Nonhyperbolic moveout analysis in VTI media using rational interpolation

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. D59-D71 ◽  
Author(s):  
Huub Douma ◽  
Alexander Calvert
Keyword(s):  
Field Data ◽  
Symmetry Axis ◽  
Interpolation Method ◽  
Transversely Isotropic ◽  
Rational Interpolation ◽  
Data Types ◽  
Computational Overhead ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Vti Media

Anisotropic velocity analysis using qP-waves in transversely isotropic media with a vertical symmetry axis (VTI) usually is done by inferring the anellipticity parameter [Formula: see text] and the normal moveout velocity [Formula: see text] from the nonhyperbolic character of the moveout. Several approximations explicit in these parameters exist with varying degrees of accuracy. Here, we present a rational interpolation approach to nonhyperbolic moveout analysis in the [Formula: see text] domain. This method has no additional computational overhead compared to using expressions explicit in [Formula: see text] and [Formula: see text]. The lack of such overhead stems from the observation that, for fixed [Formula: see text] and zero-offset two-way traveltime [Formula: see text], the moveout curve for different values of [Formula: see text] can be calculated by simple stretching of the offset axis. This observation is based on the assumptions that the traveltimes of qP-waves in transversely isotropic media mainly depend on [Formula: see text] and [Formula: see text], and that the shear-wave velocity along the symmetry axis has a negligibleinfluence on these traveltimes. The accuracy of the rational interpolation method is as good as that of these approximations. The method can be tuned accurately to any offset range of interest by increasing the order of the interpolation. We test the method using both synthetic and field data and compare it with the nonhyperbolic moveout equation of Alkhalifah and Tsvankin (1995) and the shifted hyperbola equation of Fomel (2004). Both data types confirm that for [Formula: see text], our method significantly outperforms the nonhyperbolic moveout equation in terms of combined unbiased parameter estimation with accurate moveout correction. Comparison with the shifted hyperbola equation of Fomel for Greenhorn-shale anisotropy establishes almost identical accuracy of the rational interpolation method and his equation. Even though the proposed method currently deals with homogeneous media only, results from application to synthetic and field data confirm the applicability of the proposed method to horizontally layered VTI media.

Kirchhoff time migration for transversely isotropic media: An application to Trinidad data

Geophysics ◽  
2006 ◽  
Vol 71 (1) ◽  
pp. S29-S35 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Data Set ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Higher Order Terms ◽  
Isotropic Media ◽  
Migration Method ◽  
Vti Media ◽  
Vertical Inhomogeneity

Using a newly developed nonhyperbolic offset-mid-point traveltime equation for prestack Kirchhoff time migration, instead of the conventional double-square-root (DSR) equation, results in overall better images from anisotropic data. Specifically, prestack Kirchhoff time migration for transversely isotropic media with a vertical symmetry axis (VTI media) is implemented using an analytical offset-midpoint traveltime equation that represents the equivalent of Cheop's pyramid for VTI media. It includes higher-order terms necessary to better handle anisotropy as well as vertical inhomogeneity. Application of this enhanced Kirchhoff time-migration method to the anisotropic Marmousi data set demonstrates the effectiveness of the approach. Further application of the method to field data from Trinidad results in sharper reflectivity images of the subsurface, with the faults better focused and positioned than with images obtained using isotropic methods. The superiority of the anisotropic time migration is evident in the flatness of the image gathers.

Rational interpolation of qP-traveltimes for semblance-based anisotropy estimation in layered VTI media

Geophysics ◽  
2008 ◽  
Vol 73 (4) ◽  
pp. D53-D62 ◽  
Author(s):  
Huub Douma ◽  
Mirko van der Baan
Keyword(s):  
Parameter Estimation ◽  
Random Noise ◽  
Anisotropy Parameter ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Rational Interpolation ◽  
Transversely Isotropic Media ◽  
Interpolation Procedure ◽  
Isotropic Media ◽  
Vti Media

The [Formula: see text] domain is the natural domain for anisotropy parameter estimation in horizontally layered media. The need to transform the data to the [Formula: see text] domain or to pick traveltimes in the [Formula: see text] domain is, however, a practical disadvantage. To overcome this, we combine [Formula: see text]-derived traveltimes and offsets in horizontally layered transversely isotropic media with a vertical symmetry axis (VTI) with a rational interpolation procedure applied in the [Formula: see text] domain. This combination results in an accurate and efficient [Formula: see text]-based semblance analysis for anisotropy parameter estimation from the moveout of qP-waves in horizontally layered VTI media. The semblance analysis is applied to the moveout to search directly for the interval values of the relevant parameters. To achieve this, the method is applied in a layer-stripping fashion. We demonstrate the method using synthetic data examples and show that it is robust in the presence of random noise and moderate statics.

Kinematics of 3-D DMO operators in transversely isotropic media

Geophysics ◽  
1997 ◽  
Vol 62 (4) ◽  
pp. 1214-1219 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Transversely Isotropic Media ◽  
Out Of Plane ◽  
Isotropic Media ◽  
The Cross ◽  
Vti Media

The 3-D dip‐moveout (DMO) operator in homogeneous transversely isotropic media with vertical symmetry axis (VTI media), unlike that in homogeneous isotropic media, has an out‐of‐plane (cross‐line) component. In general, this component has a shape that is opposite to that of the cross‐line component of the isotropic v(z) saddle‐shaped operator. The width of the cross‐line component of the VTI operator is also smaller overall than that associated with isotropic v(z) media. When both typical anisotropy and inhomogeneity are combined, the net result is an operator with a smaller cross‐line component and a shape that is influenced more by the v(z) behavior. The large cost of a 3-D DMO operator, as well as the generally small cross‐line components associated with the DMO operator in VTI media, suggests the possibility, as is usually done for isotropic v(z) media, of just ignoring the contribution of the out‐of‐plane portion of the operator.

Transverse isotropy of the upper mantle in the vicinity of Pacific fracture zones

1969 ◽  
Vol 59 (1) ◽  
pp. 59-72
Author(s):  
Robert S. Crosson ◽  
Nikolas I. Christensen
Keyword(s):  
Upper Mantle ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Seismic Wave Propagation ◽  
Fracture Zones ◽  
Mantle Material ◽  
Transversely Isotropic Media ◽  
Horizontal Symmetry ◽  
Isotropic Media

Abstract Several recent investigations suggest that portions of the Earth's upper mantle behave anisotropically to seismic wave propagation. Since several types of anisotropy can produce azimuthal variations in Pn velocities, it is of particular geophysical interest to provide a framework for the recognition of the form or forms of anisotropy most likely to be manifest in the upper mantle. In this paper upper mantle material is assumed to possess the elastic properties of transversely isotropic media. Equations are presented which relate azimuthal variations in Pn velocities to the direction and angle of tilt of the symmetry axis of a transversely isotropic upper mantle. It is shown that the velocity data of Raitt and Shor taken near the Mendocino and Molokai fracture zones can be adequately explained by the assumption of transverse isotropy with a nearly horizontal symmetry axis.

scholarly journals The offset-midpoint traveltime pyramid in 3D transversely isotropic media with a horizontal symmetry axis

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. T51-T62 ◽  
Author(s):  
Qi Hao ◽  
Alexey Stovas ◽  
Tariq Alkhalifah
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
P Wave ◽  
Analytic Representation ◽  
Velocity Inversion ◽  
Transversely Isotropic Media ◽  
Horizontal Symmetry ◽  
Time Domains ◽  
Isotropic Media ◽  
Hti Media

Analytic representation of the offset-midpoint traveltime equation for anisotropy is very important for prestack Kirchhoff migration and velocity inversion in anisotropic media. For transversely isotropic media with a vertical symmetry axis, the offset-midpoint traveltime resembles the shape of a Cheops’ pyramid. This is also valid for homogeneous 3D transversely isotropic media with a horizontal symmetry axis (HTI). We extended the offset-midpoint traveltime pyramid to the case of homogeneous 3D HTI. Under the assumption of weak anellipticity of HTI media, we derived an analytic representation of the P-wave traveltime equation and used Shanks transformation to improve the accuracy of horizontal and vertical slownesses. The traveltime pyramid was derived in the depth and time domains. Numerical examples confirmed the accuracy of the proposed approximation for the traveltime function in 3D HTI media.

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.

Reflection and transmission responses in a layered transversely isotropic medium with horizontal symmetry axis

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. C143-C157 ◽  
Author(s):  
Song Jin ◽  
Alexey Stovas
Keyword(s):  
Good Accuracy ◽  
Symmetry Axis ◽  
Local Property ◽  
Transversely Isotropic ◽  
Weak Anisotropy ◽  
Reflection And Transmission ◽  
Numerical Tests ◽  
Transversely Isotropic Media ◽  
Horizontal Symmetry ◽  
Isotropic Media

Seismic wave reflection and transmission (R/T) responses characterize the subsurface local property, and the widely spread anisotropy has considerable influences even at small incident angles. We have considered layered transversely isotropic media with horizontal symmetry axes (HTI), and the symmetry axes were not restricted to be aligned. With the assumption of weak contrast across the interface, linear approximations for R/T coefficients normalized by vertical energy flux are derived based on a simple layered HTI model. We also obtain the approximation with the isotropic background medium under an additional weak anisotropy assumption. Numerical tests illustrate the good accuracy of the approximations compared with the exact results.

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