Computed waveforms in transversely isotropic media

Geophysics ◽  
1982 ◽  
Vol 47 (5) ◽  
pp. 771-783 ◽  
Author(s):  
J. E. White
Keyword(s):  
Asymptotic Theory ◽  
Transversely Isotropic ◽  
Inversion Method ◽  
S Wave ◽  
Fourier Inversion ◽  
P Waves ◽  
Arrival Times ◽  
S Waves ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Radiation of elastic waves from a point force or from a localized torque into a transversely isotropic medium has been formulated in terms of displacement potentials, and transient waveforms have been computed by numerical Fourier inversion. For isotropic sandstone, this procedure yields P‐ and S‐wave pulses whose arrival times and magnitudes agree with theory. For a range of anisotropic rocks, arrival times of quasi‐P‐waves and quasi‐S‐waves agree with asymptotic theory. For extreme anisotropy, some quasi‐S‐wave pulses arrive at times which are not predicted by asymptotic theory. Magnitudes have not been compared with results of asymptotic theory, but decrease with distance appears to be in agreement. This Fourier inversion method gives near‐source changes in waveform which are not obtainable from the asymptotic theory.

scholarly journals An acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis

Geophysics ◽  
2017 ◽  
Vol 82 (1) ◽  
pp. C9-C20 ◽  
Author(s):  
Qi Hao ◽  
Tariq Alkhalifah
Keyword(s):  
Eikonal Equation ◽  
Wave Attenuation ◽  
Transversely Isotropic ◽  
S Wave ◽  
P Waves ◽  
The Earth ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Vti Media ◽  
Complex Valued

Seismic-wave attenuation is an important component of describing wave propagation. Certain regions, such as gas clouds inside the earth, exert highly localized attenuation. In fact, the anisotropic nature of the earth induces anisotropic attenuation because the quasi P-wave dispersion effect should be profound along the symmetry direction. We have developed a 2D acoustic eikonal equation governing the complex-valued traveltime of quasi P-waves in attenuating, transversely isotropic media with a vertical-symmetry axis (VTI). This equation is derived under the assumption that the complex-valued traveltime of quasi P-waves in attenuating VTI media are independent of the S-wave velocity parameter [Formula: see text] in Thomsen’s notation and the S-wave attenuation coefficient [Formula: see text] in Zhu and Tsvankin’s notation. We combine perturbation theory and Shanks transform to develop practical approximations to the acoustic attenuating eikonal equation, capable of admitting an analytical description of the attenuation in homogeneous media. For a horizontal-attenuating VTI layer, we also derive the nonhyperbolic approximations for the real and imaginary parts of the complex-valued reflection traveltime. These equations reveal that (1) the quasi SV-wave velocity and the corresponding quasi SV-wave attenuation coefficient given as part of Thomsen-type notation barely affect the ray velocity and ray attenuation of quasi P-waves in attenuating VTI media; (2) combining the perturbation method and Shanks transform provides an accurate analytic eikonal solution for homogeneous attenuating VTI media; (3) for a horizontal attenuating VTI layer with weak attenuation, the real part of the complex-valued reflection traveltime may still be described by the existing nonhyperbolic approximations developed for nonattenuating VTI media, and the imaginary part of the complex-valued reflection traveltime still has the shape of nonhyperbolic curves. In addition, we have evaluated the possible extension of the proposed eikonal equation to realistic attenuating media, an alternative perturbation solution to the proposed eikonal equation, and the feasibility of applying the proposed nonhyperbolic equation for the imaginary part of the complex-valued traveltime to invert for interval attenuation parameters.

scholarly journals An efficient Helmholtz solver for acoustic transversely isotropic media

Geophysics ◽  
2018 ◽  
Vol 83 (2) ◽  
pp. C75-C83 ◽  
Author(s):  
Zedong Wu ◽  
Tariq Alkhalifah
Keyword(s):  
Wave Equation ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
P Wave ◽  
Seismic Exploration ◽  
S Wave ◽  
P Waves ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Separable Approximation

The acoustic approximation, even for anisotropic media, is widely used in current industry imaging and inversion algorithms mainly because P-waves constitute most of the energy recorded in seismic exploration. The resulting acoustic formulas tend to be simpler, resulting in more efficient implementations, and they depend on fewer medium parameters. However, conventional solutions of the acoustic-wave equation with higher-order derivatives suffer from S-wave artifacts. Thus, we separate the quasi-P-wave propagation in anisotropic media into the elliptic anisotropic operator (free of the artifacts) and the nonelliptic anisotropic components, which form a pseudodifferential operator. We then develop a separable approximation of the dispersion relation of nonelliptic-anisotropic components, specifically for transversely isotropic media. Finally, we iteratively solve the simpler lower-order elliptical wave equation for a modified source function that includes the nonelliptical terms represented in the Fourier domain. A frequency-domain Helmholtz formulation of the approach renders the iterative implementation efficient because the cost is dominated by the lower-upper decomposition of the impedance matrix for the simpler elliptical anisotropic model. In addition, the resulting wavefield is free of S-wave artifacts and has a balanced amplitude. Numerical examples indicate that the method is reasonably accurate and efficient.

A fast-marching eikonal solver for tilted transversely isotropic media

Geophysics ◽  
2020 ◽  
Vol 85 (6) ◽  
pp. S385-S393
Author(s):  
Umair bin Waheed
Keyword(s):  
Transversely Isotropic ◽  
Fast Marching Method ◽  
Fast Marching ◽  
P Waves ◽  
Eikonal Equations ◽  
Transversely Isotropic Media ◽  
Fast Sweeping ◽  
Fast Marching Algorithm ◽  
Isotropic Media ◽  
And Migration

Fast and accurate traveltime computation for quasi-P waves in anisotropic media is an essential ingredient of many seismic processing and interpretation applications such as Kirchhoff modeling and migration, microseismic source localization, and traveltime tomography. Fast-sweeping methods are widely used for solving the anisotropic eikonal equation due to their flexibility in solving general equations compared to the fast-marching method. However, it has been observed that fast sweeping can be much less efficient than fast marching for models with curved characteristics and practical grid sizes. By representing a tilted transversely isotropic (TTI) equation as a sequence of elliptically isotropic (EI) eikonal equations, we determine that the fast-marching algorithm can be used to compute fast and accurate traveltimes for TTI media. The tilt angle is absorbed into the description of the effective EI model; therefore, the adopted approach does not compromise on the solution accuracy. Through tests on benchmark synthetic models, we test our fast-marching algorithm and discover considerable improvement in accuracy by using factorization and a second-order finite-difference stencil. The adopted methodology opens the door to the possibility of using the fast-marching algorithm for a wider class of anisotropic eikonal equations.

scholarly journals Localization by Acoustic Emission in Transversely Isotropic Slate

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Bjorn Debecker ◽  
André Vervoort
Keyword(s):  
Acoustic Emission ◽  
Transversely Isotropic ◽  
Arrival Times ◽  
Localization Method ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Load Tests ◽  
Good Identification ◽  
Actual Test ◽  
Visual Observations

A method for localization by acoustic emission in transversely isotropic media is developed and validated. Velocities are experimentally measured and then used to calculate a database of theoretical arrival times for a large number of positions. During an actual test, positions are assigned by comparing measured arrival times with the database's arrival times. The method is applied during load tests on slate samples and compared with visual observations of fractures. The localization method allowed for a good identification of the regions of fracturing at different stages during the test.

Nonlinear traveltime inversion scheme for crosshole seismic tomography in tilted transversely isotropic media

Geophysics ◽  
2008 ◽  
Vol 73 (4) ◽  
pp. D17-D33 ◽  
Author(s):  
Bing Zhou ◽  
Stewart Greenhalgh ◽  
Alan Green
Keyword(s):  
Seismic Tomography ◽  
Velocity Structure ◽  
Body Wave ◽  
Transversely Isotropic ◽  
The Body ◽  
Body Waves ◽  
Inversion Method ◽  
Perturbation Equation ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Crosshole seismic tomography often is applied to image the velocity structure of an interwell medium. If the rocks are anisotropic, the tomographic technique must be adapted to the complex situation; otherwise, it leads to a false interpretation. We propose a nonlinear kinematic inversion method for crosshole seismic tomography in composite transversely isotropic media with known dipping symmetry axes. This method is based on a new version of the first-order traveltime perturbation equation. It directly uses the derivative of the phase velocity rather than the eigenvectors of the body-wave modes to overcome the singularity problem for application to the two quasi-shear waves. We applied an iterative nonlinear solver incorporating our kinematic ray-tracing scheme and directly compute the Jacobian matrix in an arbitrary reference medium. This reconstructs the five elastic moduli or Thomsen parameters from the first-arrival traveltimes of the three seismic body waves (qP, qSV, qSH) in strongly and weakly anisotropic media. We conducted three synthetic experiments that involve determining anisotropic parameters for a homogeneous rock, reconstructing a fault embedded in a strongly anisotropic background, and imaging a complicated four-layer model containing a small channel and a buried dipping interface. We compared results of our nonlinear inversion method with isotropic tomography and the traditional linear anisotropic inversion scheme, which showed the capability and superiority of the new scheme for crosshole tomographic imaging.

Analytic calculation of phase and group velocities of P-waves in orthorhombic media

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. C79-C97 ◽  
Author(s):  
Qi Hao ◽  
Alexey Stovas
Keyword(s):  
Transversely Isotropic ◽  
P Wave ◽  
P Waves ◽  
Type Approximation ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Potential Applications ◽  
Phase And Group Velocities ◽  
And Migration ◽  
Group Velocities

We have developed an approximate method to calculate the P-wave phase and group velocities for orthorhombic media. Two forms of analytic approximations for P-wave velocities in orthorhombic media were built by analogy with the five-parameter moveout approximation and the four-parameter velocity approximation for transversely isotropic media, respectively. They are called the generalized moveout approximation (GMA)-type approximation and the Fomel approximation, respectively. We have developed approximations for elastic and acoustic orthorhombic media. We have characterized the elastic orthorhombic media in Voigt notation, and we can describe the acoustic orthorhombic media by introducing the modified Alkhalifah’s notation. Our numerical evaluations indicate that the GMA-type and Fomel approximations are accurate for elastic and acoustic orthorhombic media with strong anisotropy, and the GMA-type approximation is comparable with the approximation recently proposed by Sripanich and Fomel. Potential applications of the proposed approximations include forward modeling and migration based on the dispersion relation and the forward traveltime calculation for seismic tomography.

Three-parameter normal moveout correction in layered anisotropic media: A stretch-free approach

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. C129-C142 ◽  
Author(s):  
Mohammad Mahdi Abedi ◽  
Mohammad Ali Riahi ◽  
Alexey Stovas
Keyword(s):  
Transversely Isotropic ◽  
Real Data ◽  
Data Sets ◽  
P Waves ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Nmo Correction ◽  
Recorded Data ◽  
Normal Moveout Correction ◽  
Vti Media

In conventional normal moveout (NMO) correction, some parts of the recorded data at larger offsets are discarded because of NMO distortions. Deviation from the true traveltime of reflections due to the anisotropy and heterogeneity of the earth, and wavelet stretching are two reasons of these distortions. The magnitudes of both problems increase with increasing the offset to depth ratio. Therefore, to be able to keep larger offsets of shallower reflections, both problems should be obviated. Accordingly, first, we have studied different traveltime approximations being in use, alongside new parameterizations for two classical functional equations, to select suitable equations for NMO correction. We numerically quantify the fitting accuracy and uncertainty of known nonhyperbolic traveltime approximations for P-waves in transversely isotropic media with vertical symmetry axis (VTI). We select three suitable three-parameter approximations for NMO in layered VTI media as the VTI generalized moveout approximation, a double-square-root approximation, and a perturbation-based approximation. Second, we have developed an extension of the earlier proposed stretch-free NMO method, using the selected moveout approximations. This method involves an automatic modification of the input parameters in anisotropic NMO correction, for selected reflections. Our anisotropic stretch-free NMO method is tested on synthetic and three real data sets from Gulf of Mexico and Iranian oil fields. The results verify the success of the method in extending the usable offsets, by generating flat and stretch-free NMO corrected reflections.

The effects of transverse isotropy on reflection amplitude versus offset

Geophysics ◽  
1987 ◽  
Vol 52 (4) ◽  
pp. 564-567 ◽  
Author(s):  
J. Wright
Keyword(s):  
Wave Velocity ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
S Wave ◽  
Reflection Amplitude ◽  
Amplitude Versus Offset ◽  
Transversely Isotropic Media ◽  
P Wave Velocity ◽  
Isotropic Media

Studies have shown that elastic properties of materials such as shale and chalk are anisotropic. With the increasing emphasis on extraction of lithology and fluid content from changes in reflection amplitude with shot‐to‐group offset, one needs to know the effects of anisotropy on reflectivity. Since anisotropy means that velocity depends upon the direction of propagation, this angular dependence of velocity is expected to influence reflectivity changes with offset. These effects might be particularly evident in deltaic sand‐shale sequences since measurements have shown that the P-wave velocity of shales in the horizontal direction can be 20 percent higher than the vertical P-wave velocity. To investigate this behavior, a computer program was written to find the P- and S-wave reflectivities at an interface between two transversely isotropic media with the axis of symmetry perpendicular to the interface. Models for shale‐chalk and shale‐sand P-wave reflectivities were analyzed.

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