Inverse extrapolation of primary seismic waves

Geophysics ◽  
1989 ◽  
Vol 54 (7) ◽  
pp. 853-863 ◽  
Author(s):  
C. P. A. Wapenaar ◽  
G. L. Peels ◽  
V. Budejicky ◽  
A. J. Berkhout
Keyword(s):  
Green’S Function ◽  
Wave Field ◽  
Green's Function ◽  
Boundary Integral ◽  
Open Boundary ◽  
Complex Conjugate ◽  
Forward Wave ◽  
Propagation Effects ◽  
The One ◽  
Kirchhoff Integral

Forward wave‐field extrapolation operators simulate propagation effects from one depth level to another. Inverse wave‐field extrapolation operators eliminate those propagation effects. Since forward wave‐field extrapolation can be described in terms of spatial convolution, inverse wave‐field extrapolation can be described in terms of spatial deconvolution. A simple approximation to a stable, spatially band‐limited deconvolution operator is obtained by taking the complex conjugate of the convolution operator. A one‐way version of the Kirchhoff integral that contains the conjugate complex Green’s function is derived. Unlike the situation with respect to the forward problem, the modification of the closed surface integral into an open boundary integral involves an approximation that is identical to the approximation in the conjugate complex deconvolution approach. This approximation neglects the evanescent field and causes a second‐order amplitude error. For a plane acquisition surface, the one‐way Kirchhoff integral is transformed into a one‐way Rayleigh integral. For media with small to moderate contrasts, the one‐way Rayleigh integral with the conjugate complex Green’s function describes true amplitude inverse extrapolation of primary waves. This is illustrated with an example, in which the Green’s function has been modeled with the Gaussian beam method.

scholarly journals Unified double- and single-sided homogeneous Green’s function representations

2016 ◽  
Vol 472 (2190) ◽  
pp. 20160162 ◽  
Author(s):  
Kees Wapenaar ◽  
Joost van der Neut ◽  
Evert Slob
Keyword(s):  
Green’S Function ◽  
Multiple Scattering ◽  
Green's Function ◽  
Time Reversal ◽  
Wave Theory ◽  
Boundary Integral ◽  
Open Boundary ◽  
Quantum Mechanical ◽  
Holographic Imaging ◽  
Scattering Time

In wave theory, the homogeneous Green’s function consists of the impulse response to a point source, minus its time-reversal. It can be represented by a closed boundary integral. In many practical situations, the closed boundary integral needs to be approximated by an open boundary integral because the medium of interest is often accessible from one side only. The inherent approximations are acceptable as long as the effects of multiple scattering are negligible. However, in case of strongly inhomogeneous media, the effects of multiple scattering can be severe. We derive double- and single-sided homogeneous Green’s function representations. The single-sided representation applies to situations where the medium can be accessed from one side only. It correctly handles multiple scattering. It employs a focusing function instead of the backward propagating Green’s function in the classical (double-sided) representation. When reflection measurements are available at the accessible boundary of the medium, the focusing function can be retrieved from these measurements. Throughout the paper, we use a unified notation which applies to acoustic, quantum-mechanical, electromagnetic and elastodynamic waves. We foresee many interesting applications of the unified single-sided homogeneous Green’s function representation in holographic imaging and inverse scattering, time-reversed wave field propagation and interferometric Green’s function retrieval.

One‐way versions of the Kirchhoff Integral

Geophysics ◽  
1989 ◽  
Vol 54 (4) ◽  
pp. 460-467 ◽  
Author(s):  
A. J. Berkhout ◽  
C. P. A. Wapenaar
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Wave Field ◽  
Green's Function ◽  
Green's Functions ◽  
Acoustic Pressure ◽  
Green’S Functions ◽  
Absorbing Boundary Conditions ◽  
Multiple Reflections ◽  
Kirchhoff Integral

The conventional Kirchhoff integral, based on the two‐way wave equation, states how the acoustic pressure at a point A inside a closed surface S can be calculated when the acoustic wave field is known on S. In its general form, the integrand consists of two terms: one term contains the gradient of a Green’s function and the acoustic pressure; the other term contains a Green’s function and the gradient of the acoustic pressure. The integrand can be simplified by choosing reflecting boundary conditions for the two‐way Green’s functions in such a way that either the first term or the second term vanishes on S. This conventional approach to deriving Rayleigh‐type integrals has practical value only for media with small contrasts, so that the two‐way Green’s functions do not contain significant multiple reflections. We present a modified approach for simplifying the integrand of the Kirchhoff integral by choosing absorbing boundary conditions for the one‐way Green’s functions. The resulting Rayleigh‐type integrals are the theoretical basis for true amplitude one‐way wave‐field extrapolation techniques in inhomogeneous media with significant contrasts.

Thermodynamic Green’s Functions and Spectral Structure

2018 ◽  
Author(s):  
Norman J. Morgenstern Horing
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Spectral Weight ◽  
Statistical Thermodynamic ◽  
Spectral Structure ◽  
Imaginary Time ◽  
Translational Invariance ◽  
The One

Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.

scholarly journals DIFFERENTIAL EQUATION APPROACH FOR ONE- AND TWO-DIMENSIONAL LATTICE GREEN'S FUNCTION

2007 ◽  
Vol 21 (02n03) ◽  
pp. 139-154 ◽  
Author(s):  
J. H. ASAD
Keyword(s):  
Differential Equation ◽  
Green’S Function ◽  
Recurrence Relation ◽  
Green's Function ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
The One ◽  
Lattice Green's Function ◽  
Lattice Green’S Function

A first-order differential equation of Green's function, at the origin G(0), for the one-dimensional lattice is derived by simple recurrence relation. Green's function at site (m) is then calculated in terms of G(0). A simple recurrence relation connecting the lattice Green's function at the site (m, n) and the first derivative of the lattice Green's function at the site (m ± 1, n) is presented for the two-dimensional lattice, a differential equation of second order in G(0, 0) is obtained. By making use of the latter recurrence relation, lattice Green's function at an arbitrary site is obtained in closed form. Finally, the phase shift and scattering cross-section are evaluated analytically and numerically for one- and two-impurities.

Integral Equation Approach for Computing Green’s Function on Doubly Connected Regions via the Generalized Neumann Kernel

2014 ◽  
Vol 71 (1) ◽  
Author(s):  
Siti Zulaiha Aspon ◽  
Ali Hassan Mohamed Murid ◽  
Mohamed M. S. Nasser ◽  
Hamisan Rahmat
Keyword(s):  
Integral Equation ◽  
Dirichlet Problem ◽  
Linear System ◽  
Green’S Function ◽  
Green's Function ◽  
Boundary Integral ◽  
Generalized Neumann Kernel ◽  
Integral Equation Approach ◽  
Gauss Elimination ◽  
Neumann Kernel

This research is about computing the Green’s function on doubly connected regions by using the method of boundary integral equation. The method depends on solving a Dirichlet problem. The Dirichlet problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the generalized Neumann kernel. The method for solving this integral equation is by using the Nystrӧm method with trapezoidal rule to discretize it to a linear system. The linear system is then solved by the Gauss elimination method. Mathematica plots of Green’s functions for several test regions are also presented.

Discussions on the Convergence of the Seakeeping Simulations Based on the Panel Methods

2018 ◽  
Author(s):  
Ivana Martić ◽  
Nastia Degiuli ◽  
Šime Malenica ◽  
Andrea Farkas
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Surface Condition ◽  
Wave Length ◽  
Boundary Integral ◽  
The Body ◽  
Numerical Code ◽  
Physical Parameters ◽  
Diffraction Radiation ◽  
Panel Methods

Numerical problems related to the convergence of the classical panel methods which are employed for the diffraction-radiation simulations are discussed. It is well known that, for the panel methods, the convergence issues are not exclusively related to the physical parameters (wave length, body shape, draught ...) but also to the one purely numerical phenomenon which occurs when the Boundary Integral Equation Method (BIEM) based on the use of Kelvin (wave) type Green’s function is used. Indeed, due to the fact that the Green’s function satisfies the free surface condition in the whole fluid domain below z = 0, the numerical solution is polluted, at some particular frequencies, by the solution of the unphysical problem inside the body. This phenomenon which is purely numerical, is known as the problem of irregular frequencies. From practical point of view, it is not always easy to distinguish if the irregularities in the final solution are coming, from the body mesh which is not fine enough, from the physical resonance of the system, from the problem of irregular frequencies or from something else!? In this paper the authors discuss these issues in the context of the evaluation of the seakeeping behavior of one typical FPSO (Floating Production Storage and Offloading). Both the linear (first order) as well as the second order quantities are of concern and the different methods for the elimination of the irregular frequencies are discussed. Special attention is given to the calculations of the different physical quantities at very high frequencies. The numerical tool used within this research is the Bureau Veritas numerical code HYDROSTAR which is based on the panel method with singularities of constant strength.

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