Strategies for imaging with Marchenko-retrieved Green’s functions

Geophysics ◽  
2017 ◽  
Vol 82 (4) ◽  
pp. Q23-Q37 ◽  
Author(s):  
Satyan Singh ◽  
Roel Snieder
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Imaging Techniques ◽  
Imaging Condition ◽  
Surface Reflection ◽  
Reflection Data ◽  
First Arrival ◽  
The One

Recent papers show that imaging with the retrieved Green’s function constructed by the Marchenko equations, called Marchenko imaging, reduces artifacts from internal and free-surface multiples compared with standard imaging techniques. Even though artifacts are reduced, they can still be present in the image, depending on the imaging condition used. We have found that when imaging with the up- and downgoing Green’s functions, the multidimensional deconvolution (MDD) imaging condition yields better images than correlation and deconvolution. “Better” in this case means improved resolution, fewer artifacts, and a closer match with the true reflection coefficient of the model. We have determined that the MDD imaging condition only uses primaries to construct the image, whereas multiples are implicitly subtracted in the imaging step. Consequently, combining the first arrival of the downgoing Green’s function with the complete upgoing Green’s function produces superior (or at least equivalent) images than using the one-way Green’s functions because the first arrival of the downgoing Green’s function excludes all the downgoing multiply reflected waves. We also find that standard imaging algorithms which use the redatumed reflection response, constructed with the one-way Green’s functions, produce images with reduced artifacts from multiples compared with standard imaging conditions, which use surface reflection data. All imaging methods that rely on the Marchenko equations require the same inputs as standard imaging techniques: the reflection response at the surface and a smooth estimate of the subsurface velocities.

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.

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.

Fast Huygens’ sweeping methods for multiarrival Green’s functions of Helmholtz equations in the high-frequency regime

Geophysics ◽  
2015 ◽  
Vol 80 (2) ◽  
pp. T91-T100 ◽  
Author(s):  
Jianliang Qian ◽  
Songting Luo ◽  
Robert Burridge
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Finite Difference Methods ◽  
Fixed Number ◽  
Point Sources ◽  
Helmholtz Equations ◽  
Frequency Regime ◽  
First Arrival

Multiarrival Green’s functions are essential in seismic modeling, migration, and inversion. Huygens-Kirchhoff (HK) integrals provide a bridge to integrate locally valid first-arrival Green’s functions into a globally valid multiarrival Green’s function. We have designed robust and accurate finite-difference methods to compute first-arrival traveltimes and amplitudes, so that first-arrival Green’s functions can be constructed rapidly. We adapted a fast butterfly algorithm to evaluate discretized HK integrals. The resulting fast Huygens’ sweeping method has the following unique features: (1) it precomputes a set of local traveltime and amplitude tables, (2) it automatically takes care of caustics, (3) it constructs Green’s functions of the Helmholtz equation for arbitrary frequencies and for many point sources, and (4) for a fixed number of points per wavelength, it constructs each Green’s function in nearly optimal complexity [Formula: see text] in terms of the total number of mesh points [Formula: see text], where the prefactor of the complexity only depends on the specified accuracy, and is independent of the frequency. The 2D and 3D examples revealed the performance of the method.

Non-Equilibrium Green’s Functions: Variational Relations and Approximations for Particle Interactions

2018 ◽  
Author(s):  
Norman J. Morgenstern Horing
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Random Phase ◽  
Integration Contour ◽  
Differential Formulation ◽  
Variational Relations ◽  
Potential Perturbation ◽  
Non Equilibrium

Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator U(t0+τ,t0). Linear conductivity is exhibited as a two-particle equilibrium Green’s function (Kubo-type formulation). Admitting particle sources (S:η,η+) and non-conservation of number, the non-equilibrium multi-particle Green’s functions are constructed with numbers of creation and annihilation operators that may differ, and they may be derived as variational derivatives with respect to sources η,η+ of a generating functional eW=TrU(t0+τ,t0)CS/TrU(t0+τ,t0)C. (In the non-interacting case this yields the n-particle Green’s function as a permanent/determinant of single-particle Green’s functions.) These variational relations yield a symmetric set of multi-particle Green’s function equations. Cumulants and the Linked Cluster Theorem are discussed and the Random Phase Approximation (RPA) is derived variationally. Schwinger’s variational differential formulation of perturbation theories for the Green’s function, self-energy, vertex operator, and also shielded potential perturbation theory, are reviewed. The Langreth Algebra arises from analytic continuation of integration of products of Green’s functions in imaginary time to the real-time axis with time-ordering along the integration contour in the complex time plane. An account of the Generalized Kadanoff-Baym Ansatz is presented.

Nonequilibrium Green’s Functions

2018 ◽  
Author(s):  
Klaus Morawetz
Keyword(s):  
Green’S Function ◽  
Quantum Transport ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Equation Of Motion ◽  
Diagrammatic Technique ◽  
Initial Correlations ◽  
Nonequilibrium Green’S Functions ◽  
Nonequilibrium Green's Functions

The method of the equation of motion is used to derive the Martin–Schwinger hierarchy for the nonequilibrium Green’s functions. The formal closure of the hierarchy is reached by using the selfenergy which provides a recipe for how to construct selfenergies from approximations of the two-particle Green’s function. The Langreth–Wilkins rules for a diagrammatic technique are shown to be equivalent to the weakening of initial correlations. The quantum transport equations are derived in the general form of Kadanoff and Baym equations. The information contained in the Green’s function is discussed. In equilibrium this leads to the Matsubara diagrammatic technique.

AN INTRODUCTORY GUIDE TO GREEN’S FUNCTION METHODS IN NUCLEAR MANY-BODY PROBLEMS

1994 ◽  
Vol 03 (02) ◽  
pp. 523-589 ◽  
Author(s):  
T.T.S. KUO ◽  
YIHARN TZENG
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Model Space ◽  
Many Body ◽  
Diagram Expansion ◽  
Nucleus Scattering ◽  
Many Body Systems ◽  
General Method

We present an elementary and fairly detailed review of several Green’s function methods for treating nuclear and other many-body systems. We first treat the single-particle Green’s function, by way of which some details concerning linked diagram expansion, rules for evaluating Green’s function diagrams and solution of the Dyson’s integral equation for Green’s function are exhibited. The particle-particle hole-hole (pphh) Green’s function is then considered, and a specific time-blocking technique is discussed. This technique enables us to have a one-frequency Dyson’s equation for the pphh and similarly for other Green’s functions, thus considerably facilitating their calculation. A third type of Green’s function considered is the particle-hole Green’s function. RPA and high order RPA are treated, along with examples for setting up particle-hole RPA equations. A general method for deriving a model-space Dyson’s equation for Green’s functions is discussed. We also discuss a method for determining the normalization of Green’s function transition amplitudes based on its vertex function. Some applications of Green’s function methods to nuclear structure and recent deep inelastic lepton-nucleus scattering are addressed.

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.

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.

Sign Properties of Green's Functions For Two Classes of Boundary Value Problems

1987 ◽  
Vol 30 (1) ◽  
pp. 28-35 ◽  
Author(s):  
P. W. Eloe
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Difference Equations ◽  
Green's Functions ◽  
Green’S Functions ◽  
Boundary Value ◽  
Partial Derivatives

AbstractLet G(x,s) be the Green's function for the boundary value problem y(n) = 0, Ty = 0, where Ty = 0 represents boundary conditions at two points. The signs of G(x,s) and certain of its partial derivatives with respect to x are determined for two classes of boundary value problems. The results are also carried over to analogous classes of boundary value problems for difference equations.

Sign in / Sign up

Export Citation Format

Share Document