Homogeneous Green's Function Research Articles

3-D Virtual Seismology

We create virtual sources and receivers in a 3D subsurface using the previously derived single-sided homogeneous Green's function representation. We employ Green's functions and focusing functions that are obtained using reflection data at the Earth's surface, a macro velocity model and the Marchenko method. The homogeneous Green's function is a Green's function superposed with its time-reversal. Unlike the classical homogeneous Green's function representation, our approach requires no receivers on an enclosing boundary, however, it does require the source signal to be symmetric in time. We demonstrate that in 3D, the single-sided representation is an improvement over the classical representation by applying the representations to numerical data. We retrieve responses to virtual point sources with an isotropic and with a double-couple radiation pattern and compare the results to a directly modeled reference result. We also demonstrate the application of the single-sided representation for retrieving the response to a virtual rupture that consists of a superposition of double-couple point sources. This is achieved by obtaining the homogeneous Green's function for each source separately, before they are transformed to the causal Green's function, time-shifted and superposed. The single-sided representation is also used to monitor the complete wavefield that is caused by a numerically modeled rupture. However, the source signal of an actual rupture is not symmetric in time and the single-sided represenation can therefore only be used to obtain the causal Green's function. This approach leaves artifacts in the final result, however, these artifacts are limited in space and time.

Focusing and Green's function retrieval in three-dimensional inverse scattering revisited: A single-sided Marchenko integral for the full wave field

The authors present a framework for Marchenko-type integrals in three-dimensional full wavefield inverse scattering by introducing the homogeneous Green's function of the second kind.

INTEGRAL REPRESENTATION OF SOLUTIONS OF HALF-SPACE HOMOGENEOUS DIRICHLET AND NEUMANN PROBLEMS FOR AN EQUATION OF FOKKER-PLANCK-KOLMOGOROV TYPE OF NORMAL MARKOV PROCESS

Solutions of a homogeneous model equation of the Fokker--Planck--Kolmogorov type of a normal Markov process are consider. They are defined in $\{(t,x_1,\dots,x_n)\in\mathbb{R}^{n+1}|0<t\le T, -\infty<x_j<\infty, j\in\{1,\dots,n-1\}, x_n>0\}$ and for $x_n=0$ satisfy the homogeneous Dirichlet or Neumann conditions and relate to special weighted Lebesgue $L_p$-spaces $L_p^{k(\cdot,a)}$. The representation of such solutions in the form of Poisson integrals is established. The kernels of these integrals are the homogeneous Green's functions of the considered problems, and their densities belong to specially constructed sets $\Phi_p^a$ of functions or generalized measures. The results obtained will be used to describe solutions of the problems from spaces $L_p^{k(\cdot,a)}$. Thus, the well-known Eidelman-Ivasyshen approach will be implemented for the considered problems. According to this approach, if the initial data are taken from the set $\Phi_p^a$, then there is only one solution to the problem from the space $L_p^{k(\cdot,a)}$. It is represented as a Poisson integral. Conversely, for any solution from the space $L_p^{k(\cdot,a)}$ there is only one element $\varphi \in\Phi_p^a$ such that this solution can be represented as a Poisson integral with density $\varphi$. In this case, it becomes clear in what sense the initial condition is satisfied.

Virtual Sources and Receivers in the Real Earth: Considerations for Practical Applications

AbstractTo enhance monitoring of the subsurface, virtual sources and receivers inside the subsurface can be created from seismic reflection data at the surface of the Earth using the Marchenko method. The response between these virtual sources and receivers can be obtained through the use of homogeneous Green's function retrieval. A homogeneous Green's function is a superposition of a Green's function and its time reversal. The main aim of this paper is to obtain accurate homogeneous Green's functions from field data. Classical homogeneous Green's function retrieval requires an unrealistic enclosing recording surface; however, by using a recently proposed single‐sided retrieval scheme, this requirement can be avoided. We first demonstrate the principles of using the single‐sided representation on synthetic data and show that different source signatures can be taken into account. Because the Marchenko method is sensitive to recording limitations of the reflection data, we study five cases of recording limitations with synthetic data and demonstrate their effects on the final result. Finally, the method is demonstrated on a preprocessed field data set that fulfills the requirements for applying the single‐sided Green's function retrieval scheme. The scheme has the potential to be used in future applications, such as source localization.

Green's theorem in seismic imaging across the scales

Abstract. The earthquake seismology and seismic exploration communities have developed a variety of seismic imaging methods for passive- and active-source data. Despite the seemingly different approaches and underlying principles, many of those methods are based in some way or another on Green's theorem. The aim of this paper is to discuss a variety of imaging methods in a systematic way, using a specific form of Green's theorem (the homogeneous Green's function representation) as a common starting point. The imaging methods we cover are time-reversal acoustics, seismic interferometry, back propagation, source–receiver redatuming and imaging by double focusing. We review classical approaches and discuss recent developments that fully account for multiple scattering, using the Marchenko method. We briefly indicate new applications for monitoring and forecasting of responses to induced seismic sources, which are discussed in detail in a companion paper.

A single-sided representation for the homogeneous Green's function of a unified scalar wave equation

A unified scalar wave equation is formulated, which covers three-dimensional (3D) acoustic waves, 2D horizontally-polarised shear waves, 2D transverse-electric EM waves, 2D transverse-magnetic EM waves, 3D quantum-mechanical waves and 2D flexural waves. The homogeneous Green's function of this wave equation is a combination of the causal Green's function and its time-reversal, such that their singularities at the source position cancel each other. A classical representation expresses this homogeneous Green's function as a closed boundary integral. This representation finds applications in holographic imaging, time-reversed wave propagation and Green's function retrieval by cross correlation. The main drawback of the classical representation in those applications is that it requires access to a closed boundary around the medium of interest, whereas in many practical situations the medium can be accessed from one side only. Therefore, a single-sided representation is derived for the homogeneous Green's function of the unified scalar wave equation. Like the classical representation, this single-sided representation fully accounts for multiple scattering. The single-sided representation has the same applications as the classical representation, but unlike the classical representation it is applicable in situations where the medium of interest is accessible from one side only.

A single-sided representation for passive and active Green's function retrieval, time-reversal acoustics, and holographic imaging

The homogeneous Green's function, defined as the superposition of the Green's function and its time-reversal, plays an important role in a variety of acoustic applications, such as passive and active acoustic Green's function retrieval, seismic interferometry, time-reversal acoustics, and holographic imaging. An exact representation of the homogeneous Green's function originates from the field of optical holographic imaging (Porter, 1970, JOSA). In this representation, the homogeneous Green's function between two points A and B is expressed as an integral along an arbitrary boundary enclosing A and B. This implies that the Green's function between A and B can be retrieved from measurements carried out at a closed boundary, or, via reciprocity, from passive observations at A and B of the responses to sources on a closed boundary. In practical situations, the closed-boundary integral usually needs to be approximated by an open-boundary integral. This can lead to significant artifacts in the retrieved Green's function. I will discuss a new, single-sided, representation of the homogeneous Green's function, which obviates the need for omnidirectional access. Like the classical closed-boundary representation, this new single-sided representation fully accounts for multiple scattering. I will indicate applications of this new representation in the aforementioned fields.

A single-sided homogeneous Green's function representation for holographic imaging, inverse scattering, time-reversal acoustics and interferometric Green's function retrieval

Abstract Green's theorem plays a fundamental role in a diverse range of wavefield imaging applications, such as holographic imaging, inverse scattering, time-reversal acoustics and interferometric Green's function retrieval. In many of those applications, the homogeneous Green's function (i.e. the Green's function of the wave equation without a singularity on the right-hand side) is represented by a closed boundary integral. In practical applications, sources and/or receivers are usually present only on an open surface, which implies that a significant part of the closed boundary integral is by necessity ignored. Here we derive a homogeneous Green's function representation for the common situation that sources and/or receivers are present on an open surface only. We modify the integrand in such a way that it vanishes on the part of the boundary where no sources and receivers are present. As a consequence, the remaining integral along the open surface is an accurate single-sided representation of the homogeneous Green's function. This single-sided representation accounts for all orders of multiple scattering. The new representation significantly improves the aforementioned wavefield imaging applications, particularly in situations where the first-order scattering approximation breaks down.

Finite-difference time-domain technique as an efficient tool for calculating the regularized Green function: applications to the local-field problem in quantum optics for inhomogeneous lossy materials

The calculation of the local density of states (LDOS) in lossy materials has long been disputed due to the divergence of the homogeneous Green function with equal space arguments. For arbitrary-shaped lossy structures, such as those of interest in nanoplasmonics, this problem is particularly challenging. A nondivergent LDOS obtained in numerical methods such as the finite-difference time-domain (FDTD) technique, at first sight appears to be wrong. Here we show that FDTD is not only an ideal choice for obtaining the regularized LDOS, but it can address the local-field problem for any lossy inhomogeneous material. We exemplify the case of a finite-size photon emitter (e.g., a single quantum dot) embedded within and outside a lossy metal nanoparticle and show excellent agreement with analytical results.

Application of Kummer's Transformation to the Efficient Computation of the 3-D Green's Function With 1-D Periodicity

The 3-D homogeneous Green's function with 1-D periodicity is commonly expressed as spatial and spectral infinite series that may show very slow convergence. In this work Kummer's transformation is applied to the spatial series in order to accelerate its convergence. By retaining a sufficiently large number of asymptotic terms in the application of Kummer's transformation, the spatial series is split into a set of series which can be accurately obtained with very low computational effort. The numerical results obtained show that, when the number of asymptotic terms retained in Kummer's transformation is large enough, the convergence acceleration method proposed in this work is always faster than existing acceleration methods such as the spectral Kummer-Poisson's method and Ewald's method.

An Efficient Approach for the Computation of 2-D Green's Functions With 1-D and 2-D Periodicities in Homogeneous Media

This paper presents an algorithm for the acceleration of the series involved in the computation of 2-D homogeneous Green's functions with 1-D and 2-D periodicities. The algorithm is based on an original implementation of the spectral Kummer-Poisson's method, and it can be applied to the efficient computation of a wide class of infinite series. In the algorithm the number of asymptotic terms retained in Kummer's transformation is externally controlled so that any of the series that has to be accelerated is split into one series with exponential convergence and another series with algebraic convergence of arbitrarily large order. Numerical simulations have shown that there is an ldquooptimumrdquo number of asymptotic terms retained in Kummer's transformation for which the CPU time needed in the summation of the series is minimized. The CPU times required by Ewald's method for the evaluation of 2-D Green's functions with 1-D and 2-D periodicities have been compared with those required by the present algorithm, and the algorithm has been found to be between 1.2 and 3 times faster than Ewald's method when working in ldquooptimumrdquo operation conditions.

Steady flows driven by sources of random strength in heterogeneous aquifers with application to partially penetrating wells

Average steady source flow in heterogeneous porous formations is modelled by regarding the hydraulic conductivity K(x) as a stationary random space function (RSF). As a consequence, the flow variables become RSFs as well, and we are interested into calculating their moments. This problem has been intensively studied in the case of a Neumann type boundary condition at the source. However, there are many applications (such as well-type flows) for which the required boundary condition is that of Dirichlet. In order to fulfill such a requirement the strength of the source must be proportional to K(x), and therefore the source itself results a RSF. To solve flows driven by sources whose strength is spatially variable, we have used a perturbation procedure similar to that developed by Indelman and Abramovich (Water Resour Res 30:3385–3393, 1994) to analyze flows generated by sources of deterministic strength. Due to the linearity of the mathematical problem, we have focused on the explicit derivation of the mean head distribution G d (x) generated by a unit pulse. Such a distribution represents the fundamental solution to the average flow equations, and it is termed as mean Green function. The function G d (x) is derived here at the second order of approximation in the variance σ2 of the fluctuation $$\varepsilon \left({{\mathbf{x}}}\right) = 1- \frac{K{\left({{\mathbf{x}}}\right)}}{K_{A}}$$ (where K A is the mean value of K(x)), for arbitrary correlation function ρ(x), and any dimensionality d of the flow domain. We represent G d (x) as product between the homogeneous Green function G (0) (x) valid in a domain with constant K A , and a distortion term Ψ d (x) = 1 + σ2ψ d (x) which modifies G (0) (x) to account for the medium heterogeneity. In the case of isotropic formations ψ d (x) is expressed via one quadrature. This quadrature can be analytically calculated after adopting specific (e.g.. exponential and Gaussian) shape for ρ(x). These general results are subsequently used to investigate flow toward a partially-penetrating well in a semi-infinite domain. Indeed, we construct a σ2-order approximation to the mean as well as variance of the head by replacing the well with a singular segment. It is shown how the well-length combined with the medium heterogeneity affects the head distribution. We have introduced the concept of equivalent conductivity K eq(r,z). The main result is the relationship $$\frac{K^{\rm eq}\left(r,z\right)} {K_{A}} = 1-\sigma^{2}\psi^{\left(w\right)}\left(r,z\right)$$ where the characteristic function ψ(w)(r,z) adjusts the homogeneous conductivity K A to account for the impact of the heterogeneity. In this way, a procedure can be developed to identify the aquifer hydraulic properties by means of field-scale head measurements. Finally, in the case of a fully penetrating well we have expressed the equivalent conductivity in analytical form, and we have shown that $$K^{({\rm efu})}\leq K^{\rm eq}\left(r\right) \leq K_{A}$$ (being $$K^{({\rm efu})}$$ the effective conductivity for mean uniform flow), in agreement with the numerical simulations of Firmani et al. (Water Resour Res 42:W03422, 2006).

Hilbert space inverse wave imaging in a planar multilayer environment

Most diffraction tomography (DT) algorithms use a homogeneous Green function (GF) regardless of the medium being imaged. This choice is usually motivated by practical considerations: analytic inversions in standard geometries (Cartesian, spherical, etc.) are significantly simplified by the use of a homogeneous GF, estimating a nonhomogeneous GF can be very difficult, as can incorporating a nonhomogeneous GF into standard DT algorithms. Devaney has circumvented these issues by developing a purely numerical DT inversion algorithm [A. J. Devaney and M. Dennison, Inverse Probl. 19, 855-870 (2003)] that is independent of measurement system geometry, number of frequencies used in the reconstruction, and GF. A planar multilayer GF has been developed for use in Devaney's "Hilbert space" algorithm and used in a proof-of-principle nondestructive evaluation (NDE) experiment to image noninvasively a flaw in an aluminum/copper planar multilayer medium using data collected from an ultrasonic measurement system. The data were collected in a multistatic method with no beamforming: all focusing through the multilayer was performed mathematically "after-the-fact," that is, after the data were collected.

Fractional diffusion and fractional heat equation

This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is formulated in terms of the Fourier transform of its spatially and temporally homogeneous Green function. The spectral density of the resulting solution is then obtained explicitly. The result implies that the solution of the fractional heat equation may possess spatial long-range dependence asymptotically.

Fractional diffusion and fractional heat equation

This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is formulated in terms of the Fourier transform of its spatially and temporally homogeneous Green function. The spectral density of the resulting solution is then obtained explicitly. The result implies that the solution of the fractional heat equation may possess spatial long-range dependence asymptotically.

A fast algorithm for the analysis of radiation and scattering from microstrip arrays on finite substrates

The analysis of radiation and scattering from a multiregion () quasiplanar structure using the conentional method of moments MoM is computationally expensie. To circument this difficulty, a computa- tional scheme that supplements the classical MoM with the steepest descent fast-multipole algorithm is presented in this paper. This proposed methodology is particularly well suited to the analysis of finite microstrip antenna arrays. Q 1999 John Wiley & Sons, Inc. Microwave Opt Technol Lett 23: 306)310, 1999. . ory costs of a naive approach . The SDFMM achieves its efficiency by casting terms in an MoM matrix)vector product in the form of discrete, inhomogeneous plane-wave expan- sions arising from a representation of the homogeneous medium Green's function in terms of a steepest descent-path integral and a two-dimensional fast-multipole expansion. The presented fast iterative solution permits the analysis of scattering and radiation from large and complex structures within realistic CPU times on laboratory workstations. These structures include large microstrip patch arrays on finite substrates and ground planes modeled using 100,000 or more MoM basis functions.

An inverse algorithm for velocity reconstruction

We have devised an inverse algorithm for the reconstruction of underground velocity from seismic waveforms. The algorithm allows for the use of multiple transmitter-receiver pairs and a wide band of frequencies. The background medium velocity can be homogeneous or heterogeneous. The algorithm is an efficient one-step imaging process and requires no matrix inversion. It may provide a good estimate of the undergound velocity as long as the scattering is weak and the scatterer is small relative to the wavelength of the excitation field. The analytical verification of the reconstruction algorithm makes use of a homogeneous Green function which is defined to be the difference between the causal Green function and its acausal counterpart. The homogeneous Green function stands for the backpropagation of the scattered field by a physical source. The inverse algorithm comprises four processing steps: data backpropagation, phase correction, low-pass filtering and imaging. The algorithm is derived for acoustic waves but has the potential to be extended to elastic waves.

Numerical and experimental studies of a helical antenna loaded by a dielectric resonator

A helical antenna loaded by a dielectric resonator (DR) is rigorously studied. Effects due to the dielectric core are analyzed by replacing it with an equivalent polarization current density so that only the homogeneous dyadic Green's function is required. Special interest is paid to the handling of the singularity of the dyadic Green's function in the source region. Two coupled integral equations are obtained by enforcing the boundary condition on the surface of the helical antenna and by equating the total electric field inside the dielectric resonator. The method of moments is used to solve the coupled equations and hence to obtain other properties of the antenna, such as the input impedance, the radiation pattern, the beam width characteristics, and the axial ratio. Experimental measurements were conducted and compared with the theoretical results. It is found that excellent agreement is observed between theoretical values and experimental measurements. Various characteristics of the dielectric resonator loaded antenna are also discussed.

Integral Transport Theory

The one-velocity time-independent neutron integro-differential transport equation is converted into an integral equation by the use of a homogeneous Green's function. The neutron flux, Green's func...

Green's Distributions Associated with the Operator [□m − (−μ2)m]l

The results of a previous paper on homogeneous Green's functions for a multi-dimensional iterated Klein-Gordon operator (□ + μ2)l are extended to include homogeneous Green's functions associated with the operator [□m (− μ2)m]l in multi-dimensional spaces. The Fourier representation of the Green's functions may be expressed, after some angular integrations, as a one-dimensional infinite integral which does not in general converge. Using the concepts of distribution analysis, it is shown how this improper integral can be evaluated directly to get explicit expressions for the Green's functions. The Green's functions themselves must then be interpreted as distributions in the sense of Schwartz. Several distributions instrumental in this treatment are introduced and their properties studied. Explicit expressions for the singularities of the Green's functions on the light cone are presented. The well-known difference between even- and odd-dimensional spaces is reflected in the nature of these singularities. The singularities appearing for odd-dimensional spaces consist of a finite linear combination of derivatives of the Dirac delta function δ(s2), where s is the space-time distance. The highest derivative appearing is of order ½(n−2m−l) with n giving the number of space dimensions. The singular part for even-dimensional spaces consists of a polynomial in 1/s of degree n − 2ml + 1. No singularities appear when the order of the operator is greater than the number of space dimensions. Finally, a complete set of homogeneous Δ-function solutions is given along with their initial conditions at zero time. All of these functions would be needed in obtaining the general solution to the Cauchy problem for the operator considered.