Parabolic Equation Method Research Articles

A Novel PE/FDTD Hybrid Model for Predicting Echo Signals of Radar Targets in Large-Scale Complex Environments

The prediction of radar target echo signal in a large-scale complex environment is of great significance in target detection, radar design and other applications. In this paper, a novel PE/FDTD hybrid model is proposed to predict monostatic radar target echo signals in large-scale complex environments. The target echo signal can be regarded as the output response of the transmitted signal which passes through a linear time invariant system composed of complex environment and target. The transport function of the complex environment is computed by the parabolic equation (PE) method and the scattering characteristics of the target are calculated by finite difference time domain method (FDTD). The combination of PE and FDTD is realized through the system response function. In combination with the “stop-go” method, the prediction of moving target echo signal is realized. In addition, the error of combining PE with FDTD is analyzed, and the result shows that the error is less than 0.1% when the target distance is more than 10km. Additional numerical examples are given to demonstrate the correctness of the method in semi-space, rain, fog and atmospheric duct environments. The calculated results are compared with those of theoretical method, time-domain shooting and bouncing ray (TDSBR), multi-level fast multi-pole method (MLFMM) and waveguide mode theory, and good agreement among them is observed. Finally, the simulation analysis of the missile echo signal in the mixed sea-land environment with surface duct is carried out. The simulation results show that this model can be used to predict the echo signal of airborne targets in a large-scale complex environment. It is a promising option for multi-scale computing involving radar target and large-scale environment.

Multisector parabolic-equation approach to compute acoustic scattering by noncanonically shaped impenetrable objects.

Parabolic equation (PE) methods have long been used to efficiently and accurately model wave phenomena described by hyperbolic partial differential equations. A lesser-known but powerful application of parabolic equation methods is to the target scattering problem. In this paper, we use noncanonically shaped objects to establish the limits of applicability of the traditional approach and introduce wide-angle and multiple-scattering approaches to allow accurate treatment of concave scatterers. The PE calculations are benchmarked against finite-element results, with good agreement obtained for convex scatterers in the traditional approach, and for concave scatterers with our modified approach. We demonstrate that the PE-based method is significantly more computationally efficient than the finite-element method at higher frequencies where objects are several or more wavelengths long.

Modeling Equation for Multiple Knife-Edge Diffraction

This article is devoted to the canonical problem of the radio-wave propagation in the presence of multiple knife-edges. The system of integral equations in a spectral domain for the scattered field is derived. On the basis of its solution diffraction and reflection effects spawned by the multiple knife-edges may be accurately modeled. The electromagnetic field is generated by a source of the monochromatic radio waves with an arbitrary beam pattern. The efficient numerical solution to the proposed equation is given. The comparison with the two-way parabolic equation (PE) method, the uniform theory of diffraction (UTD) method, and the finite element method (FEM) is presented.

A variationally separable splitting for the generalized‐α method for parabolic equations

AbstractWe present a variationally separable splitting technique for the generalized‐α method for solving parabolic partial differential equations. We develop a technique for a tensor‐product mesh which results in a solver with a linear cost with respect to the total number of degrees of freedom in the system for multidimensional problems. We consider finite elements and isogeometric analysis for the spatial discretization. The overall method maintains user‐controlled high‐frequency dissipation while minimizing unwanted low‐frequency dissipation. The method has second‐order accuracy in time and optimal rates (hp+1 in L2 norm of u and hp in L2 norm of ∇u) in space. We present the spectral analysis on the amplification matrix to establish that the method is unconditionally stable. Various numerical examples illustrate the performance of the overall methodology and show the optimal approximation accuracy.

Hybrid FDTD–PE method for Loran‐C ASF prediction with near‐source complex topography

The parabolic equation (PE) method and finite-difference time-domain (FDTD) method are combined to predict long-range Loran-C additional secondary factors (ASFs) in situations with near-source complex topography. The non-uniform FDTD method is employed in the complex source region, while the PE method is used for propagation over smooth terrain to a long distance. By using the advantages of each algorithm, the hybrid FDTD-PE approach is capable of modelling the near-source topographic details, while saving much computational resources as compared to a full FDTD solution. Numerical results along several paths are used to validate the hybrid algorithm. Furthermore, the hybrid method is applied to analyse the effect of near-source topographic complexities, including non-flat land, land-sea transition, and island (cliff)-sea transition, on the navigational coverage for the planning of the Loran-C systems.

Composite Refractive Index on Electromagnetic Wave Propagating in Spherical Medium

Calculating how electromagnetic waves propagate in spherical media is currently approached in two ways: by using eikonal equations based on a spherical coordinate system or by using parabolic equations based on a Cartesian coordinate system. However, eikonal equations can produce problems of accuracy and stability, while the modified refractive index used for parabolic equations is only suitable for low-altitude conditions. In this article, a comparison of the differential form of Bouguer’s law and Snell’s law is used to develop a composite refractive index. This enables the problem of how electromagnetic waves propagate in spherical medium to be converted into one of how they propagate in cuboid medium. By providing a way of solving eikonal equations according to a Cartesian coordinate system, the problems associated with traditional methods are overcome. It is proved that the modified refractive index widely used in the field of radio is the approximation of the composite refractive index, which breaks the height limit of parabolic equation method based on modified refractive index.

Convergent numerical methods for parabolic equations with reversed time via a new Carleman estimate

The key tool of this paper is a new Carleman estimate for an arbitrary parabolic operator of the second order for the case of reversed time data. This estimate works on an arbitrary time interval. On the other hand, the previously known Carleman estimate for the reversed time case works only on a sufficiently small time interval. First, a stability estimate is proven. Next, the quasi-reversibility numerical method is proposed for an arbitrary time interval for the linear case. This is unlike a sufficiently small time interval in the previous work. The convergence rate for the quasi-reversibility method is established. Finally, the quasilinear parabolic equation with reversed time is considered. A weighted globally strictly convex Tikhonov-like functional is constructed. The weight is the Carleman weight function which is involved in that Carleman estimate. The global convergence of the gradient projection method to the exact solution is proved for this functional.

A combined finite element method for parabolic equations posted in domains with rough boundaries

In this paper, we consider the solution of parabolic equations posted in domains with rough boundaries using the combined finite element procedure proposed in Xu et al. [A combined finite element method for elliptic problems posted in domains with rough boundaries, J. Comput. Appl. Math. 336 (2018), pp. 235–248] in space and backward-Euler scheme in time. We first prove error estimates for elliptic projection operator for both and by the dual argument which depend on the elliptic auxiliary problem, and then energy error estimates of semi-discrete and fully discrete schemes which have convergence rates about and , where s>0, respectively. Numerical results are provided for parabolic equations in domains with non-oscillating or oscillating boundaries to verify the theoretical findings.

HOMOM/PE Hybrid Algorithm Based Calculation on Ground-Wave of the HF Antennas over Irregular Terrain

The parabolic equation model usually uses a gaussian pattern to fit the actual antenna’s pattern in free space to set the initial field, and then uses the image theory to deal with the effect of ground on the antennas. However, this method of setting the initial field cannot consider the specific HF antenna, and the image theory is only applicable to the radio wave propagation of antennas working in the microwave bands and above. The HF antenna is erected low, and the irregular terrain or the buildings around the antenna and their media parameters will affect the current distribution on the antenna, resulting in a large difference with the radiation pattern in the free space. The traditional method of fitting the pattern is no longer accurate. Aiming at the problem of radio wave propagation of HF antennas in the actual installation environment by parabolic equation method, a hybrid algorithm based on higher order-method of moment and parabolic equation method (HOMOM-PE) is proposed. The hybrid algorithm divides the whole calculation region into two regions, which is the inner zone and the outer zone. The inner zone is the zone that has a great influence on the antenna current distribution. In this area, HOMOM is used to model the antenna and the terrain together, and the influence of the complex terrain near the antenna on the antenna is strictly calculated. The PE algorithm is used to calculate the long-distance propagation, and the initial field data is derived from the electromagnetic vector distribution calculated in the inner region. The hybrid algorithm combines the antenna radiation and the wave propagation directly, which solves the problem of the initial field setting of the actual HF antenna under the complicated terrain conditions.

Application of a multi-sector parabolic equation approach to compute acoustic scattering by non-canonically shaped impenetrable objects

While parabolic equation (PE) methods have seen great use in ocean acoustic waveguide propagation, it is not as widely known that a multi-sector PE method, originally proposed by Levy and Zaporozhets [J. Acoust. Soc. Am. 103, 735–741 (1998)], can be applied to compute the scattered field about an object. Previous work was only able to benchmark the method with canonical geometries. In this talk, we demonstrate favorable agreement between free-field PE-based and finite-element based computations, the latter being taken as a benchmark, in two and three dimensions for non-canonically shaped impenetrable objects. Run-time scaling comparisons are also presented, which demonstrate the great advantage of the PE method in the high-frequency limit when a target is several or more wavelengths long. We also show how wide-angle PE and multiple-scattering corrections can be incorporated to treat concave scatterers. [Work sponsored by the Office of Naval Research.]While parabolic equation (PE) methods have seen great use in ocean acoustic waveguide propagation, it is not as widely known that a multi-sector PE method, originally proposed by Levy and Zaporozhets [J. Acoust. Soc. Am. 103, 735–741 (1998)], can be applied to compute the scattered field about an object. Previous work was only able to benchmark the method with canonical geometries. In this talk, we demonstrate favorable agreement between free-field PE-based and finite-element based computations, the latter being taken as a benchmark, in two and three dimensions for non-canonically shaped impenetrable objects. Run-time scaling comparisons are also presented, which demonstrate the great advantage of the PE method in the high-frequency limit when a target is several or more wavelengths long. We also show how wide-angle PE and multiple-scattering corrections can be incorporated to treat concave scatterers. [Work sponsored by the Office of Naval Research.]

Variable Step Size Solution of Standard Parabolic Equation in Complex Environment

Aiming at the problem that it is difficult to balance the accuracy and velocity of the fixed-step solution of standard parabolic equation in the study of radio wave propagation in a wide range of complex environments, a variable-step solution method of standard parabolic equation is proposed. Firstly, after deducing the relationship between the error of SSFT solution and step size, frequency and other factors; And the basic selection range of step size for SPE variable step size solution is given. Secondly, the action mechanism of different environmental factors and the requirement of changing trend for step size are expounded through simulation, and the complex environment of SPE application is classified according to the requirement of error. Finally, the method is used to simulate the characteristics of the typical complex environment. The simulation results show that the method can save up to 71.4% of the time compared with the fixed-step method of parabolic equation under the condition of ensuring the calculation accuracy. The reliability and efficiency of the proposed method are verified, and the calculation efficiency of the parabolic equation radio wave prediction can be greatly improved. Therefore, the parabolic equation method with variable step size can ensure the accuracy of calculation, reduce the memory and time required for calculation, greatly improves the efficiency of calculation, and has practical significance in the application of real-time prediction of electromagnetic wave propagation in a wide range of complex environments.

Levin's collocation method for predicting sound fields in the presence of sound speed gradients

Sound propagation in a stratified medium has been studied for decades. The Ray tracing method, which is one of the most popular methods, calculates the ray path and propagating angle of a sound ray. However, the ray method is not accurate under many circumstances compared with the wave-based numerical schemes. The Fast field program (FFP), an efficient method for numerical evaluation of the Fourier integral, is accurate but requires significant computational resources for high frequency sound fields because of the highly oscillatory functions found in the integrand. In this paper, a Levin’s collocation method is introduced and used in the evaluation of the sound fields above a locally reacting ground with the presence of the sound speed gradient. The method is far more efficient than most other available numerical techniques such as the FFP and Parabolic Equation method.

LF Groundwave Propagation Modeling Over Irregular Terrain by the Hybrid Two-Way Parabolic Equation Method

The traditional two-way (2W) parabolic equation (PE) (2W-PE) method has been applied for modeling groundwave propagation over irregular terrain. However, based on a staircase terrain model (STM), the method is not sufficient in sloping facets modeling. Although more accurate one-way PE method based on conformal mapping model (CMM) is available, it handles only forward-propagating waves and neglects the backward-propagating waves, which become significant especially for steep terrain. In this paper, a hybrid 2W-PE method is presented to model the low-frequency (LF) groundwave propagation over irregular terrain. The method combines the STM and CMM to improve the prediction accuracy of both forward and backward propagations. The numerical results are compared with those of the conventional 2W-PE and finite-difference time-domain (FDTD) methods. While calibrated against the FDTD method, the proposed hybrid 2W-PE method has higher accuracy than the traditional 2W-PE method at the same computational cost.

Application of elastic parabolic equation solutions to acoustic reverberation in ice-covered underwater environments

Parabolic equation solutions can be obtained for underwater acoustic environments where elastic boundaries affect the acoustic propagation. One example of this situation is in the Arctic where a (potentially rough) ice layer is on top of the water column. The parabolic equation method shown here rigorously applies the zero-traction and fluid-elastic interface boundary conditions to obtain full field Green’s function estimates near the ice-water interface. This solution is crucial for calculation of recently derived reverberation estimates that require both horizontal and vertical derivatives of the complex acoustic field. Monostatic reverberation estimates for a rough under-ice interface are calculated for an upward refracting water sound speed profile and an elastic ocean bottom with a nearly fluid mud layer. These calculations could be used, for example, to estimate areal distribution of the ice layer roughness from azimuth-time dependence of acoustic reverberation measured in Arctic environments. Performance of the Pade rational function approach in the presence of thin ice cover and low shear speed ocean bottom layers will be addressed.

Near shore atmospheric acoustic transmission loss: Numerical predictions

This work presents advances in the development of a numerical model of atmospheric acoustic propagation designed for a littoral or riverine environment with a near-shore acoustic source and on-shore receivers. A parallel experimental effort uses a measurement system that records concurrent atmospheric and acoustic transmission loss data. The numerical model uses a generalized terrain parabolic equation method to account for sea surface roughness and ground topography along the propagation path. The model accounts for vertical temperature and wind speed profiles as well as surface impedance at the shore. For the comparison with experimental data, wind speed is used to predict the sea state. The propagation range consists of three segments: over water, over swash and sandy shore, and over complex dune with highly varied vegetation of up to 2 m height. Surface impedance estimates for the sandy shore are made based on grain size distribution and impedance tube measurements of excised samples. Existing models for surface impedance of vegetation were used for the dune portion of the propagation range.This work presents advances in the development of a numerical model of atmospheric acoustic propagation designed for a littoral or riverine environment with a near-shore acoustic source and on-shore receivers. A parallel experimental effort uses a measurement system that records concurrent atmospheric and acoustic transmission loss data. The numerical model uses a generalized terrain parabolic equation method to account for sea surface roughness and ground topography along the propagation path. The model accounts for vertical temperature and wind speed profiles as well as surface impedance at the shore. For the comparison with experimental data, wind speed is used to predict the sea state. The propagation range consists of three segments: over water, over swash and sandy shore, and over complex dune with highly varied vegetation of up to 2 m height. Surface impedance estimates for the sandy shore are made based on grain size distribution and impedance tube measurements of excised samples. Existing models ...

Use of the parabolic equation method for local infrasound propagation predictions

Local infrasound (<150 km) propagation is strongly influenced by range- and time-dependent meteorological conditions. The Parabolic Equation computational method provides an efficient and flexible tool for predicting infrasound propagation in this regime. Over these longer distances, the influence of spatial and temporal variations in the meteorological conditions can be significant. Meteorological conditions extracted from simulations using the Weather Research and Forecasting (WRF) model are incorporated by updating the effective sound speed profile as a function of propagation range. By examining horizontal grid increments between 1 and 15 km and temporal resolutions between 4 and 60 seconds, we begin to quantify the potential variability in predictions at ranges of up to 150 km, for frequencies between 1 and 20 Hz. This procedure will inform a broader discussion of the application of the PE method to infrasound prediction. [Work funded by the Assistant Secretary of the Army (Acquisition, Logistics, and Technology) [ASA(ALT)] under 62784/T40. Distribution Statement A: Approved for public release; Distribution is unlimited.]Local infrasound (<150 km) propagation is strongly influenced by range- and time-dependent meteorological conditions. The Parabolic Equation computational method provides an efficient and flexible tool for predicting infrasound propagation in this regime. Over these longer distances, the influence of spatial and temporal variations in the meteorological conditions can be significant. Meteorological conditions extracted from simulations using the Weather Research and Forecasting (WRF) model are incorporated by updating the effective sound speed profile as a function of propagation range. By examining horizontal grid increments between 1 and 15 km and temporal resolutions between 4 and 60 seconds, we begin to quantify the potential variability in predictions at ranges of up to 150 km, for frequencies between 1 and 20 Hz. This procedure will inform a broader discussion of the application of the PE method to infrasound prediction. [Work funded by the Assistant Secretary of the Army (Acquisition, Logistics, an...

Modified Method for Parabolic Equations in one Dimensional with Nonlocal time Weighting Initial Condition

In this paper, we develop a modified version of explicit scheme based on finite difference method for the one-dimensional parabolic partial differential equations with nonlocal time weighting initial conditions. The dominancy of the Saulyev’s schemes based on finite difference, over the previous explicit FTCS, Duke-Frankel, as well as implicit BTCS, Crandal’s technique and Crank Nicholson’s scheme has already been established, which proved to be unconditionally stable, use less CPU time and computational effort. However, in this paper a modification of Saulyev’s first kind formula is developed. Main focus was to reduce error of the Saulyev’s formula using proposed method. The comparison has been carried out between both methods to observe errors in different conditions and step sizes. The new modified scheme is proved to be satisfactory and unconditionally stable.

Local accuracy of cross-term corrections of three-dimensional parabolic-equation models.

This paper deals with accuracy assessment for cross-derivative (depth and azimuth) term corrections to a three-dimensional (3D) parabolic equation (PE). The focus is on local errors, involving comparison of the two sides of the PE at insertion of (scaled) Helmholtz-equation reference solutions. For media with a particular type of lateral sound-speed variation, mode expansion together with wavenumber integration to compute modal expansion coefficients produces very accurate Helmholtz solutions for the field and its spatial derivatives. There are explicit expressions for the wavenumber integrands in terms of Airy and exponential functions, and accuracy improvements by PE cross-derivative terms are easy to assess. For a relevant example, with 3D effects similar to those in a 3D Acoustical Society of America wedge benchmark, inclusion of a leading-order as well as an additional, higher-order, cross-derivative term in the PE is favorable. The additional term provides a fourth-order accurate approximation of the PE square-root operator. A fifth-order accurate Padé approximation yields further improvement, approaching the numerical accuracy limit set by the PE method itself. The adiabatic approximation is exact for the particular media under study, but the local PE errors are similar for related 3D wedge examples with mode coupling.

Three-dimensional boundary fitted parabolic-equation model of underwater sound propagation.

A three-dimensional (3-D) parabolic-equation (PE) method utilizing a higher-order split-step Padé algorithm and a boundary fitted grid has been developed to accurately solve 3-D underwater sound propagation problems with non-planar or tilted boundaries. At each PE marching step, the split-step Padé algorithm enables the method of alternating directions to implement the square-root Helmholtz operator by carrying out its one-dimensional (1-D) derivative components alternately, and it also allows a straightforward application of the 1-D non-uniform Galerkin method to discretize the solution mesh. The advantage of the boundary fitted grid to improve PE solution accuracy is most profound in the case of fitting to a pressure release surface as its boundary condition is a scalar and has no direction. This method can also be applied to a sloping interface by rotating the grid to align with the interface. Numerical problems of semi-circular waveguide and tilted wedge were solved using this boundary fitted PE method, and benchmark reference solutions were used to examine and confirm the accuracy of the PE solutions. Future applications include modeling 3-D acoustic scattering from a rough sea surface and 3-D sound propagation in beach environments.

On deep-water reception of pseudorandom acoustical signals propagating from the shelf zone to the deep ocean

The results of an experiment conducted in September 2017 to substantiate the applicability of the “acoustic mudslide” effect when solving the positioning problems of autonomous underwater vehicles in cases of their operation at depths substantially exceeding the depth of the axis of the underwater sound channel are discussed. The results of the experimental studies and numerical analysis of the effect of an acoustic energy focusing in the near-bottom layer of the shelf and its transition into deep-water (up to 500 m) layers of the Sea of Japan for summer-autumn hydrological conditions are presented. Mathematical modelling of the propagation of acoustic waves in a model waveguide reproducing the experimental conditions using the parabolic equation method has been performed. The observed effective signal propagation speeds are explained using the normal wave method.