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.

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.

Acoustic anisotropic wavefields through perturbation theory

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. WC41-WC50 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Perturbation Theory ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Symmetry Axis ◽  
Finite Difference Methods ◽  
Transversely Isotropic ◽  
Direct Access ◽  
Acoustic Wave Equation ◽  
Anisotropy Parameters ◽  
The Cost

Solving the anisotropic acoustic wave equation numerically using finite-difference methods introduces many problems and media restriction requirements, and it rarely contributes to the ability to resolve the anisotropy parameters. Among these restrictions are the inability to handle media with [Formula: see text] and the presence of shear-wave artifacts in the solution. Both limitations do not exist in the solution of the elliptical anisotropic acoustic wave equation. Using perturbation theory in developing the solution of the anisotropic acoustic wave equation allows direct access to the desired limitation-free solutions, that is, solutions perturbed from the elliptical anisotropic background medium. It also provides a platform for parameter estimation because of the ability to isolate the wavefield dependency on the perturbed anisotropy parameters. As a result, I derive partial differential equations that relate changes in the wavefield to perturbations in the anisotropy parameters. The solutions of the perturbation equations represented the coefficients of a Taylor-series-type expansion of the wavefield as a function of the perturbed parameter, which is in this case [Formula: see text] or the tilt of the symmetry axis. The expansion with respect to the symmetry axis allows use of an acoustic transversely isotropic media with a vertical symmetry axis (VTI) kernel to estimate the background wavefield and the corresponding perturbation coefficients. The VTI extrapolation kernel is about one-fourth the cost of the transversely isotropic model with a tilt in the symmetry axis kernel. Thus, for a small symmetry axis tilt, the cost of migration using a first-order expansion can be reduced. The effectiveness of the approach was demonstrated on the Marmousi model.

ADAPTIVE FINITE ELEMENT TECHNIQUES FOR THE ACOUSTIC WAVE EQUATION

2001 ◽  
Vol 09 (02) ◽  
pp. 575-591 ◽  
Author(s):  
WOLFGANG BANGERTH ◽  
ROLF RANNACHER
Keyword(s):  
Finite Element ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Efficient Computation ◽  
Acoustic Wave Equation ◽  
Grid Adaptation ◽  
Local Cell ◽  
Error Indicators

We present an adaptive finite element method for solving the acoustic wave equation. Using a global duality argument and Galerkin orthogonality, we derive an identity for the error with respect to an arbitrary functional output of the solution. The error identity is evaluated by solving the dual problem numerically. The resulting local cell-wise error indicators are used in the grid adaptation process. In this way, the space-time mesh can be tailored for the efficient computation of the quantity of interest. We give an overview of the implementation of the proposed method and illustrate its performance by several numerical examples.

Simulation of acoustic wavefields in heterogeneous media: A robust method for automatic suppression of numerical dispersion

Geophysics ◽  
2010 ◽  
Vol 75 (3) ◽  
pp. T99-T110 ◽  
Author(s):  
Dinghui Yang ◽  
Guojie Song ◽  
Biaolong Hua ◽  
Henri Calandra
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Large Scale ◽  
Heterogeneous Media ◽  
Numerical Dispersion ◽  
Acoustic Wave Equation ◽  
Discrete Method ◽  
Computational Costs ◽  
Refined Method

Numerical dispersion limits the application of numerical simulation methods for solving the acoustic wave equation in large-scale computation. The nearly analytic discrete method (NADM) and its improved version for suppressing numerical dispersion were developed recently. This new method is a refinement of two previous methods and further increases the ability of suppressing numerical dispersion for modeling acoustic wave propagation in 2D heterogeneous media, which uses acoustic wave displacement, particle velocity, and their gradients to restructure the acoustic wavefield via the truncated Taylor expansion and the high-order interpolation approximate method. For the method proposed, we investigate its implementation and compare it with the higher-order Lax-Wendroff correction (LWC) scheme, the original nearly analytic discrete method (NADM) and its im-proved version with regard to numerical dispersion, computational costs, and computer storage requirements. The numerical dispersion relations provided by the refined algorithm for 1D and 2D cases are analyzed, as well as the numerical results obtained by this method against the exact solution for the 2D acoustic case. Numerical results show that the refined method gives no visible numerical dispersion for very large spatial grid increments. It can simulate high-frequency wave propagation for a given grid interval and automatically suppress the numerical dispersion when the acoustic wave equation is discretized, when too few samples per wavelength are used, or when models have large velocity contrasts. Unlike the high-order LWC methods, our present method can save substantial computational costs and memory requirements because very large grid increments can be used. The refined method can be used for the simulation of large-scale wavefields.

An acoustic wave equation for anisotropic media

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1239-1250 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Computation Time ◽  
Transversely Isotropic ◽  
Acoustic Medium ◽  
Acoustic Wave Equation ◽  
P Waves ◽  
Kinematic Approximation ◽  
Vti Media

A wave equation, derived using the acoustic medium assumption for P-waves in transversely isotropic (TI) media with a vertical symmetry axis (VTI media), yields a good kinematic approximation to the familiar elastic wave equation for VTI media. The wavefield solutions obtained using this VTI acoustic wave equation are free of shear waves, which significantly reduces the computation time compared to the elastic wavefield solutions for exploding‐reflector type applications. From this VTI acoustic wave equation, the eikonal and transport equations that describe the ray theoretical aspects of wave propagation in a TI medium are derived. These equations, based on the acoustic assumption (shear wave velocity = 0), are much simpler than their elastic counterparts, yet they yield an accurate description of traveltimes and geometrical amplitudes. Numerical examples prove the usefulness of this acoustic equation in simulating the kinematic aspects of wave propagation in complex TI models.

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.

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