Parabolic Equation Calculations Research Articles

Evidence of 3D effects for along shelf acoustic propagation during SBCEX21

The 2021 Seabed Characterization Experiment (SBCEX21) was performed on the New England Shelf Break (NESB) in May 2021 with the main objective of characterizing the NESB sediment geoacoustic properties. In this talk, we focus on low-frequency propagation (f<100 Hz) along the 200 m isobath line. Time-frequency dispersion of normal modes is estimated using a SUS charge signal recorded on a distant single hydrophone TOSSIT system. Geoacoustic inversion is performed, but two-dimensional (2D) acoustic propagation models fail to capture the observed behavior of higher order acoustic modes, and predict much shorter travel times than are seen in the data. By introducing a three-dimensional (3D) model environment, both parabolic equation and modal ray calculations confirm that the gradual shelf slope has a nontrivial influence on modal travel times. This comparison between 2D and 3D simulations of SBCEX21 acoustics and experimental data emphasizes the importance of 3D effects from sloped bathymetry on understanding propagation in a shelfbreak environment. These effects will need to be accounted for when doing geoacoustic inversion of SBCEX21 along-shelf low-frequency data.

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.

Wind turbine low frequency and infrasound propagation and sound pressure level calculations at dwellings.

This study was developed to estimate wind turbine low frequency and infrasound levels at 1238 dwellings in Health Canada's Community Noise and Health Study. In field measurements, spectral peaks were identifiable for distances up to 10 km away from wind turbines at frequencies from 0.5 to 70 Hz. These measurements, combined with onsite meteorology, were in agreement with calculations using Parabolic Equation (PE) and Fast Field Program (FFP). Since onsite meteorology was not available for the Health Canada study, PE and FFP calculations used Harmonoise weather classes and field measurements of wind turbine infrasound to estimate yearly averaged sound pressure levels. For comparison, infrasound propagation was also estimated using ISO 9613-2 (1996) calculations for 63 Hz. In the Health Canada study, to a distance of 4.5 km, long term average FFP calculations were highly correlated with the ISO based calculations. This suggests that ISO 9613-2 (1996) could be an effective screening method. Both measurements and FFP calculations showed that beyond 1 km, ISO based calculations could underestimate sound pressure levels. FFP calculations would be recommended for large distances, when there are large numbers of wind turbines, or when investigating specific meteorological classes.

Internal tides and deep diel fades in acoustic intensity.

A mechanism is presented by which the observed acoustic intensity is made to vary due to changes in the acoustic path that are caused by internal-tide vertical fluid displacements. The position in range and depth of large-scale caustic structure is determined by the background sound-speed profile. Internal tides cause a deformation of the background profile, changing the positions of the caustic structures-which can introduce intensity changes at a distant receiver. Gradual fades in the acoustic intensity occurring over timescales similar to those of the tides were measured during a low-frequency (284-Hz) acoustic scattering experiment in the Philippine Sea in 2009 [White et al., J. Acoust. Soc. Am. 134(4), 3347-3358 (2013)]. Parabolic equation and Hamiltonian ray-tracing calculations of acoustic propagation through a plane-wave internal tide environmental model employing sound-speed profiles taken during the experiment indicate that internal tides could cause significant gradual changes in the received intensity. Furthermore, the calculations demonstrate how large-scale perturbations to the index of refraction can result in variation in the received intensity.

Two-directional Synchronous Visual Sensing and Image Processing of Weld Pool in Aluminum Alloy Twin Arc Pulsed MIG Welding

Visual sensing of weld pool in pulsed twin arc welding of aluminum alloy is very difficult because of strong arc and reflection. A two-directional synchronous visual sensing system is established consisting of hardware and software system. By near infrared filter technology and exposure time of charge-coupled device(CCD) shutter controlling, clear weld pool images from rear and side directions are obtained. A new butterworth low-pass filtering approach is proposed to remove the horizontal stripes interference of high frequency and desired result is obtained. Through image processing methods such as gray stretch, threshold segmentation and edge detection, the profiles of weld pools from the two directions have been acquired. By analyzing the geometry of the weld pool images, it is found that the rear part of the weld pool is like parabola and the front and middle part is like ellipse. The geometric parameters of the weld pool such as the width, length, area, perimeter, rear factor and other parameters are defined and calculated on the basis of parabolic and elliptic curve equation calculation.

Modeling propagation of infrasound signals observed by a dense seismic network

The long-range propagation of infrasound from a surface explosion with an explosive yield of about 17.6 t TNT that occurred on June 16, 2008 at the Utah Test and Training Range (UTTR) in the western United States is simulated using an atmospheric model that includes fine-scale layered structure of the wind velocity and temperature fields. Synthetic signal parameters (waveforms, amplitudes, and travel times) are calculated using parabolic equation and ray-tracing methods for a number of ranges between 100 and 800 km from the source. The simulation shows the evolution of several branches of stratospheric and thermospheric signals with increasing range from the source. Infrasound signals calculated using a G2S (ground-to-space) atmospheric model perturbed by small-scale layered wind velocity and temperature fluctuations are shown to agree well with recordings made by the dense High Lava Plains seismic network located at an azimuth of 300° from UTTR. The waveforms of calculated infrasound arrivals are compared with those of seismic recordings. This study illustrates the utility of dense seismic networks for mapping an infrasound field with high spatial resolution. The parabolic equation calculations capture both the effect of scattering of infrasound into geometric acoustic shadow zones and significant temporal broadening of the arrivals.

A parabolic equation method for predicting sound propagation above and below a layered ground

A numerical model based on a wide-angle parabolic equation (PE) technique has been developed for calculating the sound fields above and below a layered ground. A finite element discretization was applied along the vertical direction instead of the classical finite difference scheme. By using the finite element approach, the boundary conditions, i.e., the continuity of pressure and velocity, can be incorporated directly at the air/ground and ground/ground interfaces. The range-dependent sound fields were obtained by marching the finite-element solutions (above and below the layered ground) in the radial direction. This paper reports a preliminary formulation and presents some initial computational results for the sound fields above and below a porous ground. The numerical results from the PE calculations were compared with the results obtained from analytical solutions for and other benchmark results. A linear and cubic Hermite interpolate function was used in the numerical formulations for the finite element model. It has been shown that use of the cubic Hermite interpolation function generally leads to more accurate numerical solutions with fewer elements. [Work partially funded by Federal Aviation Administration and the China Research Scholarship Council.]

Metamodel predictions for long range speech intelligibility using parabolic equation calculations.

The Crank–Nicholson parabolic equation (CNPE) method was used to calculate speech intelligibility for long range propagation of voice messages. How intelligible a voice message will be at long range is very sensitive to atmospheric conditions and parameters such as source height and background noise levels. In order to identify this sensitivity, four independent variables were identified and varied throughout the study. These independent variables are the atmospheric refraction index, which takes into account both temperature and wind gradients within the atmosphere, ambient temperature, source height, and background noise level. CNPE computations were performed for 31 optimally chosen cases where all four of the independent variables were random variables and were inputted into a metamodel. This allows for curve fitting and extrapolation of the existing data to determine intermediate results for any operation scenario defined within the bounds of the current study. Upon doing this, it was determined that the atmospheric refraction index is the most sensitive variable. In addition, there is virtually no dependence on ambient surface temperature.

Time-domain simulations of sound propagation in a stratified atmosphere over an impedance ground

Finite-difference time-domain simulations of broadband sound propagation in a stratified atmosphere are presented. A method recently proposed to obtain an impedance time-domain boundary condition is implemented in a linearized Euler equations solver, which enables to study long range sound propagation over an impedance ground. Some features of the pressure pulse evolution with time are analyzed in both upward-and downward-refracting conditions, and the time-domain simulations are compared to parabolic equation calculations in the frequency domain to show the effectiveness of the proposed impedance boundary condition.

An exact point source starting field for the Fourier parabolic equation in outdoor sound propagation

A method for exactly representing a point source starting field in a Fourier parabolic equation calculation is presented. The formulation is based on an exact, analytic expression for the field in vertical wave number space (k space). The field in vertical coordinate space (z space) is obtained via a Fourier transform of the k-space field. Thus, one can directly control the Fourier components of the starting field, so that nonpropagating components are excluded. The relation of the exact starting field to the standard Gaussian starting field is demonstrated analytically. Examples of the numerical implementation of the exact starting field are given.

The range evolution of the mean wavefront intensity for the long‐range ocean acoustic propagation experiment (LOAPEX) off‐axis source transmissions

One of the main objectives of the NPAL 2004 experiment, LOAPEX (Long‐range Ocean Acoustic Propagation EXperiment), which was conducted between 10 September and 10 October 2004, was to better understand the roles of scattering and diffraction in general. The LOAPEX measurement provided acoustic transmission data for ranges of 50, 250, 500, 1000, 1600, 2300, and 3200 km. By placing the source off‐axis in order to avoid exciting low‐order modes, we are able to study phenomena of the significant in‐filling of acoustic energy into the finale region. Our focus will be on the transmissions for the off‐axis source location (nominally 350‐m depth), and the acoustic receptions as recorded on the 1400‐m‐long axial receiving array. The observation of the mean intensity of the wavefront arrival pattern at each range will be compared to deterministic ray and parabolic equation calculations. The following questions will be addressed here: (1) How does high angle acoustic energy from an off‐axis source transfer energy to low angles in the axial region of the waveguide? (2) What are the relative contributions from diffraction and scattering? (3) How does this energy transfer scale with range? [Work supported by ONR.]

An exact point-source starting field for the Green’s function parabolic equation in outdoor sound propagation

A method for exactly representing a point source in a Green’s function parabolic equation calculation is presented. The method is no more difficult to use than standard starting-field formulations and has the several advantages: (1) the point-source solution is essentially exact; (2) only the propagating Fourier components are used; and (3) there is no limitation on the source height, which can be taken to be at z=0, if necessary. The method is derived analytically and examples of the numerical implementation are given. [Research supported by U.S. Army Armament, Research Development and Engineering Center (ARDEC).]

Downslope propagation of adiabatic and stepwise coupled modes

Downslope propagation, from a shallow source, redirects steeply propagating modes toward the horizontal. Slope conversion is of particular interest for noise sources near the shelf break. Noise modeling procedures may rely on adiabatic modes for efficiency, but their ability to accurately handle downslope conversion is not well tested. The adiabatic approximation is derived from continuously coupled modes by assuming the coupling matrices are the identity matrix and by using an asymptotic approximation to solve the uncoupled ordinary differential equations for the modal coefficients. A simpler interpretation of the twofold adiabatic approximation is obtained in the context of stepwise coupled modes when the continuous differential equations are replaced by a single matrix recursion. The stepwise interpretation of the adiabatic approximation is used to evaluate its validity, for a shelf-break environment, in a side-by-side comparison with stepwise coupled modes. The adiabatic approximation partially accounts for downslope conversion, but vertical directivity, computed in with adiabatic modes and stepwise coupled modes, is not comparable. Parabolic equation calculations, which corroborate the stepwise coupled mode results, provide a practical alternative to adiabatic modes for ambient noise modeling in shelf-break environments.

Experimental evidence of three-dimensional acoustic propagation caused by nonlinear internal waves

The 1995 SWARM experiment collected high quality environmental and acoustic data. One goal was to investigate nonlinear internal wave effects on acoustic signals. This study continues an investigation of broadband airgun data from the two southwest propagation tracks. One notable feature of the experiment is that a packet of nonlinear internal waves crossed these tracks at two different incidence angles. Observed variations for the lower angle track were modeled using two-dimensional parabolic equation calculations in a previous study. The higher incidence angle is close to critical for total internal reflection, suggesting that acoustic horizontal refraction occurs as nonlinear internal waves traverse this track. Three-dimensional adiabatic mode parabolic equation calculations reproduce principal features of observed acoustic intensity variations. The correspondence between data and simulation results provides strong evidence of the actual occurrence of horizontal refraction due to nonlinear internal waves.

Highway noise levels in a suburban environment under inversion conditions

Noise levels were measured in Scottsdale, AZ during a two week period in March, 2004 to identify the reasons for increased noise levels near a major highway during early morning hours. The noise levels were accompanied by meteorological measurements as well as traffic counts to fully describe the problem. The noise levels were measured in one-third octave bands and ranged 100 ft to 2620 ft (30–800 m) from the highway and included data on both sides of the highway. The meteorological data indicated inversion conditions or downward refraction during the times of interest and model results from a Parabolic Equation (PE) calculation indicated good results with the data. The data and model indicated an approximately 10–15 dB increase in levels during inversion conditions which rapidly transitioned to neutral and lapsed conditions shortly after sunrise. The results of the modeling effort as well as the data will be presented. [Work supported by Arizona Dept. of Transportation.]

Frechet-derivative calculations of the environmental sensitivity of the arrival structure in shallow water

The expected sensitivity of a high-resolution tomographic inversion of a 10-km slice of shallow water has been studied. Parabolic equation calculations of the Frechet derivative of simulated data at 10-km range for measured sound-speed profiles were made. The computations are based on using an existing 29-element transmitter array together with a 32-element receive array separated by 10 km. The frequency bandwidth was 3.4 kHz. The Fourier syntheses of these results produce a sensitivity map for each temporal segment of the arrival structure.

Solutions to the discrete Airy equation: application to parabolic equation calculations

In the case of the equidistant discretization of the Airy differential equation (discrete Airy equation) the exact solution can be found explicitly. This fact is used to derive a discrete transparent boundary condition (TBC) for a Schrödinger-type equation with linear varying potential, which can be used in “parabolic equation” simulations in (underwater) acoustics and for radar propagation in the troposphere. We propose different strategies for the discrete TBC and show an efficient implementation. Finally a stability proof for the resulting scheme is given. A numerical example in the application to underwater acoustics shows the superiority of the new discrete TBC.

Analysis of measured broadband acoustic propagation using a parabolic equation approach

A broadband parabolic equation (PE) approach is employed to simulate data taken from two Shallow Water Acoustic Measurement Instrument (SWAMI) bottom mounted horizontal line array (HLA) experiments in shallow water environments off the east coast of the U.S. and in the Gulf of Mexico. In both experiments the HLA was deployed along an isobath. Light bulbs were imploded at known depths and ranges in both the range-independent (array end fire) and range-dependent (array broadside) directions. For the east coast experimental data, the PE model is used to infer a seabed geoacoustic description in both the range-dependent and range-independent directions. Also, comparisons of modeled time series were made for the range-independent case with a broadband normal mode model to validate the PE calculations. In the Gulf of Mexico experiment, the sediment geoacoustic profile is well known from previous inversions and geophysical measurements. This known seabed description was used to simulate the range-dependent data. A broadband energy-conserving coupled mode approach is also employed to model the range-dependent propagation. This allows the physical mechanisms associated with range-dependent propagation to be examined in a quantitative manner for this shallow water environment. [Work supported by ONR.]

Chaos and wave propagation regimes

Ray chaos theory and parabolic equation numerical modeling were two thrusts of Fred Tappert’s research that were perpetually in tension. Fred was interested in the problem of identifying wave propagation regimes, most notably the strong focusing caustic regime and its evolution into the saturation regime. On the one hand, chaos theory held the seed of the complexity Fred believed existed in ocean acoustic wavefields; on the other hand ocean acoustic ray chaos theory (which Fred helped to pioneer) was a disdainful approximation to the full wave treatments offered by parabolic equation calculations. Fred was convinced that the saturation limit could not be obtained using ray theory and therefore he examined a new field of inquiry: a blend of chaotic ray insight and full wave dynamics called wave chaos. This talk will discuss some of Fred’s insights on this topic and how they relate to observations from basin scale acoustic transmissions.

Measurement of sound transmission and signal gain in the complex Strait of Korea

The experiment, The Acoustic Characterization Test III, was conducted in the oceanographically complex Strait of Korea to accurately measure the sound transmission under known environmental conditions. Geoacoustic profiles derived from geophysical measurements, measured bathymetry, and sound-speed profiles were the basis for range dependent parabolic equation (PE) calculations. Agreement between measured and calculated transmission loss was obtained with an attenuation profile in the near water-sediment interface layer with a dependence on frequency to the 1.8 power consistent with measurements in other sand-silt areas. Since the environment was oceanographically complex and the shipping noise levels were high, the coherency of the sound transmission was estimated using relative signal gain (RSG). RSG was taken as the difference between the gain calculated with PE and measured with the array and at longer ranges and higher frequencies was found to be approximately -2 dB with a signal gain coefficient of variation of 5%. This RSG degradation, attributed to the random signal phase fluctuations resulting from scattering from the surfaces and volume of the waveguide, yielded using a Gaussian coherence function a spatial coherence length of 30/spl lambda/ @ 400 Hz-40 km. In addition, high resolution imaging of five targets with two bottom mounted arrays illustrate the achievable performance of low-to-mid frequency active sonar in this environment.