Parabolic Wave Equation Research Articles

Comparison of 2D and 3D Electromagnetic Approaches to Predict Tropospheric Turbulence Effects in Clear Sky Conditions

Two-dimensional electromagnetic simulations are often used to evaluate the atmospheric turbulence effects on radiowave propagation in clear sky conditions. However, turbulence is clearly a three-dimensional atmospheric process. Therefore, errors potentially introduced by 2D propagation schemes to predict 3D scintillation effects have to be quantitatively assessed. On the one hand, as part of an analytical approach and starting from the Kolmogorov-von Karman turbulent spectrum, 2D formulations for log-amplitude and phase variances and for log-amplitude and phase temporal power spectra are derived from the 2D scalar propagation equation. They are compared asymptotically to their classical 3D counterparts. On the other hand, as part of a numerical approach, the scintillation effects are evaluated from 3D and 2D parabolic wave equation (PWE) approaches associated with 2D and 1D multiple phase screen (MPS), respectively. It is then shown that 2D propagation schemes underestimate by a factor 1.86 the log-amplitude variances in Fresnel regime and can lead to significant errors in predicting log-amplitude and phase temporal spectra at low frequencies. It is then suggested that the dimensional reduction should be limited to the prediction of log-amplitude and phase variances in Fraunhofer configurations, or to the evaluation of log-amplitude and phase power spectra at high frequencies.

Toward the diffraction limit with transmissive x-ray lenses in astronomy

We develop an analytical approach to refractive, blazed diffractive, and achromatic x-ray lenses of scalable dimensions for energies from 1 to 20 keV. Based on the parabolic wave equation, their wideband imaging properties are compared and optimized for a given spectral range. Low-Z lens materials for massive cores and rugged alternatives, such as polycarbonate or Si for flat Fresnel components, are investigated with respect to their suitability for diffraction-limited high-energy astronomy. Properly designed "hybrid" combinations can serve as an approach to x-ray telescopes with an enhanced efficiency throughout the whole considered band, nearly regardless of their inherent absorption.

Simulating the impacts of ionospheric scintillation on L band SAR image formation

We develop a phase screen model called the Synthetic Aperture Radar (SAR) Scintillation Simulator (SAR‐SS) for predicting the impacts of ionospheric scintillation on SAR image formation. SAR‐SS consists of a phase screen generator and a propagator. The screen generator creates a 2‐D random realization of spatial phase fluctuations resulting from the traversal of small‐scale field‐aligned irregularities in the ionosphere. It accounts for the motion of the radar platform, the drift of the ionospheric irregularities, and the oblique angle of propagation, all of which determine the scale sizes of the irregularities sampled by the radar beam. The propagator solves the 3‐D parabolic wave equation using the split step technique to compute the ionospheric transfer function for two‐way propagation. This ionospheric transfer function is used to modulate the SAR signal due to terrestrial features in order to assess the ionospheric impact on SAR image formation in the small target approximation. We compare simulated and observed PALSAR imagery over Brazil during disturbed ionospheric conditions. We demonstrate that SAR‐SS can reproduce the field‐aligned streaks in PALSAR imagery caused by irregularities in the equatorial ionosphere that have been observed by previous authors. The field‐aligned streaks exhibited a dominant wavelength larger than the Fresnel break scale, which suggests that refractive scatter was dominant over diffraction as the physical mechanism responsible for the scintillation of the radar signal in this case. The spectral index of phase fluctuations in the screen was quite large (9.0), suggesting that these irregularities were possibly associated with bottomside sinudoidal irregularities rather than equatorial plasma bubbles.

Three-dimensional attenuation structure of intrinsic absorption and wide-angle scattering of S waves in northeastern Japan

SUMMARY Intrinsic absorption and random velocity inhomogeneities are important medium properties for studies on wave propagation at high frequency (>1 Hz) in the lithosphere. Recent studies have shown that the spatial distribution of random velocity inhomogeneities can be estimated by an inversion analysis of peak delay times, where the peak delay time is the time lag between direct wave onset and its maximal amplitude arrival. This study estimates the 3-D structure of the S-wave attenuation quality factor Q−1, after we corrected the maximal amplitude attenuation due to multiple forward scattering on the basis of the Markov approximation for parabolic wave equation. Even though this estimate of Q−1 includes both intrinsic absorption and wide-angle scattering, the contribution of wide-angle scattering can be identified by taking into account random inhomogeneities estimated in the peak delay time analysis. The estimated Q−1 structure in northeastern Japan shows weak attenuation (Q−1(f) ∼ 10−2.5f−0.5) on the forearc side of the volcanic front, and strong attenuation (Q−1(f) ∼ 10−2.3f−0.3) beneath the Quaternary volcanoes and near the collision zone between the Honshu and Kuril arcs. According to the Born approximation, the estimated Q−1 at 0–40 km depth consists mainly of the intrinsic absorption if the characteristic scale of random inhomogeneities is 5 km or longer. Beneath the Quaternary volcanoes of northeastern Japan, Q−1(f) shows weak frequency dependence and the power spectral density functions of random velocity inhomogeneities are characterized by a weak spectral gradient that means rich in small-scale inhomogeneities. These results demonstrate the importance of the frequency dependence of wave propagation in the lithosphere. That is, the scale dependence of velocity fluctuation or small-scale inhomogeneities is an important property of the lithosphere underlying Quaternary volcanoes.

Spatiotemporal variability of underwater sound fields near steep slopes

Studies using idealized bathymetry have shown that steep slopes can have important effects on the geometric structure of ocean sound fields. Other studies have provided insight into the temporal variability of sound fields in the moving ocean. There is a general understanding that both effects occur together in the ocean, but their relative importance is not fully known, nor is the way they interact. The processes are often linked because slopes can generate strong internal tides and waves. Here, the coupled spatial and temporal variability is investigated using time-stepped three-dimensional (3.5-D) sound propagation modeling. The needed ocean sound-speed information is generated using modern regional ocean modeling. Solutions of the 3D parabolic acoustic wave equation are generated for a few hundred hertz. The spatial patterns of sound reveal that temporal intensity variation (scintillation) caused by internal tides can be enhanced at areas where sound encounters steep slopes. The magnitude of the effect is measured by repeating the simulations with a smoothed seafloor substituted for the realistic seafloor. The effect of internal tide strength on the interaction is quantified similarly. The effects of tide strength and seafloor steepness on spatial patterns of the sound field horizontal correlation length are also investigated.

The sea surface directional wave spectrum and forward scattering from the sea surface

An influence of the directional wave spectrum on acoustic forward scattering from the sea surface is difficult to measure. Here we present results of an experiment to measure vertical spatial coherence from an acoustic path interacting once with the sea surface at two different angles with respect to the wave direction. The measurements were part of the Shallow-Water 2006 program that took place off the coast of New Jersey in August 2006. An acoustic source was deployed at depth 40 m, and signals were recorded on a moored receiving system consisting of two, 1.4 m long vertical line arrays centered at depths 25 and 50 m. Measurements were made over four source-receiver bearing angles separated by 90°, during which sea surface conditions remained stable and characterized by an rms waveheight of 0.17 m and a mixed swell, and wind-wave field originating from different directions. The measurements show a statistically significant difference depending on source-receiver bearing when the acoustic frequency is less than about 10 kHz; a result not observed at higher frequencies. This paper will present field observations along with modeling based on a rough surface parabolic wave equation utilizing synthetic sea surfaces. [Research supported by ONR Ocean Acoustics].

Modeling low elevation GPS signal propagation in maritime atmospheric ducts

Using the parabolic wave equation (PWE) method, we model low elevation GPS L1 signal propagation in maritime atmospheric ducts. To consider sea surface impedance, roughness, and the effects of earth's curvature, we propose a new initial field model for the GPS PWE split-step solution. On the basis of the comparison between the proposed model and the conventional initial field model for a smooth, perfectly conducting sea surface on a planar earth, we conclude that both the amplitude and phase of the initial field are influenced by surface impedance and roughness, and that the interference behavior between direct and reflected GPS rays is affected by earth's curvature. The performance of the proposed model is illustrated with examples of low elevation GPS L1 signal propagation in three types of ducts: an evaporation duct, a surface-based duct, and an elevated duct. The GPS PWE is numerically implemented using the split-step discrete mixed Fourier transform algorithm to enforce impedance-type boundary conditions at the rough sea surface. Because the GPS signal is right hand circularly polarized, we calculate its power strength by combining the propagation predictions of the horizontally and the vertically polarized components. The effects of the maritime atmospheric ducts on low elevation GPS signal propagation are demonstrated according to the presented examples, and the potential applications of the GPS signals affected by ducts are discussed.

An Angle-Dependent Impedance Boundary Condition for the Split-Step Parabolic Equation Method

A formulation of the impedance boundary condition (IBC) is derived, which can be used in split-step parabolic wave equation solvers to accurately represent an interface with arbitrary dielectric properties. The approach relies on ensuring that the plane-wave decomposition of the field satisfies the appropriate IBC for each spectral component. Numerical experiments illustrating the robustness of the approach for low-contrast interfaces and angles-of-incidence near the Brewster angle are presented.

Thin-Layer Solutions of the Helmholtz and Related Equations

This paper concerns a certain class of two-dimensional solutions to four generic partial differential equations—the Helmholtz, modified Helmholtz, and convection-diffusion equations, and the heat conduction equation in the frequency domain—and the connections between these equations for this particular class of solutions. Specifically, we consider “thin-layer” solutions, valid in narrow regions across which there is rapid variation, in the singularly perturbed limit as the coefficient of the Laplacian tends to zero. For the well-studied Helmholtz equation, this is the high-frequency limit and the solutions in question underpin the conventional ray theory/WKB approach in that they provide descriptions valid in some of the regions where these classical techniques fail. Examples are caustics, shadow boundaries, whispering gallery, and creeping waves and focusing and bouncing ball modes. It transpires that virtually all such thin-layer models reduce to a class of generalized parabolic wave equations, of which the heat conduction equation is a special case. Moreover, in most situations, we will find that the appropriate parabolic wave equation solutions can be derived as limits of exact solutions of the Helmholtz equation. We also show how reasonably well-understood thin-layer phenomena associated with any one of the four generic equations may translate into less well-known effects associated with the others. In addition, our considerations also shed some light on the relationship between the methods of matched asymptotic, WKB, and multiple-scales expansions.

A NOVEL WAVELET-GALERKIN METHOD FOR MODELING RADIO WAVE PROPAGATION IN TROPOSPHERIC DUCTS

In this paper, a novel Wavelet-Galerkin Method (WGM) is presented to model the radio-wave propagation in tropospheric ducts. Galerkin method, with Daubechies scaling functions, is used to discretize the height operator. Later, a marching algorithm is developed using Crank-Nicolson (CN) method. A new \flctitious domain is also developed for parabolic wave equation to incorporate the impedance boundary conditions in WGM. In the end, results are compared with those from Advance Refractive Efiects Prediction System (AREPS). Results show that the wavelet based methods are indeed feasible to model the radio wave propagation in troposphere as accurately as AREPS and proposed method can be a good alternative to other conventional methods.

Numerical simulation of wave-induced currents combined with parabolic mild-slope equation in curvilinear coordinates

Researches on breaking-induced currents by waves are summarized firstly in this paper. Then, a combined numerical model in orthogonal curvilinear coordinates is presented to simulate wave-induced current in areas with curved boundary or irregular coastline. The proposed wave-induced current model includes a nearshore current module established through orthogonal curvilinear transformation form of shallow water equations and a wave module based on the curvilinear parabolic approximation wave equation. The wave module actually serves as the driving force to provide the current module with required radiation stresses. The Crank-Nicolson finite difference scheme and the alternating directions implicit method are used to solve the wave and current module, respectively. The established surf zone currents model is validated by two numerical experiments about longshore currents and rip currents in basins with rip channel and breakwater. The numerical results are compared with the measured data and published numerical results.

Envelope synthesis of short-period seismograms in 3-D random media for a point shear dislocation source based on the forward scattering approximation: Application to small strike-slip earthquakes in southwestern Japan

[1] Even though high-frequency seismograms of local earthquakes are complex because of the excitation of scattered waves caused by random lithospheric heterogeneities, their envelopes are well reproduced by the forward scattering approximation on the basis of stochastic characterization of random media. We present a method to synthesize three-component vector wave envelopes for a point shear dislocation source by way of the stochastic raypath method using the Markov approximation for the parabolic wave equation. Because of multiple forward scattering, the directionality of the synthesized envelope shows a smoothed quadrant pattern compared to the original radiation pattern; the scattered or diffracted energy appears even in the direction of the null axis and on the nodal planes. The quadrant pattern becomes smooth as hypocentral distance increases. To examine the availability of the synthesized envelope, we analyze seismograms of two small strike-slip earthquakes that occurred in southwestern Japan. First, we estimate the statistical parameters characterizing the randomness and the intrinsic absorptions in frequencies 2–4, 4–8, and 8–16 Hz assuming a von Karman-type random medium, and then we examine the directionality of the envelope. The maximum energy density of the envelope shows a quadrant pattern for both P and S waves with a 45 degree shift. The directionality appears in the frequencies up to 16 Hz and at distances up to 200 km. The directionality also appears on the S wave envelope duration. The observed directionality is well explained by the synthesized envelope, but it is still difficult to explain the partition of energy into three components by the synthesized envelopes.

Transition between free-space Helmholtz equation solutions with plane sources and parabolic wave equation solutions

The slowly varying envelope approximation is applied to the radiation problems of the Helmholtz equation with a planar single-layer and dipolar sources. The analyses of such problems provide procedures to recover solutions of the Helmholtz equation based on the evaluation of solutions of the parabolic wave equation at a given plane. Furthermore, the conditions that must be fulfilled to apply each procedure are also discussed. The relations to previous work are given as well.

Acoustic radiation during marine pile driving.

Pile driving in water produces extremely high sound levels in the surrounding under water environment. Sound levels as high as 220 dB re 1 μPa are not uncommon 10 m away from a steel pile as it is driven into the sediment with an impact hammer. The primary source of underwater sound originating from pile driving is associated with compression of the pile. The pile is struck and the Poisson effect produces a radial displacement motion in the pile that will propagate downward at a computed speed comparable to but less than the longitudinal wave speed in steel. It is shown, using both finite element analysis and modeling based on the parabolic wave equation, that this radial motion of the pile is responsible for the ensuing high underwater sound pressures. It is also shown that the radial motion of the pile is transmitted into the water, either directly from the pile or indirectly via the bottom sediment that is in contact with the pile. A dominant feature of the resulting sound field is an axisymmetric Mach cone with apex traveling along with the pile deformation wave front.

Acoustic transmission loss in industrial pile driving.

Industrial pile driving is a source of high-level underwater noise and new understanding of its effects on the behavior and health of marine mammals and fish has motivated numerous regulations intended to limit these effects. Of primary importance is to identify a suitable transmission loss model to predict where, given a certain source level, the noise produced by the pile driving reaches regulatory thresholds. In November 2009, data were collected from a marine construction site in the Puget Sound. Measurements at two ranges (8 and 12 m) from the pile being driven were taken using a nine hydrophone vertical line array (VLA). Concurrently, at a range of approximately 120 m, there was also a single hydrophone at a depth of 5 m (sensitive to frequencies greater than 10 kHz). By comparing the levels at the VLA to the more distant hydrophone across a number of pile strikes (each forming a identifiable short- and far-range pair), the transmission loss can be estimated. These results are in turn modeled using an approach based on the parabolic wave equation. [Work was funded by the Washington Department of Transportation.]

Observations and parabolic wave modeling of underwater pile driving impact noise.

Pile driving in water produces extremely high sound levels in both surrounding air and underwater environments. In a companion work [Reinhall and Dahl] it is shown using finite element simulation that for underwater case the primary sound signal originates from a compression wave traveling down the pile at a speed in excess of Mach 3. In this work, we present measurements pile driving impact noise made from a marine construction site in Puget Sound using a vertical line array (VLA) positioned at ranges 8–15 m from full-scale impact pile driving. The measurements are modeled using the parabolic wave equation approach for which synthetic time series are generated (bandwidth 50–2050 Hz). The simulation is achieved by way of a phased array of point sources, representing one source traveling down the pile at supersonic speed. Pile end reflections are included and the process is repeated with both an up- and down-traveling time-delayed sources. With the field computed in this manner, excellent agreement is achieved between model and observations of peak pressure level, and the compression wave speed is also confirmed by way of arrival angle estimation using the VLA. Implications on transmission loss are also discussed. [Work supported by the Washington State Department of Transportation.]

Cross focusing of two coaxial cosh-Gaussian laser beams in a parabolic medium

In this paper authors have presented the cross focusing of two coaxial cosh-Gaussian laser beams of different frequencies in a parabolic medium. Starting from the expression for field distribution of cosh-Gaussian beams, differential equations for beam-width parameters of two beams have been established through parabolic wave equation approach and analytical solution for weak beam has been obtained. The effect of decentred parameter of strong beam on the behavior of beam-width parameter with the normalized distance of propagation for weak beam has been specifically considered. The results have been presented graphically and discussed.

Spectral wave transformation model for simulating refraction-diffraction with strongly reflecting coastal structures

On the basis of the wave action balance equation which incorporates refraction, diffraction, reflection and wave-current interaction, a directional spectral wave transformation model WABED is developed for predicting the irregular wave refraction-diffraction with strongly reflecting structures in coastal regions. In the model, diffraction is taken into account by introducing a term formulated from a parabolic approximation wave equation, and reflection is calculated through a back-marching numerical approach at the reflecting boundary. Two experimental data sets are used to examine the performance of present model with regard to wave characteristics around reflecting coastal structures. One is from a physical experiment at idealized inlet with parallel jetties, while the other is from a laboratory study on a coastal project of the concave breakwater. Reasonably good agreements are found for both cases, revealing the applicability of the present model for predicting combined wave refraction-diffraction processes with strongly reflecting coastal structures.

Exact transparent boundary condition for the parabolic equation in a rectangular computational domain

In this paper, an exact three-dimensional transparent boundary condition for the parabolic wave equation in a rectangular computational domain is reported. It is a generalization of the well-known two-dimensional Basakov-Popov-Papadakis transparent boundary condition. It relates the boundary transversal derivative of the wave field at any given longitudinal position to the field values at all preceding computational steps. Several examples demonstrate propagation of light along simple structured optical fibers as well as in x-ray guiding structures. The proposed condition is simple and robust and can help to reduce the size of the computational domain considerably.

Theoretical analysis and numerical experiments of variational adjoint approach for refractivity estimation

This paper puts forward possibilities of refractive index profile retrieval using field measurements at an array of radio receivers in terms of variational adjoint approach. The derivation of the adjoint model begins with the parabolic wave equation for a smooth, perfectly conducting surface and horizontal polarization conditions. To deal with the ill‐posed difficulties of the inversion, the regularization ideas are introduced into the establishment of the cost function. Based on steepest descent iterations, the optimal value of refractivity could be retrieved quickly at each point over height. Numerical experiments demonstrate that the method works well for low‐distance signals, while it is not accurate enough for long‐distance propagations. Through curve fitting processing, however, giving a good initial refractivity profile could generally improve the inversions.