Asymptotic Green's Function Research Articles

An efficient illumination compensation method for reverse time migration

AbstractBy directly solving the full two‐way wave equation, reverse time migration has superiority over other imaging algorithms in handling steeply dipping structures and other complicated geological models. Moreover, by incorporating the asymptotic inversion operator into reverse time migration imaging condition, the imaging algorithm is able to give a quantitative estimation of parameter perturbation in high‐frequency approximation sense. However, because conventional asymptotic inversion only accounts for geometrical spreading, uneven illumination due to irregular acquisition geometry and inhomogeneous subsurface at each image point is neglected. The omit of illumination compensation significantly affects the imaging quality. Wave‐equation‐based illumination compensation methods have been extensively studied in the past. However, the traditional wave‐equation‐based illumination compensation methods usually require high computational cost and huge storage. In this paper, we propose an efficient wave‐equation‐based illumination compensation method. Under high‐frequency approximation, we first define a Jacobian determinant to measure the regularity of subsurface illumination, and then illumination compensation operators are proposed based on the Jacobian. Through boundary integration, we further express the illumination compensation operators through extrapolated wavefields; the explicit computation of asymptotic Green's functions is thus avoided, and an efficient illumination compensation implementation for reverse time migration is achieved. Numerical results with both synthetic and field data validate the effectiveness and efficiency of the presented method.

An asymptotic Green's function method for time-dependent Schrödinger equations with application to Kohn-Sham equations

We present an effective asymptotic Green's function method for solving the time-dependent Schrödinger equation with scalar and vector potentials, and apply it to the Kohn-Sham equations for electronic structure calculation within the time-dependent density functional theory, where the perfectly matched layer approach is incorporated such that the computation can be performed in a bounded domain of physical interest. The method, which extends the approach in Leung et al. (2014) [40], combines the Huygens' principle or Feynman's path integral for propagating the wavefunction and the semi-classical approximations for approximating the retarded Green's function. Once the analytic approximations for the phase and amplitudes of the asymptotic retarded Green's function are obtained through short-time Taylor series expansions, a short-time propagator for the wavefunction is derived, and the resulting integral can be evaluated by fast Fourier transform after appropriate lowrank approximations. Numerical experiments are presented for demonstration.

An asymptotic Green's function method for the wave equation

We present an effective asymptotic Green's function method for propagating the waves through the linear scalar wave equation. The wave is first split into its forward-propagating and backward-propagating parts. Following that, the method, which combines the Huygens' principle and the geometrical optics approximations, is designed to propagate the forward-propagating and backward-propagating waves, where an integral with Green's functions that is based on the Huygens' principle is used to propagate the waves, and the Green's functions are approximated by the geometrical optics approximations. Upon obtaining analytic approximations for the phase and amplitude in the geometrical optics approximations through short-time Taylor series expansions, a short-time propagator for the waves is derived and the resulting integral can be evaluated efficiently by fast Fourier transform after appropriate lowrank approximations. The short-time propagator can be applied repeatedly to propagate the waves for a long time. In order to restrict the computation onto a bounded domain of interest, the perfectly matched layer technique with complex coordinate stretching transformation is incorporated. The method is first-order accurate, and has complexity O(tϵNlog⁡N) per time step with N the number of spatial points and tϵ the low rank for a prescribed accuracy requirement ϵ>0. Numerical experiments are presented to demonstrate the proposed method.

Primal–Dual Optimization Strategy With Total Variation Regularization for Prestack Seismic Image Deblurring

Seismic image, especially for the prestack image, performs a blurred version of the reflectivity image due to spatial aliasing, poor acquisition aperture, and nonuniform illumination. The blurring effects can be quantified by the point spread function (PSF). We herein adopt an explicit space-variant PSF formula, which can be defined as a sequential application of the modeling and migration operators with the asymptotic Green's function. The deblurred images are restored using the nonstationary deconvolution with total variation regularization in which the blurred images are described by the convolution between the space-variant PSF and the reflectivity image. However, nonstationary deconvolution is computationally challenging. We introduce an extending primal-dual hybrid gradient (E-PDHG) method to decompose the complex problem into a sequence of simple subproblems that have closed-form solutions. Numerical results on synthetic data and field data demonstrate that the proposed E-PDHG method outperforms the basic PDHG method in the prestack seismic image deblurring.

Scattering pattern analysis and true‐amplitude generalized Radon transform migration for acoustic transversely isotropic media with a vertical axis of symmetry

ABSTRACTIn multi‐parameter ray‐based anisotropic migration/inversion, it is essential that we have an understanding of the scattering mechanism corresponding to parameter perturbations. Because the complex nonlinearity in the anisotropic inversion problem is intractable, the construction of true‐amplitude linearized migration/inversion procedures is needed and important. By using the acoustic medium assumption for transversely isotropic media with a vertical axis of symmetry and representing the anisotropy with P‐wave normal moveout velocity, Thomsen parameter δ and anelliptic parameter η, we formalize the linearized inverse scattering problem for three‐dimensional pseudo‐acoustic equations. Deploying the single‐scattering approximation and an elliptically anisotropic background introduces a new linear integral operator that connects the discontinuous perturbation parameters with the multi‐shot/multi‐offset P‐wave scattered data. We further apply the high‐frequency asymptotic Green's function and its derivatives to the integral operator, and then the scattering pattern of each perturbation parameter can be explicitly presented. By naturally establishing a connection to generalized Radon transform, the pseudo‐inverse of the integral operator can be solved by the generalized Radon transform inversion. In consideration of the structure of this pseudo‐inverse operator, the computational implementation is done pointwise by shooting a fan of rays from the target imaging area towards the acquisition system. Results from two‐dimensional numerical tests show amplitude‐preserving images with high quality.

Spectra of Collective Excitations and Low-Frequency Asymptotics of Green’s Functions in Uniaxial and Biaxial Ferrimagnetics

The paper studies the dynamic description of uniaxial and biaxial ferrimagnetics with spin s=1/2 in alternative external field. The nonlinear dynamic equations with sources are obtained, on basis on which low-frequency asymptotics of two-time Green functions in the uniaxial and biaxial cases of the ferrimagnet are obtained. Energy models are constructed that are specific functions of Casimir invariants of the algebra of Poisson brackets for magnetic degrees of freedom. On their basis, the question of the stable magnetic states has been solved for the considered systems. These equations were linearized, an explicit form of the collective excitations spectra was found, and their character was analyzed. The article studies the uniaxial case of a ferrimagnet, as well as biaxial cases of an antiferromagnet, easy-axis and easy-plane ferrimagnets. It is shown that for a uniaxial antiferromagnet the spectrum of magnetic excitations has a Goldstone character. For biaxial ferrimagnetic materials, it was found that the spectrum has either a quadratic character or a more complex dependence on the wave vector. It is shown that in the uniaxial case of an antiferromagnet the Green function of the type Gsα,sβ(k,0), Gsα,nβ(k,0) and Gsα,sβ(0,ω) have regular asymptotic behavior, and the Green function of type Gnα,nβ(k,0)≈1/k2 and Gsα,nβ(0,ω)≈1/ω, Gnα,nβ(0,ω)≈1/ω2 have a pole feature in the wave vector and frequency. Biaxial ferrimagnetic states have another type of the features of low-frequency asymptotics of the Green's functions. In the case of a ferrimagnet, the “easy-axis” of the asymptotic behavior of the Green functions Gsα,sβ(0,ω), Gsα,nβ(0,ω), Gnα,nβ(0,ω), Gsα,sβ(k,0), Gsα,nβ(k,0), Gnα,nβ(k,0) have a pole character. For the case of the “easy-plane” type ferrimagnet, the asymptotics of the Green functions Gsα,nβ(0,ω), Gnα,nβ(0,ω), Gsα,nβ(k,0), Gnα,nβ(k,0), have a pole character, and the Green function Gsα,sβ(k,ω) contains both the pole component and the regular part. A comparative analysis of the low-frequency asymptotics of Green functions shows that the nature of magnetic anisotropy significantly effects the structure of low-frequency asymptotics for uniaxial and biaxial cases of ferrimagnet. Separately, we note the non-Bogolyubov character of the Green function asymptotics for ferrimagnet with biaxial anisotropy Gnα,nβ(k,0)≈1/k4.

An optimized space-time Gaussian beam migration method with dynamic parameter control

In the Gaussian beam (GB) method, initial beam parameters are principal factors influencing the accuracy and computational efficiency of seismic depth imaging. Various optimized beam parameter strategies for Gaussian beam migration (GBM) have been proposed to improve imaging quality as well as computational efficiency, while optimized space-time Gaussian beam schemes for seismic migration have still not been fully investigated. In this paper, an optimized space-time Gaussian beam approach with dynamic parameter control for seismic depth imaging is developed. We first provide an expression for dynamic beam parameter by taking in account the effect of velocity field variation on the beam forming. Based on dynamic beam parameters, the new space-time adaptive Gaussian beam generated by an arbitrary source wavelet is obtained, which can adaptively calculate the beam width to make the seismic beam energy better focused in the central ray neighborhood. Then, the forward wavefield is constructed in two-dimensional (2D) acoustic media by space-time adaptive Gaussian beam for the implementation of migration. Adhering to the framework of conventional space-time Gaussian beam method, we perform the up-going ray tracing from subsurface imaging points to the receiver surface to compute the asymptotic Green function for the construction of the backward wavefield. Numerical experiments demonstrate that the new presented approach has a superior accuracy for seismic depth imaging in both shallow and deep regions compared to the conventional space-time Gaussian beam migration scheme.

Modeling methanol transfer in the mesoporous catalyst for the methanol-to-olefins reaction by the time-fractional diffusion equation

The solutions of the time-fractional diffusion equation for the short and long times are obtained via an application of the asymptotic Green's functions. The derived solutions are applied to analysis of the methanol mass transfer through H-ZSM-5/alumina catalyst grain. It is demonstrated that the methanol transport in the catalysts pores may be described by the obtained solutions in a fairly good manner. The measured fractional exponent is equal to 1.20 ± 0.02 and reveals the super-diffusive regime of the methanol mass transfer. The presence of the anomalous transport may be caused by geometrical restrictions and the adsorption process on the internal surface of the catalyst grain's pores.

Internal multiple attenuation by Kirchhoff extrapolation

ABSTRACTSurface‐related multiple elimination is the leading methodology for surface multiple removal. This data‐driven approach can be extended to interbed multiple prediction at the expense of a huge increase of the computational burden. This cost makes model‐driven methods still attractive, especially for the three dimensional case. In this paper we present a methodology that extends Kirchhoff wavefield extrapolation to interbed multiple prediction. In Kirchhoff wavefield extrapolation for surface multiple prediction a single round trip to an interpreted reflector is added to the recorded data. Here we show that interbed multiples generated between two interpreted reflectors can be predicted by applying the Kirchhoff wavefield extrapolation operator twice. In the first extrapolation step Kirchhoff wavefield extrapolation propagates the data backward in time to simulate a round trip to the shallower reflector. In the second extrapolation step Kirchhoff wavefield extrapolation propagates the data forward in time to simulate a round trip to the deeper reflector. In the Kirchhoff extrapolation kernel we use asymptotic Green's functions. The prediction of multiples via Kirchhoff wavefield extrapolation is possibly sped up by computing the required traveltimes via a shifted hyperbola approximation. The effectiveness of the method is demonstrated by results on both synthetic and field data sets.

Uniform asymptotic Green's functions for efficient modeling of cracks in elastic layers with relative shear deformation controlled by linear springs

AbstractWe present a uniform asymptotic solution (UAS) for a displacement discontinuity (DD) that lies within the middle layer of a three‐layer elastic medium in which relative shear deformation between parallel interfaces is controlled by linear springs. The DD is assumed to be normal to the two interfaces between the elastic media. Using the Fourier transform method we construct a leading term in the asymptotic expansion for the spectral coefficient functions for a DD in a three‐layer‐spring medium. Although a closed‐form solution will require a solution in terms of an infinite series, we demonstrate how this UAS can be used to construct highly efficient and accurate solutions even in the case in which the DD actually touches the interface. We compare the results using the Green's function UAS solution for a crack crossing a soft interface with results obtained using a multi‐layer boundary element method. We also present results from an implementation of the UAS Green's function approach in a pseudo‐3D hydraulic fracturing simulator to analyze the effect of interface shear deformation on the fracture propagation process. These results are compared with field measurements. Copyright © 2008 John Wiley & Sons, Ltd.

Asymptotic Green's function in homogeneous anisotropic viscoelastic media

An asymptotic Green's function in homogeneous anisotropic viscoelastic media is derived. The Green's function in viscoelastic media is formally similar to that in elastic media, but its computation is more involved. The stationary slowness vector is, in general, complex valued and inhomogeneous. Its computation involves finding two independent real-valued unit vectors which specify the directions of its real and imaginary parts and can be done either by iterations or by solving a system of coupled polynomial equations. When the stationary slowness direction is found, all quantities standing in the Green's function such as the slowness vector, polarization vector, phase and energy velocities and principal curvatures of the slowness surface can readily be calculated.The formulae for the exact and asymptotic Green's functions are numerically checked against closed-form solutions for isotropic and simple anisotropic, elastic and viscoelastic models. The calculations confirm that the formulae and developed numerical codes are correct. The computation of theP-wave Green's function in two realistic materials with a rather strong anisotropy and absorption indicates that the asymptotic Green's function is accurate at distances greater than several wavelengths from the source. The error in the modulus reaches at most 4% at distances greater than 15 wavelengths from the source.

Sensitivity kernels for finite-frequency surface waves

Sensitivity kernels for fundamental mode surface waves at finite frequency for 2-D phase speed and 3-D shear wave speed are constructed based on the Born and Rytov approximations working with a potential representation for surface waves. The use of asymptotic Green's functions for scalar wave equations provides an efficient way to calculate the Born or Rytov kernels. The 2-D sensitivity kernels enable us to incorporate the finite-frequency effects of surface waves, as well as off-great-circle propagation, in tomographic inversions for phase-speed structures. We derive examples of the 2-D sensitivity kernels both for a homogeneous background model (or a spherically symmetric model), and for a laterally heterogeneous model. The resulting distortions of the shape of the sensitivity kernels for a heterogeneous background model indicate the importance of the use of proper kernels to account of the heterogeneity in the real Earth. By combining a set of 2-D sensitivity kernels with 1-D vertical sensitivity kernels for a particular frequency range and taking the inverse Fourier transform, we can derive 3-D sensitivity kernels for surface waves in the time domain. Such 3-D kernels are useful for efficient forward modelling of surface waveforms incorporating finite-frequency effects, and will also enable us to perform direct inversion of surface waveforms into 3-D structure taking account of finite-frequency effects.

High-frequency Green's function for a semi-infinite array of electric dipoles on an infinite grounded stratified dielectric slab: part III $phase-matched wave interactions and numerical results

The particular effects of phase matching in interactions between different wave species and interaction mechanisms pertaining to the asymptotic Green's function for a semi-infinite phased array of parallel dipoles on an infinite grounded stratified dielectric slab are addressed. The two previous parts have lead to a grouping of certain asymptotic terms, which provides physically appealing interpretations of interacting wave processes that involve slab-modulated propagating (radiating) Floquet waves (FW), slab-guided leaky waves (LW), evanescent (nonradiating) FWs and surface waves (SW), together with their associated conical edge-coupled wave constituents. Special attention is now given to the role, in these wave interaction functions,of the difference between the propagating wave numbers (i.e., the degree of phase matching) that characterizes any pair of propagating FWs and LWs, and evanescent FWs and SWs. Cutoff transitions of FWs from the evanescent to the propagating regime are described also in terms of a second wave interaction function, structured similarly to the first. In connection with applications, the interaction between SWs and evanescent FWs, as well as FW transitions from evanescent to propagating, are discussed from the perspective of scan blindness in actual arrays. Various numerical studies highlight the role of phase matching in different problem scenarios.

Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients

We give the asymptotics at infinity of a Green function for an elliptic equation with periodic coefficients on R d . Basic ingredients in establishing the asymptotics are an integral representation of the Green function and the saddle point method. We also completely determine the Martin compactification of R d with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the Green function.

Time-Domain Poroelastic Green's Functions

A method previously developed for asymptotic solution of systems of integro-differential equations with singular memory is applied to the determination of the time-domain asymptotic Green's function of Biot's poroelasticity. Asymptotic time-domain Green's functions are constructed in a neighborhood of the wavefronts. The general anisotropic medium as well as the isotropic case are considered.

On the applicability of high-frequency approximations to Lilley's equation

Three forms of the high-frequency asymptotic Green function for Lilley's equation are reviewed and compared to the exact solution over a wide range of Strouhal numbers. The asymmetric approximation, which applies to sources away from the jet axis, and the quasi-symmetric approximation, which uses a near-axis source assumption, are both obtained for parallel round jets from a formal Fourier-transform solution. The latter of the two is the basis of the so-called MGB computer code (High velocity jet noise source location and reduction, FAA-RD-76-96-II, 1978) and its derivatives. The ray-theory solution, which is the only high-frequency approximation that can be applied to more general mean flows, follows from a WKB ansatz and is shown to be closely related to the asymmetric approximation. The comparisons show that the best overall prediction of the exact Green function is given by the asymmetric approximation which remains accurate down to a Strouhal number of 1/2. The close relationship between the asymmetric and ray-theory approximations suggests that the high-frequency asymptotic Green function for more general mean flows would be similarly successful.

High‐frequency approximation for mutual coupling calculations between apertures on a perfect electric conductor circular cylinder covered with a dielectric layer: Nonparaxial region

An asymptotic Green's function technique is presented for the calculation of the surface fields on a perfect electric conductor circular cylinder covered with a dielectric layer. The sources are apertures located on the perfect electric conductor surface. This method is very efficient and valid for large cylinders in terms of wavelengths. The new representation is obtained via Watson's transformation and integral contour deformations. The efficiency of the method results from the circumferentially propagating representation of the Green's function as well as the efficient numerical evaluation of various integrals. Numerical results are presented showing good agreement when compared to other solutions as well as measurements.

A PRACTICAL APPROACH TO MODELING DOUBLY CURVED CONFORMAL MICROSTRIP ANTENNAS - ABSTRACT

Designers are increasingly integrating conformal microstrip antennas into the curved structures of either air or land vehicles. Quite often, these structures are doubly curved (e.g. curved along two orthogonal surface directions). This practice necessitates the development of accurate codes versatile enough to model conformal antennas with arbitrarily shaped apertures radiating from doubly curved surfaces. Traditional planar-structure-based design techniques are not well suited for this application. A hybrid finite element-boundary integral formulation appropriate for the high-frequency analysis and design of doubly curved conformal antennas is introduced in this paper. The novelty of this approach lies in its use of an asymptotic prolate spheroidal dyadic Green's function to model the physics of curved surface diffraction. To demonstrate the utility of this approach, the effects of curvature on the resonant frequency and input impedance of both a doubly curved conformal square and circular patch antenna are investigated. Different feed positions are also considered. Due to a paucity of published experimental data, the numerical results are benchmarked by comparison with the results for planar square and circular patch antennas. The planar results are obtained by using an experimentally validated planar finite element-boundary integral code.

Asymptotic Green functions: a generalized semiclassical approach for scattering by multiple barrier potentials

We show how quantum mechanical barrier reflection and transmission coefficients and can be obtained from asymptotic Green functions. We exemplify our results by calculating such coefficients for the Rosen-Morse (RM) potential. For multiple barrier potentials, V(x) = ∑jV (j)(x), where each V (j) goes to zero for x→±∞, we derive the asymptotic Green functions by a generalized semiclassical approximation, which is based on the usual sum over classical paths considered only in the classically allowed regions and includes local quantum effects through the individual (j) and (j). The approach is applied to double RM potentials and to Woods-Saxon barriers. We obtain analytical expressions for the transmission and reflection probabilities of these potentials which are very accurate when compared with exact numerical calculations, being much better than the usual WKB approximation. Finally we briefly discuss how to extend the present method to other kinds of potential.

A Hamiltonian approach to asymptotic seismic reflection and diffraction modelling

A model‐independent Hamiltonian formulation of paraxial and diffracted ray‐tracing equations is presented. It is applied to asymptotic Green's function computations. The medium can have an arbitrary number of interfaces, possibly intersecting at diffracting edges and vertices. Continuously varying model parameters and anisotropy are allowed. The algorithm for elastic waves, involving accuracy control and amplitude computation, is implemented in a platform‐independent object‐orientated C++ package. Numerical tests and modelling examples are presented.