Three-dimensional Parabolic Equation Research Articles

Exploiting environmental asymmetry for vessel localization from the vertical coherence of radiated noise.

A method to determine the range and bearing of a moving broadband acoustic source, such as a surface vessel, using the coherence measured on two omni-directional, vertically separated hydrophones is demonstrated using acoustic data recorded near Alvin Canyon on the New England shelf break. To estimate the vessel's range, two theoretical approaches, a half-space model and a Pekeris waveguide model based on normal modes, establish simple relationships between the broadband signal coherence and frequency, source range, and the vertical separation of the receiver hydrophones. A brute force inversion produces a passive acoustic estimate of vessel range. Rapidly changing bathymetry with large features, such as that near Alvin Canyon, produces azimuthal asymmetry in the plan-view coherence pattern about the receivers due to horizontal refraction, focussing, and the up- (down-) slope compression (extension) of modal interference patterns. For vessels with a constant speed and heading, this generates an asymmetry in the received power and vertical coherence fringing pattern. This effect is first demonstrated using reciprocal three-dimensional parabolic equation and raytracing models in an idealized Gaussian canyon, then observed in Alvin Canyon measurements. By comparing the experimental observations to the modeled coherence, the vessel's bearing and range relative to the receivers are obtained.

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.

A HYBRID NUMERICAL TECHNIQUE FOR SOLVING THREE-DIMENSIONAL SECOND-ORDER PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

In this paper, a hybrid approach is presented for the numerical solution of three-dimensional parabolic partial differential equations. This new approach is applicable to both linear and nonlinear parabolic problems including systems. This hybrid numerical technique is based on the Haar wavelet collocation technique and the finite difference method. In this technique, the space derivative is approximated by truncated Haar wavelet series whereas the time derivative is approximated by finite difference method. The aforementioned proposed algorithms are very simple and can easily be implemented in any computer-oriented language efficiently. In order to demonstrate the efficiency and better accuracy of the newly developed numerical technique it is applied to some well-known examples from previous literature that comprises linear and nonlinear three-dimensional parabolic equations including systems. The obtained results affirm better accuracy and widespread applicability of the newly proposed numerical technique for a range of benchmark problems.

Mapping of surface-generated noise coherence

The performance of a hydrophone array can be evaluated by its coherent gain, which depends on the spatial correlation of both the signal of interest and the background noise between different array elements, where one hopes to maximize the former while minimizing the latter with array signal processing. In this paper, a computational vertical noise coherence map of the first zero-crossing is generated near Alvin Canyon, south of Martha’s Vineyard, Massachusetts, to study its dependence on the spatial variation in bathymetry, water column sound speed and sediment type. A two and three-dimensional Parabolic Equation propagation model based on reciprocity theory were used for the simulation. The results showed that the seabed parameters have the greatest impact on vertical noise coherence at the array location in the Alvin Canyon area, when compared to 3-D bathymetric and water column sound speed profile variability, especially in the shallower water. The analysis reveals the ideal spacing for a vertical hydrophone array for better signal detection in acoustic experiments. In the continental shelf and slope regions, the ideal spacing lies between 3λ⁄8 in deep water and λ⁄2 in shallow water, and for areas with strong bathymetric variations the ideal spacing can be determined by comprehensive numerical models.

Megameter propagation and correlation of T-waves from Kermadec Trench and Islands

On 18 June 2020 and 4 March 2021, very energetic low-frequency underwater T-wave signals (2 to 25 Hz) were recorded at the Comprehensive Nuclear-Test-Ban Treaty (CTBT) International Monitoring System (IMS) hydrophone stations in the Pacific Ocean (Stations HA11 and HA03) and the South Atlantic Ocean (Station HA10). This work investigates the long-range (megameters) propagation of these T-waves. Their sources were three powerful submarine earthquakes in the Kermadec Trench and Islands, located at approximately 6000, 8800, and 15100 km from Stations HA11, HA03, and HA10, respectively. Arrival time and back azimuth of the recorded T-waves were estimated using the Progressive Multi-Channel Correlation algorithm installed on the CTBT Organization (CTBTO) virtual Data Exploitation Centre (vDEC). Different arrivals within the duration of the earthquake signals were identified, and their correlations were also analyzed. The data analysis at HA03 and HA10 revealed intriguing T-wave propagation paths reflecting, refracting, or even transmitting through continents, as well as T-wave excitation along a chain of seamounts. The analysis also showed much higher transmission loss (TL) in the propagation paths to HA11 than to HA03 and HA10. Moreover, strong discrepancies between expected and measured back azimuths were observed for HA11, and a three-dimensional (3D) parabolic equation model was utilized to identify the cause of these differences. Numerical results revealed the importance of 3D effects induced by the Kermadec Ridge, Fiji archipelago, and Marshall Islands on T-wave propagation to HA11. This analysis can guide future improvements in underwater event localization using the CTBT-IMS hydroacoustic sensor network.

Learning coordinate systems for fast and accurate acoustic modeling

Many popular numerical methods, such as finite difference and spectral methods, rely on simple domain geometry and environmental smoothness. Unfortunately, these features are rarely found in real-world simulations. We propose learning coordinate transformations with deep neural networks to facilitate acoustic modeling in complex media. Using automatic differentiation, we obtain new coordinate systems by solving various optimization problems that map the computational domain to a rectangular grid. Different choices of the objective function are utilized to attain different goals, including (i) mapping complicated boundaries such as the seafloor to straight lines and (ii) reducing the rank of two-dimensional slices of the refractive index. The first choice allows for domain decomposition to accurately model sharp discontinuities in density and sound speed. The second drastically accelerates a non-intrusive reduced-order modeling technique referred to as the dynamical low-rank approximation. Using realistic ocean test cases, we compare the performance of our learned coordinate systems with domain decomposition to the classic approach of smoothing sharp discontinuities, and we compare full-rank, low-rank, and coordinate-system-accelerated low-rank solutions of the three-dimensional parabolic wave equation.

The Clutter Simulation of a Known Terrain by the 3D Parabolic Equation and RCS Computation

Ground clutter data are usually generated using a statistical model, but they cannot effectively reflect the spatial distribution characteristics of ground objects. It is important in practical projects to effectively predict ground clutter when terrain data are known. In this paper, a scheme of ground clutter simulation based on the three-dimensional parabolic equation (3DPE) model is proposed. Radio wave propagation was modeled by the PE model and the spatial field distribution was solved. After the radar cross section (RCS) calculation based on space cells, the ground clutter information data were obtained. Then the radar echo was obtained and the clutter map was simulated. The simulation experimental results show that the ground clutter map simulated by the proposed method has a good reference value, meets the demand of strong clutter area prediction under known terrain conditions, and provides a theoretical basis for radar location and optimal deployment.

Solutions to Three-Phase-Field Model for Solidification

Our research on applied mathematics is in line with the research scope of the journal. The phase-field model is applied to simulate the material and other areas. A phase-field model describing the non-isothermal solidification of an ideal multi-component alloy system is proposed in this paper. The time and space variation of a three-phase-field function and the governing equations of the temperature field are established. The global existence of weak solutions for three-dimensional parabolic differential equations is proved by the Faedo–Galerkin method. The existence of a maximum theorem is also extensively studied.

Comparative analysis of the numerical methods for 3D continuation problem for parabolic equation with data on the part of the boundary

The problem of continuation of the solution of a three-dimensional parabolic equation with data given on a time-like surface is investigated. Two numerical methods for solving the continuation problem are compared: the finite-difference scheme inversion and the solution of inverse problem by gradient method. The functional gradient formula is obtained. The results of numerical calculations are presented.

Atmospheric parabolic equation with Galerkin discretization and boundary fitted grid for modeling infrasound propagation over topography

Studies utilizing the parabolic equation (PE) to model infrasound propagation over topography have traditionally implemented them as 2D or Nx2D PE models. These methods, to one extent or another, trade-off accuracy for enhanced computational speed. Three-dimensional parabolic equation (3DPE) models are seldom seen in atmospheric acoustic propagation studies compared to their 2D and Nx2D counterparts, despite the fact that theta coupling and proper modeling of horizontal refraction and diffraction in a 3DPE model should enable greater accuracy. In this work a 3DPE in cylindrical coordinates is developed under the alternating direction implicit scheme, with a Galerkin discretization and boundary fitted grid, such that the PE is capable of handling arbitrary and irregular topography features. Finally, 2D, Nx2D, and 3D implementations of the aforementioned PE model are examined for cases of propagation over a simple hill in homogeneous, downward refracting, and upward refracting atmospheres. Insights are drawn with regards to the accuracy at a variety of positions and ranges, and the computational speeds of the various methods.

Three-dimensional parabolic equation for infrasound propagation above topography

The parabolic equation (PE) is one of the most popular methods for the modeling of low-frequency sound in the atmosphere. Over the past decades, considerable efforts have been dedicated to the inclusion of medium inhomogeneities, such as wind and turbulence, extending the PE to more realistic atmospheric conditions. On the other hand, few models take topography into account, while its effects on infrasound propagation are important in some cases. A new development has enabled the inclusion of irregular terrain in the narrow-angle 3DPE for a moving inhomogeneous medium [Khodr et al., JASA (2020)]. A coordinate transform similar to the Beilis-Tappert mapping used in 2D [Parakkal et al., JASA (2012)] is introduced to account for the variation in topography. The numerical solution relies on a Crank–Nicolson implicit scheme along the propagating direction. The pressure field at every step is computed with an iterative fixed-point algorithm which consists of a succession of tridiagonal systems that can be efficiently solved. A validation is performed against 3D Boundary Element simulations for propagation above a Gaussian hill in a homogeneous atmosphere. The existence of transversal scattering in the shadow zone, not represented by 2D models, is highlighted. Further developments include the extension of the method to a split-step wide-angle formulation. [This work was conducted at the University of Bristol in partnership with AWE Blacknest.]

Sobolev orthogonal Legendre rational spectral methods for exterior problems

The purpose of this paper is to develop the diagonalized Legendre rational spectral method for exterior problems. We first consider the exterior problems of two-dimensional elliptic and parabolic equations in polar coordinates, construct the Sobolev orthogonal Legendre rational basis functions, and then propose the diagonalized Legendre rational spectral methods. Then we consider the exterior problems of three-dimensional elliptic and parabolic equations in spherical coordinates, construct the Sobolev orthogonal Legendre rational basis functions, and then propose the diagonalized Legendre rational spectral methods. The main advantages of the suggested approaches are that the discrete systems are diagonal and the numerical solutions can be represented as truncated Fourier series. The numerical results show their effectiveness and accuracy.

Skeletonization Accelerated Solution of Crank-Nicolson Method for Solving Three-Dimensional Parabolic Equation

Parabolic equation models discretized with the finite difference method have been extensively studied for a long time. However, several explicit and implicit schemes exist in the literature. The advantage in explicit schemes is its simplicity, while its disadvantage is conditional stability. On the other hand, implicit schemes are unconditionally stable but require special treatment for a fast and accurate solution such as the Crank-Nicolson (CN) method. This method becomes computationally intensive for problems with dense meshes. The resulting matrix from the CN in two and three-dimensional cases requires high computational resources. This paper applies hierarchical interpolative factorization (HIF) to reduce the computational cost of the CN method. Numerical experiments are conducted to validate the proposed HIF acceleration.

Efficient solution of Noye–Hayman implicit finite‐difference method for modelling wave propagation in tunnels

Parabolic equation models discretised with the finite-difference method have been a research topic for a long time. Through the simulation of electromagnetic wave propagation characteristics in typical scenarios, it is pointed out that the fast algorithm needs to be developed for the solution of two-level Noye–Hayman (NH2) implicit method to meet the accuracy and computational requirements of electromagnetic wave propagation problems in three-dimensional environments. In this Letter, hierarchical skeletonisation is used for an efficient inversion of three-dimensional parabolic equation discretised by the NH2 implicit method which is the modification of conventional Crank–Nicolson (CN) method. Numerical results show that the NH2 implicit method is more accurate than the CN method at no additional cost for a given numerical example.

The unconditional stability and mass‐preserving S‐DDM scheme for solving parabolic equations in three dimensions

AbstractIn this paper, the unconditional stability and mass‐preserving splitting domain decomposition method (S‐DDM) for solving three‐dimensional parabolic equations is analyzed. At each time step level, three steps (x‐direction, y‐direction, and z‐direction) are proposed to compute the solutions on each sub‐domains. The interface fluxes are first predicted by the semi‐implicit flux schemes. Second, the interior solutions and fluxes are computed by the splitting implicit solution and flux coupled schemes. Last, we recompute the interface fluxes by the explicit schemes. Due to the introduced z‐directional splitting and domain decomposition, the analysis of stability and convergence is scarcely evident and quite difficult. By some mathematical technique and auxiliary lemmas, we prove strictly our scheme meet unconditional stability and give the error estimates in L2‐norm. Numerical experiments are presented to illustrate the theoretical analysis.

An iterative three-dimensional parabolic equation solver for propagation above irregular boundaries.

This paper describes the development of an iterative three-dimensional parabolic equation solver that takes into account the effects of irregular boundaries and refraction from a layered atmosphere. A terrain-following coordinate transformation, based on the well-known Beilis-Tappert mapping, is applied to the narrow-angle parabolic equation in an inhomogeneous media. The main advantage of this approach, which has been used in two dimensions in the past, is the simplification of the impedance boundary condition at the earth's surface. The transformed initial-boundary value problem is discretized using the Crank-Nicholson marching scheme in the propagating direction and second-order finite-differences in the transversal plane. The proposed method relies on an efficient iterative fixed-point solver, which involves the inversion of tridiagonal matrices only. The accuracy of the method is evaluated through a comparison with boundary element simulations in a homogeneous atmosphere above a Gaussian hill. Results show that transversal scattering occurs in the shadow zone of the obstacle where the two-dimensional parabolic equation underestimates the pressure amplitude. The model is particularly suited for the simulation of infrasound in a three-dimensional environment with realistic topographies.

The effect of submarine canyon bathymetry on range estimation using cross-correlated ship noise field

Ambient noise data collected on a pair of omni-directional vertically separated hydrophones at the head of Alvin Canyon, a shelf-break submarine canyon, is used to estimate the range and bearing of a passing vessel. Comparison of the vertical coherence data against a Pekeris normal-mode model may be used to provide the range of the ship along bearing lines that do not cross the canyon’s rapidly changing bathymetry. To investigate the effect of bathymetry on the range estimate, a reciprocal three-dimensional Parabolic Equation (3-D PE) model is used to generate a map of the vertical coherence field for all possible ship positions over the domain of a Gaussian canyon, demonstrating that horizontal reflection and refraction lead to focusing of ship noise along the canyon axis. The same method is used to obtain vessel range and bearing information from the data recorded at Alvin Canyon. The vessel bearing relative to the pair of vertically spaced omni-directional receivers is obtained by exploiting the canyon bathymetry.

Local accuracy of cross-term corrections of three-dimensional parabolic-equation models.

This paper deals with accuracy assessment for cross-derivative (depth and azimuth) term corrections to a three-dimensional (3D) parabolic equation (PE). The focus is on local errors, involving comparison of the two sides of the PE at insertion of (scaled) Helmholtz-equation reference solutions. For media with a particular type of lateral sound-speed variation, mode expansion together with wavenumber integration to compute modal expansion coefficients produces very accurate Helmholtz solutions for the field and its spatial derivatives. There are explicit expressions for the wavenumber integrands in terms of Airy and exponential functions, and accuracy improvements by PE cross-derivative terms are easy to assess. For a relevant example, with 3D effects similar to those in a 3D Acoustical Society of America wedge benchmark, inclusion of a leading-order as well as an additional, higher-order, cross-derivative term in the PE is favorable. The additional term provides a fourth-order accurate approximation of the PE square-root operator. A fifth-order accurate Padé approximation yields further improvement, approaching the numerical accuracy limit set by the PE method itself. The adiabatic approximation is exact for the particular media under study, but the local PE errors are similar for related 3D wedge examples with mode coupling.

Mathematical Modeling of the Process of Humid Absorption in a Porous Medium on the Example of Moisture of Lime Composites

A mathematical description of the dependence of the equilibrium moisture content of the porous material of the lime composite on the relative humidity of the air - curves of moisture adsorption in the material under study is considered. Mathematical modeling of the moisture absorption process was performed on the basis of the representation of a certain elementary volume of a lime composite in the form of a volume-porous medium with averaged physicochemical parameters. Mathematical modeling is used for theoretical study of the laws of the process of wetting dry construction mixes. Comparison of calculated and experimental data is carried out, their good consistency is shown. The moistened material is a material with a developed surface throughout the volume and can be considered as a porous medium with effective characteristics distributed throughout the volume, such as porosity, thermal conductivity, mass diffusivity, adsorption characteristics, etc. In the process of moistening the material of the medium at each point of the porous space, an adsorption reaction occurs moisture, which has its macro and microkinetic patterns. Moist air is supplied to the internal, hard-to-reach areas of the porous material due to diffusion and air flow caused by various reasons. The speed of the microscopic reaction of wetting the surface of the composite granules should be taken into account in accordance with the theory of the dependence of the moisture content of the material on the relative humidity of the air. Analysis of the type of sorption curves makes it possible to notice that for a mathematical description of experimental dependence, it is advisable to distinguish two sections on each curve: a section with a bulge up and a segment with a bulge down. The basic equation of moisture transfer in a porous medium is a three-dimensional parabolic differential equation. Under certain conditions, the process of moistening a porous material has the same regularities on any straight line perpendicular to the air-medium interface, which makes it possible to reduce the equation to one-dimensional in spatial variables. The presented calculation results show that the value of the diffusion coefficient significantly affects the process of moistening a porous material and can be used as a measure of the possible moisture content of the composite.

Underwater acoustic energy fluctuations during strong internal wave activity using a three-dimensional parabolic equation model.

The three-dimensional Monterey-Miami parabolic equation model is used to simulate a nonlinear internal wave (NIW) crossing the sound field in a shallow water environment. The impetus for this research stems from acoustic measurements taken during the Shallow Water '06 (SW06) field experiment, where a NIW traversed the water column such that soliton wavecrests were nearly parallel to the source-receiver path. Horizontal refraction effects are important in this scenario. A sound speed profile adapted from experimental SW06 data is used to simulate the NIW, assuming variations along the wavecrests (e.g., curvature) are negligible. Broadband and modal energy metrics show acoustic fluctuations due to internal wave activity. Repeated model runs simulate the NIW crossing the parabolic equation (PE) field over space and time. Statistical analysis shows the PE data are best fit by a lognormal distribution but tends to an exponential distribution during certain scenarios. Small angle differences between the acoustic track and the propagating NIW cause substantial differences in energy distribution throughout the PE field. While refraction effects due to the leading edge of the NIW's arrival are important in all cases, the impacts of focusing and defocusing in the perfectly parallel case dominate the field fluctuations. In the non-parallel case, the strong fluctuations introduced by the passage of the NIW are of similar order to the refraction off the leading edge.