Elastic wave propagation in heterogeneous anisotropic media using the lumped finite‐element method

Geophysics ◽  
2002 ◽  
Vol 67 (2) ◽  
pp. 625-638 ◽  
Author(s):  
Jianfeng Zhang ◽  
D. J. Verschuur
Keyword(s):  
Wave Propagation ◽  
Symmetry Axis ◽  
Numerical Technique ◽  
Computational Cost ◽  
Vertical Plane ◽  
Grid Cell ◽  
Transversely Isotropic ◽  
Isotropic Case ◽  
Lamb’S Problem ◽  
Anisotropic Structures

A numerical technique for wave‐propagation simulation in 2‐D heterogeneous anisotropic structures is presented. The scheme is flexible in incorporating arbitrary surface topography, inner openings, liquid/solid boundaries, and irregular interfaces, and it naturally satisfies the free‐surface conditions of complex geometrical boundaries. The algorithm, based on a discretization mesh of triangles and quadrilaterals, solves the problem using integral equilibrium around each node instead of satisfying elastodynamic differential equations at each node as in the finite‐difference method. This study is an extension of previous work for the elastic‐isotropic case. Besides accounting for anisotropy, a simplified quadrilateral grid cell with low computational cost is introduced. The transversely isotropic medium with a symmetry axis on the horizontal or vertical plane, as typically caused by a system of parallel cracks or fine layers, is discussed in detail. A 2‐D algorithm is presented that can handle the situation where the symmetry axis of the anisotropy does not lie in the 2‐D plane. The proposed scheme is successfully tested against an analytical solution for Lamb's problem with a symmetry axis normal to the surface and agrees well with a numerical solution of the reflectivity method for a plane‐layered model in the isotropic case. Computed radiation patterns show characteristics such as shear‐wave splitting and triplications of quasi‐SV wavefronts, as predicted by the theory. Examples of surface‐wave propagation in an anisotropic half‐space with a semicylindrical pit on the surface and mixed liquid/(anisotropic) solid model with an inclined liquid/solid interface are presented. Moreover, seismograms are modeled for dome‐layered and plane‐layered anisotropic structures.

Modeling and inversion of PS-wave moveout asymmetry for tilted TI media: Part 2 — Dipping TTI layer

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. D123-D134 ◽  
Author(s):  
Pawan Dewangan ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Vertical Plane ◽  
Transversely Isotropic ◽  
Estimation Algorithm ◽  
Velocity Model ◽  
Inversion Algorithm ◽  
S Wave ◽  
Weak Anisotropy ◽  
Symmetry Direction ◽  
Tti Media

Dipping transversely isotropic layers with a tilted symmetry axis (TTI media) cause serious imaging problems in fold-and-thrust belts and near salt domes. Here, we apply the modified [Formula: see text] method introduced in Part 1 to the inversion of long-offset PP and PS reflection data for the parameters of a TTI layer with the symmetry axis orthogonal to the bedding. The inversion algorithm combines the time- and offset-asymmetry attributes of the PSV-wave with the hyperbolic PP- and SS-wave moveout in the symmetry-axis plane (i.e., the vertical plane that contains the symmetry axis). The weak-anisotropy approximations for the moveout-asymmetry attributes, verified by numerical analysis, indicate that small-offset (leading) terms do not contain independent information for the inversion. Therefore, the parameter-estimation algorithm relies on PS data recorded at large offsets (the offset-to-depth ratio has to reach at least two), which makes the results generally less stable than those for a horizontal TTI layer in Part1. The least-resolved parameter is Thomsen’s coefficient [Formula: see text]that does not directly influence the moveout of either pure or converted modes. Still, the contribution of the PS-wave asymmetry attributes helps to constrain the TTI model for large tilts [Formula: see text] of the symmetry axis [Formula: see text]. The accuracy of the inversion for large tilts can be improved further by using wide-azimuth PP and PS reflections. With high-quality PS data, the inversion remains feasible for moderate tilts [Formula: see text], but it breaks down for models with smaller values of [Formula: see text] in which the moveout asymmetry is too weak. The tilt itself and several combinations of the medium parameters (e.g., the ratio of the P- and S-wave velocities in the symmetry direction), however, are well constrained for all symmetry-axis orientations. The results of Parts 1 and 2 show that 2D measurements of the PS-wave asymmetry attributes can be used effectively in anisotropic velocity analysis for TTI media. In addition to providing an improved velocity model for imaging beneath TTI beds, our algorithms yield information for lithology discrimination and structural interpretation.

Effect of anisotropy on Love wave propagation

1968 ◽  
Vol 58 (1) ◽  
pp. 259-266
Author(s):  
Janardan G. Negi ◽  
S. K. Upadhyay
Keyword(s):  
Wave Propagation ◽  
Frequency Dependence ◽  
Characteristic Frequency ◽  
Love Wave ◽  
Transversely Isotropic ◽  
Frequency Equation ◽  
Isotropic Case ◽  
Rigid Base ◽  
Isotropic Layer ◽  
Transversely Isotropic Layer

abstract A study on Love wave propagation in a transversely isotropic layer with stress free upper surface and underlying rigid base, is presented. The characteristic frequency equation is obtained and frequency dependence of the velocity parameters for different modes is analysed in detail. Several distinctive propagation phenomena which differ considerably from those in isotropic case are listed below:

Numerical test of the Schoenberg-Muir theory

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. C27-C35 ◽  
Author(s):  
José M. Carcione ◽  
Stefano Picotti ◽  
Fabio Cavallini ◽  
Juan E. Santos
Keyword(s):  
Wave Propagation ◽  
Numerical Simulations ◽  
Layered Medium ◽  
Symmetry Axis ◽  
Numerical Test ◽  
Transversely Isotropic ◽  
Thin Layers ◽  
Strong Anisotropy ◽  
Full Wave ◽  
Points Of View

The Schoenberg-Muir theory states that an equivalent, homogeneous and anisotropic medium can be constructed from a layered medium composed of several thin layers, each anisotropic, under the assumption of stationarity. To test the theory we considered single transversely isotropic layers with different orientations of the symmetry axis and performed numerical simulations of wave propagation with a full-wave solver. The equivalent media have orthorhombic and monoclinic symmetries, respectively. The theory performed very well from the kinematical and dynamical points of view, even for strong anisotropy and layers described by media whose symmetry axes have different orientations.

Scalar generalized‐screen algorithms in transversely isotropic media with a vertical symmetry axis

Geophysics ◽  
2001 ◽  
Vol 66 (5) ◽  
pp. 1538-1550 ◽  
Author(s):  
Jéro⁁me H. Le Rousseau ◽  
Maarten V. de Hoop
Keyword(s):  
Symmetry Axis ◽  
Complex Structure ◽  
Numerical Algorithms ◽  
Transversely Isotropic ◽  
Isotropic Case ◽  
Polarization Angle ◽  
Screen Method ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
The One

The scalar generalized‐screen method in isotropic media is extended here to transversely isotropic media with a vertical symmetry axis (VTI). Although wave propagation in a transversely isotropic medium is essentially elastic, we use an equivalent “acoustic” system of equations for the qP‐waves which we prove to be accurate for both the dispersion relation and the polarization angle in the case of “mild” anisotropy. The enhanced accuracy of the generalized‐screen method as compared to the split‐step Fourier methods allows the extension to VTI media. The generalized‐screen expansion of the one‐way propagator follows closely the method used in the isotropic case. The medium is defined in terms of a background and a perturbation. The generalized‐screen expansion of the vertical slowness is based upon an expansion of the medium parameters simultaneously into magnitude and smoothness of variation. We cast the theory into numerical algorithms, and assess the accuracy of the generalized‐screen method in a particular VTI medium with complex structure (the BP Amoco Valhall model) in which multipathing is significant.

TRIANGLE-QUADRANGLE GRID METHOD FOR POROELASTIC, ELASTIC, AND ACOUSTIC WAVE EQUATIONS

2001 ◽  
Vol 09 (02) ◽  
pp. 681-702 ◽  
Author(s):  
JIANFENG ZHANG
Keyword(s):  
Wave Propagation ◽  
Wave Equations ◽  
Complex Geometry ◽  
Numerical Technique ◽  
Grid Method ◽  
Equilibrium Equations ◽  
Lamb’S Problem ◽  
Poroelastic Media ◽  
Acoustic Media ◽  
Biot’S Equations

A new numerical technique is developed for wave propagation in heterogeneous poroelastic media and mixed poroelastic, elastic and acoustic media. The scheme, based on a first-order hyperbolic Biot's system and a discretization mesh of triangles and quadrangles, solves the problem using integral equilibrium equations around each node, instead of satisfying Biot's differential equations at each node as in the finite-difference method. The surface topography and complex geometrical interfaces can be accurately modeled with the proposed algorithm by making the nodes of triangles and quadrangles follow the curved interfaces. The elastic (acoustic)/poroelastic interface conditions of complex geometry are introduced using the integral equilibrium equations around nodes at the interface based on the continuities of total stresses and velocities between the interface. The free-surface conditions of complex geometrical boundaries are satisfied naturally for the scheme. This work is an extension of the grid method for the heterogeneous elastic media to the heterogeneous poroelastic one. The proposed algorithm is successfully tested against an analytical solution for Lamb's problem when the algorithm is reduced to handle the elastic limit of the Biot's equations. Examples of wave propagation in a poroelastic half-space with a semi-cylindrical pit on the surface and mixed acoustic-poroelastic and elastic-poroelastic models with inclined interfaces are presented.

Simulation of anisotropic wave propagation based upon a spectral element method

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1251-1260 ◽  
Author(s):  
Dimitri Komatitsch ◽  
Christophe Barnes ◽  
Jeroen Tromp
Keyword(s):  
Wave Propagation ◽  
Spectral Element Method ◽  
Mass Matrix ◽  
Computational Cost ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Spectral Element ◽  
Weak Formulation ◽  
Element Mesh ◽  
Element Method

We introduce a numerical approach for modeling elastic wave propagation in 2-D and 3-D fully anisotropic media based upon a spectral element method. The technique solves a weak formulation of the wave equation, which is discretized using a high‐order polynomial representation on a finite element mesh. For isotropic media, the spectral element method is known for its high degree of accuracy, its ability to handle complex model geometries, and its low computational cost. We show that the method can be extended to fully anisotropic media. The mass matrix obtained is diagonal by construction, which leads to a very efficient fully explicit solver. We demonstrate the accuracy of the method by comparing it against a known analytical solution for a 2-D transversely isotropic test case, and by comparing its predictions against those based upon a finite difference method for a 2-D heterogeneous, anisotropic medium. We show its generality and its flexibility by modeling wave propagation in a 3-D transversely isotropic medium with a symmetry axis tilted relative to the axes of the grid.

scholarly journals New acoustic approximation for transversely isotropic media with a vertical symmetry axis

Geophysics ◽  
2019 ◽  
Vol 85 (1) ◽  
pp. C1-C12 ◽  
Author(s):  
Shibo Xu ◽  
Alexey Stovas ◽  
Tariq Alkhalifah ◽  
Hitoshi Mikada
Keyword(s):  
Wave Propagation ◽  
Seismic Data ◽  
Eikonal Equation ◽  
Symmetry Axis ◽  
Transversely Isotropic ◽  
P Wave ◽  
Seismic Data Processing ◽  
S Wave ◽  
P Waves ◽  
Acoustic Approximation

Seismic data processing in the elastic anisotropic model is complicated due to multiparameter dependency. Approximations to the P-wave kinematics are necessary for practical purposes. The acoustic approximation for P-waves in a transversely isotropic medium with a vertical symmetry axis (VTI) simplifies the description of wave propagation in elastic media, and as a result, it is widely adopted in seismic data processing and analysis. However, finite-difference implementations of that approximation are plagued with S-wave artifacts. Specifically, the resulting wavefield also includes artificial diamond-shaped S-waves resulting in a redundant signal for many applications that require pure P-wave data. To derive a totally S-wave-free acoustic approximation, we have developed a new acoustic approximation for pure P-waves that is totally free of S-wave artifacts in the homogeneous VTI model. To keep the S-wave velocity equal to zero, we formulate the vertical S-wave velocity to be a function of the model parameters, rather than setting it to zero. Then, the corresponding P-wave phase and group velocities for the new acoustic approximation are derived. For this new acoustic approximation, the kinematics is described by a new eikonal equation for pure P-wave propagation, which defines the new vertical slowness for the P-waves. The corresponding perturbation-based approximation for our new eikonal equation is used to compare the new equation with the original acoustic eikonal. The accuracy of our new P-wave acoustic approximation is tested on numerical examples for homogeneous and multilayered VTI models. We find that the accuracy of our new acoustic approximation is as good as the original one for the phase velocity, group velocity, and the kinematic parameters such as vertical slowness, traveltime, and relative geometric spreading. Therefore, the S-wave-free acoustic approximation could be further applied in seismic processing that requires pure P-wave data.

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