Prestack phase‐shift migration of separate offsets

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1179-1194 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Phase Shift ◽  
Analytical Solutions ◽  
Stationary Point ◽  
Anisotropic Media ◽  
Phase Method ◽  
Velocity Models ◽  
Analytical Approximations ◽  
Isotropic Media ◽  
Point Solution ◽  
Zero Offset

Prestack phase‐shift migration is implemented by evaluating the offset‐wavenumber ([Formula: see text]) integral using the stationary‐phase method. Thus, the stationary point along [Formula: see text] must be calculated prior to applying the phase shift. This type of implementation allows for migration of separate offsets, as opposed to migration of the whole prestack data when using the original formulas. For zero‐offset data, the stationary point ([Formula: see text]) is known in advance, and, therefore, the phase‐shift migration can be implemented directly. For nonzero‐offset data, we first evaluate the [Formula: see text] that corresponds to the stationary point solution either numerically or through analytical approximations. The insensitivity of the phase to [Formula: see text] around the stationary point solution (its maximum) implies that even an imperfect [Formula: see text] obtained analytically can go a long way to getting an accurate image. In transversely isotropic media, the analytical solutions of the stationary point ([Formula: see text]) include more approximations than those corresponding to isotropic media (i.e., approximations corresponding to weaker anisotropy). Nevertheless, the resultant equations, obtained using Shanks transform and perturbation theory, produce accurate migration signatures for strong anisotropy (η ≈ 0.3) and even large offset‐to‐depth ratios (>2). The analytical solutions are particularly accurate in predicting the nonhyperbolic moveout behavior associated with anisotropic media, a key ingredient to performing an accurate nonhyperbolic moveout inversion for strongly anisotropic media. Although the prestack correction achieved using the phase‐shift method can also be obtained using a cascade of NMO correction, dip‐moveout (DMO) correction, and zero‐offset time migration, the prestack approach can handle sharper velocity models more efficiently. In addition, the resulting operator is sharper than that obtained from the DMO method. Synthetic, including the Marmousi, data applications of the proposed prestack migration demonstrate its usefulness.

scholarly journals Transformation to zero offset in transversely isotropic media

Geophysics ◽  
1996 ◽  
Vol 61 (4) ◽  
pp. 947-963 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Ray Tracing ◽  
Impulse Response ◽  
Homogeneous Medium ◽  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Amplitude Distribution ◽  
Isotropic Media ◽  
Zero Offset ◽  
Vertical Inhomogeneity

Nearly all dip‐moveout correction (DMO) implementations to date assume isotropic homogeneous media. Usually, this has been acceptable considering the tremendous cost savings of homogeneous isotropic DMO and considering the difficulty of obtaining the anisotropy parameters required for effective implementation. In the presence of typical anisotropy, however, ignoring the anisotropy can yield inadequate results. Since anisotropy may introduce large deviations from hyperbolic moveout, accurate transformation to zero‐offset in anisotropic media should address such nonhyperbolic moveout behavior of reflections. Artley and Hale’s v(z) ray‐tracing‐based DMO, developed for isotropic media, provides an attractive approach to treating such problems. By using a ray‐tracing procedure crafted for anisotropic media, I modify some aspects of their DMO so that it can work for v(z) anisotropic media. DMO impulse responses in typical transversely isotropic (TI) models (such as those associated with shales) deviate substantially from the familiar elliptical shape associated with responses in homogeneous isotropic media (to the extent that triplications arise even where the medium is homogeneous). Such deviations can exceed those caused by vertical inhomogeneity, thus emphasizing the importance of taking anisotropy into account in DMO processing. For isotropic or elliptically anisotropic media, the impulse response is an ellipse; but as the key anisotropy parameter η varies, the shape of the response differs substantially from elliptical. For typical η > 0, the impulse response in TI media tends to broaden compared to the response in an isotropic homogeneous medium, a behavior opposite to that encountered in typical v(z) isotropic media, where the response tends to be squeezed. Furthermore, the amplitude distribution along the DMO operator differs significantly from that for isotropic media. Application of this anisotropic DMO to data from offshore Africa resulted in a considerably better alignment of reflections from horizontal and dipping reflectors in common‐midpoint gather than that obtained using an isotropic DMO. Even the presence of vertical inhomogeneity in this medium could not eliminate the importance of considering the shale‐induced anisotropy.

The migrator’s equation for anisotropic media

Geophysics ◽  
1990 ◽  
Vol 55 (11) ◽  
pp. 1429-1434 ◽  
Author(s):  
N. F. Uren ◽  
G. H. F. Gardner ◽  
J. A. McDonald
Keyword(s):  
Physical Model ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Original Structure ◽  
Model Data ◽  
P Waves ◽  
Reflection Seismic ◽  
Isotropic Media ◽  
Zero Offset

The migrator’s equation, which gives the relationship between real and apparent dips on a reflector in zero‐offset reflection seismic sections, may be readily implemented in one step with a frequency‐domain migration algorithm for homogeneous media. Huygens’ principle is used to derive a similar relationship for anisotropic media where velocities are directionally dependent. The anisotropic form of the migrator’s equation is applicable to both elliptically and nonelliptically anisotropic media. Transversely isotropic media are used to demonstrate the performance of an f-k implementation of the migrator’s equation for anisotropic media. In such a medium SH-waves are elliptically anisotropic, while P-waves are nonelliptically anisotropic. Numerical model data and physical model data demonstrate the performance of the algorithm, in each case recovering the original structure. Isotropic and anisotropic migration of anisotropic physical model data are compared experimentally, where the anisotropic velocity function of the medium has a vertical axis of symmetry. Only when anisotropic migration is used is the original structure recovered.

Prestack phase-shift migration of separate offsets in laterally inhomogeneous media

Geophysics ◽  
2010 ◽  
Vol 75 (3) ◽  
pp. S95-S101 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Wave Equation ◽  
Phase Shift ◽  
Synthetic Data ◽  
Real Data ◽  
Inhomogeneous Media ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Fast Wave ◽  
Lateral Inhomogeneity ◽  
Zero Offset

Using the stationary-phase method, prestack phase-shift migration is implemented one offset at a time. This separate-offset implementation allows for a Fourier (reasonably fast) wave-equation-type migration on data with irregular offset sampling. However, the separate-offset phase-shift migration, like its zero-offset counterpart, handles only vertically inhomogeneous media. Using the combination of the split-step and phase-shift-plus-interpolation (PSPI) approaches, the separate-offset phase-shift migration is extended to handle laterally inhomogeneous media. The cost of the separate-offset implementation is practically equivalent to that of the conventional zero-offset version. However, due to the lack of exact source and receiver ray-trajectory information in the separate-offset implementation, the combined split-step and PSPI handles only smooth lateral inhomogeneity. Specifically, it produces images equivalent to those resulting from smoothing the velocity model laterally over a window equal to the half offset. Thus, for zero-offset or laterally homogeneous media, the separate-offset migration is equivalent to any wave-equation-based migration. Errors might occur for finite-offset data in laterally inhomogeneous media. Such errors depend primarily on the strength of lateral inhomogeneity. Using this separate-offset phase-shift migration, accurate images of synthetic data of a model with large reflector dips and good images from real data from offshore Trinidad are obtained.

Solution of Anisotropic Problems of First Class by Coordinate-Transformation

1979 ◽  
Vol 101 (2) ◽  
pp. 340-345 ◽  
Author(s):  
K. C. Poon ◽  
R. C. H. Tsou ◽  
Y. P. Chang
Keyword(s):  
Heat Conduction ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Analytical Solutions ◽  
Coordinate Transformation ◽  
Anisotropic Media ◽  
Anisotropic Problems ◽  
Isotropic Media

In an earlier paper, it was suggested to divide heat conduction problems in anisotropic media into three classes for the systematic presentation of their analytical solutions. It is shown in this paper that a large number of the first-class problems governed by differential equations of hyperbolic, parabolic and elliptic types with boundary conditions of the first, second, third and fourth kinds can be transformed into those for isotropic media. Solutions of two illustrative problems are shown while those of more complicated problems will be reported in subsequent papers.

A Simple Analytical Tool for Legged Robot Design

2012 ◽  
Author(s):  
Zhuohua Shen ◽  
Justin Seipel
Keyword(s):  
Analytical Solutions ◽  
Inverted Pendulum ◽  
Legged Locomotion ◽  
Robot Design ◽  
Slip Model ◽  
Legged Robot ◽  
Biologically Relevant ◽  
Analytical Approximations ◽  
Energy Conserving ◽  
Spring Loaded Inverted Pendulum

Although legged locomotion is better at tackling complicated terrains compared with wheeled locomotion, legged robots are rare, in part, because of the lack of simple design tools. The dynamics governing legged locomotion are generally nonlinear and hybrid (piecewise-continuous) and so require numerical simulation for analysis and are not easily applied to robot designs. During the past decade, a few approximated analytical solutions of Spring-Loaded Inverted Pendulum (SLIP), a canonical model in legged locomotion, have been developed. However, SLIP is energy conserving and cannot predict the dynamical stability of real-world legged locomotion. To develop new analytical tools for legged robot designs, we first analytically solved SLIP in a new way. Then based on SLIP solution, we developed an analytical solution of a hip-actuated Spring-Loaded Inverted Pendulum (hip-actuated-SLIP) model, which is more biologically relevant and stable than the canonical energy conserving SLIP model. The analytical approximations offered here for SLIP and the hip actuated-SLIP solutions compare well with the numerical simulations of each. The analytical solutions presented here are simpler in form than those resulting from existing analytical approximations. The analytical solutions of SLIP and the hip actuated-SLIP can be used as tools for robot design or for generating biological hypotheses.

Quasi‐shear waves in inhomogeneous weakly anisotropic media by the quasi‐isotropic approach: A model study

Geophysics ◽  
2001 ◽  
Vol 66 (1) ◽  
pp. 308-319 ◽  
Author(s):  
Ivan Pšenčík ◽  
Joe A. Dellinger
Keyword(s):  
Order Approximation ◽  
Anisotropic Media ◽  
Ray Method ◽  
Weak Anisotropy ◽  
S Waves ◽  
Model Study ◽  
Arbitrary Strength ◽  
Isotropic Media ◽  
Synthetic Models ◽  
Ray Methods

In inhomogeneous isotropic regions, S-waves can be modeled using the ray method for isotropic media. In inhomogeneous strongly anisotropic regions, the independently propagating qS1- and qS2-waves can similarly be modeled using the ray method for anisotropic media. The latter method does not work properly in inhomogenous weakly anisotropic regions, however, where the split qS-waves couple. The zeroth‐order approximation of the quasi‐isotropic (QI) approach was designed for just such inhomogeneous weakly anisotropic media, for which neither the ray method for isotropic nor anisotropic media applies. We test the ranges of validity of these three methods using two simple synthetic models. Our results show that the QI approach more than spans the gap between the ray methods: it can be used in isotropic regions (where it reduces to the ray method for isotropic media), in regions of weak anisotropy (where the ray method for anisotropic media does not work properly), and even in regions of moderately strong anisotropy (in which the qS-waves decouple and thus could be modeled using the ray method for anisotropic media). A modeling program that switches between these three methods as necessary should be valid for arbitrary‐strength anisotropy.

Normal moveout from dipping reflectors in anisotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 268-284 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Weak Anisotropy ◽  
Anisotropic Models ◽  
Transversely Isotropic Media ◽  
Wide Range ◽  
Isotropic Media ◽  
The Difference

Description of reflection moveout from dipping interfaces is important in developing seismic processing methods for anisotropic media, as well as in the inversion of reflection data. Here, I present a concise analytic expression for normal‐moveout (NMO) velocities valid for a wide range of homogeneous anisotropic models including transverse isotropy with a tilted in‐plane symmetry axis and symmetry planes in orthorhombic media. In transversely isotropic media, NMO velocity for quasi‐P‐waves may deviate substantially from the isotropic cosine‐of‐dip dependence used in conventional constant‐velocity dip‐moveout (DMO) algorithms. However, numerical studies of NMO velocities have revealed no apparent correlation between the conventional measures of anisotropy and errors in the cosine‐of‐dip DMO correction (“DMO errors”). The analytic treatment developed here shows that for transverse isotropy with a vertical symmetry axis, the magnitude of DMO errors is dependent primarily on the difference between Thomsen parameters ε and δ. For the most common case, ε − δ > 0, the cosine‐of‐dip–corrected moveout velocity remains significantly larger than the moveout velocity for a horizontal reflector. DMO errors at a dip of 45 degrees may exceed 20–25 percent, even for weak anisotropy. By comparing analytically derived NMO velocities with moveout velocities calculated on finite spreads, I analyze anisotropy‐induced deviations from hyperbolic moveout for dipping reflectors. For transversely isotropic media with a vertical velocity gradient and typical (positive) values of the difference ε − δ, inhomogeneity tends to reduce (sometimes significantly) the influence of anisotropy on the dip dependence of moveout velocity.

P–S converted wave: conversion point and zero-offset energy in anisotropic media

2004 ◽  
Vol 56 (3) ◽  
pp. 155-163 ◽  
Author(s):  
Fredy A.V. Artola ◽  
Ricardo Leiderman ◽  
Sergio A.B. Fontoura ◽  
Mércia B.C. Silva
Keyword(s):  
Anisotropic Media ◽  
Converted Wave ◽  
Conversion Point ◽  
Zero Offset ◽  
Wave Conversion
Sign in / Sign up

Export Citation Format

Share Document