Time-domain Parabolic Equation Research Articles

Efficient FDTD-PE Method for GNSS Propagation in Airport to Support DCPS

To support Differentially Corrected Positioning Service (DCPS) in airport, the Global Navigation Satellite System (GNSS) receiver’s anti-multipath ability should be validated carefully to airport environment. This paper discusses the quantification of GNSS signal characteristics in airport based on Finite Difference Time Domain-Parabolic Equation (FDTD-PE) method. The proposed method can deal with complex environment with different objects and large scale clear space of airport efficiently. An equivalent process is introduced to deal with the problem of non-uniform sampling grid. By comparing with FDTD method, the validation of this method is proved. The simulation results demonstrate the effectiveness and accuracy of this method for predicting the GNSS signal propagation in airport environment.

Pulse-Compression Signal Propagation and Parameter Estimation in the Troposphere With Parabolic Equation

Wireless channel analysis is essential in the design, performance evaluation, and error correction of radar system. In this paper, an efficient parabolic equation (PE) method, which employs the split-step Fourier transform (SSFT) solution and Fourier synthesis technique, is developed for the propagation and parameter estimation of pulse-compression signals in the troposphere considering anomalous propagation conditions. A sliding window method is applied to reduce computational loads for long-distance propagation in time-domain PE. The signal delay is obtained via searching the peak of the correlation function of the received signal and a known reference signal according to the autocorrelation of the signals. The numerical examples indicate that the presented method is well suited for pulse-compression signals. Beyond that, a multiple signal classification (MUSIC) algorithm with spatial smoothing technique is introduced to obtain the signal direction of arrival (DOA) in PE model, where the covariance matrix is constructed via the array fields obtained from PE and the curvature of wavefronts due to the atmospheric refraction is considered in the array steering vector. The numerical examples verify the accuracy of the presented method. The simulation experiments in a typical sea-to-land scenario are presented to analyze the sensitivity of pulse-compression signals to evaporation ducts, including pulse waveform, time delay, and DOA, utilizing the presented methods.

A Novel Marching-on-in-Degree Solver of Time Domain Parabolic Equation for Transient EM Scattering Analysis

A novel marching-on-in-degree (MOD) solver of 3-D time domain parabolic equation is proposed to solve the transient electromagnetic scattering from electrically large perfect electric conductor (PEC) targets. The finite difference (FD) scheme is applied to the spatial discretization, while the weighted Laguerre polynomials are used as the temporal basis functions. In this way, a large number of computational resources can be saved by using the FD scheme along the paraxial direction and the late-time stability can be guaranteed by the MOD method. Both the FD schemes of Crank–Nicolson and the alternating direction implicit types are discussed in this communication. Numerical results are given to demonstrate the accuracy and efficiency of the proposed method.

Fast analysis of wide-band scattering from electrically large targets with time-domain parabolic equation method

An efficient three-dimensional time domain parabolic equation (TDPE) method is proposed to fast analyze the narrow-angle wideband EM scattering properties of electrically large targets. The finite difference (FD) of Crank–Nicolson (CN) scheme is used as the traditional tool to solve the time-domain parabolic equation. However, a huge computational resource is required when the meshes become dense. Therefore, the alternating direction implicit (ADI) scheme is introduced to discretize the time-domain parabolic equation. In this way, the reduced transient scattered fields can be calculated line by line in each transverse plane for any time step with unconditional stability. As a result, less computational resources are required for the proposed ADI-based TDPE method when compared with both the traditional CN-based TDPE method and the finite-different time-domain (FDTD) method. By employing the rotating TDPE method, the complete bistatic RCS can be obtained with encouraging accuracy for any observed angle. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method.

Modeling the generation of infrasound from earthquakes

Earthquakes can generate complex seismo-acoustic wavefields, consisting of seismic waves, epicenter-coupled infrasound, and secondary infrasound. We report on the development of a numerical seismo-acoustic model for the generation of infrasound from earthquakes. We model the generation of seismic waves using a 3D finite difference algorithm that accounts for the earthquake moment tensor, source time function, depth, and local geology. The resultant acceleration-time histories (on a 2D grid at the surface) provide the initial conditions for modeling the near-field infrasonic pressure wave using the Rayleigh integral. Finally, we propagate the near-field source pressure through the Ground-to-Space atmospheric model using a time-domain parabolic equation technique. The modeling is applied to an earthquake of MW 4.6, that occurred on January 3, 2011 in Circleville, Utah; the ensuing predictions are in good agreement with observations made at the Utah network of infrasonic arrays, which are unique and indicate that the signals recorded at 6 arrays are from the epicentral region. These results suggest that measured infrasound from the Circleville earthquake is consistent with the generation of infrasound from body waves in the epicentral region.

A seismoacoustic study of the 2011 January 3 Circleville earthquake

SUMMARY Wereportonauniquesetofinfrasoundobservationsfromasingleearthquake,the2011January 3Circlevilleearthquake(Mw 4.7,depthof8km),whichwasrecordedbynineinfrasoundarrays inUtah.Basedonananalysisofthesignalarrivaltimesandbackazimuthsateacharray,wefind thattheinfrasoundarrivalsatsixarrayscanbeassociatedtothesamesourceandthatthesource location is consistent with the earthquake epicentre. Results of propagation modelling indicate that the lack of associated arrivals at the remaining three arrays is due to path effects. Based on these findings we form the working hypothesis that the infrasound is generated by body waves causing the epicentral region to pump the atmosphere, akin to a baffled piston. To test this hypothesis, we have developed a numerical seismoacoustic model to simulate the generation of epicentral infrasound from earthquakes. We model the generation of seismic waves using a 3-D finite difference algorithm that accounts for the earthquake moment tensor, source time function, depth and local geology. The resultant acceleration–time histories on a 2-D grid at the surface then provide the initial conditions for modelling the near-field infrasonic pressure wave using the Rayleigh integral. Finally, we propagate the near-field source pressure through the Ground-to-Space atmospheric model using a time-domain Parabolic Equation technique. By comparing the resultant predictions with the six epicentral infrasound observations from the 2011 January 3, Circleville earthquake, we show that the observations agree well with our predictions. The predicted and observed amplitudes are within a factor of 2 (on average, the synthetic amplitudes are a factor of 1.6 larger than the observed amplitudes). In addition, arrivalsarepredictedatallsixarrayswheresignalsareobserved,andimportantlynotpredicted at the remaining three arrays. Durations are typically predicted to within a factor of 2, and in some cases much better. These results suggest that measured infrasound from the Circleville earthquake is consistent with the generation of infrasound from body waves in the epicentral region.

Topographic effects on infrasound propagation

Infrasound data were collected using portable arrays in a region of variable terrain elevation to quantify the effects of topography on observed signal amplitude and waveform features at distances less than 25 km from partially contained explosive sources during the Frozen Rock Experiment (FRE) in 2006. Observed infrasound signals varied in amplitude and waveform complexity, indicating propagation effects that are due in part to repeated local maxima and minima in the topography on the scale of the dominant wavelengths of the observed data. Numerical simulations using an empirically derived pressure source function combining published FRE accelerometer data and historical data from Project ESSEX, a time-domain parabolic equation model that accounted for local terrain elevation through terrain-masking, and local meteorological atmospheric profiles were able to explain some but not all of the observed signal features. Specifically, the simulations matched the timing of the observed infrasound signals but underestimated the waveform amplitude observed behind terrain features, suggesting complex scattering and absorption of energy associated with variable topography influences infrasonic energy more than previously observed.

Infrasound radiated by the Gerdec and Chelopechene explosions: propagation along unexpected paths

SUMMARY Infrasound propagation paths through the atmosphere are controlled by the temporally and spatially varying sound speed and wind speed amplitudes. Because of the complexity of atmospheric acoustic propagation it is often difficult to reconcile observed infrasonic arrivals with the sound speed profiles predicted by meteorological specifications. This paper provides analyses of unexpected arrivals recorded in Europe and north Africa from two series of accidental munitions dump explosions, recorded at ranges greater than 1000 km: two explosions at Gerdec, Albania, on 2008 March 15 and four explosions at Chelopechene, Bulgaria, on 2008 July 3. The recorded signal characteristics include multiple pulsed arrivals, celerities between 0.24 and 0.34 km s−1 and some signal frequency content above 1 Hz. Often such characteristics are associated with waves that have propagated within a ground-to-stratosphere waveguide, although the observed celerities extend both above and below the conventional range for stratospheric arrivals. However, state-of-the-art meteorological specifications indicate that either weak, or no, ground-to-stratosphere waveguides are present along the source-to-receiver paths. By incorporating realistic gravity-wave induced horizontal velocity fluctuations into time-domain Parabolic Equation models the pulsed nature of the signals is simulated, and arrival times are predicted to within 30 s of the observed values (<1 per cent of the source-to-receiver transit time). Modelling amplitudes is highly dependent upon estimates of the unknown acoustic source strength (or equivalent chemical explosive yield). Current empirical explosive yield relationships, derived from infrasonic amplitude measurements from point-source chemical explosions, suggest that the equivalent chemical yield of the largest Gerdec explosion was of the order of 1 kt and the largest Chelopechene explosion was of the order of 100 t. When incorporating these assumed yields, the Parabolic Equation simulations predict peak signal amplitudes to within an order of magnitude of the observed values. As gravity wave velocity perturbations can significantly influence both infrasonic arrival times and signal amplitudes they need to be accounted for in source location and yield estimation routines, both of which are important for explosion monitoring, especially in the context of the Comprehensive Nuclear-Test-Ban Treaty.

ELECTROMAGNETIC PULSE PROPAGATION OVER NONUNIFORM EARTH SURFACE: NUMERICAL SIMULATION

Parabolic equation method proposed by Leontowich and Fock [1,2] is an efficient simulation approach to VHF propagation over the earth surface. Deep physical analysis and advanced mathematical methods [3,4] turned Leontovich’s PE into a universal tool of diffraction theory. Its applications go far beyond the initial problem circle – e. g. [5-8]. The key role in this development played the decisive turn to straightforward numerical techniques pioneered by Malyuzhinets and Tappert [9,10]. In radio wave propagation, PE was used first to derive explicit analytical formulae for the EM field strength in model environments. A simplification has been reached by introducing the impedance boundary condition (BC) [11]. Taking into account tropospheric refraction ducts required the use of sophisticated asymptotic methods [12]. Further development (almost exclusively towards numerical implementation) was aimed at refined PE modifications [13-15], account for irregular terrain [16], introducing artificial transparent boundaries [17,18] and nonlocal BC to describe rough interfaces [19]. A non-stationary PE counterpart and a finite-difference (FD) scheme for its solution have been proposed by Claerbout and applied to seismic problems [13]. Afterwards, this “time-domain parabolic equation” (TDPE) was used to calculate acoustic propagation in ocean [20]. At the same time, little attempts of using TDPE to simulate EM pulse propagation in realistic environments are known. In this paper, we consider computational aspects of EM pulse propagation along the nonuniform earth surface. For ultrawide-band pulses without carrier, TDPE results directly from the exact wave equation written in a narrow vicinity of the wave front. To solve it by finite differences, we introduce a time-domain analog of the impedance BC and a nonlocal BC of transparency reducing the open computational domain to a strip of finite width. Numerical examples demonstrate the influence of soil conductivity on the received pulse waveform which can be used in remote sensing.

Infrasonic scattering studies using time‐domain parabolic equation (TDPE) and gravity wave models

The time‐domain parabolic equation (TDPE) model is useful in predicting infrasonic waveforms, as it can account for diffraction and scattering mechanisms. In this study, a split‐step Fourier (SSF) implementation of the TDPE is used to study infrasonic propagation. Scattering is modeled using a spectral gravity wave model, and multiple gravity wave realizations are used to quantify the uncertainty in waveform properties introduced by scattering from the gravity wave inhomogeneities. TDPE predictions are compared to specific infrasonic events and conclusions regarding the dominant propagation mechanisms are presented.

Parabolic equation (PE) model approximations and implications for infrasound

The continuous-wave parabolic equation (PE) model is widely used in the prediction of atmospheric propagation. In this study, the effects of several PE approximations are evaluated in the context of long-range infrasonic propagation. Specifically, the focus is on quantifying: phase errors resulting from different split-step Fourier (SSF) implementations, solution stability with respect to step size, and prediction sensitivity to the choice of reference sound speed. The tradeoff between improved performance gain and increased computational loading will also be considered. The study will include comparison of PE waveform predictions with measurements from infrasonic events. These comparisons are of interest in assessing the PE modeling performance, applicability, and limitations. Waveform predictions are made by integrating the continuous-wave PE model into a Fourier-synthesis Time-domain PE (TDPE).

Modelling EM transient propagation over irregular dispersive boundary

On the basis of the time-domain parabolic equation, a numerical simulation of single pulse electromagnetic (EM) propagation over a smoothly non-uniform terrain is performed. Ground conductivity is taken into account via a generalised impedance boundary condition (BC). A time-domain version of a perfectly transparent BC is used to simulate radiation into free space.

Numerical simulation of time domain ocean acoustic propagation

The time domain parabolic equation is studied for ocean acoustic long-range propagation, including wide-angle propagation, without involving Fourier synthesis. Instead of solving the wave equation with boundary conditions in the whole ocean domain, the governing equations are written in a moving coordinate system. Since the studied frame is moving with acoustic perturbations, dependent variables evolve more gradually, which increases time integration accuracy. The number of space points and thus CPU time are greatly reduced because the pulse occupies a narrow range window. Numerical tests were made to study first the headwave problem by handling discontinuities of sound speed and density at a layer interface. Media with continuously variable celerity profile were also studied: there was a particular interest in studying wave propagation in media with multiple deep sound channels. Results were compared with those worked out with other codes based on Bessel–Fourier analysis (SPARC developed by M. B. Porter), requiring much CPU time. An excellent agreement was observed. The originality of the interactive program developed is to display a movie of wave propagation. It is an efficient tool to solve the direct problem and consider the inverse problem.

Nonlinear wide-angle paraxial acoustic propagation in shallow-water channels

A time-domain model that describes wide-angle paraxial propagation of acoustic pulses in shallow water is developed. This model incorporates weak nonlinear effects and depth variability in both ambient density and sound speed. Derivations of paraxial approximations are based on an iterative approach, in which the wide-angle approximation is obtained by using a narrow-angle equation to approximate the second range derivative in the two-way equation. Scaling arguments are used to obtain a more tractable simplification of the equation. The wide-angle equation is solved numerically by splitting into components representing distinct physical processes and using a modified version of the time-domain parabolic equation (TDPE) code [M. D. Collins, J. Acoust. Soc. Am. 84, 2114–2125 (1988)]. A high-order upwind flux-correction method is modified to handle the nonlinear component. Numerical results are presented for adiabatic propagation in a shallow, isospeed channel. It is demonstrated that nonlinear effects are significant, even at small ranges, if the peak source pressure is high enough. Both nonlinear and wide-angle effects are illustrated, and their differences are discussed.

Comparison of time-domain parabolic equation and measured ocean impulse responses in a range-dependent environment

Ocean impulse response functions computed with the time-domain parabolic equation model are shown to be consistent with measured response functions in a shallow-water range-dependent environment off the east coast of the United States. Impulse response functions are measured by correlating the source signature of a 20- to 150-Hz linear, modulated signal with signals received on a 15 element vertical line array. Environmental inputs to the model are derived from Ocean Drilling Project (ODP) sites near the experiment location. Correlation coefficients as a function of depth are used to compare measured and modeled responses at three separate ranges. [Work supported by The Office of Naval Research under the Naval Research Laboratory/Stennis Space Center Acoustic Transients Project.]

Effects of environmental degradation on higher-order correlation detectors for deterministic signals

Higher-order correlation detectors are investigated using essentially band-limited deterministic finite energy signals. These signals have been propagated through a simulated range-dependent ocean environment using a time domain parabolic equation model. The detectors are analyzed using receiver operating characteristic (ROC) curves. Correlation coefficients [R. L. Field, J. H. Leclere, and E. J. Yoerger, submitted to J. Underwater Acoust.] and limiting signal-to-noise ratios for which the detectors exhibit good performance are given. Previously reported results for undegraded signals [J. Acoust. Soc. Am. Suppl. 1 86, S118 (1989)] are extended to include rectification [W. B. Moseley (private communication)] as part of the bicorrelation detection scheme. The bicorrelation detector not only benefits from rectification, but rectification permits its use in cases where it might otherwise be inapplicable. [Work supported by the Naval Oceanic and Atmospheric Research Laboratory through the U. S. Navy-ASEE Summer Faculty Research Program and through Grant No. N00014-89-J-6002.]

Applications and time-domain solution of higher-order parabolic equations in underwater acoustics

A higher-order parabolic equation and the corresponding higher-order time-domain parabolic equation are derived from a Padé series and solved numerically. These models provide accurate solutions for problems involving very-wide-angle propagation (e.g., propagation in the nearfield or over a hard ocean bottom); propagation in domains in which sound-speed variations are large (e.g., propagation in deep water, deep within the ocean bottom, in high-speed ocean bottoms, and possibly of different wave types); and propagation out to very long ranges. The possibility of modeling elastic wave propagation with a similar approach is considered.

A parabolic equation model for scattering in the ocean

The small-angle-of-propagation limit and the method of matched asymptotics are applied to derive an efficient model for solving realistic underwater acoustics problems involving both propagation and scattering from a submerged object. The propagation and scattering aspects of the waveguide scattering problem are decoupled by approximating the waveguide Green’s function on the surface of the scatterer. For low frequencies, the small-angle limit also allows one to approximate the incident field with a horizontally propagating plane wave and the scattered field with an azimuthally specular point-source field. With these approximations, scattering calculations can be performed efficiently in the time domain. Calculations involving the three-dimensional parabolic equation and the time-domain parabolic equation are presented.

The time-domain solution of the wide-angle parabolic equation including the effects of sediment dispersion

The wide-angle time-domain parabolic equation (TDPE), which is the inverse Fourier transform of the wide-angle parabolic equation (PE), is derived. A numerical solution for the model is described and a benchmark calculation is presented. The narrow-angle TDPE is also considered and its error is analyzed and compared with the error of the narrow-angle PE. The TDPE is compared with the progressive wave equation, which is shown to be restricted to narrow-angle propagation for practical purposes. In the sediment, attenuation is assumed to depend linearly on frequency and the corresponding causal dispersion law is assumed. The model is used to show that the effect of sediment dispersion on pulse propagation in the ocean can be significant.

The time-domain parabolic wave equation and its path integral representation

A path integral representation for the solution of a time-domain parabolic wave equation (TDPE) was developed. It shows promise for describing propagation of sound pulses in the ocean when boundary interactions can be neglected. The TDPE is a linearized version of McDonald and Kuperman's nonlinear progressive wave equation [B. E. McDonald and W. A. Kuperman, J. Acoust. Soc. Am. 81, 1406 (1987)]. Using the techniques developed by these authors one can derive the TDPE from the acoustic wave equation thus showing that the TDPE's solution is an approximate solution to the acoustic wave equation [M.D. Collins, J. Acoust. Soc. Am. Suppl. 82, S122 (1987)]. The TDPE has the mathematical structure of a time-dependent Schrödinger equation and a phase space path integral representation for its solution can be constructed by well-known methods. Applications to be discussed include propagation in a sound channel and in randomly fluctuating media. [Work supported by U.S. Naval Research Laboratory.]