Parabolic Wave Equation Research Articles

Range-dynamical low-rank split-step Fourier method for the parabolic wave equation.

Numerical solutions to the parabolic wave equation are plagued by the curse of dimensionality coupled with the Nyquist criterion. As a remedy, a new range-dynamical low-rank split-step Fourier method is developed. The integration scheme scales sub-linearly with the number of classical degrees of freedom in the transverse directions. It is orders of magnitude faster than the classic full-rank split-step Fourier algorithm and saves copious amounts of storage space. This enables numerical solutions of the parabolic wave equation at higher frequencies and on larger domains, and simulations may be performed on laptops rather than high-performance computing clusters. Using a rank-adaptive scheme to optimize the low-rank equations further ensures the approximate solution is highly accurate and efficient. The methodology and algorithms are demonstrated on realistic high-resolution data-assimilative ocean fields in Massachusetts Bay for two three-dimensional acoustic configurations with different source locations and frequencies. The acoustic pressure, transmission loss, and phase solutions are analyzed in the two geometries with seamounts and canyons across and along Stellwagen Bank. The convergence with the rank of the subspace and the properties of the rank-adaptive scheme are demonstrated, and all results are successfully compared with those of the full-rank method when feasible.

Dominance of polarization modes and absorption on self-focusing of laser beams in collisionless magnetized plasma

This work investigates the polarization modes i.e., Extraordinary (E-mode) and Ordinary (O-mode) on self-focusing of higher-order mode of elegant Hermite-cosh-Gaussian (EHChG) laser beams within collisionless magnetized plasma medium. In this study, the authors have taken transverse electromagnetic ([Formula: see text]) mode with mode indices (0, 4) and explored. The ponderomotive force is primarily responsible for the dielectric constant’s nonlinearity in this context. Also, WKB and paraxial approximations via the parabolic wave equation approach are used to establish the general form of differential equations for beam-width parameters in both transverse dimensions of the beam. In this paper, the authors have theoretically investigated the dominance of polarization modes and absorption coefficient on self-focusing. The final results show good enhancement of laser propagation through collisionless plasma medium.

Asymptotic Ray Method for the Double Square Root Equation

The parabolic wave equation describes wave propagation in a preferable direction, which is usually horizontal in underwater acoustics and vertical in seismic applications. For dense receiver arrays (receiver spacing is less than signal wavelength), this equation can be used for propagating the recorded wavefield back into the medium for imaging sources and scattering objects. Similarly, for multiple source and receiver array acquisition, typical for seismic exploration and potentially beneficial for ocean acoustics, one can model data in one run using an extension of the parabolic equation—the pseudo-differential Double Square Root (DSR) equation. This extended equation allows for the modeling and imaging of multi-source data but operates in higher-dimensional space (source, receiver coordinates, and time), which makes its numerical computation time-consuming. In this paper, we apply a faster ray method for solving the DSR equation. We develop algorithms for both kinematic and dynamic ray tracing applicable to either data modeling or true-amplitude recovery. Our results can be used per se or as a basis for the future development of more elaborated asymptotic techniques that provide accurate and computationally feasible results.

Self-focusing/defocusing of skew-cosh-Gaussian laser beam for collisional plasma

The inverse relationship between the linear increase in skewness parameter s and the domain’s width of the order of skewness n plays a vital role in both critical beam radius and propagation dynamics of skew-cosh-Gaussian (skew-chG) laser beams. The interplay between the skewness parameter s and the order of skewness n is explored analytically and graphically in the current study to unveil the complexity of the propagation dynamics of the skew-chG laser beam. Naturally, the intensity’s complexity considerably affects the dielectric constant of the medium. Basically, the nonlinearity in the dielectric function of collisional plasma is attributed to the non-uniform heating of energy carriers along the wavefront of the laser beam, which becomes important and is used in the current study. By following Akhmanov’s parabolic wave equation approach under Wentzel–Kramers–Brillouin and paraxial approximations, the nonlinear differential equations are set up for the beam width parameters f 1 and f 2 and solved numerically. The present work analytically investigates the domains of the order n of skew-chG laser beams for a given set of skewness parameter s to investigate their effects on the self-focusing and defocusing of skew-chG laser beams. The critical curve also gets affected significantly due to the choice of domains for n. Finally, the numerical results are presented in the form of graphs and discussed in this study.

Anisotropic Airy beams

Anisotropic Airy beams are presented as solutions to the parabolic wave equation. We obtain the cross polarization terms in the new beams using the elegant Mukunda–Simon–Sudarshan matrix substitution rule for transition from Helmholtz scalar wave optics to the Maxwell vector wave optics. Other properties such as the intensity distribution are also presented.

Physics‐based wave propagation model assisted vehicle localisation in tunnels

AbstractConnected and autonomous vehicles systems can provide a safe and convenient transportation solution through wireless communication between vehicles and roadside communication infrastructure. The vehicle's ability to establish its location plays an important role in such systems. This article presents an efficient localisation framework by combining physics‐based wireless propagation models and fingerprint localisation algorithms. To that end, a parabolic wave equation (PWE)‐based wireless channel simulator is utilised to obtain the direction of arrival (DOA) values that construct the fingerprint dataset. A detailed procedure on how to obtain the DOA information using PWE, as well as analysis and guidelines on the selection of the key parameters of the localisation framework are provided. Numerical results are compared with theoretical data in terms of accuracy, and the selection of parameters during the localisation process is investigated in actual road tunnel cases. Besides, various numbers of wireless access points and schemes of fingerprinting algorithms have been studied, showing the capability of the proposed approach to be employed for the pre‐design of wireless communication systems.

Embedding antennas with tilted beam patterns into parabolic wave equation‐based models for tunnel propagation

AbstractAccurate radio wave prediction models are necessary for the innovation of mobile communication technology and the deployment of wireless sensor networks in railways. The parabolic wave equation (PWE) method is one commonly used method to predict the electromagnetic characteristic of long railway tunnels. However, the advent of novel antennas and technologies (e.g., beam steering, beamforming) has posed new challenges to PWE‐based models such as effective means of taking tilted beam patterns into account. This study addresses such a challenge by using tilted Gaussian beams to embed antennas with oriented patterns or tilted beam directions into PWE‐based models. The proposed approach does not reduce the simulation efficiency of PWE itself. The validity and practicability of the technique are demonstrated by comparing against ray‐tracing as well as experimental measurements in various tunnel environments.

Influence of Magnetic field and Decentred Parameter on Self-Focusing of CoshGaussian Laser Beam in Collisional Magnetized Plasma

In the present investigation, Three-dimensional Cosh-Gaussian laser beam is introduced. The self-focusing and defocusing of Cosh-Gaussian laser beam in collisional magnetized plasma have been investigated theoretically. The final Differential equation for the beam width parameter is derived by following Wentzel-Kramer’s-Brillouin (WKB) and paraxial approximation through standard Akhmanov`s parabolic wave equation. The final results of numerical computation are presented in the plot of beam width parameters (f1 & f2) versus normalized propagation distance ( ). In present investigation the author shows nonlinear effect due to magnetic field (B0), decentered parameter (b) and plasma density on self-focusing and defocusing in collisional magnetized plasma. The results show well enhancement in beam of self-focusing.

Methods and Algorithms of Subsurface Holographic Sounding

In our experiments, we develop and test portable multi-element receiver antenna arrays, electrically scanned in order to immediately obtain a recognizable image of subsurface objects. Two quadrature components of the radar return signal are processed with a Kirchhoff backward migration algorithm. Physical theory is used to assess the quality of the holographic image, and the synthetic aperture approach is developed and tested. The parabolic wave equation and Gaussian beam technique are used in order to take into account refraction effects and to suppress specular reflection from the air-ground interface. Laboratory and field tests confirmed the predicted device parameters.

Learning coordinate systems for fast and accurate acoustic modeling

Many popular numerical methods, such as finite difference and spectral methods, rely on simple domain geometry and environmental smoothness. Unfortunately, these features are rarely found in real-world simulations. We propose learning coordinate transformations with deep neural networks to facilitate acoustic modeling in complex media. Using automatic differentiation, we obtain new coordinate systems by solving various optimization problems that map the computational domain to a rectangular grid. Different choices of the objective function are utilized to attain different goals, including (i) mapping complicated boundaries such as the seafloor to straight lines and (ii) reducing the rank of two-dimensional slices of the refractive index. The first choice allows for domain decomposition to accurately model sharp discontinuities in density and sound speed. The second drastically accelerates a non-intrusive reduced-order modeling technique referred to as the dynamical low-rank approximation. Using realistic ocean test cases, we compare the performance of our learned coordinate systems with domain decomposition to the classic approach of smoothing sharp discontinuities, and we compare full-rank, low-rank, and coordinate-system-accelerated low-rank solutions of the three-dimensional parabolic wave equation.

Evaluation of COAMPS Boundary Layer Refractivity Forecast Accuracy for 2–40 GHz Electromagnetic Wave Propagation

AbstractIn this paper, the performance of the Coupled Ocean‐Atmosphere Mesoscale Prediction System (COAMPS®) electromagnetic (EM) propagation forecast is assessed using numerous metrics such as forecast lengths, frequencies, ranges, and spatial resolutions. The modeled PL (Propagation Loss) for 2–40 GHz based on the COAMPS ®predicted refractivity profiles is evaluated by 1 month of range‐dependent measurements, which were collected during Coupled Air‐Sea Processes and Electromagnetic ducting Research (CASPER) East Campaign. PL data were collected using a shore‐to‐ship radio link, near the coast of Duck, NC. Mesoscale atmospheric refractivity predictions for the CASPER East operational area are provided by COAMPS®. Propagation loss predictions are computed from these refractivity profiles via a parabolic wave equation propagation code. The results show a good agreement between the COAMPS®‐predicted and the measured propagation loss for the northwest Atlantic region. The analyses show that the prediction error varies from 3 to 8 dB as the frequency increases from 2 to 40 GHz. In addition, COAMPS® successfully tracked the changes in the refractivity field even when the forecast length increased from 0 to 48 hr, with only a marginal degradation in EM prediction accuracy within the first 12 hr. Higher horizontal range resolution does not result in significant improvement in EM predictions as long as there are at least 2 M‐profiles that allows the EM propagation code to capture the first order horizontal variation. Low‐resolution, high order nested COAMPS® fields still produce good EM forecasts for this particular geometry.

Analytical Study of Skew-cosh-Gaussian Laser Beam Propagation through Collisionless Plasma

In the present theoretical investigation the domains of the order of skew-cosh-Gaussian (skew-chG) laser beams for agiven set of skewness parameters and its significant effect on the self-focusing/defocusing in collisionless plasma has beenstudied. The nonlinearity in the dielectric function of collisionless plasma is mainly due to the ponderomotive force. Byfollowing parabolic wave equation approach under Wentzel-Kramers-Brillouin (WKB) and paraxial approximations,nonlinear coupled differential equations are established for the beam width parameters. These equations have been solvednumerically by using fourth-order Runge-Kutta method. The results obtained are presented graphically and discussed.

Frequency Diversity in Electromagnetic Remote Sensing of Lower Atmospheric Refractivity

This article explores the effects of system frequency on electromagnetic (EM) remote sensing of atmospheric refractivity. A frequency range of 2–40 GHz covering S- to Ka-bands is analyzed. A multifrequency inversion algorithm is developed to estimate the evaporation duct heights using a parabolic wave equation propagation model. The capabilities of a theoretical multifrequency system are explored for high and low signal-to-noise ratios for varying numbers of frequencies and bands. Experimental data were collected using the lower atmospheric propagation ultrawideband (LATPROP-UWB) system deployed during the Coupled Air-Sea Processes and EM Ducting Research (CASPER) East Campaign. Range-dependent propagation loss measurements at 53 frequencies ranging from 2 to 40 GHz were made. The evaporation duct refractivity profiles estimated from the multifrequency LATPROP-UWB data were compared to the profiles inferred from the concurrent meteorological and oceanographic measurements. Inversion results show that the optimal refractivity retrieval can be achieved using ten frequencies and results in a fourfold reduction in the estimation RMSE compared to a single-frequency inversion.

Optimal reduced-order solution to the 3D deterministic parabolic wave equation

In weakly scattering media, the parabolic wave equation is often utilized to develop approximate solutions to the Helmholtz equation. However, in three dimensions, at high frequencies, and when the computational domain becomes very large, computational cost grows quickly. We propose using a reduced-order modeling approach for the deterministic 3D parabolic wave equation via the dynamical low-rank approximation. This approach is similar in spirit to the normal-mode technique; however, instead of using fixed modes for the entire computational domain, we instantaneously optimally evolve the modes in range according to the PDE dynamics. This dynamic order reduction provides a more accurate low-rank approximation to the parabolic wave equation. We demonstrate the efficacy of this technique on realistic ocean acoustic test cases.

Accurate Computation of Scintillation in Tropospheric Turbulence With Parabolic Wave Equation

Parabolic wave equation (PWE) propagation models using multiple phase screens (MPS) are widely used to predict the log-amplitude variance of the signal propagating in tropospheric turbulence. Previous literature demonstrates the limitations of the MPS technique, usable only when the signal frequency is low (typically below 5 GHz for radio-wave propagation in the troposphere). In this article, we demonstrate that the accuracy of the MPS technique depends not only on the signal frequency but also on the outer scale length ( L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">os</sub> ) of the turbulent eddy. For example, when L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">os</sub> is on the order of tens of meters (e.g., in most terrestrial communication links), the MPS method is indeed unusable at high frequencies, but, when L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">os</sub> is on the order of hundreds of meters (e.g., in air-ground and satellite links), the method can be used even at millimeter-wave frequencies. Next, a correlated phase screen (CPS) implementation within the PWE is introduces, which avoids the condition of statistical noncorrelation of the phase screens in the MPS technique, and as such, it can be used for any signal frequency. Finally, an ad hoc error metric is provided, allowing for the selection of the appropriate phase screen model based on the signal frequency, turbulence outer scale length, and the electromagnetic (EM) link distance. The simulated signal log-amplitude variances as a function of range at microwave and millimeter-wave frequencies using the CPS method are reported here for the first time and are shown to be in excellent agreement with the analytically obtained values.

О применении конечно-разностной аппроксимации Паде псевдодифференциального параболического уравнения в задаче тропосферного распространения радиоволн

Рассматривается задача численного моделирования распространения электромагнитных волн в неоднородной тропосфере на основе широкоугольных обобщений метода параболического уравнения. Используется конечно-разностная аппроксимация Паде оператора распространения. Существенно, что в предлагаемом подходе указанная аппроксимация осуществляется одновременно по продольной и поперечной координатам. При этом допускается моделирование произвольного коэффициента преломления тропосферы. Метод не накладывает ограничений на максимальный угол распространения. Для различных условий распространения радиоволн проведено сравнение с методом расщепления Фурье и методом геометрической теории дифракции. Показаны преимущества предлагаемого подхода. This paper is devoted to the numerical simulation of electromagnetic wave propagation in an inhomogeneous troposphere. The study is based on the wide-angle generalizations of the parabolic wave equation. The finite-difference Padé approximation is used to approximate the propagation operator. It is important that, within the proposed approach, the Padé approximation is carried out simultaneously along with the longitudinal and transverse coordinates. At the same time, the proposed approach gives an opportunity to model an arbitrary tropospheric refractive index. The method does not impose restrictions on the maximum propagation angle. The comparison with the split-step Fourier method and the geometric theory of diffraction is discussed. The advantages of the proposed approach are shown.

Wind farm layout optimization with special attention on noise radiation

This work presents a new wind farm layout design methodology that calculates the noise radiation using an advanced sound propagation model. Firstly, the wake velocity in wind farm is calculated via an improved engineering wake model. Then, the aero-acoustic noise source of wind turbine is obtained using the BPM method. To simulate the noise level at far field, the basis of propagation model is built on the parabolic wave equation (PE) method. The cost of energy for wind farm is assessed via an engineering model. Finally, based on the genetic algorithm (GA) optimization method, and taking the noise radiation level into account, a new wind farm layout design methodology is developed to solve a trade-off between cost and noise for wind farm.

WIDEBAND SATCOM MODEL: EVALUATION OF NUMERICAL ACCURACY AND EFFICIENCY

Abstract. The spectral method is typically applied as a simple and efficient method to solve the parabolic wave equation in phase screen scintillation models. The critical factors that can greatly affect the spectral method accuracy is the uniformity and smoothness of the input function. This paper observes these effects on the accuracy of the finite difference and the spectral methods applied to a wideband SATCOM signal propagation model simulated in the ultra-high frequency (UHF) band. The finite difference method uses local pointwise approximations to calculate a derivative. The spectral method uses global trigonometric interpolants that achieve remarkable accuracy for continuously differentiable functions. The differences in accuracy are presented for a Gaussian lens and Kolmogorov phase screen. The results demonstrate loss of accuracy in each method when a phase screen is applied, despite the spectral method's computational efficiency over the finite difference method. These results provide meaningful insights when discretizing an interior domain and solving the parabolic wave equation to obtain amplitude and phase of a signal perturbation.

Self-Focusing of Hermite–Cosh–Gaussian Laser Beams in a Plasma Under a Weakly Relativistic and Ponderomotive Regime

In the theoretical investigation of the self-focusing of elegant Hermite–Cosh–Gaussian (EHChG) beams, the crucial role of decentered parameters has been explored thoroughly in the case of the weak relativistic and ponderomotive regime of interaction. In the present study, the Cartesian coordinate system has been employed, which enables us to study the evolution of two transverse beam-width parameters simultaneously. The differential equations for the beam-width parameters are set up through the parabolic wave equation approach by following WKB and paraxial approximations.

Multivariable Extremum Seeking for PDE Dynamic Systems

In this article, we address the design and analysis of multivariable Newton-based extremum seeking (ES) for static locally quadratic maps subject to actuation dynamics governed by diffusion partial differential equations (PDEs). Multi-input systems with distinct diffusion coefficients in each individual input channel are dealt with. The phase compensation of the dither signals is handled as a trajectory generation problem and the inclusion of a multivariable diffusion feedback controller with a perturbation-based (averaging-based) estimate of the Hessian's inverse allow to obtain local exponential convergence results to a small neighborhood of the optimal point. The stability analysis is carried out using backstepping transformation and averaging in infinite dimensions, capturing the infinite-dimensional state due the diffusion PDEs. In addition, the generalization of the results for different classes of parabolic (reaction-advection-diffusion) PDEs, wave equations and/or first-order hyperbolic (transport-dominated) PDEs is also discussed. As for ordinary differential equations case, the proposed Newton approach removes the dependence of the algorithm's convergence rate on the unknown Hessian of the nonlinear map to be optimized, being user-assignable unlike the gradient algorithm. A numerical example illustrates the performance of our multivariable ES for compensating PDE-based systems.