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 An acoustic wave equation for orthorhombic anisotropy

Geophysics ◽  
2003 ◽  
Vol 68 (4) ◽  
pp. 1169-1172 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Transversely Isotropic ◽  
Order Equation ◽  
Sixth Order ◽  
Acoustic Medium ◽  
Complex Conjugate ◽  
Acoustic Wave Equation ◽  
P Waves ◽  
Isotropic Media

Using a dispersion relation derived under the acoustic medium assumption, I obtain an acoustic wave equation for orthorhombic media. Although an acoustic wave equation does not strictly describe a wave in anisotropic media, it accurately describes the kinematics of P‐waves. The orthorhombic acoustic wave equation, unlike the transversely isotropic one, is a sixth‐order equation with three sets of complex conjugate solutions. Only one set of these solutions are perturbations of the familiar acoustic wavefield solution for isotropic media for incoming and outgoing P‐waves and, thus, are of interest here. The other two sets of solutions are simply the result of this artificially derived sixth‐order equation.

Hybrid Fourier finite-difference 3D depth migration for anisotropic media

Geophysics ◽  
2008 ◽  
Vol 73 (2) ◽  
pp. S27-S34 ◽  
Author(s):  
Tong W. Fei ◽  
Christopher L. Liner
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Seismic Data ◽  
Transversely Isotropic ◽  
Acoustic Wave Equation ◽  
Near Surface ◽  
Depth Migration ◽  
Surface Distortions ◽  
Vti Media

When a subsurface is anisotropic, migration based on the assumption of isotropy will not produce accurate migration images. We develop a hybrid wave-equation migration algorithm for vertical transversely isotropic (VTI) media based on a one-way acoustic wave equation, using a combination of Fourier finite-difference (FFD) and finite-difference (FD) approaches. The hybrid method can suppress an additional solution that exists in the VTI acoustic wave equation, and it offers speed and other advantages over conventional FFD or FD methods alone. The algorithm is tested on a synthetic model involving log data from onshore eastern Saudi Arabia, including estimates of both intrinsic and layer-induced VTI parameters. Results indicate that VTI imaging in this region offers some improvement over isotropic imaging, primarily with respect to subtle structure and stratigraphy and to image continuity. These benefits probably will be overshadowed by perennial land seismic data issues such as near-surface distortions and multiples.

Computational methods for large-scale 3D acoustic finite-difference modeling: A tutorial

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. SM223-SM230 ◽  
Author(s):  
John T. Etgen ◽  
Michael J. O’Brien
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Large Scale ◽  
Data Flow ◽  
Three Dimensions ◽  
Acoustic Medium ◽  
Ocean Bottom ◽  
Acoustic Wave Equation ◽  
P Waves ◽  
Core Method

We present a set of methods for modeling wavefields in three dimensions with the acoustic-wave equation. The primary applications of these modeling methods are the study of acquisition design, multiple suppression, and subsalt imaging for surface-streamer and ocean-bottom recording geometries. We show how to model the acoustic wave equation in three dimensions using limited computer memory, typically using a single workstation, leading to run times on the order of a few CPU hours to a CPU day. The structure of the out-of-core method presented is also used to improve the efficiency of in-core modeling, where memory-to-cache-to-memory data flow is essentially the same as the data flow for an out-of-core method. Starting from the elastic-wave equation, we develop a vector-acoustic algorithm capable of efficiently modeling multicomponent data in an acoustic medium. We show that data from this vector-acoustic algorithm can be used to test upgoing/downgoing separation of P-waves recorded by ocean-bottom seismic acquisition.

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.

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.

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 Approximate Acoustic Wave Equation for VTI Media

2011 ◽  
Vol 54 (4) ◽  
pp. 599-607
Author(s):  
Yi-Jian HUANG ◽  
Guang-Ming ZHU ◽  
Chi-Yang LIU
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Acoustic Wave Equation ◽  
Vti Media
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