Three-dimensional Parabolic Equation Model Research Articles

Underwater acoustic energy fluctuations during strong internal wave activity using a three-dimensional parabolic equation model.

The three-dimensional Monterey-Miami parabolic equation model is used to simulate a nonlinear internal wave (NIW) crossing the sound field in a shallow water environment. The impetus for this research stems from acoustic measurements taken during the Shallow Water '06 (SW06) field experiment, where a NIW traversed the water column such that soliton wavecrests were nearly parallel to the source-receiver path. Horizontal refraction effects are important in this scenario. A sound speed profile adapted from experimental SW06 data is used to simulate the NIW, assuming variations along the wavecrests (e.g., curvature) are negligible. Broadband and modal energy metrics show acoustic fluctuations due to internal wave activity. Repeated model runs simulate the NIW crossing the parabolic equation (PE) field over space and time. Statistical analysis shows the PE data are best fit by a lognormal distribution but tends to an exponential distribution during certain scenarios. Small angle differences between the acoustic track and the propagating NIW cause substantial differences in energy distribution throughout the PE field. While refraction effects due to the leading edge of the NIW's arrival are important in all cases, the impacts of focusing and defocusing in the perfectly parallel case dominate the field fluctuations. In the non-parallel case, the strong fluctuations introduced by the passage of the NIW are of similar order to the refraction off the leading edge.

Three-dimensional parabolic equation model for seismo-acoustic propagation: Theoretical development and preliminary numerical implementation**Project supported by the National Nature Science Foundation of China (Grant Nos. 11234002 and 11704337) and the National Key esearch Program of China (Grant No. 2016YFC1400100).

A three-dimensional (3D) parabolic equation (PE) model for sound propagation in a seismo-acoustic waveguide is developed in Cartesian coordinates, with x, y, and z representing the marching direction, the longitudinal direction, and the depth direction, respectively. Two sets of 3D PEs for horizontally homogenous media are derived by rewriting the 3D elastic motion equations and simultaneously choosing proper dependent variables. The numerical scheme is for now restricted to the y-independent bathymetry. Accuracy of the numerical scheme is validated, and its azimuthal limitation is analyzed. In addition, effects of horizontal refraction in a wedge-shaped waveguide and another waveguide with a polyline bottom are illustrated. Great efforts should be made in future to provide this model with the ability to handle arbitrarily irregular fluid–elastic interfaces.

This Submission is for Special Issue on Underwater Acoustics: Perfectly Matched Layer Technique for Parabolic Equation Models in Ocean Acoustics

In ocean waveguides, the ocean bottom is usually approximated as a half-space. Thus, there exist no reflection waves at the half-space bottom and condition of radiation at infinity should be satisfied. In numerical solutions like parabolic equation methods, the depth domain has to be truncated, which can generate reflection waves from the truncated ocean bottom. To reduce the effect of reflection waves and to simulate an unbounded ocean bottom accurately, an artificial absorbing layer (ABL) was used. As was demonstrated, an ABL meets well the demand of accuracy in sound field calculation. However, both the sea-bottom layer and the artificial absorbing layer are needed to be set quite thick by using an ABL technique. Fortunately, a PML with several wavelengths can keep similar calculation accuracy with an ABL with dozens of wavelengths. In this paper, perfectly matched layer (PML) techniques for three parabolic equation (PE) models RAM, RAMS and a three-dimensional PE model in underwater acoustics are presented. A key technique of PML “complex coordinate stretching” is used to truncate unbounded domains and to simulate infinity radiation conditions instead of the ABL in those models. The numerical results illustrate that the PML technique is of higher efficiency than the ABL technique at truncating the infinity domain with minimal spurious reflections in PE models.

Three-dimensional parabolic equation modeling of mesoscale eddy deflection.

The impact of mesoscale oceanography, including ocean fronts and eddies, on global scale low-frequency acoustics is examined using a fully three-dimensional parabolic equation model. The narrowband acoustic signal, for frequencies from 2 to 16 Hz, is simulated from a seismic event on the Kerguellen Plateau in the South Indian Ocean to an array of receivers south of Ascension Island in the South Atlantic, a distance of 9100 km. The path was chosen for its relevance to seismic detections from the HA10 Ascension Island station of the International Monitoring System, for its lack of bathymetric interaction, and for the dynamic oceanography encountered as the sound passes the Cape of Good Hope. The acoustic field was propagated through two years (1992 and 1993) of the eddy-permitting ocean state estimation ECCO2 (Estimating the Circulation and Climate of the Ocean, Phase II) system. The range of deflection of the back-azimuth was 1.8° with a root-mean-square of 0.34°. The refraction due to mesoscale oceanography could therefore have significant impacts upon localization of distant low-frequency sources, such as seismic or nuclear test events.

Three-dimensional parabolic equation model for low frequency sound propagation in irregular urban canyons.

A three-dimensional wide-angle parabolic equation (3DPE) is used to model low frequency sound propagation in irregular urban canyons at low computational cost. This one-way wave equation is solved using the Alternating Direction Implicit method. A finite difference scheme adapted to the geometry of the urban environment is then developed. Abrupt variations of the street width are treated as a single scattering problem using the Kirchhoff approximation. Numerical results are compared with experimental data obtained on a scale model of a street. Comparisons show the ability of the 3DPE model to provide reliable transmitted fields even for large irregularities.

A three-dimensional parabolic equation model of sound propagation using higher-order operator splitting and Padé approximants

An alternating direction implicit (ADI) three-dimensional fluid parabolic equation solution method with enhanced accuracy is presented. The method uses a square-root Helmholtz operator splitting algorithm that retains cross-multiplied operator terms that have been previously neglected. With these higher-order cross terms, the valid angular range of the parabolic equation solution is improved. The method is tested for accuracy against an image solution in an idealized wedge problem. Computational efficiency improvements resulting from the ADI discretization are also discussed.

THE USE OF TESSELLATION IN THREE-DIMENSIONAL PARABOLIC EQUATION MODELING

A full three-dimensional parabolic equation model (MONM3D) has been developed that incorporates techniques that reduce the required number of model grid points and reduces computation time. The concept of tessellation is implemented in MONM3D, which allows the number of radial paths in the model grid to vary with range from the source, reducing the number of computational points in the horizontal plane. This design establishes a grid layout that is both numerically and computationally desirable. A benchmark test case is used to illustrate the accuracy and efficiency of the model.

Surface and deep ducting of sound parallel to strong nonlinear internal wave fronts.

Acoustic ducting effects observed in a nonlinear internal wave field on the shelf of the South China Sea northeast of Tung-Sha Island (water depth of ∼120 m) are investigated. Environmental measurements from stationary instruments in the area show that the nonlinear internal waves have large amplitudes (over 40 m) and strong curvature. A principal component interpolation method is used to reconstruct the time-dependent three-dimensional internal wave field. A three-dimensional parabolic equation model is then employed to simulate monochromatic 400-Hz acoustic propagation parallel to the internal wave fronts for comparison to field data. The passing internal waves alternately elevate and depress the thermocline to form strong three-dimensional near-bottom acoustic ducts between the wave troughs and strong surface ducts of mixed layer water in the wave troughs. These alternating, depth-dependent acoustic ducts create asynchronous, vertically-bifurcated sweeping beams of ducted acoustic energy. The three-dimensional acoustic propagation modeling results are shown to mimic the field observations to a strong degree and elucidate the propagation physics in this highly complex environment.

A three-dimensional parabolic equation model that includes the effects of rough boundaries

A three-dimensional parabolic equation (3DPE) model that handles wide propagation angles in depth, narrow propagation angles in azimuth, and rough boundaries is derived and solved numerically with the method of alternating directions. A homogeneous boundary condition, which is easily incorporated into the numerical solution of the 3DPE, is applied at the ocean surface to approximate the effects of rough boundaries in terms of a reflection coefficient that depends on grazing angle. Calculations are presented to demonstrate the accuracy of the 3DPE and to demonstrate that horizontal coupling, the effects due to the term involving azimuth derivatives, can be important in shallow water (previous work has focused on deep-water applications). The model is applied to problems involving rough boundaries and range dependence.

A mathematical model for the three-dimentional ocean sound propagation

A model is developed mathematically to represent sound propagation in a three-dimensional ocean. The complete development is based on characteristics of the physical environment, mathematical theory, and computational accuracy. While the two-dimentional underwater acoustic wave propagation problem is not yet solved completely for range-dependent environments,three-dimentional environmental effects, such as fronts and eddies, often cannot be neglected. To predict underwater sound propagation, one usually deals with the solution of the Helmholtz (reduced wave) equation. This elliptical equation, along with a set of boundary conditions including a wall condition at the maximum range, forms a well-posed problem, which is pure boundary-value problem. An existing approach to economically solve this three-dimensional range-dependent problem is by means of a two-dimensional parabolic partial differential equation. This parabolic approximation approach, within the limitation of mathematical and acoustical approximations, offers efficient solutions to a class of long-range propagation problems. The parabolic wave equation is much easier to solve than the elliptic equation; one major saving is the removal of the wall boundary condition at the maximum range. The application of the two-dimensional parabolic wave equation to a number of realistic problems has been successful. We discuss the extension of the parabolic equation approach to three-dimensional problems. This paper begins with general considerations of the three-dimensional elliptic wave equation and shows how to transform this equation into parabolic equations which are easier to solve. The development of this paper focuses on wide angle three-dimensional underwater acoustic propagation and accommodates as a special case prevoius developments by other authors. In the course of our development, the physical properties, mathematical validity, and computational accuracy are the primary factors considered. We describe how parabolic wave equations are derived and how wide angle propagation is taken into consideration. Then, a discussion of the limitations and the advantages of the parabolic equation approximation is highlighted. These provide the background for the mathematical formulation of three-dimensional underwater acoustic wave propagation models. Modelling the mathematical solution to three-dimensional underwater acoustic wave propagation involves difficulties both in describing the theoretical acoustics and in performing the large scale computations. We have used the mathematical and physical properties of the problem to simplify considerably. Simplications allow us to introduce a three-dimensional mathematical model for underwater acoustic propagation predictions. Our wide angle three-dimensional parabolic equation model is theoretically justifiable and computationally accurate. This model offers a variety of capabilities to handle a class of long-range propagation problems under acoustical environments with three-dimensional variations.