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

1991 ◽  
Author(s):  
Christopher Liner
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Constant Density ◽  
Acoustic Wave Equation

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.

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.

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.

scholarly journals Extracting angle domain common image gather with variable density acoustic-wave equation

2021 ◽  
Vol 18 (2) ◽  
pp. 1-8
Author(s):  
Yuzhu Liu ◽  
Weigang Liu ◽  
Jizhong Yang ◽  
Liangguo Dong
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Velocity Model ◽  
Variable Density ◽  
Constant Density ◽  
Acoustic Wave Equation ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Density Perturbations

Abstract Angle domain common image gathers (ADCIGs), commonly regarded as important prestacked gathers, provide the information required for velocity model construction and the phase and amplitude information needed for subsurface structures in oil/gas exploration. Based on the constant-density acoustic-wave equation assumption, the ADCIGs generated from reverse time migration ignore the fact that the subsurface density varies with location. Consequently, the amplitude versus angle (AVA) analysis extracted from these ADCIGs is not accurate. To partially solve this problem and to improve the accuracy of the AVA analysis, we developed amplitude-preserving ADCIGs suitable for density variations with the assumption of acoustic approximation. The Poynting vector approach, which is efficient and computationally inexpensive, was used to calculate the high-resolution wavefield propagation. The ADCIGs generated from the velocity and density perturbations match the theoretical AVA relationship better than ADCIGs with constant density. The extraction of the AVA analysis of the various combinations of the subsurface medium indicates that the density is non-negligible, especially when the density contrast is sharp. Numerical examples based on a layered model verify our conclusions.

An analytic signal based accurate time-domain visco-acoustic wave equation from the Constant-Q theory

Geophysics ◽  
2021 ◽  
pp. 1-58
Author(s):  
Hongwei Liu ◽  
Yi Luo
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Time Domain ◽  
Seismic Waves ◽  
Heterogeneous Media ◽  
Relaxation Property ◽  
Analytic Signal ◽  
Acoustic Wave Equation ◽  
Real Component ◽  
Frequency Components

We present a concise time-domain wave equation to accurately simulate wave propagation in visco-acoustic media. The central idea behind this work is to dismiss the negative frequency components from a time-domain signal by converting the signal to its analytic format. The negative frequency components of any analytic signal are always zero, meaning we can construct the visco-acoustic wave equation to honor the relaxation property of the media for positive frequencies only. The newly proposed complex-valued wave equation (CWE) represents the wavefield with its analytic signal, whose real part is the desired physical wavefield, while the imaginary part is the Hilbert transform of the real component. Specifically, this CWE is accurate for both weak and strong attenuating media in terms of both dissipation and dispersion and the attenuation is precisely linear with respect to the frequencies. Besides, the CWE is easy and flexible to model dispersion-only, dissipation-only or dispersion-plus-dissipation seismic waves. We have verified these CWEs by comparing the results with analytical solutions, and achieved nearly perfect matching. Except for the homogeneous Q media, we have also extended the CWEs to heterogeneous media. The results of the CWEs for heterogeneous Q media are consistent with those computed from the nonstationary operator based Fourier Integral method and from the Standard Linear Solid (SLS) equations.

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