Green’s Function Parabolic Equation Research Articles

A Green’s function parabolic equation description of infrasound propagation in an inhomogeneous and moving atmosphere

Accurate atmospheric infrasound propagation models must account for variations in local sound speed and ambient flow. This paper describes and demonstrates a method to extend the Green’s function parabolic equation (GFPE) to account for 3D high Mach number ambient flow in addition to an inhomogeneous atmosphere. Predictions of infrasonic propagation using the resulting GFPE model are then compared with predictions using other models.

Using simplified terrain and weather mapping in outdoor sound propagation predictions

A complementary experimental and computational study was undertaken to assess the variability due to model sensitivities when predicting propagation in realistic outdoor environments by using a Green’s Function Parabolic Equation (GFPE) method and including realistic weather and terrain profiles. In order to test the validity of the basic and enhanced model including real terrain and weather data, field measurements were conducted at Hogan’s Mountain, North Carolina. By incorporating USGS terrain and measured weather profiles, this project further developed a Hogan’s Mountain specific GFPE to include a stair-step terrain mapping capability and linearly interpolated sound speed profiles. This Hogan’s Mountain GFPE uses matching conditions for comparison to the measured received levels from the field test data. Due to the vast data set acquired, which allows for comparison of the model to measurement in a multitude of propagation situations, several comments are made regarding the choice of calculation parameters as well as the validity of the basic and altered Hogan’s Mountain GFPE for different propagation prediction settings. Sample results will illustrate the agreement based on frequency, specific terrain and weather measured along the propagation path for individual source-receiver pairs of interest. [Work supported by Spawar Systems Center Pacific.]

Prediction of noise levels and annoyance from aircraft run-ups at Vancouver International Airport

Annoyance complaints resulting from engine run-ups have been increasing at Vancouver International Airport for several years. To assist the Airport in managing run-up noise levels, a prediction tool based on a Green's function parabolic equation (GFPE) model has been consolidated, evaluated, and applied. It was extended to include more realistic atmospheric and ground input parameters. Measurements were made of the noise-radiation characteristics of a CRJ200 jet aircraft. The GFPE model was validated by comparing predictions with results in the literature. A sensitivity analysis showed that predicted levels are relatively insensitive to small variations in geometry and ground impedance, but relatively sensitive to variations in wind speed, atmosphere type, and aircraft heading and power setting. Predicted noise levels were compared with levels measured at noise monitoring terminals. For the four cases for which all input information was available, agreement was within 10 dBA. For events for which some information had to be estimated, predictions were within 20 dBA. The predicted annoyance corresponding to the run-up events considered ranged from 1.8% to 9.5% of people awoken, suggesting that noise complaints can be expected.

Parameter selection in the Green’s function parabolic equation

The accuracy of the Green’s function parabolic equation (GFPE) has already been confirmed for outdoor sound propagation over flat ground with a slowly varying sound speed profile and/or atmospheric turbulence. However, use of parabolic equation methods for prediction is generally limited to experts because of their dependence on numerous algorithm parameters that have significant impact on the accuracy of prediction. The present work offers a set of GFPE parameters, outlined in Table 1, that will provide accurate results for a variety of physical situations involving atmospheric propagation. The guidelines are found by comparing GFPE results to analytical results for a variety of situations and parameter choices and noting which combinations of parameters lead to accurate (within 3 dB of the analytical solution) results and which do not. A significant source of error in GFPE results is inaccuracy in the starting field or inappropriate starting field selection. The selection criteria for starting field and other parameters are discussed here.

Application of the Green’s function parabolic equation (GFPE) method to realistic outdoor sound propagation scenarios

The Green’s function parabolic equation (GFPE) is a powerful method for outdoor sound propagation prediction. Given appropriate inputs, GFPE predictions can include the effects of terrain, ground cover, sound speed profile, and turbulence on atmospheric propagation. In many situations, however, access to such input information can be limited. The input data resolution and accuracy can have a significant impact on model output accuracy. In turn, variability of terrain and sound speed profile and statistical aspects of the turbulence impact the model parameter requirements. For example, in the presence of turbulence, lower limits on computational grid height are determined by turbulence correlation scales as well as acoustic wavelength. Input data resolution requirements and some techniques that may be employed to account for insufficient input data will be discussed. Model results for realistic situations will be presented.

Discussion of errors in the Green’s function parabolic equation for outdoor sound propagation

As is to be expected of any numerical method, use of the Green’s function parabolic equation (GFPE) does result in small errors that can propagate and increase. The main causes of errors in using the GFPE for outdoor sound propagation with constant sound speeds appear to be inadequate spatial sampling, approximated starting fields, and errors in the numerical integration used to compute the surface wave. The relative significance of these errors as well as a set of appropriate computational parameters for avoiding or minimizing them will be discussed. With the proper parameter selection (for step sizes, attenuation layer depth, etc.), the GFPE produces reliably accurate results for outdoor sound propagation even over finite impedance grounds and beyond barriers or changes in ground height.

The Lamb wave in parabolic equation models

The inclusion of the Lamb wave (i.e., the inclusion of gravity) in two existing atmospheric parabolic equation (PE) models is discussed. The two models are based on, respectively, the Green’s function parabolic equation (GFPE) of Gilbert and Di and the split-step pade parabolic equation (SSPPE) of Collins. While the SSPPE results essentially duplicate recent research by Lingevitch et al. [J. Acoust. Soc. Am. 105, 3049–3056 (1999)], the GFPE research is novel and gives physical insight into the nature of the Lamb wave. Each of the two PE models considered has its own particular advantages and limitations. The GFPE, for example, has an explicit term for the Lamb wave so that physical insight is enhanced. However, including gravity in the GFPE requires an assumption of an exponentially stratified density, an approximation that is valid only to a height of 50–100 km. The SSPPE has the advantage of treating gravity exactly for an arbitrary density stratification. Unfortunately, separating the Lamb wave contribution from the total acoustic field is not straightforward with the SSPPE. This paper outlines the theories for the GFPE and SSPPE and presents numerical comparisons between the two models for a number of infrasound propagation scenarios. [Work supported by the U.S. Army Space and Missile Defense Command.]

Long-range acoustic propagation in the nocturnal boundary layer

It is a common observation that sound travels to long distances in the nocturnal boundary layer (NBL). For such long-range propagation, the cumulative effects of turbulence can be significant even though NBL turbulence is generally much weaker than typical daytime turbulence. In this paper a 3D version is used of the Green’s function parabolic equation (GF-PE) to investigate the effects of turbulence on average levels and cross-range correlation in the NBL at distances of up to 10 km. A comparison is made between 2D and 3D predictions for 50 and 100 Hz. In addition, the cross-range coherence is computed at various ranges for the two frequencies. The implications for beamforming on horizontal arrays in the NBL are discussed. [Work supported by the Defense Advanced Research Projects Agency (DARPA).]

Improved Green’s function parabolic equation method for atmospheric sound propagation

The numerical implementation of the Green’s function parabolic equation (GFPE) method for atmospheric sound propagation is discussed. Four types of numerical errors are distinguished: (i) errors in the forward Fourier transform; (ii) errors in the inverse Fourier transform; (iii) errors in the refraction factor; and (iv) errors caused by the split-step approximation. The sizes of the errors depend on the choice of the numerical parameters, in particular the range step and the vertical grid spacing. It is shown that this dependence is related to the stationary phase point of the inverse Fourier integral. The errors of type (i) can be reduced by increasing the range step and/or decreasing the vertical grid spacing, but can be reduced much more efficiently by using an improved approximation for the forward Fourier integral. The errors of type (ii) can be reduced by using a numerical filter in the inverse Fourier integral. The errors of type (iii) can be reduced slightly by using an improved refraction factor. The errors of type (iv) can be reduced only by reducing the range step. The reduction of the four types of errors is illustrated for realistic test cases, by comparison with analytic solutions and results of the Crank–Nicholson PE (CNPE) method. Further, optimized values are presented for the parameters that determine the computational speed of the GFPE method. The computational speed difference between GFPE and CNPE is discussed in terms of numbers of floating point operations required by both methods.

Atmospheric propagation modelling

With the developments in modeling the propagation of sound through the atmosphere, it is now possible to merge these models with sensor models to determine the impact of the atmosphere on acoustic sensor performance. Early numerical models such as the Fast-Field Program (FFP) allowed users to predict the effects of refraction, geometric spreading, diffraction, and complex ground impedance on sound propagation in the atmosphere. Coupling these numerical models with models for acoustic arrays, gave researchers the ability to predict the effects of the mean atmosphere on array performance. The newer numerical models like the Parabolic Equation (PE) and Green’s Function PE (GFPE) have allowed researchers to incorporate homogeneous and isotropic turbulence and complex terrain into propagation models. These new capabilities will lead to determining the effects of turbulence and terrain on acoustic arrays. With the development of high-speed/low-power/low-cost microcomputers, it is now possible to even integrate simple propagation models into an acoustic array processor. This will allow the development of smart acoustic arrays which can determine the impact of the atmosphere on their performance and possibly modify how the array processes the data.

Some intrinsic differences between atmospheric and ocean acoustic parabolic equation models

It is often assumed that propagation models developed for ocean acoustics can be applied with minimum modification to atmospheric sound propagation. Experience shows that one often finds subtle but significant problems that can lead to serious numerical (prediction) errors, even in cases where the atmospheric environmental inputs seem similar to those in ocean acoustics. In this paper three sources of error and methods for reducing the errors to acceptable levels are discussed. The first error was recently discovered by Erik Salomons [E. M. Salomons, ‘‘Improved Green’s function parabolic equation (GFPE) method for atmospheric sound propagation,’’ J. Acoust. Soc. Am. (submitted)] and concerns the failure of standard FFT algorithms when applied to atmospheric split-step parabolic equation calculations for propagation over hard ground surfaces. The second problem concerns accurate numerical treatment of the surface wave that is excited by a point source over a locally reacting ground surface. The last issue discussed is the need for improved approaches to implementing an outgoing-wave boundary condition in calculations of long-range outdoor sound propagation. Numerical examples are presented showing the effect of the errors on the predicted signal. [Work supported by the Army Research Laboratory and the Pennsylvania State University Applied Research Laboratory.]

The effect of turbulence drift on acoustical scattering into a shadow region

The turbulence structures responsible for acoustical scattering in the atmosphere are not static and evolve in time. As a first approximation, a uniform drift of the turbulence with the prevailing winds, consistent with Taylor’s hypothesis, can be anticipated. This drift can be accommodated through an extension of the Green’s function parabolic equation (GF‐PE) approach. Normal implementations of the GF‐PE compute the sound field assuming a static realization of turbulence. Such calculations can be repeated, though, for successive realizations of a turbulent atmosphere, with each realization corresponding to an additional shift in space of the initial turbulent structure. The resulting series of sound field ‘‘snapshots’’ shows the evolution of the sound field as the turbulence drifts. Recent simulations using the GF‐PE have examined scattering by drifting turbulence into an acoustical shadow, formed during upward refracting conditions, and the time evolution of the scattered field will be presented. The nature of these time variations is relevant to the use of beamforming arrays in an acoustic shadow. Some observations about predicted array responses will be discussed.

Sensitivity analysis of the Green’s function parabolic equation model for atmospheric sound propagation

A sensitivity analysis was performed on the Green’s function parabolic equation (GFPE) model to determine parameter bounds which maintain model validity. To determine these bounds, inverse theory was used to determine the ‘‘best’’ combination of input parameters over a two-dimensional domain using a normalized sum of squared residuals. A two-dimensional version of the fast field program (FFP), a widely accepted atmospheric propagation model, was used as comparison. Plots of the error surface were then made to show sensitivity bounds. Upward and downward refracting atmospheres were considered for a range of frequencies. The GFPE model was found to require a height increment parameter of about 0.05 wavelengths while the range increment parameter extended from 10 to 120 wavelengths depending on atmospheric profile and frequency. The thickness of the attenuation layer was found to be frequency dependent and ranged from 50 to over 200 wavelengths. The surface wave integral height was found to be a minimum of 30 and 13 wavelengths for the upward and downward atmospheres, respectively. The CPU time for the GFPE model ranged from 10 to 60 s, depending on frequency, and was approximately 60–1000× faster than the two-dimensional FFP.

Scattering of sound by turbulence into a shadow region

The sound field above ground due to a point source in an upwardly refracting atmosphere has been computed to determine the nature of turbulent scattering into an acoustic shadow region. A Green’s function parabolic equation (GFPE) method that includes turbulence via a phase screen approach [X. Di and K. E. Gilbert, J. Acoust. Soc. Am. 92, 2405(A) (1992)] is used for the simulations. Flat ground with finite surface impedance and a logarithmic sound-speed profile are assumed, and propagation ranges of 1 km are considered. Different realizations of a turbulent atmosphere were generated employing a Gaussian spectrum to describe the turbulence. For 500-Hz source tones, it was found that the components of the turbulence spectrum that contributed most to the scattered field in the acoustic shadow region corresponded to dimensions of the order of 2–5 m; this scale of structure can be understood in terms of a Bragg reflection condition. The sound pressure at a receiver position in the acoustic shadow appears to be dominated by contributions from a small number of localized scattering regions.