scholarly journals Velocity analysis using both reflections and refractions in seismic interferometry

Geophysics ◽  
2011 ◽  
Vol 76 (5) ◽  
pp. SA83-SA96 ◽  
Author(s):  
Simon King ◽  
Andrew Curtis
Keyword(s):  
Green’S Function ◽  
Half Space ◽  
Green's Function ◽  
Green's Functions ◽  
Seismic Velocity ◽  
Green’S Functions ◽  
Oil Field ◽  
Velocity Analysis ◽  
Space Model ◽  
Receiver Array

The Green’s function between two receiver locations can be estimated by crosscorrelating and summing the recorded Green’s functions from sources on a boundary that surrounds the receiver pair. We demonstrate that when two receivers are positioned far from the source boundary in a marine-type acquisition geometry, the crosscorrelations (the Green’s functions before summation over the source boundary) are dominated by reflected energy which can be used in a semblance analysis to determine the seismic velocity and thickness of the first layer. When these crosscorrelations are summed over the boundary of sources, the resulting Green’s function estimates along a receiver array contain nonphysical or spurious refracted energy. We illustrate that by using a further semblance analysis, the most prominent nonphysical refracted energy occurs prior to the direct arrival and determines the remaining refraction velocities of deeper layers (or interval velocities in the case of a subsurface with homogeneous layers). We demonstrate the velocity analysis procedure on a single layer over half-space model, a three layer over a half-space model, and a more realistic model based on a North Sea oil field.

The Three-Dimensional Infinite Space and Half-Space Green's Functions for Orthotropic Materials

2014 ◽  
Vol 31 (1) ◽  
pp. 21-28
Author(s):  
V.-G. Lee
Keyword(s):  
Green’S Function ◽  
Half Space ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Mathematical Concept ◽  
Orthotropic Materials ◽  
Superposition Method ◽  
Infinite Domain

ABSTRACTCommon materials, ranging from natural wood to modern composites, have been recognized as ortho-tropic materials. The elastic properties of such materials are governed by nine elastic constants. In this paper the complete set of Green's functions for an infinite medium and a half space is given, which were not reported completely before. Analytic expressions for the infinite Green's functions are derived through the explicit form of the sextic equation given explicitly. For an orthotopic half space, the Green's function is derived by a superposition method. The mathematical concept is based on the addition of a complementary term to the Green's function in an orthotropic infinite domain to fulfill the boundary condition on the free surface. Both solutions are illustrated in certain directions to demonstrate the nature of orthotropy.

Three-Dimensional Green’s Functions in an Anisotropic Half-Space With General Boundary Conditions

2003 ◽  
Vol 70 (1) ◽  
pp. 101-110 ◽  
Author(s):  
E. Pan
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Half Space ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Different Boundary Conditions

This paper derives, for the first time, the complete set of three-dimensional Green’s functions (displacements, stresses, and derivatives of displacements and stresses with respect to the source point), or the generalized Mindlin solutions, in an anisotropic half-space z>0 with general boundary conditions on the flat surface z=0. Applying the Mindlin’s superposition method, the half-space Green’s function is obtained as a sum of the generalized Kelvin solution (Green’s function in an anisotropic infinite space) and a Mindlin’s complementary solution. While the generalized Kelvin solution is in an explicit form, the Mindlin’s complementary part is expressed in terms of a simple line-integral over [0,π]. By introducing a new matrix K, which is a suitable combination of the eigenmatrices A and B, Green’s functions corresponding to different boundary conditions are concisely expressed in a unified form, including the existing traction-free and rigid boundaries as special cases. The corresponding generalized Boussinesq solutions are investigated in details. In particular, it is proved that under the general boundary conditions studied in this paper, the generalized Boussinesq solution is still well-defined. A physical explanation for this solution is also offered in terms of the equivalent concept of the Green’s functions due to a point force and an infinitesimal dislocation loop. Finally, a new numerical example for the Green’s functions in an orthotropic half-space with different boundary conditions is presented to illustrate the effect of different boundary conditions, as well as material anisotropy, on the half-space Green’s functions.

Implementation of Different Formulas for Special Green’s Functions in a 3D Half-Space BEM

2019 ◽  
Vol 27 (01) ◽  
pp. 1850051 ◽  
Author(s):  
Rafael Piscoya ◽  
Martin Ochmann
Keyword(s):  
Green’S Function ◽  
Boundary Element ◽  
Half Space ◽  
Green's Function ◽  
Numerical Stability ◽  
Green's Functions ◽  
Green’S Functions ◽  
Correction Term ◽  
Second Derivatives ◽  
Derivatives Of

The numerical stability of different formulas for the correction term of the half-space Green’s function is investigated. The formula with complex monopoles is taken as a reference. The expressions of Koh and Yook, tested in a previous publication [R. Piscoya and M. Ochmann, Acoustical Green’s function and boundary element techniques for 3D half-space problems, J. Comput. Acoust. (2017), https://doi.org/10.1142/S0218396X17300018 ], are rewritten to improve their range of application. The formulas of Sommerfeld and Thomasson are analyzed and its suitability for a BEM implementation is evaluated by comparing their accuracy against our reference. For the sake of completeness, the first and second derivatives of the formulas are explicitly written.

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.

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.

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.

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.

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