Comparison of weights in prestack amplitude‐preserving Kirchhoff depth migration

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1812-1816 ◽  
Author(s):  
Christian Hanitzsch
Keyword(s):  
Point Source ◽  
Green’S Function ◽  
Ray Tracing ◽  
Green's Function ◽  
Weight Functions ◽  
Depth Migration ◽  
Theoretical Approaches ◽  
Geometric Spreading ◽  
Dynamic Ray Tracing ◽  
Kirchhoff Depth Migration

Three different theoretical approaches to amplitude‐preserving Kirchhoff depth migration are compared. Each of them suggests applying weights in the diffraction stack migration to correct for amplitude loss resulting from geometric spreading. The weight functions are given in different notations, but as is shown, all of these expressions are similar. A notation that is well suited for implementation is suggested: entirely in terms of Green's function quantities (amplitudes or point‐source propagators). For the most common prestack configurations (common‐shot and common‐offset) and 3-D, 2.5-D, and 2-D migrations, expressions of the weights are given in this notation. The quantities needed for calculation of the weights can be computed easily, e.g., by dynamic ray tracing.

True-amplitude Kirchhoff depth migration in anisotropic media: The traveltime-based approach

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. WC33-WC39 ◽  
Author(s):  
Claudia Vanelle ◽  
Dirk Gajewski
Keyword(s):  
Ray Tracing ◽  
Reservoir Characterization ◽  
Seismic Anisotropy ◽  
Anisotropic Media ◽  
Weight Functions ◽  
Depth Migration ◽  
Dynamic Ray Tracing ◽  
Isotropic Media ◽  
Structural Image ◽  
Kirchhoff Depth Migration

True-amplitude Kirchhoff depth migration is a classic tool in seismic imaging. In addition to a focused structural image, it also provides information on the strength of the reflectors in the model, leading to estimates of the shear properties of the subsurface. This information is a key feature not only for reservoir characterization, but it is also important for detecting seismic anisotropy. If anisotropy is present, it needs to be accounted for during the migration. True-amplitude Kirchhoff depth migration is carried out in terms of a weighted diffraction stack. Expressions for suitable weight functions exist in anisotropic media. However, the conventional means of computing the weights is based on dynamic ray tracing, which has high requirements on the smoothness of the underlying model. We developed a method for the computation of the weight functions that does not require dynamic ray tracing because all necessary quantities are determined from traveltimes alone. In addition, the method led to considerable savings in computational costs. This so-called traveltime-based strategy was already introduced for isotropic media. We extended the strategy to incorporate anisotropy. For verification purposes and comparison to analytic references, we evaluated 2.5D migration examples for [Formula: see text] and [Formula: see text] reflections. Our results confirmed the high image quality and the accuracy of the reconstructed reflectivities.

Amplitude, Fresnel zone, and NMO velocity for PP and SS normal-incidence reflections

Geophysics ◽  
2006 ◽  
Vol 71 (2) ◽  
pp. W1-W14 ◽  
Author(s):  
Einar Iversen
Keyword(s):  
Phase Shift ◽  
Ray Tracing ◽  
Explicit Knowledge ◽  
Fresnel Zone ◽  
Normal Wave ◽  
Normal Incidence ◽  
Geometric Spreading ◽  
Dynamic Ray Tracing ◽  
The One ◽  
Paraxial Ray

Inspired by recent ray-theoretical developments, the theory of normal-incidence rays is generalized to accommodate P- and S-waves in layered isotropic and anisotropic media. The calculation of the three main factors contributing to the two-way amplitude — i.e., geometric spreading, phase shift from caustics, and accumulated reflection/transmission coefficients — is formulated as a recursive process in the upward direction of the normal-incidence rays. This step-by-step approach makes it possible to implement zero-offset amplitude modeling as an efficient one-way wavefront construction process. For the purpose of upward dynamic ray tracing, the one-way eigensolution matrix is introduced, having as minors the paraxial ray-tracing matrices for the wavefronts of two hypothetical waves, referred to by Hubral as the normal-incidence point (NIP) wave and the normal wave. Dynamic ray tracing expressed in terms of the one-way eigensolution matrix has two advantages: The formulas for geometric spreading, phase shift from caustics, and Fresnel zone matrix become particularly simple, and the amplitude and Fresnel zone matrix can be calculated without explicit knowledge of the interface curvatures at the point of normal-incidence reflection.

A Green’s Function Formulation of Anticracks and Their Interaction With Load-Induced Singularities

1989 ◽  
Vol 56 (3) ◽  
pp. 550-555 ◽  
Author(s):  
John Dundurs ◽  
Xanthippi Markenscoff
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Edge Dislocation ◽  
Weight Functions ◽  
Bimaterial Interface ◽  
Elastic Fields ◽  
Function Formulation ◽  
Concentrated Forces ◽  
Concentrated Couple ◽  
Working Out

This paper provides a Green’s function formulation of anticracks (rigid lamellar inclusions of negligible thickness that are bonded to the surrounding elastic material). Apart from systematizing several previously known solutions, the article gives the pertinent fields for concentrated forces, dislocations, and a concentrated couple applied on the line of the anticrack. There is a reason for working out these results: In contrast to concentrated forces, a concentrated couple approaching the tip of an anticrack makes the elastic fields explode. Finite limits can be achieved, however, by appropriately diminishing the magnitude of the couple, which then leads to fields that are intimately connected with the weight functions for the anticrack. An edge dislocation going to the tip of an anticrack puts a net force on the lamellar inclusion, which is shown to be related to a previously known feature of dislocations near a bimaterial interface.

Etude numérique des modes et fréquences propres d'une cavité à l'aide de la fonction de Green

1979 ◽  
Vol 57 (2) ◽  
pp. 208-217
Author(s):  
Jacques A. Imbeau ◽  
Byron T. Darling
Keyword(s):  
Point Source ◽  
Green’S Function ◽  
Green's Function ◽  
Normal Modes ◽  
Preceding Paper ◽  
Double Precision ◽  
Prolate Spheroidal ◽  
Normal Functions ◽  
Normal Frequency ◽  
High Degree

We apply the methods developed in our preceding paper (J. A. Imbeau and B. T. Darling. Can. J. Phys. 57, 190(1979)) for calculating the Green's function of a cavity to obtain the normal modes and normal frequencies of the cavity. As the frequency of the driving point source approaches that of a normal frequency the response (Green's function) of the cavity becomes infinite, and the form of the Green's function is dominated by the normal mode. There is also a 180° reversal of phase in passing through a resonance. In this way, we are able to calculate the normal frequencies of prolate spheroidal cavities to the full precision employed in the calculations (16 significant digits for double precision of the IBM-370). The Green's functions and the normal functions are also obtainable to a high degree of precision, except in the immediate vicinity of the surface of the cavity where they suffer a well-known discontinuity.

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