scholarly journals Green’s Functions for the Wave Equation

2015 ◽  
pp. 161-281
Keyword(s):  
Wave Equation ◽  
Green's Functions ◽  
Green’S Functions

Parabolic Representations of Scattering Green’s Functions

1999 ◽  
Author(s):  
Paul E. Barbone
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Green’S Function ◽  
Inhomogeneous Medium ◽  
Green's Function ◽  
Free Space ◽  
Green's Functions ◽  
Green’S Functions

Abstract We derive a one-way wave equation representation of the “free space” Green’s function for an inhomogeneous medium. Our representation results from an asymptotic expansion in inverse powers of the wavenumber. Our representation takes account of losses due to scattering in all directions, even though only one-way operators are used.

On: “Green's function implementation of common‐offset wave equation migration,” (A. Ehinger, P. Lailly and K. J. Marfurt, November‐December 1996 GEOPHYSICS, 61, p. 1813–1821)

Geophysics ◽  
1997 ◽  
Vol 62 (4) ◽  
pp. 1333-1335
Author(s):  
Kees Wapenaar
Keyword(s):  
Wave Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Complex Media

The paper by Ehinger and co‐authors deals with migration in complex media in which severe multi‐pathing may occur. I wholeheartedly agree with their remark that the use of complete Green's functions is preferred over approximate Green's functions containing first arrivals only.

On seismic deghosting using integral representation for the wave equation: Use of Green’s functions with Neumann or Dirichlet boundary conditions

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. T89-T98 ◽  
Author(s):  
Lasse Amundsen ◽  
Arthur B. Weglein ◽  
Arne Reitan
Keyword(s):  
Wave Equation ◽  
Integral Representation ◽  
Boundary Conditions ◽  
Representation Theorem ◽  
Green's Functions ◽  
Dirichlet Boundary ◽  
Green’S Functions ◽  
Dirichlet Boundary Conditions ◽  
Sea Surface ◽  
Receiver Side

The recent interest in broadband seismic technology has spurred research into new and improved seismic deghosting solutions. One starting point for deriving deghosting methods is the representation theorem, which is an integral representation for the wave equation. Recent research results show that by using Green’s functions with Dirichlet boundary conditions in the representation theorem, source-side deghosting of already receiver-side deghosted wavefields can be achieved. We found that the choice of Green’s functions with Neumann boundary conditions on the sea surface and the plane that contains the sources leads to an identical but simpler solution with fewer processing steps. In addition, we found that pressure data can be receiver-side deghosted by introducing Green’s functions with Dirichlet boundary conditions on the sea surface and the plane containing the receivers into a modified representation theorem. The deghosting methods derived from the representation theorem are wave-theoretic algorithms defined in the frequency-space domain and can accommodate streamers of any shape (e.g., slanted). Our theoretical analysis of deghosting is performed in the frequency-wavenumber domain where analytical deghosting solutions are well known and thus are available for verifying the solutions. A simple numerical example can be used to show how source-side deghosting can be performed in the space domain by convolving data with Green’s functions.

Adaptive stabilization for Q-compensated reverse time migration

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. S15-S32 ◽  
Author(s):  
Yufeng Wang ◽  
Hui Zhou ◽  
Hanming Chen ◽  
Yangkang Chen
Keyword(s):  
Wave Equation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Numerical Instability ◽  
Adaptive Stabilization ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Gas Chimney ◽  
Low Pass

Reverse time migration (RTM) for attenuating media should take amplitude compensation and phase correction into consideration. However, attenuation compensation during seismic propagation suffers from numerical instability because of the boosted high-frequency ambient noise. We have developed a novel adaptive stabilization method for [Formula: see text]-compensated RTM ([Formula: see text]-RTM), which exhibits superior properties of time variance and [Formula: see text] dependence over conventional low-pass filtering-based method. We derive the stabilization operator by first analytically deriving [Formula: see text]-space Green’s functions for a constant-[Formula: see text] wave equation with decoupled fractional Laplacians and its compensated equation. The time propagator of Green’s function for the viscoacoustic wave equation decreases exponentially, whereas that of the compensated equation is exponentially divergent at a high wavenumber, and it is not stable after the wave is extrapolated for a long time. Therefore, the Green’s functions theoretically explain how the numerical instability existing in [Formula: see text]-RTM arises and shed light on how to overcome this problem pertinently. The stabilization factor required in the proposed method can be explicitly identified by the specified gain limit according to an empirical formula. The [Formula: see text]-RTM results for noise-free data using low-pass filtering and adaptive stabilization are compared over a simple five-layer model and the BP gas chimney model to verify the superiority of the proposed approach in terms of fidelity and stability. The [Formula: see text]-RTM result for noisy data from the BP gas chimney model further demonstrates that our method enjoys a better antinoise performance and helps significantly to enhance the resolution of seismic images.

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