Cancellation of spurious arrivals in Green's function retrieval of multiple scattered waves

The Green's function for wave propagation can be extracted by cross-correlating field fluctuations excited on a closed surface that surrounds the employed receivers. This study treats an acoustic multiple scattering medium with discrete scatterers and shows that for a given source the cross-correlation of waves propagating along most combinations of scattering paths gives unphysical arrivals. Because theory predicts that the true Green's function is retrieved, such unphysical arrivals must cancel after integration over all sources. This cancellation occurs because the scattering amplitude of each scatterer satisfies the generalized optical theorem. The cross-correlation of scattered waves with themselves does not lead to the correct retrieval of scattered waves, because the cross-terms between the direct and scattered waves is essential.     

Small-scale seismic inversion using surface waves extracted from noise cross correlation

Green's functions can be retrieved between receivers from the correlation of ambient seismic noise or with an appropriate set of randomly distributed sources. This principle is demonstrated in small-scale geophysics using noise sources generated by human steps during a 10-min walk in the alignment of a 14-m-long accelerometer line array. The time-domain correlation of the records yields two surface wave modes extracted from the Green's function between each pair of accelerometers. A frequency-wave-number Fourier analysis yields each mode contribution and their dispersion curve. These dispersion curves are then inverted to provide the one-dimensional shear velocity of the near surface.     

A representation for Green's function retrieval by multidimensional deconvolution

Green's function retrieval by crosscorrelation may suffer from irregularities in the source distribution, asymmetric illumination, intrinsic losses, etc. Multidimensional deconvolution (MDD) may overcome these limitations. A unified representation for Green's function retrieval by MDD is proposed. From this representation, it follows that the traditional crosscorrelation method gives a Green's function of which the source is smeared in space and time. This smearing is quantified by a space-time point-spread function (PSF), which can be retrieved from measurements at an array of receivers. MDD removes this PSF and thus deblurs and deghosts the source of the Green's function obtained by correlation.     

Retrieving the elastodynamic Green's function of an arbitrary inhomogeneous medium by cross correlation

A correlation-type reciprocity theorem is used to show that the elastodynamic Green's function of any inhomogeneous medium (random or deterministic) can be retrieved from the cross correlation of two recordings of a wave field at different receiver locations at the free surface. Unlike in other derivations, which apply to diffuse wave fields in random media or irregular finite bodies, no assumptions are made about the diffusivity of the wave field. In a second version, it is assumed that the wave field is diffuse due to many uncorrelated sources inside the medium.     

Scalar Green's functions in an Euclidean space with a conical-type line singularity

In an Euclidean space with a conical-type line singularity, we determine the Green's function for a charged massive scalar field interacting with a magnetic flux running through the line singularity. We give an integral expression of the Green's function and a local form in the neighbourhood of the point source, where it is the sum of the usual Green's function in Euclidean space and a regular term. As an application, we derive the vacuum energy-momentum tensor in the massless case for an arbitrary magnetic flux.Supported by a grant from CNPq (Brazilian government agency FA)     

利用环境噪声互相关实现散射体无源成像

最近研究表明利用环境噪声的互相关可以恢复两点之间的时域格林函数(声脉冲响应),这一原理在文献中被称为格林函数恢复。基于此原理,通过对多个传声器所接收的环境噪声进行互相关处理,获取与散射体相关联的散射波的到达时延信息,结合最小二乘反演算法和改进克希霍夫移位算法,分别获得道路交通噪声场中石柱以及海浪噪声场中塑料桶的空间位置,且其估计结果与实际测量相一致。实验结果表明将环境噪声作为探测信号进行散射体无源成像是可行的。这为设计室内无源声监测系统以及通过海洋环境噪声实现对水中静默目标成像提供了新思路和有益参考。      

Green's functions for a volume source in an elastic half-space

Green's functions are derived for elastic waves generated by a volume source in a homogeneous isotropic half-space. The context is sources at shallow burial depths, for which surface (Rayleigh) and bulk waves, both longitudinal and transverse, can be generated with comparable magnitudes. Two approaches are followed. First, the Green's function is expanded with respect to eigenmodes that correspond to Rayleigh waves. While bulk waves are thus ignored, this approximation is valid on the surface far from the source, where the Rayleigh wave modes dominate. The second approach employs an angular spectrum that accounts for the bulk waves and yields a solution that may be separated into two terms. One is associated with bulk waves, the other with Rayleigh waves. The latter is proved to be identical to the Green's function obtained following the first approach. The Green's function obtained via angular spectrum decomposition is analyzed numerically in the time domain for different burial depths and distances to the receiver, and for parameters relevant to seismo-acoustic detection of land mines and other buried objects.     

Fluctuations in diffuse field-field correlations and the emergence of the Green's function in open systems

Recent intense interest in diffuse field correlation functions, with applications to passive imaging in underwater acoustics and seismology, has raised questions about the degree with which a retrieved waveform can be expected to conform to the Green's function, and in particular the degree with which a ray arrival may be discerned. On considering a simple scalar wave model consisting of fields with distributed random sources, the difffuse field-field correlation function R is defined as a sum of correlation integrals, one for each of the many distinct distributed sources. It is then shown that this ensemble of fields has a correlation function with expectation (R) equal to the Green's function. This model also lends itself to calculations of the variance of R, and thus to estimates of the degree to which an R calculated using finite amounts of data will conform to the Green's function. The model predicts that such conformation is strongest at low frequencies. Ray arrivals are detectable if sufficient data have been collected, but the amount of data needed scales in three dimensions with the square of the source-receiver separation, and the square of the frequency. Applications to seismology are discussed.     

The complex equivalent source method for sound propagation over an impedance plane

The sound field caused by a monopole source above an impedance plane can be calculated by using a superposition of equivalent point sources located along a line in the mirror space below the plane. Originally, such an approach for representing the half-space Green's function was described by Sommerfeld at the beginning of the last century, in order to treat half-space problems of heat conduction. However, the representation converges only for masslike impedances and cannot be used for the more important case of reflecting planes with springlike surface impedances. The singular part of the line integral can be transformed into a Hankel function, which shows that surface waves are contained in the whole solution. Unfortunately, this representation suffers from the lack of validity at certain receiver points and from restrictions on wave number and impedance range to ensure the necessary convergence. The main idea of the present method is to use also a superposition of equivalent point sources, but to allow that these sources can be located at complex source points. The corresponding form of the half-space Green's function is suitable for both masslike and springlike surface impedances, and can be used as a cornerstone for a boundary element method.     

Acoustical cavity excitation.

A tutorial exposition is given of cavity excitations and it is shown that relatively simple analytical solutions for the excitation of acoustic fields in cavities with lossy walls can be obtained for a broad class of such cavities. The standard analytic procedure whereby one expresses the acoustic field as a sum over the eigenmodes of the cavity is reviewed and its limitations are pointed out. An alternate procedure is to seek to make use of Green's functions depending on a single coordinate and to not express such Green's functions as sums over eigenfunctions. Instead, one expresses the Green's function as a closed form function that has a discontinuous slope at the value of the coordinate corresponding to the source location. Although the class of cavities for which this approach is possible is limited, it is sufficiently broad to be of intrinsic interest and the results may be helpful to those seeking benchmark solutions and analytic insight. Displayed examples are for cavities whose shapes are spheres and cylinders, with interior monopole, dipole, line, and distributed sources. The formulation yields simple expressions for the total power absorbed by the cavity walls.     

Extended discrete-ordinate method considering full polarization state

This paper presents an extension to the standard discrete-ordinate method (DOM) to consider generalized sources including: beam sources which can be placed at any (vertical) position and illuminate in any direction, thermal emission from the atmosphere and angularly distributed sources which illuminate from a surface as continuous functions of zenith and azimuth angles. As special cases, the thermal emission from the surface and deep space can be implemented as angularly distributed sources. Analytical-particular solutions for all source types are derived using the infinite medium Green's function. Radiation field zenith angle interpolation using source function integration is developed for all source types. The development considers the full state of polarization, including the sources (as applicable) and the (BRDF) surface, but the development can be reduced easily to scalar problems and is ready to be implemented in a single set of code for both scalar and vector radiative transfer computation.     

Closed form solutions for the acoustical impulse response over a masslike or an absorbing plane

The transient sound field caused by a Dirac delta impulse function above an infinite locally reacting plane can be calculated by applying the inverse Fourier transform of the corresponding half-space Green's function in frequency domain. As a starting point, the representation given by Ochmann [J. Acoust. Soc. Am. 116(6), 3304-3311 (2004)] is used, which consists of discrete and continuous superposition of point sources. For a locally reacting plane with masslike character and also with pure absorbing behavior, it is possible to express the resulting impulse response in closed form. Such a result is surprising, because corresponding formulations in the frequency domain are not available yet. Hence, the first main result is the closed form solution Eq. (22) for an impulse response over an infinite plane with a pure imaginary impedance. The second main result is the closed form solution Eq. (53) for an impulse response over an infinite plane with a pure real impedance. As a particular application of both main results, a convolution technique is used for deriving formulas for point sources with a general time dependency. For special signals like an exponentially decaying time signal or a triangular shaped impulse, the resulting sound field can be presented in terms of elementary functions.     

Derivation of a Closed Expression of the B-S Interaction Kernel for Quark-Antiquark Bound States

The interaction kernel in the Bethe-Salpeter (B-S) equation for quark-antiquark bound states is derived from B-S equations satisfied by the quark-antiquark four-point Green's function. The latter equations are established based on the equations of motion obeyed by the quark and antiquark propagators, the four-point Green's function and some other kinds of Green's functions, which follow directly from the QCD generating functional. The derived B-S kernel is given by a closed and explicit expression which contains only a few types of Green's functions. This expression is not only convenient for perturbative calculations, but also applicable for nonperturbative investigations. Since the kernel contains all the interactions taking place in the quark-antiquark bound states, it actually appears to be the most suitable starting point of studying the QCD nonperturbative effect and quark confinement.     

Derivation of a Closed Expression of the B-S Interaction Kernel for Quark-Antiquark Bound States

The interaction kernel in the Bethe-Salpeter (B-S) equation for quark-antiquark bound states is derivedfrom B-S equations satisfied by the quark-antiquark four-point Green‘s function. The latter equations are establishedbased on the equations of motion obeyed by the quark and antiquark propagators, the four-point Green‘s function andsome other kinds of Green‘s functions, which follow directly from the QCD generating functional. The derived B-Skernel is given by a closed and explicit expression which contains only a few types of Green‘s functions. This expressionis not only convenient for perturbative calculations, but also applicable for nonperturbative investigations. Since thekernel contains all the interactions taking place in the quark-antiquark bound states, it actually appears to be the mostsuitable starting point of studying the QCD nonperturbative effect and quark confinement.     

Adiabatic charge and spin pumping through interacting quantum dots

In this paper we investigate adiabatic charge and spin pumping through interacting quantum dots using non-equilibrium Green's function techniques and the equation-of-motion method. We treat the electronic correlations inside the dot using a Hartree-Fock approximation and succeed in obtaining closed analytic expressions for the Keldysh Green's functions. These allow us to compute charge and spin currents through the quantum dot. Depending on the parameters of the quantum dot and its coupling to the reservoirs, we show that it can be found in two different regimes: the magnetic regime and the non-magnetic regime. In the magnetic regime we find a non-vanishing spin current in addition to the charge current present in both cases.     

Green's function of radial inhomogeneous spheres excited by internal sources

Green's function in the interior of penetrable bodies with inhomogeneous compressibility by sources placed inside them is evaluated through a Schwinger-Lippmann volume integral equation. In the case of a radial inhomogeneous sphere, the radial part of the unknown Green's function can be expanded in a double Dini's series, which allows analytical evaluation of the involved cumbersome integrals. The simple case treated here can be extended to more difficult situations involving inhomogeneous density as well as to the corresponding electromagnetic or elastic problem. Finally, numerical results are given for various inhomogeneous compressibility distributions.     

Green's functions for Maxwell's equations: application to spontaneous emission

We present a new formalism for calculating the Green's function for Maxwell's equations. As our aim is to apply our formalism to light scattering at surfaces of arbitrary materials, we derive the Green's function in a surface representation. The only requirement on the material is that it should have periodicity parallel to the surface. We calculate this Green's function for light of a specific frequency and a specific incident direction and distance with respect to the surface. The material properties entering the Green's function are the reflection coefficients for plane waves at the surface. Using the close relationship between the Green's function and the density of states (DOS), we apply our method to calculate the spontaneous emission rate as a function of the distance to a material surface. The spontaneous emission rate can be calculated using Fermi's Golden Rule, which can be expressed in terms of the DOS of the optical modes available to the emitted photon. We present calculations for a finite slab of cylindrical rods, embedded in air on a square lattice. It is shown that the enhancement or suppression of spontaneous emission strongly depends on the frequency of the light. This revised version was published online in November 2006 with corrections to the Cover Date.     

Vector Green's function algorithm for radiative transfer in plane-parallel atmosphere

Green's function is a widely used approach for boundary value problems. In problems related to radiative transfer, Green's function has been found to be useful in land, ocean and atmosphere remote sensing. It is also a key element in higher order perturbation theory. This paper presents an explicit expression of the Green's function, in terms of the source and radiation field variables, for a plane-parallel atmosphere with either vacuum boundaries or a reflecting (BRDF) surface. Full polarization state is considered but the algorithm has been developed in such way that it can be easily reduced to solve scalar radiative transfer problems, which makes it possible to implement a single set of code for computing both the scalar and the vector Green's function.     

格林函数在计算部分电容中的应用

在工业上和在实验室中,我们都会遇到两个物体之间的电磁屏蔽的问题。在许多应用中,我们只须注意到两个物体之间的静电屏蔽就够了,因而它们之间的相互作用可以从计算它们之间的相互电容来定出。当干扰物体的尺寸很小因而可以认为是一个点电源时,则当它与另一个接地导体(即问题中的电磁屏蔽)共同存在时所生的效应即可由这一个接地导体的格林函数表出。关于格林函数的知识已有很丰富的积累,因而本文中所提出的方法是可以解决多种多样的问题的。文中讨论了扁椭球坐标和长椭球坐标中的格林函数,并对带虚数自变数的勒讓特函数的若干个公式作了推导,因为这些有用的公式在流行的文献中还未见到。导体表面任意形状的小孔的问题是值得讨论的,特别是有限大导体表面上的小孔问题,本文从理论上验证了文献上已经提出来的实验结果。最后我们给出如下两个物体之间的相互电容公式:其中一个是在带小孔的闭合电磁屏蔽体的另一个之内。     

Ocean acoustic interferometry

Ocean acoustic interferometry refers to an approach whereby signals recorded from a line of sources are used to infer the Green's function between two receivers. An approximation of the time domain Green's function is obtained by summing, over all source positions (stacking), the cross-correlations between the receivers. Within this paper a stationary phase argument is used to describe the relationship between the stacked cross-correlations from a line of vertical sources, located in the same vertical plane as two receivers, and the Green's function between the receivers. Theory and simulations demonstrate the approach and are in agreement with those of a modal based approach presented by others. Results indicate that the stacked cross-correlations can be directly related to the shaded Green's function, so long as the modal continuum of any sediment layers is negligible.