Kirchhoff Method Research Articles

Application of the Kirchhoff Method to the Problem of Wave Diffraction on a Ribbon at Small Sliding Incidence Angles

A new approach to the problem of electromagnetic wave diffraction on a conducting ribbon at small sliding incidence angles is developed within the framework of the Fresnel-Kirchhoff method. The wave diffraction in this system has been theoretically and experimentally studied using the proposed method. The experimental data agree with the results of theoretical analysis.

Residual dip moveout in VTI media

ABSTRACTDip‐moveout (DMO) correction is often applied to common‐offset sections of seismic data using a homogeneous isotropic medium assumption, which results in a fast execution. Velocity‐residual DMO is developed to correct for the medium‐treatment limitation of the fast DMO. For reasonable‐sized velocity perturbations, the residual DMO operator is small, and thus is an efficient means of applying a conventional Kirchhoff approach. However, the shape of the residual DMO operator is complicated and may form caustics. We use the Fourier domain for the operator development part of the residual DMO, while performing the convolution with common‐offset data in the space–time domain. Since the application is based on an integral (Kirchhoff) method, this residual DMO preserves all the flexibility features of an integral DMO. An application to synthetic and real data demonstrates effectiveness of the velocity‐residual DMO in data processing and velocity analysis.

Application of Compact Schemes to Large Eddy Simulation of Turbulent Jets

We present 3-D large eddy simulation (LES) results for a turbulent Mach 0.9 isothermal round jet at a Reynolds number of 100,000 (based on jet nozzle exit conditions and nozzle diameter). Our LES code is part of a Computational Aeroacoustics (CAA) methodology that couples surface integral acoustics techniques such as Kirchhoff's method and the Ffowcs Williams-- Hawkings method with LES for the far field noise estimation of turbulent jets. The LES code employs high-order accurate compact differencing together with implicit spatial filtering and state-of-the-art non-reflecting boundary conditions. A localized dynamic Smagorinsky subgrid-scale (SGS) model is used for representing the effects of the unresolved scales on the resolved scales. A computational grid consisting of 12 million points was used in the present simulation. Mean flow results obtained in our simulation are found to be in very good agreement with the available experimental data of jets at similar flow conditions. Furthermore, the near field data provided by the LES is coupled with the Ffowcs Williams--Hawkings method to compute the far field noise. Far field aeroacoustics results are also presented and comparisons are made with experimental measurements of jets at similar flow conditions. The aeroacoustics results are encouraging and suggest further investigation of the effects of inflow conditions on the jet acoustic field.

Investigation of integral surface formulations for acoustic post-processing of unsteady aerodynamic jet simulations

The acoustic post-processing of unsteady aerodynamic jet simulations using Kirchhoff or Ffowcs Williams and Hawkings (FW–H) surface integral methods is investigated from the theoretical and practical points of view. This analysis is carried out for a supersonic hot jet, starting from the flow fields provided by an unsteady aerodynamic simulation whose characteristics are recalled. Both acoustic integral methods are first compared from a theoretical point of view. In particular, the role of the ‘entropic’ sources on the control surface is pointed out. Various acoustic calculations are then carried out for this hot jet to assess the influence of the type of method and of the position, extent and nature (open or closed) of the control surfaces on the noise predictions. This parametric study shows that the Kirchhoff method using density as input data is poorly suited for acoustic predictions of hot jets. The Kirchhoff method using pressure as input data provides results similar to those obtained with the FW–H surface integral but the latter appears to be more reliable. The study also shows that the use of a downstream closed control surface does not present a real interest, whatever the acoustic method is.

The Choice of the Migration Method for Subsurface Radiolocation Signals Processing

There is a comparison four basic migration transform methods in the article, these are method of diffraction summation, Gazdagphase shift method, Stolt method and Kirchhoff method. The objective is the choice of the best method for practice application ofgeoradar signal processing. The methods comparison is based on the level methodical interference, noise immunity and PC timeconsumption. Chosen methods are Stolt method and method of diffraction summation. Stolt method is quick enough and has small levelof methodical interference. Method of diffraction summation is quickest. Both methods increase S/N ratio after migration. An exampleof real georadar profile migration by Stolt method is given. Ill. 11, bibl. 7 (in Russian; summaries in Lithuanian, English and Russian).

Numerical analysis of the airfoil flow fields and transonic blade-vortex interaction noise

The flow fields around NACA0012, NACA23012 and OA207 airfoils can be calculated by solving the Euler-Navier-Stokes equations. The far-field noise generated by NACA0012 airfoil- vortex interaction is predicted using the combined Euler-Kirchhoff method. The Kirchhoff surface is selected as one of the grid surfaces. Firstly, the result of flow field under airfoil-vortex interaction can be obtained by solving the Euler equations. Then the result of far-field noise can be obtained by the Kirchhoff method. Finally, the combined Euler-Kirchhoff method developed here is proved to be good for studying blade-vortex interaction noise through sample calculations.

Light scattering in cholesteric liquid crystals with a large pitch

Optical properties of cholesteric liquid crystals with a pitch larger than the wavelength of light are considered. Normal waves of the medium and the Green function of the electromagnetic field are analyzed. A general algorithm based on the application of the Kirchhoff method is proposed for calculating the scattered light intensity in media with a one-dimensional periodic structure. The WKB vector method is used for calculating the spatial correlation function of thermal fluctuations of the director. It is found that the transformation of two fluctuation modes takes place is some regions. The angular and polarization dependences of the intensity of light scattered from fluctuations of the director are calculated. It is found, in particular, that the intensity of scattering is a nonmonotonic function of the size of the system.

Analysis of a hard strip-loaded conical horn by the method of generalized scattering matrices

A hard-walled conical horn fed in the TE/sub 11/-mode from a regular circular waveguide is analyzed. The hard wall is formed by a dielectric layer with cylindrical inner surface loaded with longitudinal conducting strips. The analysis is carried out by using a stepwise representation of the horn with subsequent application of the method of generalized scattering matrices in combination with the method of factorization. The approach used refines values of the limiting aperture efficiency by 5 to 15% for diameters up to 3.2 wavelength in comparison with the underrated data based on the Kirchhoff method, and reveals the effect of the sharp worsening of the hard horn performance at some values of the horn length caused by resonances of the strip-line mode excited in the loaded section. Beyond the resonance points, the aperture efficiency remains relatively high (from 80 to 85%), and the crosspolar level is a few decibels lower than that of a standard smooth-walled horn.

Study of the Electromagnetic Scattering from the Very Rough Fractal Sea Surface**The project supported by National Natural Science Foundation of China under Grant No. 60101001 and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China

In this paper, based on the fundamental formulae of the first-order and second-order Kirchhoff approximation and with consideration of the shadowing effect, the backscattering enhancement of the one-dimensional very rough fractal sea surface with Pierson-Moskowitz spectrum is studied under the second-order Kirchhoff approximation at microwave frequency. The numerical results are compared with those of the first-order Kirchhoff approximation and integral equation method. The dependencies of the bistatic scattering cross section and the backscattering enhancement on the incident angle, fractal dimension, and windspeed over the sea surface are analyzed in detail.

Surface Integral Methods in Computational Aeroacoustics—From the (CFD) Near-Field to the (Acoustic) Far-Field

A review of recent advances in the use of surface integral methods in Computational AeroAcoustics (CAA) for the extension of near-field CFD results to the acoustic far-field is given. These integral formulations (i.e. Kirchhoff's method, permeable (porous) surface Ffowcs-Williams Hawkings (FW-H) equation) allow the radiating sound to be evaluated based on quantities on an arbitrary control surface if the wave equation is assumed outside. Thus only surface integrals are needed for the calculation of the far-field sound, instead of the volume integrals required by the traditional acoustic analogy method (i.e. Lighthill, rigid body FW-H equation). A numerical CFD method is used for the evaluation of the flow-field solution in the near field and thus on the control surface. Diffusion and dispersion errors associated with wave propagation in the far-field are avoided. The surface integrals and the first derivatives needed can be easily evaluated from the near-field CFD data. Both methods can be extended in order to include refraction effects outside the control surface. The methods have been applied to helicopter noise, jet noise, propeller noise, ducted fan noise, etc. A simple set of portable Kirchhoff/FW-H subroutines can be developed to calculate the far-field noise from inputs supplied by any aerodynamic near/mid-field CFD code.

Velocity model building by wavefield-continuation imaging in the deepwater Gulf of Mexico

Wavefield-continuation-based migration algorithms that downward extrapolate the 3D prestack wavefield (commonly known as “wave-equation migration”) have been recently shown to produce better imaging results than Kirchhoff migration in many synthetic and real data cases (Popovici, 2000). Wavefield-continuation methods are potentially more accurate and robust because they are based on the full wave equation and not on an asymptotic solution based on ray theory. In addition, wavefield-continuation methods handle multipathing naturally in contrast to Kirchhoff methods, focusing and defocusing effects of velocity variations are correctly modeled, antialiasing is handled implicitly, and amplitudes are consistent with the wave equation.

Finite difference 3D common-shot migration of data from the West Delta area of the Gulf of Mexico

The advent of the network, or “cluster,” of modern personal computers has enabled prestack 3D “wave equation” methods, such as common-shot migration, to become practical alternatives to standard Kirchhoff depth migration. Clusters have overcome hardware limitations (cost-effective memory and speed) that previously limited the widespread use of these wave equation algorithms. Prestack Kirchhoff migration is efficient and easily adapts to various data acquisition geometries, while its steep-dip and high-resolution imaging capabilities are unparalleled. Traveltime calculations, typically by ray tracing, are fundamental to conventional Kirchhoff migration. However, propagating wavefronts become complicated when traveling through complex geology; for example, a simple spherical wave propagating near its source eventually separates into several (or many) distinct “branches” as the wave encounters large lateral velocity variations. Computing 3D traveltime fields that faithfully represent such wave propagation and actually using these in 3D Kirchhoff migration is difficult and computationally expensive. Conventional Kirchhoff migration instead uses simplified, single-valued traveltime fields produced by criteria such as first arrival or maximum energy. Where several raypaths encounter a grid node, for example, the maximum-energy criterion selects the traveltime from the raypath with largest amplitude. The single-valued traveltime approximation may be poor in complex geology, where the traveltime fields really are multibranched or multivalued. Prestack wave equation imaging, on the other hand, such as finite difference common-shot migration, uses numerical modeling of the one-way wave equation instead of ray tracing. These wave equation methods can be more accurate than ray methods for modeling wave propagation in complex media. The result can be a superior image in areas of complex geology, such as beneath salt bodies. The most serious limitation of these one-way wave equation methods is that they are dip-limited relative to Kirchhoff methods, especially for strong lateral velocity variations. Most of today's prestack wave equation methods use accurate “dual-domain” techniques …

Seismic data reconstruction by inversion to common offset

Preserving amplitude information when imaging 3‐D multichannel seismic data is a challenging task. The imaging process can be described as seeking the inverse for a large matrix that relates the data to a reflectivity model. Due to irregular coverage at the surface and uneven illumination of the subsurface, the matrix is often ill‐conditioned and its coefficients are badly scaled. We present a new approach for regularizing the coverage of 3‐D surveys by partial imaging before migration. After regularization, 3‐D data become better suited for prestack migration using either “true amplitude” Kirchhoff methods or wave extrapolation techniques.Posing partial stacking before migration as an inverse problem, we develop a new optimization technique named “inversion to common offset” (ICO) that solves for optimal partial stacks from multifold 3‐D data. The inversion approach takes advantage of the redundancy of information present in multichannel recording to recover densely and regularly sampled common‐offset common‐azimuth volumes. The reconstruction of each of these volumes combines data from multiple incomplete experiments over a common earth model to simulate the ideal data that would have been recorded from a single common‐offset common‐azimuth experiment. The method is based on the inversion of a partial prestack migration operator that maps seismic data from a given acquisition geometry to an arbitrary one. This data mapping operator is the azimuth moveout operator (AMO). The main advantage of ICO is that the AMO operator is compact and consequently cheaper to apply than other imaging operators, such as full 3‐D prestack migration.We present a cost‐effective implementation of ICO based on a log‐stretch transformation of the time axis. After this transformation, AMO becomes time invariant, so the inversion can be performed in the Fourier domain on each temporal‐frequency slice independently. To accelerate the convergence of the iterative solution, we precondition the system of linear equations by two diagonal transformations that scale the rows and columns of the modeling operator. A stable solution is achieved by adding a penalty function that constrains the spatial variability of the model.Results of applying ICO to a field 3‐D land survey show that we can recover the high‐frequency features of the reflectivity function from a severely undersampled data set obtained by decimating a densely sampled prestack data set. The images produced by prestack migration after regularization are superior to those produced by direct migration of the decimated data.

Dipole-wave theory of electromagnetic diffraction

A theory of diffraction is presented which systematically employs the wave approach used in the Kirchhoff method and, unlike the Kirchhoff integral, does not have severe applicability limitations. The diffraction problem is solved by using the Hertz vector instead of the field vector used in the Kirchhoff integral. The basic diffraction problems for linearly, radially, and azimuthally polarized radiation are solved analytically. The key qualitative feature of the solutions is the presence of 'poles' — zero field points within the diffraction pattern of light and dark fringes. The poles lie along the direction of the electric field vector. The solutions obtained satisfy the Maxwell equations and the reciprocity principle.

Series approximation of the Kirchhoff method for wavefield computation on a rough subsurface

ABSTRACT Computation of the wavefield due to reflection from an irregular surface is carried out for subsurfaces with large radii of curvature. The Kirchhoff approximation is proved to be sufficiently accurate provided that the acoustic wavelength is sufficiently small with respect to the asperities of the rough surface. For cases where the irregular surface does not fulfil this condition, a series solution is proposed. The first term of this series appears to be the result obtained by conventional Kirchhoff approximation. The series, initially developed in the space–wavenumber domain by Meecham, is transformed into the space–time domain, and the general expression for the series is obtained by calculation of the normal derivative of the field function. The series solution, restricted to the first two terms, is illustrated by application to three synthetic examples. Applications show that the series approximation obtained by the Kirchhoff method contributes significantly to the modelling of narrow, steep and deep structures and consequently it appears that the second term in the series cannot be ignored in the computation of the wavefields arising from a rough surface.

Far-Field Calculations for Turbofan Noise

The development of variable-order mapped infinite wave envelope elements for finite element modeling of acoustic radiation from turbofan nacelles is reviewed. These elements can be used outside of the near field as a nonreflecting boundary condition and as a finite-dimensional discretization of the infinite exterior domain for calculation of far-field acoustic radiation. The Kirchhoff method previously has been investigated in the context of the turbofan inlet radiation problem to extend a near-field solution to the far field. When the results of the mapped infinite wave envelope element analysis are compared with results of the Kirchhoff method, it is found that the mapped element formulation efficiently provides the acoustic far field, obviating the need to carry out an additional Kirchhoff integral formulation. Relative efficiencies are considered, and it is found that finite-dimensional discretization of the far field using mapped infinite elements is superior to the Kirchhoff formulation when radiation directivity in the entire far field is required.

Efficient prediction methods for the micro-pressure wave from a high-speed train entering a tunnel using the Kirchhoff formulation.

The compression wave generated by a high-speed train emerging from the exit portal of a tunnel gives rise to an impulsive noise called a micro-pressure wave. In this study, new methods for the prediction of sonic-boom noise are proposed. The first method combines acoustic monopole analysis and the method of characteristics with the Kirchhoff method. The compression wave from a train entering a tunnel is calculated by an approximate compact Green's function, and the resultant noise at the tunnel exit is predicted by a linear Kirchhoff formulation. The second method couples the Kirchhoff formulation with the Euler equation, which is solved numerically for the generation and propagation of the compression wave. Numerical prediction of the compression wave, the propagation in the tunnel, and the micro-pressure wave obtained by the present methods are compared with measured data. The numerical results exhibit a reasonable agreement with the experimental data. The proposed methods in this study are shown to be very useful design tools for the nose shape of trains and the geometry of tunnels, and they can be utilized to minimize the pressure fluctuation in the tunnel and the corresponding booming noise.

Large Eddy Simulations of Sound Radiation from Subsonic Turbulent Jets

Large eddy simulations (LES) of subsonic turbulent jets, including the near-field flow and far-field sound radiation, were performed using two different subgrid-scale turbulence models. Two different cases were considered. As a validation case, predictions from a Mach 0.9, Reynolds number 3.6 × 10 3 randomly forced turbulent jet were performed. Both the near-field velocity statistics and far-field sound directivity were found to be in excellent agreement with previous experimental data and direct numerical simulation (DNS) results. The second case involved a Mach 0.4, Reynolds number 5 x 10 3 harmonically forced jet. Both axisymmetric and azimuthal inlet disturbances were imposed to facilitate detailed comparisons to recent DNS of axisymmetric jet sound, and to highlight the effects of small-scale turbulence. As part of this second case, the validity of Kirchhoff's method for far-field sound prediction was also assessed in conjunction with LES and the effect of the subgrid-scale turbulence model on sound radiation was examined. The sound source location and levels were different between the three-dimensional turbulent jets and similar two-dimensional axisymmetric laminar jets. Far-field sound radiation predictions, obtained using Kirchhoff's method, were in good agreement with the directly predicted LES results. LES predictions obtained with the dynamic Smagorinsky and the dynamic mixed subgrid-scale turbulence models were similar, although the mixed model resulted in higher turbulence and sound levels. The directivity was not uniform as predicted by Lighthill's theory for a fully turbulent jet, but exhibited a preferred radiation angle between 35 and 70 deg in agreement with reported experimental data from an acoustically excited jet.

Large eddy simulations of sound radiation from subsonic turbulent jets

Large eddy simulations (LES) of subsonic turbulent jets, including the near-field flow and far-field sound radiation, were performed using two different subgrid-scale turbulence models. Two different cases were considered. As a validation case, predictions from a Mach 0.9, Reynolds number 3.6 × 10 3 randomly forced turbulent jet were performed. Both the near-field velocity statistics and far-field sound directivity were found to be in excellent agreement with previous experimental data and direct numerical simulation (DNS) results. The second case involved a Mach 0.4, Reynolds number 5 x 10 3 harmonically forced jet. Both axisymmetric and azimuthal inlet disturbances were imposed to facilitate detailed comparisons to recent DNS of axisymmetric jet sound, and to highlight the effects of small-scale turbulence. As part of this second case, the validity of Kirchhoff's method for far-field sound prediction was also assessed in conjunction with LES and the effect of the subgrid-scale turbulence model on sound radiation was examined. The sound source location and levels were different between the three-dimensional turbulent jets and similar two-dimensional axisymmetric laminar jets. Far-field sound radiation predictions, obtained using Kirchhoff's method, were in good agreement with the directly predicted LES results. LES predictions obtained with the dynamic Smagorinsky and the dynamic mixed subgrid-scale turbulence models were similar, although the mixed model resulted in higher turbulence and sound levels. The directivity was not uniform as predicted by Lighthill's theory for a fully turbulent jet, but exhibited a preferred radiation angle between 35 and 70 deg in agreement with reported experimental data from an acoustically excited jet.

ACOUSTIC RADIATION FROM ROTATING BLADES—THE KIRCHHOFF METHOD IN AEROACOUSTICS

This paper reviews the current status of discrete frequency noise prediction for rotating blade machinery in the time domain. There are two major approaches both of which can be classified as the Kirchhoff method. These methods depend on the solution of two linear wave equations called the K and FW–H equations. The solutions of these equations for subsonic and supersonic surfaces are discussed and some important results of the research in the past years are presented. This paper is analytical in nature and emphasizes the work of the author and co-workers at NASA Langley Research Center.