method and Sec. VI presents the results. A conclusion is presented in Sec. VII.

During Seabed Characterization Experiment (SBCEXP) 2022, the acoustic data were collected on a VLA deployed on the 200 m depth contour near the shelf break. Figure 1 shows the track of the RV Armstrong during the acoustic measurements. The VLA was located at 40.04847° N 70.88417° W. The data of interest are time recordings from a towed source on Julian day 148 from 1216 to 1800 UTC, which emitted chirp signals in the 500–1250 Hz band, and a second source (on the same tow body and clock), which emitted chirp signals in the 1500–4500 Hz band. The ship speed was about 2.0 kn. The CPA occurred at about 1626 Z, and the CPA range was about 390 m.

SSPs, also displayed in Fig. 1, were derived from a conductivity, temperature, and depth (CTD) measurement near the VLA. Starting at about 120 m, the depth variability of the sound speed is small. For this specific experiment, the placement of the source and receiver mitigates range-dependent effects due to spatial SSP variability.

Time-frequency spectrograms are depicted in Fig. 2(a) for channel 03 and Fig. 2(b) for channel 04, which were processed by short-time Fourier analyses of the recorded time series. The processed spectrograms have a frequency spacing of about 0.19 Hz from 500 to 4500 Hz and a time spacing of about 8.9 s. At CPA, the peak and null levels in the acoustic intensity are about 110 and 80 dB re 1 μ $Pa 2 / Hz$⁠, respectively.

The SSP and source depth combined with the position of the VLA affords an opportunity to analyze the data using a ray analysis as depicted in Fig. 3. The reflection coefficient, $R$⁠, is a total plane wave reflection coefficient that includes specular reflection at the water sediment interface and all multiple reflection within the h₂ layer.

The seabed model, depicted in Fig. 4, assumes a sediment layer of finite thickness, h₂, overlying a halfspace. Each sediment layer is parameterized in terms of the geophysical parameters in the VGS model (Buckingham, 2000, 2007). The compressional sound speed and attenuations in both layers, which are input into a normal mode model, are computed from the VGS model. The first sediment layer allows for a frequency-dependent response that can occur when the acoustic wavelength is on the order of a layer thickness.

The selection of a seabed model is based on a variety of factors, including model accuracy, the ability to describe a wide class of sediments such as those characterized by the Wentworth scale (Wentworth, 1922), and the ability to determine model parameter values without overfitting the data (Goodfellow , 2016). A useful form of a seabed model is one that obeys causality. Causal models restrict the frequency dispersion of the sound speed and attenuation to a Kramers–Kronig relationship (Toll, 1956; Jackson, 1975), which acts to reduce the dimensionality of the parameter space. Examples of such causal models include visco-eleastic models such as the silt suspension (Brown , 2020) and VGS formulations. Another class of causal models are poro-elastic models, which assume a sediment frame with pore water between the sediment structure. Such examples include the Biot-Stoll model (Biot, 1956, 1962; Stoll, 2002) and variants such as the Biot-squirt model (Kimura, 2011). These models are plagued by a large number of parameters whose values are difficult to determine and, as such, are susceptible to overfitting data. Another issue with poro-elastic models is that they typically are not valid for soft sediments. Chotiros (2021) has attempted to correct this issue by introducing a poro-elastic model for mud. Again, the issue with such extensions of a poro-elastic model lies with the large number of unknown parameters. In contrast, the VGS model is based on a simple version Zener model (Carcione, 2014) with a mechanical framework consisting of a Hookean spring and Newtonian dashpot. This lends itself to a modicum number of parameters and, as such, are less susceptible to overfitting data as compared to visco-elastic models.

Table I shows the parameter space for the two-layer model, which includes the assumed upper and lower parameter bounds. The upper and lower bounds for $ϕ$ are representative of a soft clay and a very coarse sand or granules, respectively. Also, the upper and lower bounds for the grain bulk modulus are representative of a very coarse sand and a soft sediment, respectively. However, there is significant uncertainty in correlating values of the $K g 3$ to sediment classification. As the sediment sound speed is highly sensitive to $K g 3$⁠, one can expect that there will be parameter correlations between $K g 3$ and $ϕ 3$⁠. The upper bound for n in Table I assumes that for a very hard sand, n is slightly higher than the value of n = 0.085, which was derived by Buckingham (2005) for a fine to medium sand sediment. Similarly, for a lower bound of n, a value of 0.05 is used because a previous study for mud sediments suggested a value of 0.06 (Knobles , 2021).

In the VGS theory, the effects of viscosity are introduced in terms of τ. From the analysis of experiments for sandy sediments, Buckingham (2005) determined a value of τ to be about 0.00012 s or log(τ/1 s)= –3.9208. This value reproduces the observed transition frequency in which the attenuation exponent transitions from a value of two to a value of 1/2 for data taken in medium sand. It was observed that this value of τ fits the reported attenuations from experiments in the Yellow Sea (fine to medium sand; Zhou , 1987) and a medium to coarse sand observed on a sand sheet on the New Jersey continental shelf (Knobles , 2008) and, more generally, the attenuation of the group of experimental data reported by Zhou (2009). If log(τ/1 s) is set to a large value (log(τ/1 s) = 4.0; grain shearing limit of VGS), the dispersion of the sound speed is very weak, which is consistent with the clayey sandy silt sediment studied during SBCEXP 2017 and other experiments with low-speed seabeds (Wilson , 2020; Boyles, 1997).

A cautionary note is that estimates for $τ ( N )$ for a sand sediment do not simultaneously fit reported values for sound speed and attenuation dispersion over the 10 Hz to 1 × 10⁶ Hz band. For example, Chotirous and Isakson (2012) have shown that sound speed and attenuation dispersion of the SAX99 data can be well fit using a variant of the Biot model. In contrast, the VGS model, while doing a good job of fitting the attenuation, predicts sound speed ratio values that fall slightly below the data for f  > 5 kHz. This is likely not a serious concern in this study because the bandwidths of the processed data are 500–4500 Hz.

The pdf for h₂ does not have a Gaussian-like shape but rather has its largest values from about 0 to 0.2 m, and then rapidly decreases to zero at a thickness of about 0.4 m. An estimate of the thickness from Table II suggests that the first layer has thickness of less than or equal to 0.20 m. The pdf for $ϕ 2$ has a broad peak centered at a value at about 0.45, which is a fine sand classification. The long tail of the pdf causes the average value to exceed 0.50. The pdfs for $K g 2$⁠, n₂, and τ₂ are essentially flat, i.e., they convey no information about the parameter values. The pdf for $ϕ 3$ has a broad peak centered around 0.43, which has a classification of fine sand. The average value and peak value of the pdf have about the same value. The pdfs for n₃ and τ₃ are approximately flat, however, the pdf for $K g 3$ suggests that $K g 3$ > 25 GPa. The pdf for the CPA range has a sharp peak at about 388 m, which is consistent with the experimental measurement.

The optimal parameter values for $H$ from Table II are used to compute the sediment sound speeds, attenuations, and bulk densities, which are then used by the ray model to predict the received levels at CPA. Table III shows, for example, the predicted sediment sound speeds and attenuations in the 500–4500 Hz processing band for layer 3. Figure 6 depicts a comparison of the ray theory results with the measured spectrograms for channels 03 and 04. The nulls of the ray theory prediction of the acoustic intensity for channels 03 and 04 are in good agreement with the observed null frequencies. The optimal values for $H$ from Table II were also used to compute the ray model at CPA for the data samples not used in the optimization, specifically, feature frequencies extracted from channels 02 and 05. Figure 7 displays a comparison of the ray theory results with the measured spectrograms for channels 02 and 05. The nulls of the ray theory prediction of the acoustic intensity for channels 02 and 05 are qualitatively in good agreement with the observed null frequencies.

Figures 8 and 9 compare the measured spectrograms with the modeled spectrograms produced by a normal mode algorithm (Westwood , 1996) that uses optimal values for $H$ from Table II for channels 03 and 04, respectively. The model source levels are those from the implicit formulation (Tollefsen , 2022; Knobles, 2015). The assumed ship speed was the reported speed of the tow ship (2.0 kn). It is important to note that the measured striation pattern is not fully symmetric about the CPA range, which may be ascribed to an unknown range dependence along the track, or nonuniform speed and source level, or a combination of all these factors.

The seabed parameters inferred using data at short ranges can be used to make predictions at significantly longer ranges. An example is shown in Fig. 10 which compares the measured spectrogram with the modeled spectrograms produced by a normal mode algorithm that uses optimal values for $H$ from Table II for channel 03. Qualitatively, the modeled and measured received levels over the 10.5 km range scale are in agreement. However, at the longer ranges, the data do not possess the time-frequency coherent striations predicted in the simulated spectrogram. This comparison suggests the existence of an unknown mechanism that causes a loss of coherence in the acoustic field with increasing range.

A SBCEXP was performed at the New England shelfbreak. The experimental setup afforded an acoustic ray analysis method with the recombination of the direct and bottom reflected paths. The analysis introduced concepts that included feature-based inversion and model dimensional reductions. This feature-based inversion identifies a unique feature in the acoustic field and then formulates a statistical approach to infer parameter values for a seabed model.

The concepts were applied to a statistical analyses of chirp tow data in the 500–4500 Hz band. Estimates of parameter values for the porosity, grain bulk modulus, strain hardening index, and viscous time constant were made for two sediment layers. The resulting probability density indicates that the most probable characterization of the seabed is a thin (<0.2 m) sand layer over a thicker amount of sand. The optimal inferred seabed model using acoustic measurements made at a range of about 390 m was successfully employed to predict the received acoustic levels over a range scale of about 10.5 km. However, there appears to be an unknown mechanism that causes a loss of intensity striations in the time/range frequency with increasing range.

This work was supported by the Office of Naval Research, Contract No.00014-22-1-2172.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from W.H.