Acoustic Normal Modes Research Articles

Extraction of Acoustic Normal Mode Depth Functions Using Range-Difference Method with Vertical Linear Array Data

Extraction of Acoustic Normal Mode Depth Functions Using Range-Difference Method with Vertical Linear Array Data

Polynomial chaos expansions for modelling the statistics of acoustic propagation in random waveguides

The use of Polynomial Chaos expansions to model the transfer of the statistical properties of the sound speed in ocean waveguides to those of acoustic waves passing through them is described. A perturbational framework approximating the interaction of acoustic normal modes with fluctuations around a background sound speed is used, with the fluctuations being represented vertically by empirical orthogonal functions and horizontally by correlation functions. The Polynomial Chaos expansions are derived to second order in the uncorrelated random variable representing the sound speed fluctuations for the generalized phase of the complex modal amplitudes, and to third order for the modal amplitudes themselves. The ability of the second order polynomial chaos expansion for the generalized model phase to accurately model acoustic propagation statistics is closely coupled to the accuracy of the adiabatic approximation in light scattering regimes. The weights of the Polynomial Chaos expansions are obtained using both direct (theoretical) and indirect (least-squares fit) methods. Divergence between these two sets of weights increases with combinations of increasing strength of sound speed fluctuations, increasing frequency, or increasing range, indicating where a higher order expansion may be necessary. Modelling results for the first and second moments of the acoustic intensity and the Scintillation Index in random waveguides are presented and compared to Monte Carlo results.

Manifestations of the weak shear rigidity of marine sediments in amplitudes and travel times of acoustic normal modes and lateral waves

Compressional-to-shear wave conversion at interfaces within unconsolidated marine sediments and interference of shear waves within thin sediment layers have been found to make significant contributions to sound attenuation [O. A. Godin, J. Acoust. Soc. Am. 149, 3586–3598 (2021)] and produce unexpectedly strong perturbations in the normal mode group speed even when the shear speed is small compared to the compressional wave speed. This paper extends the previous analysis to the lateral waves and their applications to characterize the compressional wave speed and attenuation in the seabed. Shear wave-induced contributions to horizontal refraction of normal modes over horizontally inhomogeneous seabed are modeled to characterize the bearing and travel time errors that arise in the fluid-bottom approximation. The acoustic travel time and amplitude effects of the shear waves are found to accumulate at different rates with the propagation range depending on the stability of the shear wave interference within the seabed. The ways to distinguish between the manifestations in acoustic observables of the weak shear rigidity in the seabed and the other environmental uncertainties are discussed for range-independent, range-dependent, and horizontally inhomogeneous shallow-water waveguides. [Work supported by ONR.]

Effects of weak shear rigidity of the seabed on sound propagation in a range-dependent ocean

The shear wave speed is often small compared to the compressional wave speed in unconsolidated marine sediments. Sediment stratification and especially mass density variation in the seabed enhance the effects of weak shear on acoustic normal modes in range-independent ocean and produce significant shear wave contributions to mode attenuation in shallow water [O. A. Godin, J. Acoust. Soc. Am., 149, 3586–3598 (2021)]. A distinctive feature of the shear-induced perturbations in the mode phase speed and attenuation is their rapid variation with sound frequency due to shear wave interference within a sediment layer. This paper investigates theoretically the shear wave-induced perturbations in normal mode phase, travel time, and attenuation when water depth and/or sediment layer thickness vary gradually with range. Frequency dependencies of the adiabatic normal mode travel time and attenuation are found to be highly sensitive to range dependence of the seabed layering. Gradual changes in the sediment layer thickness have an effect similar to frequency averaging or artificially increased shear wave attenuation. Implications of these findings for geoacoustic inversions will be discussed. Contributions of shear rigidity and cross-range gradients of the sediment layer thickness to horizontal refraction of normal modes will be also examined. [Work supported by ONR.]

The Prufer transform for the calculation of acoustic normal modes

The PrüferTransform for the Sturm-Liouvile (SL) equation was introduced by Prüferin 1923. An excellent reference is Numerical Solutions of Sturm-Liouville Problems by Pryce. Porter in Computational Ocean Acoustics introduced it to ocean acoustics. This author stumbled upon working on state variables at CMRE on 1978. The method derives first order equations for the phase and log-magnitude for second order equation in a phase plane. The essential feature is the phase equation is not coupled to the magnitude, so it may be solved as a nonlinear, first order equation and the surface boundary condition leads to a unique solution. New results for normal modes using the Prüfermethod are introduced. i) equations for the phase shifts and log-magnitudes at layer boundaries such as at the seabed; ii) state variable transformations based upon Ricatti equations and requiring evaluating one transcendental quantity, important for numerical methods; iii) issues of numerical stability for coupling the phase solution across the potential barriers of local sound speed maxima and poorly coupled boundaries; iv) an equation for computing the imaginary part leading to modal attenuation’s. The method is fast has survived many students using it, even those writing Matlab versions.

Modal-MUSIC extensions

Modal-MUSIC [Akins and Kuperman, JASA Express Lett. 2(7), 074802 (2022)] is a variant of the well-known MUSIC plane wave algorithm that extracts direction of arrivals (DOA’s) from data containing multiple sources. Modal-MUSIC, rather than extract the DOA’s of the multiple sources, extracts the acoustic normal modes from incoherently summed data from multiple and/or moving sources received on a short vertical array. The method uses the local sound speed profile but does not require bottom geophysical information. We discuss extensions of the method that include using shorter arrays, geophysical-inversion and the additional application of synthetic source aperture [Akins and Kuperman, J. Acoust. Soc. Am. 150(1), 270–289 (2021)] to enhance source localization.

Wind, sea swell, and excitation of atmospheric waveguides by underwater earthquakes

Powerful underwater sources, such as volcano eruptions and large explosions, can generate infrasonic waves, which remain detectable at distant receivers on shore. Evers et al. [Geophys. Res. Lett. 41, 1644–1650 (2014)] described observations of acoustic signals in air from the 2004 Macquarie Ridge submarine earthquake, by the infrasonic array IS05AU of the International Monitoring System. Infrasound propagated in the tropospheric and stratospheric waveguides and was received 1325 km from the epicenter. Specific physical mechanisms of the excitation of guided infrasonic waves by underwater sources are not fully understood. Here, we emphasize the potential role of local meteorological conditions. Scattering of ballistic waves from the earthquake epicenter by wind waves and sea swell on the ocean surface is a potent mechanism of excitation of normal modes in hydroacoustic waveguides (T-waves) [Godin, J. Acoust. Soc. Am. 150, 3999–4017 (2021)]. We investigate the hypothesis that the scattering of ballistic waves at the ocean surface also explains the earthquake excitation of acoustic normal modes in the tropospheric and stratospheric waveguides. Relative significance of the surface scattering and the previously proposed evanescent coupling mechanism is studied as a function of the wind speed, wave frequency, and the source depth.

Modal Analysis of Acoustic Energy Periodic Fluctuations Due to Internal Tides in the Yellow Sea

Measurement data of the internal-wave and acoustic-fluctuation experiments conducted in the Yellow Sea in summer 2009 are analyzed herein. The internal-wave activity dominated by semidiurnal tides is observed. For the source and 16-element vertical linear array with fixed distance, the received energy fluctuations of 260-Hz continuous wave signals reach 20 dB in a single channel of the receiving array, and the fluctuation period is about 2–3 h across 10-h continuous acoustic measurement. The energy depth distribution varies with time, and the energy fluctuations are less than 10 dB post-depth-integration, evidencing that part of the energy transfers between channels. Acoustic normal modes are calculated according to the temperature chain observation results obtained at the receiving station and subsequently used to analyze the acoustic energy fluctuations: the difference between the average energy depth distribution in 0–5 h and 5–10 h indicates that the energy fluctuations of the single channel may be related to the dominant-mode alternation occurring between the first- and second-order modes; the modal decomposition results of the received acoustic field prove such alternation and explain the single-channel and depth-integral energy fluctuations caused by the first two modes’ amplitude variations.

Mode coupling and intensity fluctuation of sound propagation over continental slope in presence of internal waves

The topographic variation underwater of the continental slope is one of the main causes for triggering off the formation of internal waves, and the continental slope internal waves are ubiquitous in the ocean. The horizontal variation of waveguide environment, caused by the internal wave and the continental slope, can lead to acoustic normal mode coupling, and then generate sound field fluctuation. Most of the existing research work focused on studying the effect of single perturbation factor of either the internal waves or the continental slope on acoustic mode coupling and intensity fluctuation, while it is hard to find some research work that takes into account both the internal waves and the topographic variations as influencing factors. In this work, numerical simulations for the sound waves to propagate through the internal waves in the downhill direction are performed by using the acoustic coupled normal-mode model in four waveguide environments: thermocline, internal wave, continental slope and continental slope internal wave. And the mode coupling and intensity fluctuation characteristics and their physical mechanisms are studied by comparing and analyzing the simulation results of the four different waveguide environment constructed. Some conclusions are obtained as follows. The intra-mode conduction coefficients are symmetric with respect to the center of the internal wave, while the inter-mode coupling coefficients are antisymmetric around it. As the sound waves propagate toward or away from the center of the internal wave, the acoustic mode coupling becomes enhanced or weakened, and the coupling coefficients curves for large mode oscillate. The influence of internal wave perturbation makes the energy transfer from the smaller modes to the larger modes, which increases the attenuation of sound field intensity. The number of the waveguide modes increases and the mode intensity attenuation decreases, when the sound waves propagate downhill. The total intensity of all modes for the continental slope internal wave environment is greater than for the internal wave environment and less than for the continental environment, and the energy transfer between mode groups is stronger than for individual effect of internal wave or continental slope, which leads more energy to transfer from the smaller to larger mode groups and the energy of the sound field above the thermocline to increase.

Contributions of gravity waves in the ocean to T-phase excitation by earthquakes.

The generation of T waves in a deep ocean by an earthquake in its epicentral region is often observed, but the mechanism of the excitation of the acoustic waves travelling horizontally with the speed of sound remains controversial. Here, the hypothesis is investigated that the abyssal T waves are generated by the scattering of ballistic sound waves by surface and internal gravity waves in the ocean. Volume and surface scattering are studied theoretically in the small perturbation approximation. In the 3-50 Hz typical frequency range of the observed T waves, the linear internal waves are found to lack the necessary horizontal spatial scales to meet the Bragg scattering condition and contribute appreciably to the T-wave excitation. In contrast, the ocean surface roughness has the necessary spatial scales at typical sea states and wind speeds. The efficiency of the acoustic normal modes' excitation at surface scattering of the ballistic body waves by wind seas and sea swell is quantified and found to be comparable to that of the established mechanism of the T-wave generation at downslope conversion at the seamounts. The surface scattering mechanism is consistent with key observational features of the abyssal T waves, including their ubiquity, low-frequency cutoff, presence on seafloor sensors, and weak dependence on the earthquake focus depth.

The fluid bottom approximation and its implications for sound attenuation in underwater waveguides

The fluid-bottom model makes forward and inverse problems more amenable to theoretical analysis and numerical simulations but has some well-known limitations. The accuracy and weaknesses of the fluid approximation are elucidated in this paper by quantifying the effects of weak shear rigidity on acoustic normal modes. Mathematically, appearance of finite shear rigidity and attendant slow shear waves is a singular perturbation of the fluid-bottom model, which standard perturbation theories fail to capture correctly. The problem is addressed here by combination of an asymptotic technique and an analysis of exact solutions for idealized seabed models and placed in the context of earlier studies of the effects of shear rigidity. Due to coupling between compressional and shear waves at the seafloor and within the seabed, shear rigidity affects phase and group speeds of normal modes. By transferring the energy from compressional to shear waves, wave coupling makes a significant contribution to sound attenuation from low up to mid-frequencies. It is found that the effects of shear rigidity on the acoustic field in water are magnified by seabed stratification and especially by density variations. Implications of the results for seabed parameterization in geoacoustic inversions will be discussed. [Work supported by ONR.]

Mixed layer acoustic propagation with small scale ocean dynamics

The effects of dynamic oceanographic processes in the upper ocean mixed layer on acoustic propagation are investigated using at-sea sound speed observations and acoustic transmission simulations. The sound speed environment is based on a 400 m deep and 1000 km long East-West salinity and temperature transect that was collected during April of 2005 in the North Pacific where there is a mixed layer acoustic duct with a mean depth of 89 m. The deterministic baseline of this sound speed measurement is first estimated from the observations using a spatial lowpass filter with a cut off length scale of 50-km. The remaining small length scale variations in sound speed are assumed to have two physically distinct contributions: one due to the vertical displacement of isopycnals caused by eddies and internal waves and another due to compensating temperature and salinity anomalies along isopycnals termed spice. These two contributions can be separated yielding three distinct sound speed fields: i(1) the deterministic background; (2) thedisplacement field; and (3) the spice field. Acoustic transmission loss calculations are carried out on these three fields for frequencies of 400 and 1000 Hz for sources both in and below the mixed layer duct. Statistical and deterministic results for acoustic normal modes and full field transmission loss are presented, which quantify the relative importance of displacement and spice for acoustic propagation in the upper ocean.

Polarization of ocean acoustic normal modes.

In ocean acoustics, shallow water propagation is conveniently described using normal mode propagation. This article proposes a framework to describe the polarization of normal modes, as measured using a particle velocity sensor in the water column. To do so, the article introduces the Stokes parameters, a set of four real-valued quantities widely used to describe polarization properties in wave physics, notably for light. Stokes parameters of acoustic normal modes are theoretically derived, and a signal processing framework to estimate them is introduced. The concept of the polarization spectrogram, which enables the visualization of the Stokes parameters using data from a single vector sensor, is also introduced. The whole framework is illustrated on simulated data as well as on experimental data collected during the 2017 Seabed Characterization Experiment. By introducing the Stokes framework used in many other fields, the article opens the door to a large set of methods developed and used in other contexts but largely ignored in ocean acoustics.

Two Chebyshev Spectral Methods for Solving Normal Modes in Atmospheric Acoustics.

The normal mode model is important in computational atmospheric acoustics. It is often used to compute the atmospheric acoustic field under a time-independent single-frequency sound source. Its solution consists of a set of discrete modes radiating into the upper atmosphere, usually related to the continuous spectrum. In this article, we present two spectral methods, the Chebyshev-Tau and Chebyshev-Collocation methods, to solve for the atmospheric acoustic normal modes, and corresponding programs are developed. The two spectral methods successfully transform the problem of searching for the modal wavenumbers in the complex plane into a simple dense matrix eigenvalue problem by projecting the governing equation onto a set of orthogonal bases, which can be easily solved through linear algebra methods. After the eigenvalues and eigenvectors are obtained, the horizontal wavenumbers and their corresponding modes can be obtained with simple processing. Numerical experiments were examined for both downwind and upwind conditions to verify the effectiveness of the methods. The running time data indicated that both spectral methods proposed in this article are faster than the Legendre-Galerkin spectral method proposed previously.

Separation of acoustic normal modes in shallow water

The normal mode method is widely used for the acoustic field presentation in the deep ocean. In shallow water, the advantages of the mode description are not obvious, since in this case the mode structure is determined by the boundary conditions at the bottom and the surface, the properties of which are most often unknown, and are also nonstationary in space and time. Under such conditions, it is not clear to what extent the mode representation obtained in numerical models corresponds to reality. The authors studied the mode structure of several shallow water bodies and the coastal shelf of the White Sea. In the case when it was possible to measure the vertical profile of the field at the receiving point, the mode profiles are easily distinguished and can be used, for example, to determine unknown boundary conditions at the bottom and surface, including in the presence of ice cover. A method has been developed for identifying modes in the case of field reception by single spaced hydrophones; the difficulty in this case is that at each moment of time the received signal contains all propagating modes that have to be selected in subsequent processing.

A Chebyshev-Tau spectral method for normal modes of underwater sound propagation with a layered marine environment

The normal mode model is one of the most popular approaches for solving underwater sound propagation problems. Among other methods, the finite difference method is widely used in classic normal mode programs. In many recent studies, the spectral method has been used for discretization. It is generally more accurate than the finite difference method. However, the spectral method requires that the variables to be solved are continuous in space, and the traditional spectral method is powerless for a layered marine environment. A Chebyshev-Tau spectral method based on domain decomposition is applied to the construction of underwater acoustic normal modes in this paper. In this method, the differential equation is projected onto spectral space from the original physical space with the help of an orthogonal basis of Chebyshev polynomials. A complex matrix eigenvalue / eigenvector problem is thus formed, from which the solution of horizontal wavenumbers and modal functions can be solved. The validity of the acoustic field calculation is tested in comparison with classic programs. The results of analysis and tests show that compared with the classic finite difference method, the proposed Chebyshev-Tau spectral method has the advantage of high computational accuracy. In addition, in terms of running time, our method is faster than the Legendre-Galerkin spectral method.

Geoacoustic parameter estimation using machine learning techniques

The Airy phase region corresponding the minimum group velocity of the dispersive acoustic normal modes in shallow water are extremely sensitive to bottom properties. The group speed minima and the associated frequency data for two lower order modes were used to train a neural network to predict the bottom parameters in a previous study (Potty et al., 2019). The geoacoustic model for the bottom consisted of a sediment layer over acoustic basement. Synthetic data generated by varying the values of the sound speeds in the sediment layer and basement were used to train and evaluate the technique. The output of the neural network was used as the background model for a linear perturbation inversion. Data collected from the Shelf break Primer experiment were used to test the algorithm and the preliminary results were promising. This study will continue the previous work by incorporating higher order modes in the training data, testing with data from other experiments and using other machine learning techniques. The performance of this hybrid approach (neural network combined with linearized inversion) will be compared with fully non-linear inversion, both in terms of accuracy and computation time. [Work supported by the Office of Naval Research.]

The Convergence of the Legendre–Galerkin Spectral Method for Constructing Atmospheric Acoustic Normal Modes

The asymptotic rate of convergence of the Legendre–Galerkin spectral approximation to an atmospheric acoustic eigenvalue problem is established, as the dimension of the approximating subspace approaches infinity. Convergence is in the [Formula: see text] Sobolev norm and is based on the existing theory [F. Chatelin, Spectral Approximations of Linear Operators (SIAM, 2011)]. The assumption is made that the eigenvalues are simple. Numerical results that help interpret the theory are presented. Eigenvalues corresponding to acoustic modes with smaller [Formula: see text] norms are especially accurately approximated, even with lower dimensioned basis sets of Legendre polynomials. The deficiencies in the potential applications of the theoretical results are noted in connection with the numerical examples.

Estimation of geoacoustic parameters using machine learning techniques

Modal dispersion of broadband acoustic data has been used extensively to estimate the geoacoustic parameters using perturbation techniques (Rajan et al., 1987), non-linear inversion techniques (Potty et al., 2000) and Bayesian inference (Warner et al., 2015). The Airy phase region corresponding the minimum group velocity of the normal modes are extremely sensitive to bottom properties in comparison with properties of the water column. This talk will explore the sensitivity of the group speed minima and the associated frequencies of the acoustic normal modes to bottom parameters. The group speed minima and the associated frequency data will be used to train a machine-learning algorithm. Synthetic data will be generated for various bottoms and it will be used as the training pool. Data collected from New England Bight, East China Sea, and New England Mud patch will be used to test the algorithm. The value added to the a priori bottom parameter information using the algorithm will be discussed in the context of computational cost associated with the algorithm. [Work supported by the Office of Naval Research.]

Effect of Shear on Modal Arrival Times

The sea bottom is generally modeled as a fluid for many of the shallow water acoustic propagation modeling applications. The inherent assumption is that the bottom does not support any shear wave propagation. This study explores the impact of this assumption on the dispersion behavior of acoustic normal modes. A sensitivity study is performed to investigate the effect of density, shear, and compressional wave speeds on modal dispersion. This study focuses on low-frequency (less than 100 Hz) and lower order modes (mode four and lower) in shallow water (70–90-m water depth). These modes and frequency bands are characterized by deeper penetration and larger depth averaging scales. This enabled the use of a simple half-space elastic model of the sea bottom for most of the analyses in this study. A depth-dependent bottom model was used to investigate the effect of a mud layer on modal dispersion. Sediment shear wave speeds were also estimated using broadband data from two shallow water experiments: Shelf Break Primer (1996) and Seabed Characterization Experiment (2017). The inversion results from these two experiments are compared and they are also compared with deep core data and prior inversions.