Modeling of subcritical penetration into sediments due to interface roughness

Testing the theoreticians

The Reflections series takes a look back on historical articles from The Journal of the Acoustical Society of America that have had a significant impact on the science and practice of acoustics.

The effect of seafloor roughness on passive estimates of the seabed reflection coefficient

In this work, a model is developed for the effect of seafloor interface roughness on passive estimates of the reflection coefficient. The main result is an expression for the total intensity reflection coefficient, with separate coherent and incoherent contributions. Assumptions of this model include constant sound speed in the ocean, stationary and Gaussian seafloor roughness, and ambient noise. Numerical examples for the coherent, incoherent, and total contributions to the intensity reflection coefficient are presented for halfspace and layered environments-all using the small slope approximation. To illustrate the potential parameter errors that results from using a flat interface wave model when roughness is present, a geoacoustic inversion is performed using the proposed model as input data. A joint roughness-geoacoustic inversion of simulated data using the proposed model was also performed. It was found that the true roughness and geoacoustic parameters can be inverted using this model, but the sensitivity to the outer scale of the rough surface has the highest error compared to the other parameters.

Acoustic Scattering Models from Rough Surfaces: A Brief Review and Recent Advances

This paper proposes a brief review of acoustic wave scattering models from rough surfaces. This review is intended to provide an up-to-date survey of the analytical approximate or semi-analytical methods that are encountered in acoustic scattering from random rough surfaces. Thus, this review focuses only on the scattering of acoustic waves and does not deal with the transmission through a rough interface of waves within a solid material. The main used approximations are classified here into two types: the two historical approximations (Kirchhoff approximation and the perturbation theory) and some sound propagation models more suitable for grazing observation angles on rough surfaces, such as the small slope approximation, the integral equation method and the parabolic equation. The use of the existing approximations in the scientific literature and their validity are highlighted. Rough surfaces with Gaussian height distribution are usually considered in the models hypotheses. Rather few comparisons between models and measurements have been found in the literature. Some new criteria have been recently determined for the validity of the Kirchhoff approximation, which is one of the most used models, owing to its implementation simplicity.

Scattering from layered seafloors: Comparisons between theory and integral equations

Acoustic scattering from layered seafloors exhibits dependence on both the mean geoacoustic layering, as well as the roughness properties of each layer. Several theoretical treatments of this environment exist, including the small roughness perturbation approximation, the Kirchhoff approximation, and three different versions of the small slope approximation. All of these models give different results for the scattering cross section and coherent reflection coefficient, and there is currently no way to distinguish which model is the most correct. In this work, an integral equation for scattering from a layered seafloor with rough interfaces is presented, and compared with small roughness perturbation method, and two of the small slope approximations. It is found that the most recent small slope approximation by Jackson and Olson [J. Acoust. Soc. Am. 147(1), 56-73 (2020)] is the most accurate when the root-mean-square (rms) roughness is large, and some models are in close agreement with each other when the rms roughness is small.

The small-slope approximation for layered, fluid seafloors

The small-slope approximation (SSA) for rough-interface scattering is most commonly applied to the upper boundary of either impenetrable media or uniform half-space media, but has been recently developed for layered media in the acoustic and electromagnetic cases. The present work gives an overview of three forms of the SSA for layered media. The first has been previously presented in the acoustics literature. The second is from the electromagnetics literature and in the present work is converted to the fluid-sediment problem. A missing proof is supplied of a key consistency condition demanded of the small-slope ansatz. As is usual, these small-slope results are expressed in k-space. A third SSA for layered seafloors follows from conversion of the usual half-space formulation from k-space to coordinate space. This form turns out to be useful for reverberation simulations. The three different approaches are compared with respect to scattering strength and the coherent reflection coefficient, but an assessment of their relative merits will require comparison with exact calculations.

The mutual scattering cross section

A generalization of the conventional interface scattering cross section is introduced. This new object will be called the mutual scattering cross section, and, like the conventional cross section, can be used in narrow-band sonar applications. It can treat both sea-surface and seafloor scattering and is useful in cases where large arrays are employed as well as in multipath environments. The application to large arrays with uniform half-space water column and seafloor is examined briefly, but the bulk of this article is devoted to multipathing in the ocean waveguide. Comparisons with more accurate, but more numerically intensive, approaches in range-independent environments show that the mutual cross section can provide an efficient solution for the reverberation intensity time series. The mutual cross section incorporates interference effects causing oscillations in the reverberation time series. Such oscillations have been reported in the literature, but previous modeling efforts have been ad hoc, not based on scattering physics. The mutual cross section is shown to model backscattering enhancement due to multipathing, another phenomenon not seen in simpler models. Expressions for the mutual cross section are derived for seafloor roughness scattering and sediment volume scattering.

A time-domain model for seafloor scattering

Bottom scattering is important for a number of underwater applications: it is a source of noise in target detection and a source of information for sediment classification and geoacoustic inversion. While current models can predict the effective interface scattering strength for layered sediments, these models cannot directly compute the ensemble averaged mean-square pressure. A model for bottom scattering due to a point source is introduced which provides a full-wave solution for mean-square scattered pressure as a function of time under first-order perturbation theory. Examples of backscatter time series from various types of seafloors will be shown, and the advantages and limitations of this model will be discussed.

Measurements of high-frequency acoustic scattering from glacially eroded rock outcrops

Measurements of acoustic backscattering from glacially eroded rock outcrops were made off the coast of Sandefjord, Norway using a high-frequency synthetic aperture sonar (SAS) system. A method by which scattering strength can be estimated from data collected by a SAS system is detailed, as well as a method to estimate an effective calibration parameter for the system. Scattering strength measurements from very smooth areas of the rock outcrops agree with predictions from both the small-slope approximation and perturbation theory, and range between -33 and -26 dB at 20° grazing angle. Scattering strength measurements from very rough areas of the rock outcrops agree with the sine-squared shape of the empirical Lambertian model and fall between -30 and -20 dB at 20° grazing angle. Both perturbation theory and the small-slope approximation are expected to be inaccurate for the very rough area, and overestimate scattering strength by 8 dB or more for all measurements of very rough surfaces. Supporting characterization of the environment was performed in the form of geoacoustic and roughness parameter estimates.

Analysis of shear-wave attenuation in unconsolidated sands and glass beads

Chotiros and Isakson [J. Acoust. Soc. Am. 135, 3264-3279 (2014)] contend that the physics-based grain-shearing (GS) theories of wave propagation in granular materials are not consistent with one particular shear-attenuation data set for water-saturated angular sand that has appeared in the literature. This provides them with the rationale for developing their own model, an extension of the empirical Biot-Stoll model, which they designate the Extended Biot (EB) model. In this article, the EB model and the grain-shearing theories are briefly reviewed, and it is demonstrated that, in fact, the original GS theory accurately matches the frequency-dependent trends of all the shear attenuation data sets that are currently available, including those for saturated angular sands after random fluctuations are suppressed by averaging over several realizations of the medium. It is also pointed out that Chotiros and Isakson's treatment of the available shear-attenuation data is highly selective, and that the format in which they present the selected data makes their comparisons with theoretical models difficult to interpret. Thus, their attempts at validating the EB model and their conclusions concerning alternative theories should be treated with caution.

Subcritical penetration into rough seafloors due to Bragg scattering

A tank experiment and theoretical analysis are carried out to study acoustic Bragg scattering by a sinusoidal surface with period 0.3 m and amplitude 2 cm between water and sand sediment. The penetrating field is measured in the frequency range from 20 to 40 kHz at grazing angles 10° to 90° in the tank. A theoretical solution for acoustic scattering by the sinusoidal surface is derived to explain the interference pattern observed in the experiment. The result shows that the minus first order Bragg scattering wave is strong enough to interfere with the refracted wave obeying Snell's law, forming interference patterns that can be detected experimentally.

Quantifying the effects of roughness scattering on reflection loss measurements

Seafloor reflection loss and roughness measurements were taken at the Experimental Validation of Acoustic Modeling Techniques experiment in 2006. The magnitude and phase of the reflection loss was measured at frequencies from 5 to 80 kHz and grazing angles from 7° to 77°. Approximately 1500 samples were taken for each angle. The roughness was measured with a laser profiler. Geoacoustic parameters such as water and sediment sound speed and density were measured concurrently. The reflection loss data were compared with three models: A flat interface elastic model based on geoacoustic measurements; a flat interface poro-elastic model based on the Biot/Stoll model; and a rough interface model based on the measured interface roughness power spectrum. The data were most consistent with the poro-elastic model including scattering. The elastic model consistently predicted values for the reflection loss which were higher than measured. The data exhibited more variability than the model due to layering and fluctuations in the propagating medium.

Evidence for a common scale O(0.1) m that controls seabed scattering and reverberation in shallow water

Analysis of the spectral content of long-range reverberation yields two observations. First, there is a remarkably similar scale, O(0.1) m, between three diverse continental shelf regions. This is surprising given the complexity and diversity of geologic processes. Second, there is strong evidence that the scale is associated with heterogeneities within the sediment. Thus, sediment volume scattering, not interface scattering, controls long-range reverberation from a few hundred hertz to several kilohertz. This is also unexpected given that at long ranges the vertical grazing angles are less than the critical angle, and hence the penetration of the acoustic field into the sub-bottom is expected to be modest. The consistency of the scale, O(0.1) m, suggests an underlying feature or mechanism that is consistent across many ostensibly diverse geological settings. Neither the feature nor mechanism is known at this time.

Application of small-roughness perturbation theory to reverberation in range-dependent waveguides

A rough-interface reverberation model is developed for range-dependent environments. First-order perturbation theory is employed, and the unperturbed background medium can be layered and heterogeneous with arbitrary range dependence. To calculate the reverberation field, two-way forward scatter due to the slowly changing unperturbed environment is handled by fast numerical methods. Backscatter due to small roughness superimposed on any of the slowly varying interfaces is handled efficiently using a Monte Carlo approach. Numerical examples are presented to demonstrate the application of the model. The primary purpose of the model is to incorporate relevant physics while improving computational speed.

Subcritical penetration of acoustic waves into inhomogenous seafloors

A generalized model is applied to estimate the incoherent penetration ratio caused by volume scattering at grazing angles below the critical grazing angle. The factors that affect volume scattering have been discussed using experimental data in literature. A two-layered model that refers to sound scattering in two-layered media is used to evaluate the incoherent penetration ratio for most typical sediments. But for special cases, such as the experiment, SAX04, a three-layered model is necessary to describe scattering features especially for grazing angles θ<30°. It is shown that subcritical penetration is enhanced when the scale of volume fluctuations is comparable with the acoustic wavelength, and the scattered waves into the seafloor dominate over evanescent waves at depths larger than a few wavelengths.

Modeling of acoustic penetration into sandy sediments: physical and geometrical aspects

Two different approaches to the problem of acoustic penetration into sandy marine sediments are considered: application of the Buckingham constitutive model for sediment with a plane surface and boundary element analysis of a rough surface of sediment represented as a homogeneous fluid. By a careful modeling of the constitutive behavior for plane seafloors, it is possible to partly reproduce some features of known experimental dependencies for acoustical pressure. However, accounting for roughness appears to be more important. Accordingly, the authors present a detailed numerical analysis of penetration into rough sediments using the boundary element method. The simulation results support conclusions reached by other investigators and demonstrate how local surface irregularities violate the evanescence condition that holds for a plane interface at subcritical incidence, thus considerably increasing penetration. The results apply to the frequency range 0.5-50 kHz and grazing angles larger than approximately 6 degrees -8 degrees at 10-50 kHz. For lower frequencies, when diffraction becomes important, the lowest possible grazing angle strongly depends on the range covered by the incident beam and is, in general, considerably larger. The authors provide several characteristic examples with frequencies 5 and 15 kHz and grazing angles 15 degrees -30 degrees illustrating the impact of roughness on penetration.

An effective density fluid model for acoustic propagation in sediments derived from Biot theory

In this paper we present an acoustic propagation model that approximates a porous medium as a fluid with a bulk modulus and effective density derived from Biot theory. Within the framework of Biot theory it is assumed here that the porous medium has low values of frame bulk and shear moduli relative to the other moduli of the medium and these low values are approximated as zero. This leads to an effective density fluid model. It is shown that, for saturated sand sediments, the dispersion, transmission, reflection, and in-water backscattering predicted with this effective density fluid model are in close agreement with the predictions of Biot theory. In this agreement we demonstrate that the frame bulk and shear moduli play only a minor role in determining several aspects of sand acoustics. Thus, for many applications the effective density fluid model is an accurate alternative to full Biot theory and is much simpler to implement.

Acoustic transmission across a roughened fluid-fluid interface

A set of tank experiments was performed to investigate acoustic transmission across a roughened fluid-fluid interface with the intention to test heuristic Bragg scattering predictions used to explain observations of anomalous transmission in field experiments. In the tank experiments, two immiscible fluids (vegetable oil floating on glycerin) formed the layers. Small polystyrene beads were floated at the interface to simulate roughness. An array of hydrophones placed in the bottom layer (glycerin) was used to measure the acoustic levels transmitted across the interface. This array was also employed as a beamformer to determine the apparent angle and sound speed of the scattered signals. Data were acquired at subcritical grazing angles in the frequency range of 100-200 kHz for three different bead diameters and for various configurations in which the locations of the beads floating on the interface were varied. Results of these measurements demonstrated that a significant amount of acoustic energy can be scattered into the bottom layer by beads floating at the interface. The scattered levels increased with increasing bead diameter. However, discrepancies occurred between observed propagation properties and the Bragg predictions. By comparing the processed tank data to a computer simulation of the same it was determined that these discrepancies are a consequence of near-field reception of the scattering by the bead array and ignoring the directionality of the scattering by the beads. Consequences to observations made in field experiments are discussed.

Acoustic scattering by a three-dimensional elastic object near a rough surface

The ensemble-averaged field scattered by a smooth, bounded, elastic object near a penetrable surface with small-scale random roughness is formulated. The formulation consists of combining a perturbative solution for modeling propagation through the rough surface with a transition (T-) matrix solution for scattering by the object near a planar surface. All media bounding the rough surface are assumed to be fluids. By applying the results to a spherical steel shell buried within a rough sediment bottom, it is demonstrated that the ensemble-averaged "incoherent" intensity backscattered by buried objects illuminated with shallow-grazing-angle acoustic sources can be well enhanced at high frequencies over field predictions based on scattering models where all environmental surfaces are planar. However, this intensity must compete with the incoherent intensity scattered back from the interface itself, which can defeat detection attempts. The averaged "coherent" component of the field maintains the strong evanescent spectral decay exhibited by flat interface predictions of shallow-angle measurements but with small deviations. Nevertheless, bistatic calculations of the coherent field suggest useful strategies for improving long-range detection and identification of buried objects.

An explanation for the anomalous ultrasonic slow wave in underwater sand

An explanation is given for the propagation time of the well-known anomalous ultrasonic slow wave observed in water-saturated sand using a three-layer elastic model. The rapid increase of elastic properties of sand with depth causes conversion of near-grazing underwater acoustic waves into multiple coupled shear and compressional waves.