Estimating SV‐wave stacking velocities for transversely isotropic solids

Geophysics ◽  
1991 ◽  
Vol 56 (10) ◽  
pp. 1596-1602 ◽  
Author(s):  
Patricia A. Berge
Keyword(s):  
Shear Wave ◽  
Isotropic Medium ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Transversely Isotropic Medium ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Snell’S Law ◽  
Snell's Law

Conventional seismic experiments can record converted shear waves in anisotropic media, but the shear‐wave stacking velocities pose a problem when processing and interpreting the data. Methods used to find shear‐wave stacking velocities in isotropic media will not always provide good estimates in anisotropic media. Although isotropic methods often can be used to estimate shear‐wave stacking velocities in transversely isotropic media with vertical symmetry axes, the estimations fail for some transversely isotropic media even though the anisotropy is weak. For a given anisotropic medium, the shear‐wave stacking velocity can be estimated using isotropic methods if the isotropic Snell’s law approximates the anisotropic Snell’s law and if the shear wavefront is smooth enough near the vertical axis to be fit with an ellipse. Most of the 15 transversely isotropic media examined in this paper met these conditions for short reflection spreads and small ray angles. Any transversely isotropic medium will meet these conditions if the transverse isotropy is weak and caused by thin subhorizontal layering. For three of the media examined, the anisotropy was weak but the shear wave-fronts were not even approximately elliptical near the vertical axis. Thus, isotropic methods provided poor estimates of the shear‐wave stacking velocities. These results confirm that for any given transversely isotropic medium, it is possible to determine whether or not shear‐wave stacking velocities can be estimated using isotropic velocity analysis.

scholarly journals Dip‐moveout error in transversely isotropic media with linear velocity variation in depth

Geophysics ◽  
1993 ◽  
Vol 58 (10) ◽  
pp. 1442-1453 ◽  
Author(s):  
Ken L. Larner
Keyword(s):  
Homogeneous Model ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Velocity Variation ◽  
Transversely Isotropic Medium ◽  
Velocity Gradients ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Homogeneous Media

Levin modeled the moveout, within common‐midpoint (CMP) gathers, of reflections from plane‐dipping reflectors beneath homogeneous, transversely isotropic media. For some media, when the axis of symmetry for the anisotropy was vertical, he found departures in stacking velocity from predictions based upon the familiar cosine‐of‐dip correction for isotropic media. Here, I do similar tests, again with transversely isotropic models with vertical axis of symmetry, but now allowing the medium velocity to vary linearly with depth. Results for the same four anisotropic media studied by Levin show behavior of dip‐corrected stacking velocity with reflector dip that, for all velocity gradients considered, differs little from that for the counterpart homogeneous media. As with isotropic media, traveltimes in an inhomogeneous, transversely isotropic medium can be modeled adequately with a homogeneous model with vertical velocity equal to the vertical rms velocity of the inhomogeneous medium. In practice, dip‐moveout (DMO) is based on the assumption that either the medium is homogeneous or its velocity varies with depth, but in both cases isotropy is assumed. It turns out that for only one of the transversely isotropic media considered here—shale‐limestone—would v(z) DMO fail to give an adequate correction within CMP gathers. For the shale‐limestone, fortuitously the constant‐velocity DMO gives a better moveout correction than does the v(z) DMO.

Aberration‐free image for SH reflection in transversely isotropic media

Geophysics ◽  
1987 ◽  
Vol 52 (11) ◽  
pp. 1563-1565 ◽  
Author(s):  
J. M. Blair ◽  
J. Korringa
Keyword(s):  
Group Velocity ◽  
Isotropic Medium ◽  
Homogeneous Medium ◽  
Transversely Isotropic ◽  
Transversely Isotropic Medium ◽  
Elastic Symmetry ◽  
Source Point ◽  
Slowness Vector ◽  
Transversely Isotropic Media ◽  
Isotropic Media

This note is intended formulate and prove a theorem about shear (S) waves in a transversely isotropic medium for which we have found no reference in the literature. The theorem states the following: SH waves emanating from a point source in a homogeneous transversely isotropic medium are reflected from a planar interface between the transversely isotropic medium and another homogeneous medium in such a way that they define a reflective image that is free of aberrations, regardless of the relative orientation of the elastic symmetry axis and the interface. It is an image for rays in the direction of the group velocity vectors, not the slowness vectors. The image is located on a line through the source point in the direction of the group velocity of a wave for which the slowness vector is perpendicular to the interface. The distance, measured along this line, of the image behind the interface is equal to that of the source point in front. An analogous theorem for slowness vectors exists only for isotropic media, where it is trivial and coincides with the above.

Crosswell tomographic estimation of elastic constants in heterogeneous transversely isotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (3) ◽  
pp. 774-783 ◽  
Author(s):  
Reinaldo J. Michelena ◽  
Jerry M. Harris ◽  
Francis Muir
Keyword(s):  
Elastic Constants ◽  
Isotropic Medium ◽  
Field Data ◽  
Transversely Isotropic ◽  
Transversely Isotropic Medium ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Axes Of Symmetry

The procedure to estimate elastic constants of a transversely isotropic medium from limited‐aperture traveltimes has two steps. First, P‐ and SV‐wave traveltimes are fitted with elliptical velocity functions around one of the axes of symmetry. Second, the parameters that describe the elliptical velocity functions are transformed analytically into elastic constants. When the medium is heterogeneous, the process of fitting the traveltimes with elliptical velocity functions is performed tomographically, and the transformation to elastic constants is performed locally at each position in space. Crosswell synthetic and field data examples show that the procedure is accurate as long as the data aperture is constrained as follows: it should not be too large otherwise the elliptical approximation may not be adequate, and it should not be too small because the tomographic estimation of elliptical velocities fails, even if the medium is actually isotropic.

Approximate polarization of plane waves in a medium having weak transverse isotropy

Geophysics ◽  
1994 ◽  
Vol 59 (10) ◽  
pp. 1605-1612 ◽  
Author(s):  
Björn E. Rommel
Keyword(s):  
Isotropic Medium ◽  
Transverse Isotropy ◽  
Full Range ◽  
Velocity Ratio ◽  
Plane Waves ◽  
Transversely Isotropic ◽  
Seismic Exploration ◽  
Transversely Isotropic Medium ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Many real rocks and sediments relevant to seismic exploration can be described by elastic, transversely isotropic media. The properties of plane waves propagating in a transversely isotropic medium can be given in an analytically exact form. Here the polarization is recast into a comprehensive form that includes Daley and Hron’s normalization and Helbig’s full range of elastic constants. But these formulas are rather lengthy and do not easily reveal the features caused by anisotropy. Hence Thomsen suggested an approximation scheme for weak transverse isotropy. His derivation of the approximate polarization, however, is based on a property that is not suitable to measure small differences between an isotropic and a weakly transversely isotropic medium. Therefore the approximation of the polarization is recast. The corrected approximation does show a dependence on weak transverse isotropy. This feature can be viewed as an additional rotation of the polarization with respect to the wavenormal. It depends on the anisotropy as well as the inverse velocity ratio. An approximate condition of pure polarization, which occurs in certain directions, is also obtained. The corrected approximation results in a better match of the approximate polarization with the exact one, providing the assumption of weak transverse isotropy is met. When comparing the additional rotation with the deviation of the (observable) ray direction from the wavenormal, the qSV‐wave indicates transverse isotropy most clearly.

Normal moveout from dipping reflectors in anisotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 268-284 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Weak Anisotropy ◽  
Anisotropic Models ◽  
Transversely Isotropic Media ◽  
Wide Range ◽  
Isotropic Media ◽  
The Difference

Description of reflection moveout from dipping interfaces is important in developing seismic processing methods for anisotropic media, as well as in the inversion of reflection data. Here, I present a concise analytic expression for normal‐moveout (NMO) velocities valid for a wide range of homogeneous anisotropic models including transverse isotropy with a tilted in‐plane symmetry axis and symmetry planes in orthorhombic media. In transversely isotropic media, NMO velocity for quasi‐P‐waves may deviate substantially from the isotropic cosine‐of‐dip dependence used in conventional constant‐velocity dip‐moveout (DMO) algorithms. However, numerical studies of NMO velocities have revealed no apparent correlation between the conventional measures of anisotropy and errors in the cosine‐of‐dip DMO correction (“DMO errors”). The analytic treatment developed here shows that for transverse isotropy with a vertical symmetry axis, the magnitude of DMO errors is dependent primarily on the difference between Thomsen parameters ε and δ. For the most common case, ε − δ > 0, the cosine‐of‐dip–corrected moveout velocity remains significantly larger than the moveout velocity for a horizontal reflector. DMO errors at a dip of 45 degrees may exceed 20–25 percent, even for weak anisotropy. By comparing analytically derived NMO velocities with moveout velocities calculated on finite spreads, I analyze anisotropy‐induced deviations from hyperbolic moveout for dipping reflectors. For transversely isotropic media with a vertical velocity gradient and typical (positive) values of the difference ε − δ, inhomogeneity tends to reduce (sometimes significantly) the influence of anisotropy on the dip dependence of moveout velocity.

scholarly journals An efficient wavefield inversion for transversely isotropic media with a vertical axis of symmetry

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. R195-R206 ◽  
Author(s):  
Chao Song ◽  
Tariq Alkhalifah
Keyword(s):  
High Resolution ◽  
Optimization Problem ◽  
Source Function ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Trade Off ◽  
Transversely Isotropic Media ◽  
Highly Nonlinear ◽  
Isotropic Media ◽  
Axis Of Symmetry

Conventional full-waveform inversion (FWI) aims at retrieving a high-resolution velocity model directly from the wavefields measured at the sensor locations resulting in a highly nonlinear optimization problem. Due to the high nonlinearity of FWI (manifested in one form in the cycle-skipping problem), it is easy to fall into local minima. Considering that the earth is truly anisotropic, a multiparameter inversion imposes additional challenges in exacerbating the null-space problem and the parameter trade-off issue. We have formulated an optimization problem to reconstruct the wavefield in an efficient matter with background models by using an enhanced source function (which includes secondary sources) in combination with fitting the data. In this two-term optimization problem to fit the wavefield to the data and to the background wave equation, the inversion for the wavefield is linear. Because we keep the modeling operator stationary within each frequency, we only need one matrix inversion per frequency. The inversion for the anisotropic parameters is handled in a separate optimization using the wavefield and the enhanced source function. Because the velocity is the dominant parameter controlling the wave propagation, it is updated first. Thus, this reduces undesired updates for anisotropic parameters due to the velocity update leakage. We find the effectiveness of this approach in reducing parameter trade-off with a distinct Gaussian anomaly model. We find that in using the parameterization [Formula: see text], and [Formula: see text] to describe the transversely isotropic media with a vertical axis of symmetry model in the inversion, we end up with high resolution and minimal trade-off compared to conventional parameterizations for the anisotropic Marmousi model. Application on 2D real data also indicates the validity of our method.

scholarly journals Frequency domain multiparameter acoustic inversion for transversely isotropic media with a vertical axis of symmetry

Geophysics ◽  
2019 ◽  
Vol 84 (1) ◽  
pp. C1-C14 ◽  
Author(s):  
Ramzi Djebbi ◽  
Tariq Alkhalifah
Keyword(s):  
Frequency Domain ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
The Other ◽  
Sensitivity Analyses ◽  
Data Set ◽  
Trade Off ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Axis Of Symmetry

Multiparameter full-waveform inversion for transversely isotropic media with a vertical axis of symmetry (VTI) suffers from the trade-off between the parameters. The trade-off results in the leakage of one parameter’s update into the other. It affects the accuracy and convergence of the inversion. The sensitivity analyses suggested a parameterization using the horizontal velocity [Formula: see text], Thomsen’s parameter [Formula: see text], and the anelliptic parameter [Formula: see text] to reduce the trade-off for surface recorded seismic data. We aim to invert for this parameterization using the scattering integral (SI) method. The available Born sensitivity kernels, within this approach, can be used to calculate additional inversion information. We mainly compute the diagonal of the approximate Hessian, used as a conjugate-gradient preconditioner, and the gradients’ step lengths. We consider modeling in the frequency domain. The large computational cost of the SI method can be avoided with direct Helmholtz equation solvers. We applied our method to the VTI Marmousi II model for various inversion strategies. We found that we can invert the [Formula: see text] accurately. For the [Formula: see text] parameter, only the short wavelengths are well-recovered. On the other hand, the [Formula: see text] parameter impact is weak on the inversion results and can be fixed. However, a good background [Formula: see text], with accurate long wavelengths, is needed to correctly invert for [Formula: see text]. Furthermore, we invert a real data set acquired by CGG from offshore Australia. We simultaneously invert all three parameters using our inversion approach. The velocity model is improved, and additional layers are recovered. We confirm the accuracy of the results by comparing them with well-log information, as well as looking at the data and angle gathers.

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