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Paul Sava - One of the best experts on this subject based on the ideXlab platform.

  • improving the efficiency of elastic wave mode separation for heterogeneous tilted transverse isotropic media
    Geophysics, 2011
    Co-Authors: Paul Sava
    Abstract:

    Wave-mode separation for TI (transversely isotropic) models can be carried out by nonstationary filtering the elastic wavefields with localized filters. These filters are constructed based on the polarization vectors obtained by solving the Christoffel Equation using local medium parameters. This procedure, although accurate, is computationally expensive, especially in 3D. We develop an efficient method for wave-mode separation, which exploits the same general idea of projecting wavefields onto polarization vectors. The method consists of two steps: (1) separate wave modes in the wavenumber domain at a number of reference models to obtain the same number of partially separated wavefields; then transform all the wavefields to the space domain; (2) interpolate the wavefields (obtained in step 1) in the space domain using the spatially-variable model parameters. The new method resembles the phaseshift plus interpolation (PSPI) technique, which interpolates the wavefields that are reconstructed at several reference velocities. Synthetic examples indicate that the separation followed by interpolation is effective for models with complex geology. The new technique has the benefits of speed and accuracy.

  • Efficient Elastic Wave-mode Separation in TTI Media
    72nd EAGE Conference and Exhibition incorporating SPE EUROPEC 2010, 2010
    Co-Authors: Paul Sava
    Abstract:

    Wave mode separation is an indispensable step in elastic wave Equation imaging. For isotropic media, the separation is typically done using Helmholtz decomposition. However, Helmholtz decomposition does not completely separate wave modes for anisotropic media. Wavefield separation operators for TI (transverse isotropic) models are constructed based on the polarization vectors evaluated at each point of the medium by solving the Christoffel Equation using local medium parameters. These polarization vectors can be represented in the space domain as localized filters, which resemble conventional derivative operators. The wave mode separation for TI media is usually implemented as non-stationary filtering with local filters. However, the accurate separation in the space domain is computationally expensive especially in 3D. In this paper, we show an efficient method for wave-mode separation, which exploits the same general idea of projection on polarization vectors. The method consists of two steps: first separate wave modes in the wavenumber domain for a number of reference models, and second interpolate the wavefields in the space domain using the spatially-variable model parameters. An example shows that the separation by interpolation works well for models with complex geology.

  • elastic wave mode separation for vti media
    Geophysics, 2009
    Co-Authors: Paul Sava
    Abstract:

    Elastic wave propagation in anisotropic media is well represented by elastic wave Equations. Modeling based on elasticwaveEquationscharacterizesbothkinematicsanddynamics correctly. However, because P- and S-modes are both propagated using elastic wave Equations, there is a need to separate P- and S-modes to efficiently apply single-mode processing tools. In isotropic media, wave modes are usually separated using Helmholtz decomposition. However, Helmholtz decomposition using conventional divergence and curl operators in anisotropic media does not give satisfactory resultsandleavesthedifferentwavemodesonlypartiallyseparated.Theseparationofanisotropicwavefieldsrequiresmore sophisticated operators that depend on local material parameters. Anisotropic wavefield-separation operators are constructedusingthepolarizationvectorsevaluatedateachpoint of the medium by solving the Christoffel Equation for local mediumparameters.Thesepolarizationvectorscanberepresented in the space domain as localized filtering operators, which resemble conventional derivative operators. The spatially variable pseudo-derivative operators perform well in heterogeneous VTI media even at places of rapid velocity/ densityvariation.Syntheticresultsindicatethattheoperators can be used to separate wavefields for VTI media with an arbitrarydegreeofanisotropy.

Sergey V. Kuznetsov - One of the best experts on this subject based on the ideXlab platform.

  • Eigensolutions for Rayleigh Wave Analysis
    IUTAM Symposium on Diffraction and Scattering in Fluid Mechanics and Elasticity, 2002
    Co-Authors: A. V. Kaptsov, Sergey V. Kuznetsov
    Abstract:

    Eigenvalues and eigenvectors of the Christoffel Equation for the problem of Rayleigh waves propagating in media with arbitrary anisotropy are analyzed on the basis of three-dimensional complex formalism.

  • Spectral properties of Christoffel Equation for Rayleigh waves
    Comptes Rendus de l'Académie des Sciences - Series IIB - Mechanics-Physics-Astronomy, 1999
    Co-Authors: A. V. Kaptsov, Sergey V. Kuznetsov
    Abstract:

    Abstract Spectral properties of the Christoffel Equation for the problem of Rayleigh waves propagating in homogeneous media with arbitrary anisotropy are analyzed on the bases of three-dimensional complex formalism.

  • Spectral properies of Christoffel Equation for Rayleigh waves
    The Journal of the Acoustical Society of America, 1999
    Co-Authors: Sergey V. Kuznetsov, A. V. Kaptsov
    Abstract:

    Spectral properties of the matrix Christoffel Equation are needed for construction of the exponentially decaying surface (Rayleigh) waves. The following analysis covers theorems on root characterization, the structure of eigenvalues, and eigenspaces of the Christoffel Equation. The main results are obtained by three‐dimensional complex formalism, which goes back to Rayleigh. For the case of anisotropic media this approach was exploited by Farnell in numerical analysis of Rayleigh waves propagating in anisotropic crystals and in analytical study by Stoneley and later by Dieulesaint and Royer. Sextic formalism for Rayleigh wave analysis was proposed by Stroh. This is based on the similarity of solutions for line dislocations in unbounded medium and the propagation of Rayleigh waves. This approach was developed by Barnett and Lothe and later by Chadwick et al., so theorems of uniqueness and existence for Rayleigh waves were proved. Substitution of a complex root in Christoffel’s Equation produces an Equation for the eigenvector (amplitude vector) determination. Propositions which flow out directly from the analysis of the structure of this Equation describe spectral properties and characterization of eigenspaces. One of the most interesting results asserts that algebraic multiplicity of the phase speed equals its geometric multiplicity. [Work was supported by INTAS 96‐2003.]

  • NUMERICAL ANALYSIS OF RAYLEIGH WAVES IN ANISOTROPIC MEDIA
    NATO Science Series II: Mathematics Physics and Chemistry, 1
    Co-Authors: A. V. Kaptsov, Sergey V. Kuznetsov
    Abstract:

    Eigenvalues and eigenvectors of the Christoffel Equation for the problem of Rayleigh waves propagating in media with arbitrary anisotropy are analyzed on the bases of three-dimensional complex formalism. Numerical algorithm for analysis of speed of Rayleigh waves is discussed.

M. D. Sharma - One of the best experts on this subject based on the ideXlab platform.

  • Inhomogeneous waves at the boundary of a generalized thermoelastic anisotropic medium
    International Journal of Solids and Structures, 2008
    Co-Authors: M. D. Sharma
    Abstract:

    The Christoffel Equation is derived for the propagation of plane harmonic waves in a generalized thermoelastic anisotropic (GTA) medium. Solving this Equation for velocities implies the propagation of four attenuating waves in the medium. The same Christoffel Equation is solved into a polynomial Equation of degree eight. The roots of this Equation define the vertical slownesses of the eight attenuating waves existing at a boundary of the medium. Incidence of inhomogeneous waves is considered at the boundary of the medium. A finite non-dimensional parameter defines the inhomogeneity of incident wave and is used to calculate its (complex) slowness vector. The reflected attenuating waves are identified with the values of vertical slowness. Procedure is explained to calculate the slowness vectors of the waves reflected from the boundary of the medium. The slowness vectors are used, further, to calculate the phase velocities, phase directions, directions and amounts of attenuations of the reflected waves. Numerical examples are considered to analyze the variations of these propagation characteristics with the inhomogeneity and propagation direction of incident wave. Incidence of each of the four types of waves is considered. Numerical example is also considered to study the propagation and attenuation of inhomogeneous waves in the unbounded medium.

  • Three-dimensional wave propagation in a general anisotropic poroelastic medium: phase velocity, group velocity and polarization
    Geophysical Journal International, 2004
    Co-Authors: M. D. Sharma
    Abstract:

    SUMMARY This is an attempt to study 3-D wave propagation in a general anisotropic poroelastic medium. Biot’s theory is used to derive a modified Christoffel Equation for the propagation of plane harmonic waves in an anisotropic fluid-saturated porous solid. This Equation is solved further to obtain a biquadratic Equation, the roots of which represent the phase velocities of all the four quasi-waves that may propagate in such a medium. These phase velocities vary with the direction of phase propagation. Expressions are derived to calculate the group velocities of all the four quasi-waves without using numerical differentiation. The eigensystem of modified Christoffel Equation is used to calculate the polarizations of all the quasi-waves. The particle motion of each wave is a function of the direction of phase propagation. Some fundamental differences between wave propagation in anisotropic poroelastic medium and anisotropic elastic medium are suggested, an interesting one is that in an anisotropic poroelastic medium, the polarizations of different quasi-waves need not be mutually orthogonal. In the anisotropic poroelastic medium, the motion of fluid particles deviates from solid particles and this deviation varies, also, with the matrix porosity. Propagation regimes for an isotropic medium, giving velocities and polarizations of both compressional and shear waves, are obtained as special cases. The variations of phase velocity, group velocity, ray direction with phase direction (in 3-D space), are plotted for a numerical model of general anisotropic poroelastic solid. The same numerical model is used to plot the deviations of polarizations from phase direction and ray direction. The deviations among the motion in fluid of the particles and solid parts of porous aggregate are also plotted.

A. V. Kaptsov - One of the best experts on this subject based on the ideXlab platform.

  • Eigensolutions for Rayleigh Wave Analysis
    IUTAM Symposium on Diffraction and Scattering in Fluid Mechanics and Elasticity, 2002
    Co-Authors: A. V. Kaptsov, Sergey V. Kuznetsov
    Abstract:

    Eigenvalues and eigenvectors of the Christoffel Equation for the problem of Rayleigh waves propagating in media with arbitrary anisotropy are analyzed on the basis of three-dimensional complex formalism.

  • Spectral properties of Christoffel Equation for Rayleigh waves
    Comptes Rendus de l'Académie des Sciences - Series IIB - Mechanics-Physics-Astronomy, 1999
    Co-Authors: A. V. Kaptsov, Sergey V. Kuznetsov
    Abstract:

    Abstract Spectral properties of the Christoffel Equation for the problem of Rayleigh waves propagating in homogeneous media with arbitrary anisotropy are analyzed on the bases of three-dimensional complex formalism.

  • Spectral properies of Christoffel Equation for Rayleigh waves
    The Journal of the Acoustical Society of America, 1999
    Co-Authors: Sergey V. Kuznetsov, A. V. Kaptsov
    Abstract:

    Spectral properties of the matrix Christoffel Equation are needed for construction of the exponentially decaying surface (Rayleigh) waves. The following analysis covers theorems on root characterization, the structure of eigenvalues, and eigenspaces of the Christoffel Equation. The main results are obtained by three‐dimensional complex formalism, which goes back to Rayleigh. For the case of anisotropic media this approach was exploited by Farnell in numerical analysis of Rayleigh waves propagating in anisotropic crystals and in analytical study by Stoneley and later by Dieulesaint and Royer. Sextic formalism for Rayleigh wave analysis was proposed by Stroh. This is based on the similarity of solutions for line dislocations in unbounded medium and the propagation of Rayleigh waves. This approach was developed by Barnett and Lothe and later by Chadwick et al., so theorems of uniqueness and existence for Rayleigh waves were proved. Substitution of a complex root in Christoffel’s Equation produces an Equation for the eigenvector (amplitude vector) determination. Propositions which flow out directly from the analysis of the structure of this Equation describe spectral properties and characterization of eigenspaces. One of the most interesting results asserts that algebraic multiplicity of the phase speed equals its geometric multiplicity. [Work was supported by INTAS 96‐2003.]

  • NUMERICAL ANALYSIS OF RAYLEIGH WAVES IN ANISOTROPIC MEDIA
    NATO Science Series II: Mathematics Physics and Chemistry, 1
    Co-Authors: A. V. Kaptsov, Sergey V. Kuznetsov
    Abstract:

    Eigenvalues and eigenvectors of the Christoffel Equation for the problem of Rayleigh waves propagating in media with arbitrary anisotropy are analyzed on the bases of three-dimensional complex formalism. Numerical algorithm for analysis of speed of Rayleigh waves is discussed.

Stefaan Cottenier - One of the best experts on this subject based on the ideXlab platform.

  • Solving the Christoffel Equation: Phase and group velocities
    Computer Physics Communications, 2016
    Co-Authors: Jan Jaeken, Stefaan Cottenier
    Abstract:

    Abstract We provide Christoffel , a Python tool for calculating direction-dependent phase velocities, polarization vectors, group velocities, power flow angles and enhancement factors based on the stiffness tensor of a solid. It is built in a modular way to allow for efficient and flexible calculations, and the freedom to select and combine results as desired. All derivatives are calculated analytically, which circumvents possible numerical sampling problems. GNUPlot scripts are provided for convenient visualization. Program summary Program title: Christoffel Catalogue identifier: AFAT_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AFAT_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU General Public License, version 3 No. of lines in distributed program, including test data, etc.: 2631 No. of bytes in distributed program, including test data, etc.: 14958 Distribution format: tar.gz Programming language: Python. Computer: Workstations. Operating system: Linux/UNIX/Windows/MacOS. Classification: 7.8. External routines: NumPy Nature of problem: Calculating acoustic phase and group velocities in homogeneous solids Solution method: Solving the Christoffel Equation eigenvalue problem by diagonalization; calculating group velocities and enhancement factors analytically as derivatives. Running time: Seconds/minutes