Parabolic Equation Simulations Research Articles

Mid-frequency convergence zone propagation

Convergence zone propagation has a cycle distance of order 50 km with most of the path spent below the thermocline. Calculations using Garrett-Munk statistics and rays computed with a smoothed background sound speed profile over this single cycle predict propagation in the partially saturated regime and an integration time of 500 seconds (time spent in a single output bin of a discrete Fourier transform). Parabolic equation simulations with synthetic fine structure and ocean dynamics validate the ray-based calculations. Data of narrowband transmissions from 1.5 to 7.5 kHz from a shallow towed source (150 m) to a shallow drifting array (150 m) depth at a range of 59 km demonstrate smoothly varying phase in a beam permitting coherent integrations of 3.5 minutes. The shorter integration time in data is consistent with non-uniform motion estimated from GPS.

Parabolic equation simulation of diffraction effects in a sound beam propagating through a flowing fluid

Ultrasonic transit time difference flow meters are today industrially accepted for custody transfer measurements of oil and for natural gas. Such meters are currently planned to be used also subsea and in remote operations, where the calibration possibilities are few. In such applications the speed of sound measured by these meters will be a powerful input for estimation of density and calorific value of the flowing oil or gas. The ultrasonic transit time measurements in such meters are carried out in a flowing oil or gas, over a range typically between 6 and 40 in. For precise transit time measurements over such ranges, diffraction corrections may be of high importance. Diffraction effects for an acoustic beam generated by a uniform piston source and propagating through a flowing fluid are therefore studied numerically. The flow direction will be perpendicular to the propagation direction of the acoustic beam. The investigation is based on a narrow-angle three-dimensional parabolic equation. Effects both on amplitude and on phase will be presented.

The broadband transport theory approach to model internal wave induced scattering across deep water acoustic time-fronts.

There are currently no models to fully predict the effects of internal wave induced scattering on acoustic pulses. Existing models, which predict time domain statistics, either use the ray-based path integral method or Monte Carlo type simulations. The path integral method fails to accurately predict all of the effects of scattering. The Monte Carlo methods base the statistics on ensemble averages and are not physics-based models. This paper overcomes these limitations by using the modes of the waveguide in a transport theory application. The transport theory equations have, thus far, been used only to explain diffusion of mode intensities and decorrelation due to internal waves at individual frequencies. This paper extends the current narrowband application predict mode correlations across different frequencies and, from that, the broadband time-front, time wander, travel time bias, and the amount of spread in intensity across time and depth. To validate these predictions, this paper uses separate parabolic equation simulations. The comparisons between the two are good, suggesting a success for the mode-based transport theory approach.

Automated acoustic arrival matching using a convolutional neural network approach informed by statistics of acoustic scattering from internal waves

A machine learning model was developed to automatically align underwater acoustic measurements taken at various depths and ranges from a transmitting source in the Philippine Sea to a reference model of long range acoustic arrival structure, simultaneously determining source-receiver range and travel-time offsets associated with multipath arrivals. Ocean sound-speed variability complicates the task as the measured arrivals may exhibit scattering not present in range-independent predictions. Monte Carlo style broadband parabolic equation simulations through random internal wave fields consistent with the Garrett-Munk internal wave energy spectrum were used to generate a large data set of simulated acoustic receptions including scattered multipath arrivals with known source-receiver ranges and imposed travel time offsets. These simulated receptions were used to train and evaluate the machine learning model for arrival pattern matching to the reference model. The inclusion of various data dimensions, such as peak amplitude and width, and contextual information, such as range and depth, were also explored as input to the model. Ranging results for the machine learning model were compared to a programmatic solution engineered for the same task.

Measurements and Spatial Distribution Simulation of Impact Pile Driving Underwater Noise Generated During the Construction of Offshore Wind Power Plant Off the Southwest Coast of Korea

Offshore wind power plants are under construction worldwide, and concerns about the adverse effects of underwater noise generated during their construction on the marine environment are increasing. As part of an environmental impact assessment, underwater noise generated by impact pile driving was measured during the construction of an offshore wind farm off the southwest coast of Korea. The sound exposure levels of impact pile driving noise were estimated as a function of distance and compared with those predicted by a damped cylindrical spreading model and broadband parabolic equation simulation. Source level at 1 m was estimated to be in a range of 183–184 dB re 1μPa2s in the sound exposure level based on the model predictions and it tended to decrease by 21log⁡r as the distance increased. Finally, the spatial distribution of impact pile driving noise was predicted. This result, if combined with noise-induced damage thresholds for marine life, may be used to assess the effects of wind farm construction on marine ecosystems.

A mode based broadband scattering model for deep water sound propagation

Internal-wave induced sound-speed perturbations cause phase and intensity fluctuations of time-fronts. This paper focuses on predictions for broadband Transmission Losses (TL), time wander, and Scintillation Index (SI), along various sections of a typical deep water time-front. These statistics are usually predicted using Monte-Carlo Parabolic Equation (PE) simulations, or the high frequency based path-integral method. A physics based model that fully connects the wave-field statistics, and the internal wave spectrum, to the broadband statistics is not yet available. This paper uses the mode-based transport theory approach to predict the pressure field scattering at different frequencies, and from that the broadband statistics. This requires calculations for cross-modal correlations over mode number and frequency, which becomes computationally intensive for large bandwidths. This paper considers the adiabatic phase coherence to simplify the computations, and tests the final predictions against PE simulations and observations from the Philippine Sea 2010 experiment.

A coupled mode model for omnidirectional three-dimensional underwater sound propagation.

A fully three-dimensional (3D) omnidirectional numerical coupled mode model of acoustic propagation is detailed. A combination of normal mode and finite element computational methods is applied to produce the numerical results. The technique is tested in a strongly range-dependent ocean environment modeled after the Hudson Canyon. Modeled sound from three source locations selected over different bathymetric depths is examined to determine capabilities and difficulties associated with varying numbers of propagating vertical modes across the horizontal domain, and variable amounts of mode coupling. Model results are compared to those from a unidirectional Cartesian 3D parabolic equation simulation, and from adiabatic (uncoupled) simulations to illustrate the capabilities of the techniques to study the influences of coupling, strong refraction, and reflection.

A Comparative Study of Cubic B-spline-Based Quasi-interpolation and Differential Quadrature Methods for Solving Fourth-Order Parabolic PDEs

In this work, we present two approaches for simulation of fourth-order parabolic partial differential equations. In the first method, cubic B-spline quasi-interpolation is used to approximate the spatial derivative of the dependent variable and forward difference to approximate the time derivative. In the second method, we have used modified cubic B-spline functions-based differential quadrature method (DQM) for space discretization to get a system of ODEs and then this system is solved by SSP-RK43 method to get the results at knots. The numerical results demonstrate the accuracy of the proposed method. The stability analysis of the methods has also been discussed. It is observed that quasi-interpolation-based method is unconditionally stable, whereas for DQM, the stability has to be checked for a large number of space points. Moreover, for the small number of grid points, DQM gives better results, while for a large number of grid points, quasi-interpolation-based method is better.

A new high-order compact finite difference scheme based on precise integration method for the numerical simulation of parabolic equations

This paper presents two high-order exponential time differencing precise integration methods (PIMs) in combination with a spatially global sixth-order compact finite difference scheme (CFDS) for solving parabolic equations with high accuracy. One scheme is a modification of the compact finite difference scheme of precise integration method (CFDS-PIM) based on the fourth-order Taylor approximation and the other is a modification of the CFDS-PIM based on a (4,4)-Padé approximation. Moreover, on coupling with the Strang splitting method, these schemes are extended to multi-dimensional problems, which also have fast computational efficiency and high computational accuracy. Several numerical examples are carried out in order to demonstrate the performance and ability of the proposed schemes. Numerical results indicate that the proposed schemes improve remarkably the computational accuracy rather than the empirical finite difference scheme. Moreover, these examples show that the CFDS-PIM based on the fourth-order Taylor approximation yields more accurate results than the CFDS-PIM based on the (4,4)-Padé approximation.

Similarities and differences in infrasound propagation effects between arctic and temperate environments

Meteorological conditions in an arctic environment differ significantly from those in a temperate environment. Atmospheric phenomena particular to polar regions, including wind patterns such as the polar vortex and low-level jets above strongly stable layers, strong temperature and humidity gradients, and density currents, could have unique impacts on infrasound propagation that are not observed in temperate locations. In this study, parabolic-equation simulations of sound propagation are performed using measured meteorological conditions for summer and winter conditions in temperate and arctic locations. The similarities and differences in environmental conditions between these two locations and their relative impact on the predicted transmission loss are examined. For summer conditions, a comparison to measured data from explosive sources is performed for both temperate and arctic locations.

Extended source models for long range wind turbine noise propagation

Wind turbine noise can be perceived at distances greater than one kilometer and is characterized by amplitude modulations at the receiver. In order to predict this noise, it is necessary to model the dominant aeroacoustic noise sources as well as the main outdoor propagation effects. In most studies from the literature, the wind turbines are modeled as point sources to simplify the coupling between source and propagation models, but this assumption is not always justified. In this study, two original methods are proposed to couple an aeroacoustic source model based on Amiet’s theory and a Split-Step Padé parabolic equation code for acoustic propagation in an inhomogeneous atmosphere. In the first method, an initial starter is obtained for each segment of the blade using the backpropagation approach. This method enables us to accurately model the directivity of the noise sources but is very computationally intensive. In the second method, the blade segments are viewed as moving monopole sources, and only a limited number of parabolic equation simulations are needed which strongly reduces the computation time. These two methods are validated using an analytical reference solution in a homogeneous medium and compared in various inhomogeneous atmospheres.

Influence of the ocean-bottom structure in the long-distance propagation by seismic survey source

A source of Multi-Channel Seismic survey system (MCS) is the air-gun array which generates a high level and short pulse sound. However, there was anxiety which has an influence on behavior of a marine mammal by very high acoustic pressure. Therefore, JAMSTEC have decided guidance of regulation along a guideline of NOAA (TTS) in safe purpose of a marine mammal. We'll hope for a MCS investigation by a new JAMSTEC R/V “KAIMEI” (5,500GT) in the Hawaii Islands offing jointly with an U.S. research institute by geological interest in the future. So, long-distance propagation (about 100 km) from air gun source was calculated by a Parabolic Equation simulation for safety ensuring of marine mammals. The simulation results are as follows: 1) When the ocean-bottom of the long-distance transmission path is basalt, for a reflected pulses to pile up from the bottom in various courses, a received pulse width becomes very long. 2) The received level was lower than the TTS level in the watch possible distance from the mari...

Infrasound detection of the Chelyabinsk meteor at the USArray

On February 15, 2013 a small asteroid entered Earth's atmosphere near Chelyabinsk, Russia. This extremely rare event was recorded by the 400-station USArray deployed in the continental United States and Alaska. These stations recorded infrasound signals from the event at distances from 6000 to 10 000 km across a sector spanning 55° that encompassed the North Pole. This dense, extensive network permitted a detailed study of long-range infrasound propagation and source characteristics. We observe long wavetrains at all stations (ranging to over 100 min) but clear variations in the character of the wavetrains across the network. Ray-tracing through a spatially and temporally varying atmospheric model indicates the source excited resonance in the thermospheric duct to all stations. Resonance was also excited in a persistent stratospheric duct between the source and stations in Alaska and along the west coast of the United States due to favorable winds at those azimuths, leading to higher group velocities and frequency content at these stations than those to the east. An attenuation formula derived from parabolic equation simulations is used to estimate infrasound transmission losses at all stations, using simplified models of the effective sound speed along each source-receiver path. Observed variations in signal energies from higher than expected at stations in the thermospheric duct in the eastern United States, to lower than expected in Alaska, at azimuths nearly orthogonal to the asteroid's Mach cone, lead us to conclude that (1) the source was dominantly isotropic and (2) the model overestimates attenuation in the thermospheric duct.

Depth dependence of the intensity vector from impact pile driving

The direction of the time-integrated acoustic intensity vector defines active intensity streamlines which are perpendicular to this vector. For impact pile driving, these intensity streamlines depend on both depth and time. The time-integrated intensity vector (active intensity) has been shown to change direction over the initial downward traveling Mach cone, the upward traveling reflection of this from the bottom of the pile, and subsequent downward traveling reflection from the top of the pile, referred to as phases in Dahl and Reinhall [J. Acoust. Soc. Am. 134, EL1 (2013)]. In this study, vertical line array data are used to demonstrate the depth dependence of the active intensity vector during each of the three phases. The vertical component of the active intensity is approximated between each pair of neighboring hydrophones by finite difference methods. A numerical approach based on parabolic wave equation simulations [Reinhall and Dahl, J. Acoust. Soc. Am. 130, 1209 (2011)] is modified here and used to construct theoretical intensity streamlines, motivated by those depicted in Zampolli et al. [Acoust. Soc. Am. 133, 72 (2013)]. These intensity streamlines are validated by the experimental measurements. Energy propagation through the water column and the sediment-water interface is also discussed.

Deep fades in intensity: Exploration of measurement-Monte Carlo parabolic equation mismatch in the Philippine Sea

The oceanography of the Philippine Sea is partially characterized by energetic mesoscale and strong locally generated internal tides. Despite the simplification of range-independence and the exclusion of internal tides from Monte Carlo parabolic equation (MCPE) simulations, predictions of scintillation index, variance of log-intensity, and the distribution of intensity for acoustic paths with upper-turning-points (UTP) below the extreme upper ocean generally agree with measurements made during an experiment in 2009. These measures of the fluctuations did not appear to be strongly influenced by the number of UTPs in the path, though a compensating effect due to differences in UTP position cannot be ruled out. Enhanced variability in the form of deep fades is observed for paths turning in the extreme upper ocean; this enhanced variability is not predicted by the MCPE model employed. Seaglider-based observations of mixed-layer depth (from 2010 to 2011) and moored measurements of internal-tide-related sound-speed perturbations are presented. A plane-wave internal-tide model and results from acoustic mode propagation through range-independent profiles measured in situ are compared with the observed character of the intensity fades.

Wavefront intensity statistics for 284-Hz broadband transmissions to 107-km range in the Philippine Sea: Observations and modeling

In the spring of 2009, broadband transmissions from a ship-suspended source with a 284-Hz center frequency were received on a moored and navigated vertical array of hydrophones over a range of 107 km in the Philippine Sea. During a 60-h period over 19,000 transmissions were carried out. The observed wavefront arrival structure reveals four distinct purely refracted acoustic paths: One with a single upper turning point near 80 m depth, two with a pair of upper turning points at a depth of roughly 300 m, and one with three upper turning points at 420 m. Individual path intensity, defined as the absolute square of the center frequency Fourier component for that arrival, was estimated over the 60-h duration and used to compute scintillation index and log-intensity variance. Monte Carlo parabolic equation simulations using internal-wave induced sound speed perturbations obeying the Garrett-Munk internal-wave energy spectrum were in agreement with measured data for the three deeper-turning paths but differed by as much as a factor of four for the near surface-interacting path.

Time-angle sensitivity kernels for sound-speed perturbations in a shallow ocean

Acoustic waves traveling in a shallow-water waveguide produce a set of multiple paths that can be characterized as a geometric approximation by their travel time (TT), direction of arrival (DOA), and direction of departure (DOD). This study introduces the use of the DOA and DOD as additional observables that can be combined to the classical TT to track sound-speed perturbations in an oceanic waveguide. To model the TT, DOA, and DOD variations induced by sound-speed perturbations, the three following steps are used: (1) In the first-order Born approximation, the Fréchet kernel provides a linear link between the signal fluctuations and the sound-speed perturbations; (2) a double-beamforming algorithm is used to transform the signal fluctuations received on two source-receiver arrays in the time, receiver-depth, and source-depth domain into the eigenray equivalent measured in the time, reception-angle and launch angle domain; and finally (3) the TT, DOA, and DOD variations are extracted from the double-beamformed signal variations through a first-order Taylor development. As a result, time-angle sensitivity kernels are defined and used to build a linear relationship between the observable variations and the sound-speed perturbations. This approach is validated with parabolic-equation simulations in a shallow-water ocean context.

Time-angle ocean acoustic tomography using sensitivity kernels: The forward problem

Broadband acoustic signals around 1 kHz propagate through shallow water oceanic waveguides of ~100 m in depth and ~2 km in range as multiple ray-like wavefronts. These acoustic arrivals can be characterized by the following observables: travel-time (TT), direction-of-arrival (DOA), and direction-of-departure (DOD). By applying double-beamforming on the point-to-point signals recorded between two source-receiver arrays, the acoustic contribution of each arrival can be separated from the multi-reverberated data and the TT, DOA, and DOD observable variations are accurately measured. This study deals with the use of time-angle sensitivity kernels (TASK) to estimate the observable variations induced by sound speed perturbations in the waveguide. This approach is based on the first order Born approximation and takes into account the finite-frequency effects associated with wave propagation. The robustness the TASK approach is analyzed and compared to numerical parabolic equation simulations involving different sound speed perturbations. For example, parameters such as the perturbation location, the value and shape of the perturbation in the waveguide are modified. The combination of several perturbations and the influence of the source-receiver array apertures on the TT, DOA, and DOD estimates are also studied.

Solving anisotropic elliptic and parabolic equations by a meshless method: simulation of the electrical conductivity of a tissue

In this paper, the extension of the generalized finite difference method to the solution of anisotropic elliptical and parabolic partial differential equations is given in the case of considering two space dimensions. The explicit finite difference formulae and the criterion of stability for the anisotropic parabolic equations are given. The stability condition has been expressed in function of the coefficients of the star equation for irregular clouds of nodes. Various anisotropic cases of simulation of cardiac tissue, including elliptic and parabolic equations, have been solved, and the results show the accuracy of the method.

The effect of bottom layering on the acoustic vector field

A signal reflected from a layered sea-bed contains information pertaining to the sediment properties. Typically, a signal intended to probe the sea-bed is designed to have a large bandwidth to allow for time separation of arrivals from the multiple layers. Depending on the geometry, it may impossible to avoid interference of these arrivals. The interference of these multiple arrivals does establish a pattern observable in the vector intensity. Measurements of the vertical complex acoustic intensity of a near-bottom source (~λ from the seafloor) collected off the coast of New Jersey in 2006 demonstrate the effect of a sub-bottom layer and the observable interference pattern between the first bottom reflection and the sub-bottom reflection. The spatial structure of the complex intensity can be used to infer bottom properties, which are in close agreement with a number of experimental studies at this location. The observable in the complex intensity can also be directly measured with a particle motion sensor. Parabolic equation simulations of the experimental site are used to demonstrate both the characteristic of the vector field and the sensitivity of these vector properties to changes in the sediment properties.