A two‐way nonreflecting wave equation

Geophysics ◽  
1984 ◽  
Vol 49 (2) ◽  
pp. 132-141 ◽  
Author(s):  
Edip Baysal ◽  
Dan D. Kosloff ◽  
J. W. C. Sherwood
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Forward Modeling ◽  
Conventional Approach ◽  
Reflection Coefficients ◽  
Constant Density ◽  
Acoustic Wave Equation ◽  
Seismic Modeling ◽  
Homogeneous Regions ◽  
Dominant Direction

In seismic modeling and in migration it is often desirable to use a wave equation (with varying velocity but constant density) which does not produce interlayer reverberations. The conventional approach has been to use a one‐way wave equation which allows energy to propagate in one dominant direction only, typically this direction being either upward or downward (Claerbout, 1972). We introduce a two‐way wave equation which gives highly reduced reflection coefficients for transmission across material boundaries. For homogeneous regions of space, however, this wave equation becomes identical to the full acoustic wave equation. Possible applications of this wave equation for forward modeling and for migration are illustrated with simple models.

Nonzero offset seismic forward modeling by one‐way acoustic wave‐equation

1998 ◽  
Author(s):  
He Zhenhua ◽  
Xiong Gaojun ◽  
Zhang Yixiang
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Forward Modeling ◽  
Acoustic Wave Equation

scholarly journals ONE-STEP EXTRAPOLATION METHOD USING CHEBYSHEV’S RECURRENCE RELATION APPLIED TO SEISMIC MODELING

2016 ◽  
Vol 34 (4) ◽  
Author(s):  
Laura Lara Ortiz ◽  
Reynam C. Pestana
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Difference Scheme ◽  
Chebyshev Polynomials ◽  
Finite Difference Scheme ◽  
Acoustic Wave Equation ◽  
Seismic Modeling ◽  
First Order ◽  
One Step

ABSTRACT. In this work we show that the solution of the first order differential wave equation for an analytical wavefield, using a finite-difference scheme in time, follows exactly the same recursion of modified Chebyshev polynomials. Based on this, we proposed a numerical...Keywords: seismic modeling, acoustic wave equation, analytical wavefield, Chebyshev polinomials. RESUMO. Neste trabalho, mostra-se que a solução da equação de onda de primeira ordem com um campo de onda analítico usando um esquema de diferenças finitas no tempo segue exatamente a relação de recorrência dos polinômios modificados de Chebyshev. O algoritmo...Palavras-chave: modelagem sísmica, equação da onda acústica, campo analítico, polinômios de Chebyshev.

Modified imaging principle: An alternative to preserved-amplitude least-squares wave-equation migration

Geophysics ◽  
2007 ◽  
Vol 72 (6) ◽  
pp. S221-S230 ◽  
Author(s):  
Denis Kiyashchenko ◽  
René-Edouard Plessix ◽  
Boris Kashtan
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Least Squares ◽  
Forward Modeling ◽  
Acoustic Wave Equation ◽  
Impedance Contrast ◽  
Least Squares Migration

Impedance contrast images can result from a least-squares migration or from a modified imaging principle. Theoretically, the two approaches should give similar results, but in practice they lead to different estimates of the impedance contrasts because of limited acquisition geometry, difficulty in computing exact weights for least-squares migration, and small contrast approximation. To analyze those differences, we compare the two approaches based on 2D synthetics. Forward modeling is either a finite-difference solver of the full acoustic wave equation or a one-way wave-equation solver that correctly models the amplitudes. The modified imaging principle provides better amplitude estimates of the impedance contrasts and does not suffer from the artifacts at-tributable to diving waves, which can be seen in two-way, least-squares migrated sections. However, because of the shot-based formulation, artifacts appear in the modified imaging principle results in shadow zones where energy is defocused. Those artifacts do not exist with the least-squares migration algorithm because all shots are processed simultane-ously.

A comparison between one‐way and two‐way wave‐equation migration

Geophysics ◽  
2004 ◽  
Vol 69 (6) ◽  
pp. 1491-1504 ◽  
Author(s):  
W. A. Mulder ◽  
R.‐E. Plessix
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Computational Cost ◽  
Real Data ◽  
Spatial Filtering ◽  
Two Dimensions ◽  
Data Migration ◽  
Constant Density ◽  
Acoustic Wave Equation ◽  
High Pass

Results for wave‐equation migration in the frequency domain using the constant‐density acoustic two‐way wave equation have been compared to images obtained by its one‐way approximation. The two‐way approach produces more accurate reflector amplitudes and provides superior imaging of steep flanks. However, migration with the two‐way wave equation is sensitive to diving waves, leading to low‐frequency artifacts in the images. These can be removed by surgical muting of the input data or iterative migration or high‐pass spatial filtering. The last is the most effective. Iterative migration based on a least‐squares approximation of the seismic data can improve the amplitudes and resolution of the imaged reflectors. Two approaches are considered, one based on the linearized constant‐density acoustic wave equation and one on the full acoustic wave equation with variable density. The first converges quickly. However, with our choice of migration weights and high‐pass spatial filtering for the linearized case, a real‐data migration result shows little improvement after the first iteration. The second, nonlinear iterative migration method is considerably more difficult to apply. A real‐data example shows only marginal improvement over the linearized case. In two dimensions, the computational cost of the two‐way approach has the same order of magnitude as that for the one‐way method. With our implementation, the two‐way method requires about twice the computer time needed for one‐way wave‐equation migration.

A brief comparison of some Kirchhoff integral formulas for migration and inversion

Geophysics ◽  
1991 ◽  
Vol 56 (8) ◽  
pp. 1164-1169 ◽  
Author(s):  
Paul Docherty
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Reflection Coefficients ◽  
Acoustic Wave Equation ◽  
Inversion Formulas ◽  
Kirchhoff Migration ◽  
Integral Formulas ◽  
The One ◽  
And Inversion ◽  
Kirchhoff Integral

Kirchhoff migration has traditionally been viewed as an imaging procedure. Usually, few claims are made regarding the amplitudes in the imaged section. In recent years, a number of inversion formulas, similar in form to those of Kirchhoff migration, have been proposed. A Kirchhoff‐type inversion produces not only an image but also an estimate of velocity variations, or perhaps reflection coefficients. The estimate is obtained from the peak amplitudes in the image. In this paper prestack Kirchhoff migration and inversion formulas for the one‐parameter acoustic wave equation are compared. Following a heuristic approach based on the imaging principle, a migration formula is derived which turns out to be identical to one proposed by Bleistein for inversion. Prestack Kirchhoff migration and inversion are, thus, seen to be the same—both in terms of the image produced and the peak amplitudes of the output.

Theory of a 2.5-D acoustic wave equation for constant density media

Geophysics ◽  
1991 ◽  
Vol 56 (12) ◽  
pp. 2114-2117 ◽  
Author(s):  
Christopher L. Liner
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Symmetry Axis ◽  
Constant Density ◽  
Efficient Computation ◽  
Acoustic Wave Equation ◽  
Out Of Plane ◽  
Acoustic Media ◽  
D Wave

The theory of 2.5-dimensional (2.5-D) wave propagation (Bleistein, 1986) allows efficient computation of 3-D wavefields in c(x, z) acoustic media when the source and receivers lie in a common y-plane (assumed to be y = 0 in this paper). It is really a method of efficiently computing an inplane 3-D wavefield in media with one symmetry axis. The idea is to raytrace the wavefield in the (x, z)-plane while allowing for out‐of‐plane spreading. In this way 3-D amplitude decay is honored without 3-D ray tracing. This theory has its conceptual origin in work by Ursin (1978) and Hubral (1978). Bleistein (1986) gives an excellent overview and detailed reference to earlier work.

3-D acoustic wave equation forward modeling with topography

2007 ◽  
Vol 4 (1) ◽  
pp. 8-15 ◽  
Author(s):  
Xiangchun Wang ◽  
Xuewei Liu
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Forward Modeling ◽  
Acoustic Wave Equation
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