Parabolic Equation Techniques Research Articles

Predicting ocean pressure field with a physics-informed neural network.

Ocean sound pressure field prediction, based on partially measured pressure magnitudes at different range-depths, is presented. Our proposed machine learning strategy employs a trained neural network with range-depth as input and outputs complex acoustic pressure at the location. We utilize a physics-informed neural network (PINN), fitting sampled data while considering the additional information provided by the partial differential equation (PDE) governing the ocean sound pressure field. In vast ocean environments with kilometer-scale ranges, pressure fields exhibit rapidly fluctuating phases, even at frequencies below 100 Hz, posing a challenge for neural networks to converge to accurate solutions. To address this, we utilize the envelope function from the parabolic-equation technique, fundamental in ocean sound propagation modeling. The envelope function shows slower variations across ranges, enabling PINNs to predict sound pressure in an ocean waveguide more effectively. Additional PDE information allows PINNs to capture PDE solutions even with a limited amount of training data, distinguishing them from purely data-driven machine learning approaches that require extensive datasets. Our approach is validated through simulations and using data from the SWellEx-96 experiment.

Subsurface acoustic ducts in the Northern California Current System

A survey of sound speed profiles obtained from archived mooring and glider observations over the continental shelf, slope, and further offshore along the U.S. Pacific Northwest coastline reveals a shallow subsurface duct that appears at depths below the mixed layer. Sound propagation simulations using parabolic equation techniques demonstrate that the presence of the duct has a strong impact on sound propagation at 3.5 kHz. The duct is more prevalent in summer to fall than in winter to spring and extends to at least 200 km offshore from the shelf break. The depth of the minimum sound speed of the duct decreases onshore from between 80 m and 100 m offshore of the continental slope to less than 60 m over the shelf. The duct is also more prevalent offshore of the shelf break than over the shelf. Two processes potentially contribute to the formation of the subsurface duct: (1) differential advection of water masses driven by large-scale circulation within the northern California Current Systemand (2) wintertime surface cooling and convective deepening of the surface mixed layer. Data analysis conducted in this study suggests that the former is likely the dominant mechanism. [Work supported by the Office of Naval Research.]

Methods and Algorithms of Subsurface Holographic Sounding

In our experiments, we develop and test portable multi-element receiver antenna arrays, electrically scanned in order to immediately obtain a recognizable image of subsurface objects. Two quadrature components of the radar return signal are processed with a Kirchhoff backward migration algorithm. Physical theory is used to assess the quality of the holographic image, and the synthetic aperture approach is developed and tested. The parabolic wave equation and Gaussian beam technique are used in order to take into account refraction effects and to suppress specular reflection from the air-ground interface. Laboratory and field tests confirmed the predicted device parameters.

Вычисление множителя ослабления при обратном рассеянии от земной поверхности методом параболического уравнения.

Вычисление множителя ослабления при обратном рассеянии от земной поверхности методом параболического уравнения.

Sound propagation from a ridge wind turbine across a valley.

Sound propagation outdoors can be strongly affected by ground topography. The existence of hills and valleys between a source and receiver can lead to the shielding or focusing of sound waves. Such effects can result in significant variations in received sound levels. In addition, wind speed and air temperature gradients in the atmospheric boundary layer also play an important role. All of the foregoing factors can become especially important for the case of wind turbines located on a ridge overlooking a valley. Ridges are often selected for wind turbines in order to increase their energy capture potential through the wind speed-up effects often experienced in such locations. In this paper, a hybrid calculation method is presented to model such a case, relying on an analytical solution for sound diffraction around an impedance cylinder and the conformal mapping (CM) Green's function parabolic equation (GFPE) technique. The various aspects of the model have been successfully validated against alternative prediction methods. Example calculations with this hybrid analytical-CM-GFPE model show the complex sound pressure level distribution across the valley and the effect of valley ground type. The proposed method has the potential to include the effect of refraction through the inclusion of complex wind and temperature fields, although this aspect has been highly simplified in the current simulations.This article is part of the themed issue 'Wind energy in complex terrains'.

Numerical evaluation of pile vibration and noise emission during offshore pile driving

When offshore structures like converter platforms or wind turbines are founded by pile driving, high noise level are emitted into the water. In this paper a numerical model will be presented and validated which is able to effectively predict the noise emission during pile driving under offshore conditions. It combines a Finite Element Method in the near field of the pile with a Parabolic Equation technique to compute the pressure spectrum, sound exposure and peak level in a certain distance of the pile. The results will be compared to measurements performed during two full scale offshore tests. Furthermore, a procedure for computing the acoustic properties of the sediment as a function of the frequency, depth and density is presented. This allows to represent the layered sediment at the construction site within the numerical methods.

Numerical Evaluations of Diffraction Formulas for the Canonical Wedge-Scattering Problem [Testing Ourselves

Numerical computations of different diffraction representations for the canonical wedge-scattering problem are discussed. Representations include total-field exact series summation, exact diffraction integral, and the Physical Optics (PO), Physical Theory of Diffraction (PTD), and Parabolic Equation (PE) techniques, and the Uniform Theory of Diffraction (UTD). Numerical challenges and advantages of each are given.

Variational analysis of the Langmuir–Hinshelwood dynamic mixed-kinetic adsorption model

The paper deals with the mathematical and numerical analysis of surfactants behavior at the air–water interface, by using a mixed-kinetic adsorption model which is known as the Langmuir–Hinshelwood model and which is consistent with the classical Langmuir isotherm at equilibrium. The model consists of a coupled nonlinear system of a parabolic equation for the surfactant concentration and an ordinary differential equation for the surface concentration. The existence and uniqueness of the weak solution is proved by applying a result for the linear parabolic equation and fixed-point techniques. Then, fully discrete approximations are introduced by using a finite element method and a hybrid combination of both backward and forward Euler schemes. An a priori error estimate result is proved from which, under adequate additional regularity conditions, the linear convergence of the algorithm is derived. Finally, numerical simulations are presented to demonstrate the accuracy of the algorithm and the behavior of the solution for a commercially available surfactant.

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.

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.]

Parabolic equation techniques for reverberation modeling

Recent developments of parabolic equations for modeling reverberation will be presented. This method is applicable to the 2-D reverberation problems with general sound speed profiles, sediment layering, and interface roughness. The environment is sub-divided into range-independent regions, and the forward and back-scattered fields are coupled at each interface with a single-scattering approximation. This method incorporates a cumulative scattering loss into the forward field, but neglects the non-local coupling terms between the forward and backward scattered fields. The back-scattered field is stored at each interface on a forward sweep through the environment and back-propagated on backward sweep. Reverberation time series are Fourier synthesized from the frequency domain solutions. This technique is fundamentally different than scattering kernel based approaches and provides an independent method for obtaining the forward and back-scattered solutions for reverberation problems. As currently implemented, this model does not include multiple scattering effects, but the algorithm may be generalized to include multiple scattering with iterative sweeps through the propagation domain. The method applied to 2-D reverberation problems from the recent Reverberation Modeling Workshop (November 2006) and results are compared with ray and normal-mode based models. [Work supported by ONR.]

Propagation in an elastic wedge using the virtual source technique

The virtual source technique, which is based on the boundary integral method, provides the means to impose boundary conditions on arbitrarily shaped boundaries by replacing them by a collection of sources whose amplitudes are determined from the boundary conditions. In this paper the virtual source technique is used to model propagation of waves in a range-dependent ocean overlying an elastic bottom with arbitrarily shaped ocean-bottom interface. The method is applied to propagation in an elastic Pekeris waveguide, an acoustic wedge, and an elastic wedge. In the case of propagation in an elastic Pekeris waveguide, the results agree very well with those obtained from the wavenumber integral technique, as they do with the solution of the parabolic equation (PE) technique in the case of propagation in an acoustic wedge. The results for propagation in an elastic wedge qualitatively agree with those obtained from an elastic PE solution.

Range-dependent seismo-acoustics benchmark problems

Developing the capability to accurately solve range-dependent problems in ocean acoustics has been a topic of great interest for many years. At the time of the ASA benchmark efforts [J. Acoust. Soc. Am. 87, 1497–1545 (1990)], the elastic parabolic equation was in its early stages of development, and it was too early to consider problems involving shear waves. In recent years, there has been much progress in the development of parabolic equation and spectral techniques for range-dependent seismo-acoustics problems. The time is now ripe for extending the ASA benchmarks to cases involving shear waves. Several types of problems will be considered, including problems involving upslopes, downslopes, multiple solid layers, mode cutoff, and interface waves. [Work supported by the ONR.]

BROADBAND SIGNAL SIMULATION IN SHALLOW WATER

Today's minimum requirements for ocean acoustic models are to be able to simulate broadband signal transmissions in 2D varying environments with an acceptable computational effort. Standard approaches comprise ray, normal mode and parabolic equation techniques. In this paper we compare the performance of four broadband models [Formula: see text] on a set of shallow-water test environments with propagation out to 10 km and a maximum signal bandwidth of 10–1000 Hz. It is shown that a computationally efficient modal approach as implemented in the [Formula: see text] model is much faster than standard, less optimized models such as [Formula: see text] and [Formula: see text]. However, the handling of range dependency in the adiabatic approximation is not always sufficiently accurate, and it is suggested that a mode coupling approach be adopted in [Formula: see text]. Moreover, the interpolation of modal properties in range could lead to a further significant speed-up of mode calculations in range-dependent environments. It is concluded that coupled modes with wavenumber interpolation in both frequency and range remain the most promising wave modeling approach for broadband signal simulations in range-dependent shallow water environments. At higher frequencies (> 1 kHz) there is currently no alternative to rays as a practical signal simulation tool.

Energy-conserving and single-scattering parabolic equation solutions for elastic media

Parabolic equation techniques are efficient for solving nonseparable wave propagation problems. When the properties of the medium vary gradually in range, parabolic equation solutions are also very accurate for many problems. The key to achieving accuracy and efficiency simultaneously is to apply energy-conservation or single-scattering corrections to account properly for range dependence. This approach has proven to be very effective for acoustic media. Some progress has been made on the elastic case [J. Acoust. Soc. Am. 94, 975–982 (1993); 94, 1815–1825 (1993)], but this problem has not been fully resolved. In this paper we will discuss some recent progress in the formulation of the elastic parabolic equation [W. Jerzak, J. Acoust. Soc. Am. (submitted)], a single-scattering approach for a vector wave problem [J. Acoust. Soc. Am. 104, 783–790 (1998)], and how they are being used to improve the accuracy of parabolic equation solutions for problems involving elastic sediments. [Work supported by ONR.]

Path loss predictions in the presence of buildings on flat terrain: a 3-D vector parabolic equation approach

Starting from a parabolic approximation to the Helmholtz equation, a three-dimensional (3-D) vector parabolic equation technique for calculating path loss in an urban environment is presented. The buildings are assumed to be polygonal in cross section with vertical sides and flat rooftops and the terrain is assumed to be flat. Both buildings and ground are allowed to be lossy and present impedance-type boundary condition to the electromagnetic field. Vector fields are represented in terms of the two components of Hertzian potentials and depolarization of the fields is automatically included in the formulation. A split-step algorithm is presented for marching the aperture fields along the range. Boundary conditions on the building surfaces are treated by using a local Fourier representation of the aperture fields. Several test cases are considered to check the boundary treatment used in the technique as well as to validate the overall approach. Comparison is shown with uniform theory of diffraction (UTD), exact solutions, as well as with measurements.

A higher-order on-surface radiation condition derived from an analytic representation of a dirichlet-to-neumann map

On-surface radiation conditions are useful for obtaining approximate solutions to scattering problems involving compact obstacles. An analytic representation of the Dirichlet-to-Neumann map for a circle is derived and used to construct a higher-order on-surface radiation condition for a generally convex perfectly conducting body in two dimensions. This approach is based on a Hankel function in which a tangential operator appears in the index. In the high-frequency limit, this analytic representation approaches the square root of a differential operator which commonly arises in the application of parabolic equation techniques to propagation problems. Treating the scattered field propagation angle relative to the surface normal and the surface curvature as independent parameters, the representation is fit to a rational function to provide an accurate and efficient on-surface radiation condition that is tested for various examples.

Predicting sound propagation in the atmosphere using an artificial neural network

The spectral content of acoustic signals is dramatically altered as sound propagates through the atmosphere due to refraction from vertical gradients in wind and temperature, ground interactions, and other effects. Parabolic equation (PE) techniques have been successfully used to calculate numerical solutions for many of these effects. The PE models generally produce accurate attenuation values, but execution time is excessive for many applications where near real-time results are required. To obtain sound level attenuation predictions more quickly, we developed an artificial neural network for a range of heights and source/receiver horizontal separations. The PE and boundary conditions were modified to obtain a nondimensional representation, which resulted in seven parameters required to specify all input combinations. This model version was then used to train the neural network, using a range of values for each parameter. The standard deviation of the errors (propagation model minus neural network simulations) was generally within 2 dB for the training data set containing approximately 15<th>000 cases. The smaller test data sets resulted in errors having standard deviations of approximately 1 dB. The neural network was a good predictor of the sound propagation model results except for small values of the nondimensional ground impedance parameter.

Parabolic Equation Techniques for Seismic Waves

Parabolic equation techniques for solving seismic wave propagation problems are discussed. These techniques provide an excellent combination of accuracy and efficiency for problems involving laterally varying media. Due to recent improvements, the parabolic equation method is applicable to problems involving wide propagation angles, large variations in the properties of the medium, and all types of waves.