Target-oriented wave-equation least-squares migration/inversion with phase-encoded Hessian

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCA95-WCA107 ◽  
Author(s):  
Yaxun Tang
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Green's Functions ◽  
Green’S Functions ◽  
Random Phase ◽  
Inverse Filtering ◽  
Data Set ◽  
Phase Encoding ◽  
Hessian Operator ◽  
Least Squares Migration

Prestack depth migration produces blurred images resulting from limited acquisition apertures, complexities in the velocity model, and band-limited characteristics of seismic waves. This distortion can be partially corrected using the model-space least-squares migration/inversion approach, where a target-oriented wave-equation Hessian operator is computed explicitly and then inverse filtering is applied iteratively to deblur or invert for the reflectivity. However, one difficulty is the cost of computing the explicit Hessian operator, which requires storing a large number of Green’s functions, making it challenging for large-scale applications. A new method to compute the Hessian operator for the wave-equation-based least-squares migration/inversion problem modifies the original explicit Hessian formula, enabling efficient computation of this operator. An advantage is that the method eliminates disk storage of Green’s functions. The modifications, however, also introduce undesired crosstalk artifacts. Two different phase-encoding schemes, plane-wave-phase encoding and random-phase encoding, suppress the crosstalk. When the randomly phase-encoded Hessian operator is applied to the Sigsbee2A synthetic data set, an improved subsalt image with more balanced amplitudes is obtained.

Least‐squares wave‐equation migration for AVP/AVA inversion

Geophysics ◽  
2003 ◽  
Vol 68 (1) ◽  
pp. 262-273 ◽  
Author(s):  
Henning Kühl ◽  
Mauricio D. Sacchi
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Inversion Algorithm ◽  
Constrained Least Squares ◽  
Data Set ◽  
Parameter Domain ◽  
First Case ◽  
Least Squares Migration ◽  
Ray Parameter ◽  
Seismic Wavefield

We present an acoustic migration/inversion algorithm that uses extended double‐square‐root wave‐equation migration and modeling operators to minimize a constrained least‐squares data misfit function (least‐squares migration). We employ an imaging principle that allows for the extraction of ray‐parameter‐domain common image gathers (CIGs) from the propagated seismic wavefield. The CIGs exhibit amplitude variations as a function of half‐offset ray parameter (AVP) closely related to the amplitude variation with reflection angle (AVA). Our least‐squares wave‐equation migration/inversion is constrained by a smoothing regularization along the ray parameter. This approach is based on the idea that rapid amplitude changes or discontinuities along the ray parameter axis result from noise, incomplete wavefield sampling, and numerical operator artifacts. These discontinuities should therefore be penalized in the inversion. The performance of the proposed algorithm is examined with two synthetic examples. In the first case, we generated acoustic finite difference data for a horizontally layered model. The AVP functions based on the migrated/inverted ray parameter CIGs were converted to AVA plots. The AVA plots were then compared to the true acoustic AVA of the reflectors. The constrained least‐squares inversion compares favorably with the conventional migration, especially when incompleteness compromises the data. In the second example, we use the Marmousi data set to test the algorithm in complex media. The result shows that least‐squares migration can mitigate kinematic artifacts in the ray‐parameter domain CIGs effectively.

Green’s function implementation of common‐offset, wave‐equation migration

Geophysics ◽  
1996 ◽  
Vol 61 (6) ◽  
pp. 1813-1821 ◽  
Author(s):  
Andreas Ehinger ◽  
Patrick Lailly ◽  
Kurt J. Marfurt
Keyword(s):  
Wave Equation ◽  
Green's Functions ◽  
Wave Equations ◽  
Green’S Functions ◽  
Data Set ◽  
Kirchhoff Migration ◽  
Migration Velocity Analysis ◽  
Standard Wave ◽  
The Cost ◽  
Wavefield Extrapolation

Common‐offset migration is extremely important in the context of migration velocity analysis (MVA) since it generates geologically interpretable migrated images. However, only a wave‐equation‐based migration handles multipathing of energy in contrast to the popular Kirchhoff migration with first‐arrival traveltimes. We have combined the superior treatment of multipathing of energy by wave‐equation‐based migration with the advantages of the common‐offset domain for MVA by implementing wave‐equation migration algorithms via the use of finite‐difference Green’s functions. With this technique, we are able to apply wave‐equation migration in measurement configurations that are usually considered to be of the realm of Kirchhoff migration. In particular, wave‐equation migration of common offset sections becomes feasible. The application of our wave‐equation, common‐offset migration algorithm to the Marmousi data set confirms the large increase in interpretability of individual migrated sections, for about twice the cost of standard wave‐equation common‐shot migration. Our implementation of wave‐equation migration via the Green’s functions is based on wavefield extrapolation via paraxial one‐way wave equations. For these equations, theoretical results allow us to perform exact inverse extrapolation of wavefields.

scholarly journals Deep convolutional neural network and sparse least-squares migration

Geophysics ◽  
2020 ◽  
Vol 85 (4) ◽  
pp. WA241-WA253 ◽  
Author(s):  
Zhaolun Liu ◽  
Yuqing Chen ◽  
Gerard Schuster
Keyword(s):  
Neural Network ◽  
Convolutional Neural Network ◽  
Least Squares ◽  
Green's Functions ◽  
Green’S Functions ◽  
Computational Cost ◽  
Feature Maps ◽  
The Neural Network ◽  
Deep Cnn ◽  
Least Squares Migration

We have recast the forward pass of a multilayered convolutional neural network (CNN) as the solution to the problem of sparse least-squares migration (LSM). The CNN filters and feature maps are shown to be analogous, but not equivalent, to the migration Green’s functions and the quasi-reflectivity distribution, respectively. This provides a physical interpretation of the filters and feature maps in deep CNN in terms of the operators for seismic imaging. Motivated by the connection between sparse LSM and CNN, we adopt the neural network version of sparse LSM. Unlike the standard LSM method that finds the optimal reflectivity image, neural network LSM (NNLSM) finds the optimal quasi-reflectivity image and the quasi-migration Green’s functions. These quasi-migration Green’s functions are also denoted as the convolutional filters in a CNN and are similar to migration Green’s functions. The advantage of NNLSM over standard LSM is that its computational cost is significantly less and it can be used for denoising coherent and incoherent noise in migration images. Its disadvantage is that the NNLSM quasi-reflectivity image is only an approximation to the actual reflectivity distribution. However, the quasi-reflectivity image can be used as an attribute image for high-resolution delineation of geologic bodies.

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.

Migration deconvolution using domain decomposition

Geophysics ◽  
2021 ◽  
pp. 1-61
Author(s):  
Luana Nobre Osorio ◽  
Bruno Pereira-Dias ◽  
André Bulcão ◽  
Luiz Landau
Keyword(s):  
Domain Decomposition ◽  
Least Squares ◽  
Iterative Algorithms ◽  
Image Resolution ◽  
Image Domain ◽  
Hessian Operator ◽  
Incomplete Coverage ◽  
And Migration ◽  
Least Squares Migration ◽  
Data Frequency

Least-squares migration (LSM) is an effective technique for mitigating blurring effects and migration artifacts generated by the limited data frequency bandwidth, incomplete coverage of geometry, source signature, and unbalanced amplitudes caused by complex wavefield propagation in the subsurface. Migration deconvolution (MD) is an image-domain approach for least-squares migration, which approximates the Hessian operator using a set of precomputed point spread functions (PSFs). We introduce a new workflow by integrating the MD and the domain decomposition (DD) methods. The DD techniques aim to solve large and complex linear systems by splitting problems into smaller parts, facilitating parallel computing, and providing a higher convergence in iterative algorithms. The following proposal suggests that instead of solving the problem in a unique domain, as conventionally performed, we split the problem into subdomains that overlap and solve each of them independently. We accelerate the convergence rate of the conjugate gradient solver by applying the DD methods to retrieve a better reflectivity, which is mainly visible in regions with low amplitudes. Moreover, using the pseudo-Hessian operator, the convergence of the algorithm is accelerated, suggesting that the inverse problem becomes better conditioned. Experiments using the synthetic Pluto model demonstrate that the proposed algorithm dramatically reduces the required number of iterations while providing a considerable enhancement in the image resolution and better continuity of poorly illuminated events.

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.

Wick-Regeln mit Zwischenzuständen und die Neue Tamm-Dancoff-Methode mit Zwischenzuständen / On Wick Rules with Intermediate States and the New Tamm-Dancoff Method with Intermediate States

1976 ◽  
Vol 31 (8) ◽  
pp. 872-886 ◽  
Author(s):  
A. Friederich ◽  
W. Gerling
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Equations Of Motion ◽  
Random Phase ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Great Success ◽  
Hartree Fock ◽  
Intermediate States ◽  
Many Body Systems

AbstractInstead of emphasizing the ground state as is done in Green's function method, we take a finite-dimensional subspace of the Hilbert space: the space of the "intermediate states". A systematic introduction of intermediate states is effected by an extension of the method of generating functionals: we combine the generating functionals of the n-point Green's functions to a "matrix functional" T, and form new matrix functionals, which are matrix functions of T. The aim of this paper is to develop the functional calculus in such a way that the transition from scalar functionals to matrix functionals is straightforward, and the method of obtaining further results becomes clear. Following the lines of Dürr and Wagner we get u η-and ζ-rules with intermediate states". Using them we define a truncation procedure for the equations of motion of the n-point Green's functions, the "New Tamm-Dancoff method with intermediate states". This extension makes it possible to treat the effect of nearby levels in many body systems with Green's functions. In ad-dition to well-known approximations, such as the Hartree-Fock and the Hartree-Bogoliubov theory, the RPA and the quasiparticle RPA, we obtain a series of new approximations. Among these are the "Hartree-Fock theory with intermediate states" and the "random-phase approximation with intermediate states", which we already applied with great success to some exactly soluble models.

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