On the fundamentals of the virtual source method

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. A13-A17 ◽  
Author(s):  
Valeri Korneev ◽  
Andrey Bakulin
Keyword(s):  
Green’S Function ◽  
Direct Measurement ◽  
Green's Function ◽  
Velocity Model ◽  
Practical Approach ◽  
Virtual Source ◽  
Processing Stage ◽  
Source Method ◽  
Complex Part ◽  
Seismic Images

The virtual source method (VSM) has been proposed as a practical approach to reduce distortions of seismic images caused by shallow, heterogeneous overburden. VSM is demanding at the acquisition stage because it requires placing downhole geophones below the most complex part of the heterogeneous overburden. Where such acquisition is possible, however, it pays off later at the processing stage because it does not require knowledge of the velocity model above the downhole receivers. This paper demonstrates that VSM can be viewed as an application of the Kirchhoff-Helmholtz integral (KHI) with an experimentally measured Green’s function. Direct measurement of the Green’s function ensures the effectiveness of the method in highly heterogeneous subsurface conditions.

scholarly journals A Green’s Function for Acoustic Problems in Pekeris Waveguide Using a Rigorous Image Source Method

2021 ◽  
Vol 11 (6) ◽  
pp. 2722
Author(s):  
Zhiwen Qian ◽  
Dejiang Shang ◽  
Yuan Hu ◽  
Xinyang Xu ◽  
Haihan Zhao ◽  
...  
Keyword(s):  
Green’S Function ◽  
Shallow Water ◽  
Green's Function ◽  
Plane Waves ◽  
Spherical Wave ◽  
Sound Field ◽  
Efficient Computation ◽  
Image Source ◽  
Source Method ◽  
Pekeris Waveguide

The Green’s function (GF) directly eases the efficient computation for acoustic radiation problems in shallow water with the use of the Helmholtz integral equation. The difficulty in solving the GF in shallow water lies in the need to consider the boundary effects. In this paper, a rigorous theoretical model of interactions between the spherical wave and the liquid boundary is established by Fourier transform. The accurate and adaptive GF for the acoustic problems in the Pekeris waveguide with lossy seabed is derived, which is based on the image source method (ISM) and wave acoustics. First, the spherical wave is decomposed into plane waves in different incident angles. Second, each plane wave is multiplied by the corresponding reflection coefficient to obtain the reflected sound field, and the field is superposed to obtain the reflected sound field of the spherical wave. Then, the sound field of all image sources and the physical source are summed to obtain the GF in the Pekeris waveguide. The results computed by this method are compared with the standard wavenumber integration method, which verifies the accuracy of the GF for the near- and far-field acoustic problems. The influence of seabed attenuation on modal interference patterns is analyzed.

scholarly journals Tutorial on seismic interferometry: Part 2 — Underlying theory and new advances

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. 75A211-75A227 ◽  
Author(s):  
Kees Wapenaar ◽  
Evert Slob ◽  
Roel Snieder ◽  
Andrew Curtis
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Seismic Interferometry ◽  
Virtual Source ◽  
Bending Waves ◽  
Complex Source ◽  
Recent Developments ◽  
Diffusion Phenomena ◽  
Quantum Mechanical Scattering ◽  
Moving Media

In the 1990s, the method of time-reversed acoustics was developed. This method exploits the fact that the acoustic wave equation for a lossless medium is invariant for time reversal. When ultrasonic responses recorded by piezoelectric transducers are reversed in time and fed simultaneously as source signals to the transducers, they focus at the position of the original source, even when the medium is very complex. In seismic interferometry the time-reversed responses are not physically sent into the earth, but they are convolved with other measured responses. The effect is essentially the same: The time-reversed signals focus and create a virtual source which radiates waves into the medium that are subsequently recorded by receivers. A mathematical derivation, based on reciprocity theory, formalizes this principle: The crosscorrelation of responses at two receivers, integrated over differ-ent sources, gives the Green’s function emitted by a virtual source at the position of one of the receivers and observed by the other receiver. This Green’s function representation for seismic interferometry is based on the assumption that the medium is lossless and nonmoving. Recent developments, circumventing these assumptions, include interferometric representations for attenuating and/or moving media, as well as unified representations for waves and diffusion phenomena, bending waves, quantum mechanical scattering, potential fields, elastodynamic, electromagnetic, poroelastic, and electroseismic waves. Significant improvements in the quality of the retrieved Green’s functions have been obtained with interferometry by deconvolution. A trace-by-trace deconvolution process compensates for complex source functions and the attenuation of the medium. Interferometry by multidimensional deconvolution also compensates for the effects of one-sided and/or irregular illumination.

scholarly journals Data-driven wavefield focusing and imaging with multidimensional deconvolution: Numerical examples for reflection data with internal multiples

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. WA107-WA115 ◽  
Author(s):  
Filippo Broggini ◽  
Roel Snieder ◽  
Kees Wapenaar
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Imaging Techniques ◽  
Velocity Model ◽  
Single Scattering ◽  
Data Driven ◽  
Multiple Reflections ◽  
Prestack Migration ◽  
Reflection Data ◽  
Internal Multiples

Standard imaging techniques rely on the single scattering assumption. This requires that the recorded data do not include internal multiples, i.e., waves that have bounced multiple times between reflectors before reaching the receivers at the acquisition surface. When multiple reflections are present in the data, standard imaging algorithms incorrectly image them as ghost reflectors. These artifacts can mislead interpreters in locating potential hydrocarbon reservoirs. Recently, we introduced a new approach for retrieving the Green’s function recorded at the acquisition surface due to a virtual source located at depth. We refer to this approach as data-driven wavefield focusing. Additionally, after applying source-receiver reciprocity, this approach allowed us to decompose the Green’s function at a virtual receiver at depth in its downgoing and upgoing components. These wavefields were then used to create a ghost-free image of the medium with either crosscorrelation or multidimensional deconvolution, presenting an advantage over standard prestack migration. We tested the robustness of our approach when an erroneous background velocity model is used to estimate the first-arriving waves, which are a required input for the data-driven wavefield focusing process. We tested the new method with a numerical example based on a modification of the Amoco model.

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 Marchenko imaging

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. WA39-WA57 ◽  
Author(s):  
Kees Wapenaar ◽  
Jan Thorbecke ◽  
Joost van der Neut ◽  
Filippo Broggini ◽  
Evert Slob ◽  
...  
Keyword(s):  
Green’S Function ◽  
Inverse Scattering ◽  
Green's Function ◽  
Seismic Interferometry ◽  
Virtual Source ◽  
Target Zone ◽  
Scattering Problems ◽  
Inverse Scattering Problems ◽  
Marchenko Equation ◽  
Internal Multiples

Traditionally, the Marchenko equation forms a basis for 1D inverse scattering problems. A 3D extension of the Marchenko equation enables the retrieval of the Green’s response to a virtual source in the subsurface from reflection measurements at the earth’s surface. This constitutes an important step beyond seismic interferometry. Whereas seismic interferometry requires a receiver at the position of the virtual source, for the Marchenko scheme it suffices to have sources and receivers at the surface only. The underlying assumptions are that the medium is lossless and that an estimate of the direct arrivals of the Green’s function is available. The Green’s function retrieved with the 3D Marchenko scheme contains accurate internal multiples of the inhomogeneous subsurface. Using source-receiver reciprocity, the retrieved Green’s function can be interpreted as the response to sources at the surface, observed by a virtual receiver in the subsurface. By decomposing the 3D Marchenko equation, the response at the virtual receiver can be decomposed into a downgoing field and an upgoing field. By deconvolving the retrieved upgoing field with the downgoing field, a reflection response is obtained, with virtual sources and virtual receivers in the subsurface. This redatumed reflection response is free of spurious events related to internal multiples in the overburden. The redatumed reflection response forms the basis for obtaining an image of a target zone. An important feature is that spurious reflections in the target zone are suppressed, without the need to resolve first the reflection properties of the overburden.

Retrieving reflections by source-receiver wavefield interferometry

Geophysics ◽  
2011 ◽  
Vol 76 (1) ◽  
pp. SA1-SA8 ◽  
Author(s):  
Oleg V. Poliannikov
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Boundary Data ◽  
Ray Theory ◽  
Geometric Description ◽  
Virtual Source ◽  
Classical Correlation ◽  
Common Path ◽  
The Common ◽  
Stationary Sources

Source-receiver wavefield interferometry has been proposed recently as an extension of classical correlation-based interferometry. Under idealized assumptions, it allows the Green’s function between a source and a receiver to be reconstructed from boundary data that are collected using two closed contours of sources and receivers, respectively. An intuitive geometric description of this method, based on ray theory, can be used to design a method for reconstructing virtual-gather events that typically are lost when conventional interferometry is employed. In classical interferometry, events in the Green’s function between two receiver locations are reconstructed by automatically finding a ray that emanates from a stationary source, passes through the virtual source, reflects off the structure, and finally is recorded by the second receiver. The common path to both receivers is then canceled by a suitable crosscorrelation. If illumination of the structure is not ideal, such a ray may not exist and the reflection is not reconstructed. Source-receiver wave interferometry can be given a similar geometric description. The reflection off the structure can be constructed using not one but multiple rays produced simultaneously by two stationary sources. This new redatuming technique proves successful in geometries in which classical interferometry fails. Extensive numerical simulations support these theoretical conclusions.

Zero-offset seismic amplitude decomposition and migration

Geophysics ◽  
2007 ◽  
Vol 72 (4) ◽  
pp. S187-S193
Author(s):  
Bjørn Ursin ◽  
Martin Tygel
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Normal Wave ◽  
Normal Incidence ◽  
Velocity Model ◽  
Transmission And Reflection Coefficients ◽  
Geometric Spreading ◽  
Map Migration ◽  
Zero Offset ◽  
And Migration

In an anisotropic medium, a normal-incidence wave is multiply transmitted and reflected down to a reflector where the phase-velocity vector is parallel to the interface normal. The ray code of the upgoing wave is equal to the ray code of the downgoing wave in reverse order. The geometric spreading, KMAH index, and transmission and reflection coefficients of the normal-incidence ray can be expressed in terms of products or sums of the corresponding quantities of the one-way normal and normal-incidence-point (NIP) waves. Here, we show that the amplitude of the ray-theoretic Green’s function for the reflected wave also follows a similar decomposition in terms of the amplitude of the Green’s function of the NIP wave and the normal wave. We use this property to propose three schemes for true-amplitude poststack depth migration in anisotropic media where the image represents an estimate of the zero-offset reflection coefficient. The first is a map migration procedure in which selected primary zero-offset reflections are converted into depth with attached true amplitudes. The second is a ray-based, Kirchhoff-type full migration. The third is a wave equation continuation algorithm to reverse-propagate the recorded wavefield in a half-velocity model with half the elastic constants and double the density. The image is formed by taking the reverse-propagated wavefield at time equal to zero followed by a geometric spreading correction.

Source-receiver Marchenko redatuming: Obtaining virtual receivers and virtual sources in the subsurface

Geophysics ◽  
2017 ◽  
Vol 82 (3) ◽  
pp. Q13-Q21 ◽  
Author(s):  
Satyan Singh ◽  
Roel Snieder
Keyword(s):  
Free Surface ◽  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Virtual Source ◽  
First Arrival

By solving the Marchenko equations, one can retrieve the Green’s function (Marchenko Green’s function) between a virtual receiver in the subsurface and points at the surface (no physical receiver is required at the virtual location). We extend the idea behind these equations to retrieve the Green’s function between any two points in the subsurface, i.e., between a virtual source and a virtual receiver (no physical source or physical receiver is required at either of these locations). This Green’s function is called the virtual Green’s function, and it includes all primary, internal, and free-surface multiples. Similar to the Marchenko Green’s function, this virtual Green’s function requires the reflection response at the surface (single-sided illumination) and an estimate of the first-arrival traveltime from the virtual locations to the surface. These Green’s functions can be used to image the interfaces from above and below.

A Short Survey on Green’s Function for Acoustic Problems

2020 ◽  
Vol 28 (02) ◽  
pp. 1950025
Author(s):  
Augustus R. Okoyenta ◽  
Haijun Wu ◽  
Xueliang Liu ◽  
Weikang Jiang
Keyword(s):  
Integral Equation ◽  
Boundary Conditions ◽  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Acoustic Medium ◽  
Acoustic Source ◽  
Impedance Boundary ◽  
Source Method

Green’s functions for acoustic problems is the fundamental solution to the inhomogeneous Helmholtz equation for a point source, which satisfies specific boundary conditions. It is very significant for the integral equation and also serves as the impulse response of an acoustic wave equation. They are important for acoustic problems that involve the propagation of sound from the source point to the observer position. Once the Green’s function, which satisfies the necessary boundary conditions, is obtained, the sound pressure at any point away from the source can be easily calculated by the integral equation. The major problem faced by researchers is in the process of constructing these Green’s functions which satisfy a specific boundary condition. The aim of this work is to review some of these fundamental solutions available in the literature for different boundary conditions for the ease of analyzing acoustics problems. The review covers the free-space Green’s functions for stationary source and rotational source, for both when the observer and the acoustic medium are at rest and when the medium is in uniform flow. The half-space Green’s functions are also summarized for both stationary acoustic source and moving acoustic source, derived using the image source method, equivalent source method and complex equivalent method in both time domain and frequency domain. Each of these methods used depends on the different impedance boundary conditions for which the Green’s function will satisfy. Finally, enclosed spaced Green’s functions for both rectangular duct and cylindrical duct for an infinite and finite duct is also covered in the review.

scholarly journals Green's function retrieval from reflection data, in absence of a receiver at the virtual source position

2014 ◽  
Vol 135 (5) ◽  
pp. 2847-2861 ◽  
Author(s):  
Kees Wapenaar ◽  
Jan Thorbecke ◽  
Joost van der Neut ◽  
Filippo Broggini ◽  
Evert Slob ◽  
...  
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Virtual Source ◽  
Source Position ◽  
Reflection Data
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