A complex variable Green's function representation of plane inelastic deformation in isotropic solids

1988 ◽  
Vol 72 (3-4) ◽  
pp. 205-221 ◽  
Author(s):  
M. Denda
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Complex Variable ◽  
Inelastic Deformation ◽  
Function Representation ◽  
Isotropic Solids

scholarly journals On the Marchenko equation for multicomponent single-sided reflection data

2014 ◽  
Vol 199 (3) ◽  
pp. 1367-1371 ◽  
Author(s):  
Kees Wapenaar ◽  
Evert Slob
Keyword(s):  
Green’S Function ◽  
Recent Work ◽  
Green's Function ◽  
Scattered Field ◽  
Iterative Solution ◽  
Virtual Source ◽  
Function Representation ◽  
Reflection Data ◽  
Marchenko Equation ◽  
Multicomponent Data

Abstract Recent work on the Marchenko equation has shown that the scalar 3-D Green's function for a virtual source in the subsurface can be retrieved from the single-sided reflection response at the surface and an estimate of the direct arrival. Here, we discuss the first steps towards extending this result to multicomponent data. After introducing a unified multicomponent 3-D Green's function representation, we analyse its 1-D version for elastodynamic waves in more detail. It follows that the main additional requirement is that the multicomponent direct arrival, needed to initiate the iterative solution of the Marchenko equation, includes the forward-scattered field. Under this and other conditions, the multicomponent Green's function can be retrieved from single-sided reflection data, and this is demonstrated with a 1-D numerical example.

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.

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