Adiabatic Mode Parabolic Equation Research Articles

Analytical Lie-algebraic solution of a 3D sound propagation problem in the ocean

The problem of sound propagation in a shallow sea with variable bottom slope is considered. The sound pressure field produced by a time-harmonic point source in such inhomogeneous 3D waveguide is expressed in the form of a modal expansion. The expansion coefficients are computed using the adiabatic mode parabolic equation theory. The mode parabolic equations are solved explicitly, and the analytical expressions for the modal coefficients are obtained using a Lie-algebraic technique.

Three-dimensional adiabatic mode parabolic equation method and its applications

Complex zone of the ocean is often characterized by horizontal variations of environmental parameters(bathymetry, sound speed profile, bottom properties etc.), initiating redistribution of the sound field in horizontal plane, which is the so-called three-dimensional (3D) effect. Based on the adiabatic mode parabolic equation method, modeling of 3D effects is carried out, in which the eigenvalues and eigenfunctions are calculated by the standard normal mode model KRAKEN, and the amplitude corresponding to each mode is computed by the wide-angle parabolic equation model RAM. The present 3D model is very efficient and can give clear physical meaning, but it can be only applied to a waveguide whose properties vary gradually with horizontal range due to the adiabatic assumption between different modes. This model is then used to analyze the horizontal refraction caused by internal waves and by a coastal wedge. The numerical results show that there are several areas in the horizontal plane, corresponding to different structures of intensity distributions. Moreover, the redistribution of the sound field in horizontal plane depends on source frequency and mode number. Frequency and modal dependences lead to variations of spectrum, distortion of signal with some spectrum, and spatiotemporal fluctuations of the sound field.

Long range acoustic measurements of an undersea volcano

A seamount 8 km southeast of Sarigan Island erupted on 29 May 2010 and was visually observed. The recordings on two sets of hydrophones, operated by International Monitoring System (IMS) of the Comprehensive Test Ban Treaty Organization (CTBTO) are analyzed. Each array is a triplet of axial single hydrophones deployed as a 2 km triangle. Measurements of acoustic intensity for the path to the southern triplet are on the order of 6 dB lower than those received on the northern triplet. Temporal cross-correlation beamforming estimation is performed and the estimated arrival angles for the two arrays, 265° and 267° were consistent with the predicted geodesic arrival of 264.6° and 267.8°, respectively. Cross-correlation between single phones on the northern and southern arrays reveals a peak at 266°, with a cross-correlation of 0.1. Nx2D parabolic equation modeling predicts complete blockage due to seamount interaction along the geodesic path. Overprediction of the seamount blockage indicates that the 2D approximation is incorrect, and three-dimensional propagation must be used to explain the observations. This is demonstrated by the computation of the Adiabatic Mode Parabolic Equation Transmission Loss, which predicts a 5-10 dB lower reception at the southern site.

Comparison of hybrid three-dimensional modeling with measurements on the continental shelf

During the CALOPS 2007 experiment, off the coast of Fort Lauderdale, Florida, three-dimensional (3D) multipath was observed using a bottom mounted horizontal line array during source tows along the 200 m isobath [Kevin D. Heaney and James J. Murray, J. Acoust. Soc. Am. 125(4), 1394-1402 (2008)]. In this paper a hybrid modeling approach is presented to model the 3D sound on the Florida shelf, nearly shaped like the canonical wedge. The hybrid approach combines vertical acoustic normal modes with the parabolic equation solution (in range/cross-range). The approach is shown to satisfy the 3D Cartesian-coordinate wave equation in the limit of adiabatic mode propagation. In the adiabatic mode parabolic equation (AMPE) approach modal phase speeds vs position are used as the input to the parabolic equation computation with dimensions of easting (km) and northing (km). Vertical adiabatic modes and horizontal rays are also computed to illustrate the 3D multipath arrival. The AMPE field is computed for all the modes for each element of the horizontal array. Beamforming vs source range is then conducted and excellent agreement with data is achieved.

Horizontal Lloyd mirror patterns from straight and curved nonlinear internal waves

Experimental observations and theoretical studies show that nonlinear internal waves occur widely in shallow water and cause acoustic propagation effects including ducting and mode coupling. Horizontal ducting results when acoustic modes travel between internal wave fronts that form waveguide boundaries. For small grazing angles between a mode trajectory and a front, an interference pattern may arise that is a horizontal Lloyd mirror pattern. An analytic description for this feature is provided along with comparisons between results from the formulated model predicting a horizontal Lloyd mirror pattern and an adiabatic mode parabolic equation. Different waveguide models are considered, including boxcar and jump sound speed profiles where change in sound speed is assumed 12 m/s. Modifications to the model are made to include multiple and moving fronts. The focus of this analysis is on different front locations relative to the source as well as on the number of fronts and their curvatures and speeds. Curvature influences mode incidence angles and thereby changes the interference patterns. For sources oriented so that the front appears concave, the areas with interference patterns shrink as curvature increases, while convexly oriented fronts cause patterns to expand.

Perth-Bermuda revisited again: Global adiabatic mode parabolic equation results

In 1960, a set of explosives were detonated off the coast of Perth Australia, and multi-pulse receptions were recorded from moored hydrophones off of Bermuda. The Perth-Bermuda experiment demonstrated the capability of trans oceanic acoustic propagation. The two-pulse arrival, separated by approximately 25 s, was explained by Heaney, etal. (JASA, 90(5), 2586–2594) in terms of two disparate paths: a northern path refracting off islands in the southern Indian Ocean and the other refracting off the shelf-break on the coast of Brazil. In this paper a global adiabatic mode-parabolic equation hybrid model is used to compute the multi mode, broadband full-field pressure response from Perth to Bermuda. The PE field demonstrates that Bermuda is in the acoustic shadow of the refracted geodesics, yet observations of arrivals were made. PE results demonstrate that two significant scattered paths, one to the north passing by the Cape of Good Hope and one to the south, passing by the coast of Brazil yield strong strong...

Horizontal Lloyd mirrors arising from propagation through straight and curved nonlinear internal wave fronts.

Experimental observations and theoretical studies show that nonlinear internal waves (NIWs) occur widely in shallow water and cause acoustic propagation effects including mode coupling and ducting. Horizontal ducting results when acoustic modes interact with NIW fronts that comprise waveguide boundaries. For small grazing angles between a mode trajectory and a front, an interference pattern may arise that is hypothesized [Lynch et al., J. Ocean Eng. 31, 33–48 (2006)] to be a horizontal Lloyd mirror. We examine acoustic formulations for this feature and benchmark calculations for the acoustic intensity with those from the adiabatic mode parabolic equation. Results using different waveguide features are compared, including continuous‐gradient and jump sound‐speed profiles of varying strengths. We focus on differences in the location of the source relative to the NIW as well as the frontal curvature. The curvature influences both incidence angles and reflection characteristics. For sources oriented inside the front, as curvature increases the areas with interference patterns shrink, while sources beyond the front cause patterns to expand. [Work supported by ONR.]

Hydroacoustic blockage prediction and measurement at Diego Garcia using the Adiabatic Mode Parabolic Equation Model

Underwater explosion monitoring with sparse sensors at long ranges relies on the efficient propagation of acoustic energy in the sound fixing and ranging (SOFAR) channel. When sound traveling in this channel encounters an island or seamount, it will either diffract, scatter, or be converted into seismic energy. Signals observed on the opposite side of these obstructions have been affected by some combination of these processes, and models of global detection and localization depend on knowing these effects. We present a study using the Adiabatic Mode Parabolic Equation (AMPE) model to predict these processes in three dimensions at the Chagos Archipelago. Predictions at 5, 10, and 20 Hz are compared with measurements of approximately 300 T-wave signals from six years of earthquakes on either side of the Chagos Archipelago. These have been recorded at the hydrophone arrays around Diego Garcia. The result of this 360-degree analysis, and the agreement with observed data, demonstrate the utility of the model in understanding the physical effects of these obstructions.

Experimental evidence of three-dimensional acoustic propagation caused by nonlinear internal waves

The 1995 SWARM experiment collected high quality environmental and acoustic data. One goal was to investigate nonlinear internal wave effects on acoustic signals. This study continues an investigation of broadband airgun data from the two southwest propagation tracks. One notable feature of the experiment is that a packet of nonlinear internal waves crossed these tracks at two different incidence angles. Observed variations for the lower angle track were modeled using two-dimensional parabolic equation calculations in a previous study. The higher incidence angle is close to critical for total internal reflection, suggesting that acoustic horizontal refraction occurs as nonlinear internal waves traverse this track. Three-dimensional adiabatic mode parabolic equation calculations reproduce principal features of observed acoustic intensity variations. The correspondence between data and simulation results provides strong evidence of the actual occurrence of horizontal refraction due to nonlinear internal waves.

Evidence of three-dimensional waveguide propagation in SWARM 95 data

During a period of the SWARM95 experiment, strong nonlinear internal waves passed across two tracks that had airgun pulses propagating along them. Environmental data for this period indicate that the angles between the tracks and the internal wave-fronts, which were roughly planar, were very different—one angle being close to zero, and the other approximately 42 deg. Two-dimensional PE simulations for these waveguides show dramatically different results for depth-averaged, pulse-integrated energy variations. Specifically, the observed levels can be reproduced for the waveguide with the large incidence angle, but not for the one with the small incidence angle. For the latter case, data show significant variations in pulse shapes and in the integrated energy (≊5 dB), while simulations show very small changes in both of these characteristics. Results from several recent computational and theoretical studies suggest that the cause may be three-dimensional effects from horizontal refraction and modal interference due to the nonlinear internal waves. The adiabatic mode parabolic equation [Collins, J. Acoust. Soc. Am. 94 (1993)] is used to quantify the three-dimensional influence of the internal waves on the integrated energy variations. The results demonstrate experimental evidence of three-dimensional effects from strong nonlinear internal waves. [Work supported by ONR.]

The coupled mode parabolic equation

The coupled mode parabolic equation (PE) is a generalization of the adiabatic mode PE that includes mode coupling terms. It is practical to apply this approach to large-scale problems involving coupling of energy between both modes and azimuths. The solution is expressed in terms of the normal modes and mode coefficients, which satisfy coupled horizontal wave equations. The coupled mode PE may be solved efficiently with the splitting method. The first step is equivalent to solving the adiabatic mode PE over one range step. The second step involves the integration of the coupling term. The coupling mode PE solution conserves energy, which is an important aspect of a range-dependent propagation model. The derivation of the coupled mode PE, which involves completing the square of an operator, is related to the derivation of an adiabatic mode PE that accounts for ambient flow. Examples are presented to illustrate the accuracy of the coupled mode PE.

The coupled mode parabolic equation

An additional term has been incorporated into the adiabatic mode parabolic equation (AMPE) to account for mode coupling. The coupled mode parabolic equation (CMPE) involves the solution of coupled horizontal wave equations for the mode coefficients. The numerical solution may be obtained efficiently using the splitting method. The first step is identical to the numerical solution of the AMPE. The second step involves Crank-Nicolson integration of the coupling term. The accuracy of the CMPE has been confirmed for two-dimensional problems involving coupling due to sloping bottoms and volume heterogeneities. The CMPE has been applied to three-dimensional problems involving both mode coupling and azimuthal coupling. It is practical to apply the CMPE to large-scale problems and possibly even global-scale problems at low frequencies. [Work supported by ONR.] a)A. T. Abawi is presently at AETC, Inc., 10975 Torreyana Rd. 300, San Diego, CA 92121.

Three‐dimensional acoustic propagation in a waveguide of variable thickness

The parabolic equation (PE) method has proven to be an efficient method for solving range‐dependent ocean acoustics problems. Recently an extension has been made to handle two‐dimensional waveguides of varying thickness. This type of modeling is necessary when the upper boundary of the waveguide is allowed to change, as for example in propagation up a beach or in an ocean with a gradually undulating surface. The adiabatic mode PE is another extension that has been developed to solve global scale, three‐dimensional propagation problems. In this paper, the three‐dimensional adiabatic mode PE is extended to handle problems involving a gradual range dependence in the overall thickness of the waveguide. The modified adiabatic PE has several applications. On relatively small scales, it may be applied to solve three‐dimensional beach acoustics problems or to model diffraction by an island. It may also be applied to solve global‐scale seismoacoustic problems in which topography on the continents plays a significant role. [Work supported by ONR.]

Application of the mode vector parabolic equation to the 3‐dimensional wedge problem

The MVPE [Abawi et al., this session] is applied to the problem of propagation of waves in a 3‐dimensional coastal wedge and the results are compared with those when horizontal coupling is neglected and with the results obtained from conventional 3‐dimensional methods. The components of the solution vector are the amplitudes of the vertical modes which are coupled in both range and azimuth. The coupling matrices for both range and azimuth are precomputed by the method summarized in [McDonald et al., 2357 (1994)]. In the absence of mode coupling, the solution reduces to the solution of the adiabatic mode parabolic equation [M. D. Collins, 2269 (1993)]. The system of equations are numerically solved by the Crank–Nicholson finite difference method. For a single vertical mode, this system of equations closely resembles a system of discretized parabolic equation in range and azimuth. For multiple modes, the matrices involved are band diagonal with the coupling matrices appearing in their diagonal. The size of these matrices are thus equal to the number of points in azimuth times the number of vertical modes. When the physics of the problem is contained in a subset of coupled modes, this method offers additional advantages in numerical computation over the conventional 3‐dimensional methods.

Jovian acoustics and Comet Shoemaker–Levy 9

A technique for solving three-dimensional ocean acoustics problems is modified to handle Jovian acoustics problems and applied to model propagation from the impact sites of the fragments of Comet Shoemaker–Levy 9. The adiabatic mode parabolic equation is generalized to media with fluid flow for the case of low Mach number. The vertical dependence of sound speed and density and the latitude dependence of the zonal winds are known from Voyager data. The acoustic model predicts that energy that radiated from the impact sites was vertically concentrated within a sound channel that includes the cloud layers and horizontally concentrated into intense caustics by refraction associated with wind shear. Although nonlinear effects are important near the impact sites, the spatial distribution of energy should be qualitatively represented by the linear acoustics results.

Island diffraction and other three-dimensional effects in global acoustics

The adiabatic mode parabolic equation (PE) is an efficient approach for solving three-dimensional ocean acoustics problems [M. D. Collins, J. Acoust. Soc. Am. 94, 2269–2278 (1993)]. This method is used to solve the outgoing wave equations for the mode coefficients, which depend on range and azimuth. The range- and azimuth-dependent wave numbers appearing in these equations correspond to the eigenvalues. The adiabatic mode PE is applied to solve global-scale problems involving diffraction around islands and other three-dimensional effects. The significance of azimuthal coupling is illustrated by comparing solutions generated both with and without azimuthal coupling. The coupling azimuth solutions predict that the shadow zone behind an island is wider than uncoupled azimuth solutions predict. Since the modal phase velocities increase as ocean depth decreases, this result is consistent with the expected result based on ray theory. Adiabatic mode PE solutions are in qualitative agreement with observations from the Perth–Bermuda experiment [Heaney etal., J. Acoust. Soc. Am. 90, 2586–2594 (1991)] and the Heard Island Feasibility Test [Baggeroer and Munk, Phys. Today 45(9), 22–30 (1992)]. a)Permanent address: Planning Systems, Inc., McLean, VA 22102.

Jovian acoustic tomography

Recent work in ocean acoustic tomography indicates that acoustic waves propagating global distances contain planetary-scale information [A. B. Baggeroer and W. Munk, Phys. Today 45(9), 22–30 (1992)]. The expected collision of the fragments of comet Shoemaker–Levy 9 with Jupiter in July [J. K. Beatty and D. H. Levy, Sky & Telescope 87(1), 40–44 (1994)] provides an opportunity to explore the use of techniques developed in ocean acoustics on a different planet. The nonlinear progressive wave equation [B. E. McDonald and W. A. Kuperman, J. Acoust. Soc. Am. 81, 1406–1417 (1987)] can be used to model the propagation of shock waves near the impact sites. Adiabatic modes can be used to determine horizontally refracted propagation paths on an oblate spheroid [Heaney etal., J. Acoust. Soc. Am. 90, 2586–2494 (1991)]. Normal-mode and parabolic equation (PE) models can be used to obtain two-dimensional (range and depth) solutions, which illustrate geometric dispersion and cycle distances. The adiabatic mode PE [M. D. Collins, J. Acoust. Soc. Am. 94, 2269–2278 (1993)], generalized to include horizontal wind fields, can be used to determine the horizontal distribution of energy and the locations of caustics. Available input data (from the Voyager missions and other sources) for making pre-impact predictions include wind velocity as a function of latitude, the locations of the impacts, and information for estimating sound speed as a function of depth.

The adiabatic mode parabolic equation

An efficient method is developed for solving problems involving three-dimensional acoustic propagation in a thin spherical shell. The method is based on the adiabatic mode approximation, which assumes that energy coupling between modes is negligible, and the parabolic equation (PE) method, which efficiently handles spatial variations in the acoustic properties. The PE method is used to solve two-dimensional wave equations for the adiabatic mode coefficients over latitude and longitude. Due to this reduction in dimension, the adiabatic mode PE should be practical for solving global-scale ocean acoustics problems. These types of problems, which are currently of great interest [Heaney et al., J. Acoust. Soc. Am. 90, 2586–2594 (1991); Baggeroer and Munk, Phys. Today 45(9), 22–30 (1992)], are too large to solve with other existing three-dimensional models.

Long‐range acoustic propagation in four dimensions

Ocean acoustic modeling is usually restricted to frequency‐domain problems in two dimensions. Since ocean acoustics problems typically involve domains that are very large relative to a wavelength, computation times often become prohibitive when a third dimension (either time or the third spatial variable) is included. The adiabatic mode parabolic equation (PE), which has been applied to solve global‐scale problems in three spatial dimensions [Collins et al., J. Acoust. Soc. Am. 93, 2321 (1993)], is practical for solving four‐dimensional problems out to long range (i.e., hundreds of wavelengths or more). In the time domain, the adiabatic mode PE solution illustrates the flow of energy in the modal coefficients, which depend on range and azimuth.