Fast and Accurate Seismic Computations in Laterally Varying Environments

The parabolic equation method is an accurate and efficient approach for solving nonseparable problems in ocean acoustics in which there are horizontal variations in the environmental parameters. Many range-dependent problems may be solved using 2-D parabolic equation models that ignore coupling of energy between planes of constant azimuth. When azimuthal coupling must be taken into account, the splitting method may be used to efficiently solve a 3-D parabolic equation that handles the depth operator to higher order but handles the azimuth operator only to leading order. Despite the fact that this approximation provides a favorable combination of accuracy and efficiency for 3-D problems, run times have generally been regarded as prohibitive for the long-range problems that are often of interest in ocean acoustics. It is demonstrated here that, when propagation paths from source to receiver are confined to a relatively narrow neighborhood of the vertical plane containing the source and receiver, it is practical to solve 3-D problems out to long ranges by using nonuniform azimuthal sampling, with fine sampling near the vertical plane and extremely coarse sampling elsewhere.

The parabolic equation (PE) method is widely used in radiowave propagation predictions. It has the advantages of high efficiency and stability, but it will lead to greater predicting errors in some situations, because the effects of transverse terrain gradients are not modeled. This problem can be solved by extending the 2D PE to the three-dimensional (3D) PE. However, the computing efficiency will degrade because of large scale matrix operations. In this paper, a new method is presented, in which the 3D PE is decomposed into two 2D PEs. It increases the computational efficiency and accuracy effectively. To verify the capability of the proposed method in radiowave propagation prediction, an experiment platform was set up. The computational results using this new method are compared with the experimental and Method of Moment (MoM) numerical computational results. Good agreements are achieved in the comparison. The study of radiowave propagation under complex environments is vitally important for improving the effective signal coverage in wireless communication and television broadcasting with plenty of urban buildings as well as mountainous terrains. It presents a big challenge due to the multiple wave propagation mechanisms involving the line of sight, diffraction and scattering propagation and the characteristic of large scale computation. The major issues refer to how to keep the computational accuracy and conducting the large scale computation at the same time. To meet the request for large scale computation of actual engineering problems, the modified empirical formulas are often used for analysis and computation with the drawback of poor computational accuracy, and the variance of the predicted propagation loss is generally 2 to 5 dB for the flat terrain, but for the complex terrain, the variance of the predicted propagation loss is usually about 5 to 10 dB. Many computational methods have been developed to improve the predicted accuracy. The commonly used methods are geometric diffraction methods, path integration methods and FDTD. It is hard to meet the requirement of engineering computation because these methods are computationally expensive and require high data accuracy of the terrain database at the same time (1). The Parabolic Equation (PE) method is a computational method for radiowave propagation loss besides the full wave method and ray tracing method, and has become a research focus. The PE method has high potential in radiowave propagation of mobile communication and broadcasting and television signal coverage as it can not only deal with complex terrain boundary conditions accurately, but also deal with complex atmosphere waveguide structure. Compared with the full wave method, the PE method deals with an unidirectional wave propagation, in which the equation can be solved by the step marching method in the propagation direction. The PE method can reduce the computational complexity of the full wave equation.

The parabolic equation (PE) method is a useful approach for solving range-dependent ocean acoustics problems. The PE method can also be used to model the propagation of internal gravity waves in the ocean. One of the possible applications of a range-dependent gravity wave model to ocean acoustics would be to generate a model-based noise spectrum that could be used as an alternative or complement to the empirical spectrum of Garrett and Munk [J. Geophys. Res. 80, 291–297 (1975)]. This could be done using the approach of Kuperman and Ingenito for modeling surface-generated acoustic noise [J. Acoust. Soc. Am. 67, 1988–1996 (1980)] for an appropriate distribution of gravity wave sources. The mathematical details of a PE for gravity waves will be discussed. Examples will be presented to illustrate the propagation of gravity waves in range-dependent environments. [Work supported by ONR.]

Solving an acoustic wave equation using a parabolic approximation is a popular approach for many existing ocean acoustic models. Commonly used parabolic equation (PE) model programs, such as the range-dependent acoustic model (RAM), are discretized by the finite difference method (FDM). Considering the idea and theory of the wide-angle rational approximation, a discrete PE model using the Chebyshev spectral method (CSM) is derived, and the code is developed. This method is currently suitable only for range-independent waveguides. Taking three ideal fluid waveguides as examples, the correctness of using the CSM discrete PE model in solving the underwater acoustic propagation problem is verified. The test results show that compared with the RAM, the method proposed in this paper can achieve higher accuracy in computational underwater acoustics and requires fewer discrete grid points. After optimization, this method is more advantageous than the FDM in terms of speed. Thus, the CSM provides high-precision reference standards for benchmark examples of the range-independent PE model.

The complex nature of the ocean environment requires advanced computational acoustic models to gain insights into the dominating physical factors controlling the underwater sound propagation and scattering in the ocean. In this study, we explore the “ab-initio” approach to solve wave equations with numerical algorithms that can be implemented in distributed memory parallel computers. The goal is to improve the calculation speed so long-range global scale hydroacoustic wave propagation can be studied more efficiently and effectively. Two major algorithms: the finite difference time domain (FDTD) method and the Parabolic Equation (PE) method are investigated. Two PE-based numerical models are considered. One is the Split-Step-Fourier Parabolic Equation (SSFPE) model using split-step Fourier schemes in three-dimensional (3D) environments. The second PE model is a multi-frequency implementation of the Range-dependent Acoustic Model (RAM) that computes two-dimensional (2D) sound pressure fields. One of the primary challenges in global-scale hydroacoustic propagation modeling with the “ab-initio” approach is the demand for significant computational resources on both memories and computational speeds. To address this, we employed parallel computing techniques using directive-based programming languages, such as OpenACC and XscalableACC, to leverage the power of multiple Graphics Processing Unit (GPU) systems and Central Processing Unit (CPU) cluster computers. Our performance evaluation revealed substantial speedup gains. For the FDTD method, we achieved approximately 3.5-fold and 4-fold speedups with four GPUs and four CPU cluster nodes, respectively. With the 3D SSFPE model, we obtained a speedup of 3.7-fold using four CPU cluster nodes. A significant result was observed with the 2D broadband multi-frequency RAM model, where we achieved a 110-fold speedup compared to one core original implementation. These results demonstrate the promising potential of massively parallel computers for ocean acoustic propagation modeling and highlight the significant performance benefits of utilizing parallel computing techniques. Our findings emphasize the importance of efficient computational strategies.

The parabolic equation method provides an excellent combination of accuracy and efficiency for range-dependent ocean acoustics and seismology problems. This approach is highly developed for problems in which the ocean bottom can be modeled as a fluid. For the elastic case, there remain some accuracy limitations for problems involving sloping interfaces. Progress on this problem has been made by combining a new formulation of the elastic parabolic equation that handles layering more effectively [W. Jerzak, ‘‘Parabolic Equations for Layered Elastic Media,’’ doctoral dissertation, Rensselaer Polytechnic Institute, Troy, NY (2001)] and a mapping approach that handles sloping interfaces accurately [J. Acoust. Soc. Am. 107, 1937–1942 (2000)]. This approach makes it possible to handle problems involving complex layering and steep slopes, but the rate of change of the slope must be small. The method and its application to data will be described. Our immediate goal is to model propagation of seismic surface waves propagating across a transition between dry and marshy terrain. We have suitable data applicable to vehicle-tracking problems from Marine Corps Base Camp, Pendleton, CA. [Work supported by ONR.]

The parabolic equation (PE) method has been the most efficient range-dependent propagation model in ocean acoustics ever since the split-step Fourier solution was developed [F. D. Tappert, ‘‘Numerical solutions of a canonical nonlinear dispersive wave equation,’’ SIAM Rev. 16, 140 (1974)]. Since the asymptotic accuracy of the split-step Fourier solution is restricted to leading order (a serious limitation for some problems), the PE method was extended to higher-order asymptotic accuracy using Padé approximations. Until recently, the higher-order PE has been solved using relatively inefficient finite-difference solutions, and the split-step Fourier solution has remained in widespread use despite the accuracy improvements. To improve efficiency without sacrificing accuracy, we apply Padé approximations to achieve higher-order accuracy in both the numerics and the asymptotics. The split-step Padé solution, which has been implemented for both fluid and elastic media, achieves both higher-order asymptotic accuracy and the efficiency of the split-step Fourier solution. This finite-difference solution is one to three orders of magnitude faster than other finite-difference solutions.

The propagation phenomena through 2D surface duct-to-elevated duct transition is described in terms of parabolic equation (PE) method. The split step marching solution of the PE method is first compared with the reference solutions obtained from Helmholtz wave equation in longitudinally invariant surface duct problem and the excellent agreement between the results is presented. Then the PE method is applied to propagation through surface duct-to-elevated duct transition problem which is non-separable in any of the known orthogonal coordinate system. The transformation of the surface trapped modes into propagating elevated beams is shown with the PE method. >

The parabolic equation (PE) method is very effective for computing the complex electromagnetic environment (EME) simulation. For the large-scale 3D EME simulation based on the PE method, the computation and time complexity is not acceptable. To solve this problem, a quasi-3D PE simulation method with low complexity is introduced. But the accuracy of this method is not high, and some methods have been proposed to improve the accuracy by compensating error, choosing appropriate slice angle, interpolating some values between slices and mixing the PE and ray tracing method. With a multi-level parallel method, the 3D EME simulation can be paralleled from the three levels: job parallel, task parallel, thread parallel, and it is used to accelerate the computation process dramatically in order to calculate faster.

The ionizable metal vapor released in the ionosphere can artificially generate space electron clouds. They can scatter radio waves at decameter and meter wavelengths so as to realize over-the-horizon propagation. Based on the parabolic equation (PE) method and using Padé-(1,1) polynomial, the electromagnetic scattering characteristics of cylindrical artificial electron clouds are preliminarily studied in this letter. The bistatic radar scattering cross section (RCS) and the ground received field distribution via artificial electron cloud are calculated for various incidence angles at different radio frequencies and distances. The PE method is used here since its advantages of high precision, fast computing and less computing resource occupation over traditional methods. The study of electromagnetic scattering is valuable to the follow-up release experiments for generating electron cloud. It throws a new light on the practical utilization of over-the-horizon radio communication via scattering channel of artificial electron clouds.

Consistent Nonlinear Mild-Slope Equation Models for Wide-Angle Water Waves Transformation.

This paper reviews some of the highlights of selected topics in ocean acoustics during the thirty years that have passed since the founding of the Journal of Theoretical and Computational Acoustics. Advances in computational methods and computers helped to make computational ocean acoustics a vibrant area of research during that period. The parabolic equation method provides an unrivaled combination of accuracy and efficiency for propagation problems in which the bathymetry, sound speed, and other environmental parameters vary in the horizontal directions. The extension of this approach to cases involving layers that support shear waves has been an active area of research throughout the thirty year period. Interest in basin-scale and global-scale propagation was stimulated by the Heard Island Feasibility Test for monitoring climate change in terms of changes in travel time that occur as the temperature of the ocean rises. Diminishing ice cover in the Arctic, which is one of the consequences of climate change, has stimulated renewed interest in Arctic acoustics during the past decade. Reverberation is a challenging problem that was the topic of a major research program during the beginning of the thirty year period. An innovative approach for making it feasible to solve such problems was applied to data for reverberation from the seafloor and from schools of fish, and some of the findings were featured in Science and Nature. Source localization is one of the core problems in ocean acoustics. When applied on a 2-D array of receivers, an approach based on the eigenvectors of the covariance matrix is capable of separating the signals from different sources from each other, determining when this partitioning step is successful, and tracking sources that cross each other in bearing; one of the advantages of this approach is that it does not require environmental information or solutions of the wave equation. Geoacoustic inversion for estimating the layer structure, wave speeds, density, and other parameters of ocean bottoms has also been a topic of interest throughout the thirty year period.

This work performs a general analysis on the propagation of electromagnetic waves through the troposphere over irregular terrain in the VHF and UHF bands. This is the main scenario of the so-called terrestrial waves, which comprise most of the widespread telecommunications services in society. The analysis begins with concepts of electromagnetics, passes through important aspects of antennas and covers most mechanisms that affects this type of propagation, such as: free-space propagation, propagation over flat and spherical Earth, Tropospheric propagation, diffraction over obstacles and propagation over irregular terrain. In addition, the parabolic equation method is developed in detail for propagation in this scenario and a number of tests and comparisons are performed to prove its operation. However, this method is intricate to implement in order to consider all aspects of propagation like the irregular terrain, for example. Therefore, a new hybrid propagation model is developed aiming to take advantage of an easier part of the parabolic equation model and combining it with the simple Bullington knife-edge diffraction model. In this case, the effects of terrain irregularities are considered explicitly. Results demonstrate the great potential of the developed model and its superiority compared to well known models such as the Recommendation of ITU-R P. 1546.

A wide‐angle parabolic equation model has been developed that is applicable to a variety of propagation problems in the atmosphere and ocean. Linear finite elements are used to discretize the vertical dependence of the acoustic field. This method of discretizatiun enables the model to efficiently handle both small‐scale and large‐scale vertical variations in complex wavenumber and density. For example, the model can be applied to a turbulent atmosphere and to a randomly layered ocean sediment. The computational grid is terminated at the top and bottom with a complex impedance condition so that a locally reacting surface for atmospheric problems and a pressure‐release surface for underwater problems can easily be incorporated. Results from selected applications will be presented to demonstrate the capabilities of the model.

An improved fractal sea surface model, which can describe the capillary waves very well, is introduced to simulate the one-dimension rough sea surface. In this model, the propagation of electromagnetic waves (EWs) is computed by the parabolic equation (PE) method using the finite-difference (FD) algorithm. The numerical simulation results of the introduced model are compared with those of the Miller-Brown model and the Elfouhaily spectrum inversion model. It has been shown that the effects of the fine structure of the sea surface on the EWs propagation in the introduced model are more apparent than those in the other two models.