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.

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

2016 ◽  
Vol 205 (1) ◽  
pp. 531-535 ◽  
Author(s):  
Kees Wapenaar ◽  
Jan Thorbecke ◽  
Joost van der Neut
Keyword(s):  
Green’S Function ◽  
Inverse Scattering ◽  
Green's Function ◽  
Time Reversal ◽  
Boundary Integral ◽  
Open Surface ◽  
Function Representation ◽  
Holographic Imaging ◽  
Scattering Time ◽  
Time Reversal Acoustics

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.

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.

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.

Mean dyadic Green's function of a randomly rough surface

1994 ◽  
Vol 72 (1-2) ◽  
pp. 20-29 ◽  
Author(s):  
Saba Mudaliar
Keyword(s):  
Integral Equation ◽  
Green’S Function ◽  
Multiple Scattering ◽  
Rough Surface ◽  
Green's Function ◽  
Diagram Method ◽  
Dyadic Green’S Function ◽  
Dyadic Green's Function ◽  
Special Cases ◽  
Randomly Rough Surface

The problem of wave propagation and scattering over a randomly rough surface is considered from a multiple-scattering point of view. Assuming that the magnitude of irregularities are small an approximate boundary condition is specified. This enables us to derive an integral equation for the dyadic Green's function (DGF) in terms of the known unperturbed DGF. Successive iteration of this integral equation yields the Neumann series. On averaging this and using a diagram method the Dyson equation is derived. Bilocal approximation to the mass operator leads to an integral equation whose kernel is of the convolution type. This is then readily solved and the results are presented in a simplified and useful form. It is observed that the coherent reflection coefficients involve infinite series of multiple scattering. Two special cases are considered the results of which are in agreement with our expectations.

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