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.

scholarly journals The offset‐midpoint traveltime pyramid in transversely isotropic media

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1316-1325 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
Fourth Power ◽  
Weak Anisotropy ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Equivalent Equation ◽  
Two Parameters ◽  
Vti Media

Prestack Kirchhoff time migration for transversely isotropic media with a vertical symmetry axis (VTI media) is implemented using an offset‐midpoint traveltime equation, Cheop’s pyramid equivalent equation for VTI media. The derivation of such an equation for VTI media requires approximations that pertain to high frequency and weak anisotropy. Yet the resultant offset‐midpoint traveltime equation for VTI media is highly accurate for even strong anisotropy. It is also strictly dependent on two parameters: NMO velocity and the anisotropy parameter, η. It reduces to the exact offset‐midpoint traveltime equation for isotropic media when η = 0. In vertically inhomogeneous media, the NMO velocity and η parameters in the offset‐midpoint traveltime equation are replaced by their effective values: the velocity is replaced by the rms velocity and η is given by a more complicated equation that includes summation of the fourth power of velocity.

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.

scholarly journals Traveltime approximations for transversely isotropic media with an inhomogeneous background

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA31-WA42 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Transversely Isotropic Media ◽  
Higher Order Terms ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Series Type ◽  
Ti Media ◽  
Taylor’S Series

A transversely isotropic (TI) model with a tilted symmetry axis is regarded as one of the most effective approximations to the Earth subsurface, especially for imaging purposes. However, we commonly utilize this model by setting the axis of symmetry normal to the reflector. This assumption may be accurate in many places, but deviations from this assumption will cause errors in the wavefield description. Using perturbation theory and Taylor’s series, I expand the solutions of the eikonal equation for 2D TI media with respect to the independent parameter [Formula: see text], the angle the tilt of the axis of symmetry makes with the vertical, in a generally inhomogeneous TI background with a vertical axis of symmetry. I do an additional expansion in terms of the independent (anellipticity) parameter [Formula: see text] in a generally inhomogeneous elliptically anisotropic background medium. These new TI traveltime solutions are given by expansions in [Formula: see text] and [Formula: see text] with coefficients extracted from solving linear first-order partial differential equations. Pade approximations are used to enhance the accuracy of the representation by predicting the behavior of the higher-order terms of the expansion. A simplification of the expansion for homogenous media provides nonhyperbolic moveout descriptions of the traveltime for TI models that are more accurate than other recently derived approximations. In addition, for 3D media, I develop traveltime approximations using Taylor’s series type of expansions in the azimuth of the axis of symmetry. The coefficients of all these expansions can also provide us with the medium sensitivity gradients (Jacobian) for nonlinear tomographic-based inversion for the tilt in the symmetry axis.

Acoustic approximations for processing in transversely isotropic media

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 623-631 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
P Wave ◽  
Elastic Model ◽  
S Wave ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Two Parameters ◽  
Vti Media

When transversely isotropic (VTI) media with vertical symmetry axes are characterized using the zero‐dip normal moveout (NMO) velocity [[Formula: see text]] and the anisotropy parameter ηinstead of Thomsen’s parameters, time‐related processing [moveout correction, dip moveout (DMO), and time migration] become nearly independent of the vertical P- and S-wave velocities ([Formula: see text] and [Formula: see text], respectively). The independence on [Formula: see text] and [Formula: see text] is well within the limits of seismic accuracy, even for relatively strong anisotropy. The dependency on [Formula: see text] and [Formula: see text] reduces even further as the ratio [Formula: see text] decreases. In fact, for [Formula: see text], all time‐related processing depends exactly on only [Formula: see text] and η. This fortunate dependence on two parameters is demonstrated here through analytical derivations of time‐related processing equations in terms of [Formula: see text] and η. The time‐migration dispersion relation, the NMO velocity for dipping events, and the ray‐tracing equations extracted by setting [Formula: see text] (i.e., by considering VTI as acoustic) not only depend solely on [Formula: see text] and η but are much simpler than the counterpart expressions for elastic media. Errors attributed to this use of the acoustic assumption are small and may be neglected. Therefore, as in isotropic media, the acoustic model arising from setting [Formula: see text], although not exactly true for VTI media, can serve as a useful approximation to the elastic model for the kinematics of P-wave data. This approximation can boost the efficiency of imaging and DMO programs for VTI media as well as simplify their description.

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.

Reverse time migration in tilted transversely isotropic media

2020 ◽  
Vol 38 (2) ◽  
Author(s):  
Razec Cezar Sampaio Pinto da Silva Torres ◽  
Leandro Di Bartolo
Keyword(s):  
Seismic Anisotropy ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Computer Clusters ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Tti Media

ABSTRACT. Reverse time migration (RTM) is one of the most powerful methods used to generate images of the subsurface. The RTM was proposed in the early 1980s, but only recently it has been routinely used in exploratory projects involving complex geology – Brazilian pre-salt, for example. Because the method uses the two-way wave equation, RTM is able to correctly image any kind of geological environment (simple or complex), including those with anisotropy. On the other hand, RTM is computationally expensive and requires the use of computer clusters. This paper proposes to investigate the influence of anisotropy on seismic imaging through the application of RTM for tilted transversely isotropic (TTI) media in pre-stack synthetic data. This work presents in detail how to implement RTM for TTI media, addressing the main issues and specific details, e.g., the computational resources required. A couple of simple models results are presented, including the application to a BP TTI 2007 benchmark model.Keywords: finite differences, wave numerical modeling, seismic anisotropy. Migração reversa no tempo em meios transversalmente isotrópicos inclinadosRESUMO. A migração reversa no tempo (RTM) é um dos mais poderosos métodos utilizados para gerar imagens da subsuperfície. A RTM foi proposta no início da década de 80, mas apenas recentemente tem sido rotineiramente utilizada em projetos exploratórios envolvendo geologia complexa, em especial no pré-sal brasileiro. Por ser um método que utiliza a equação completa da onda, qualquer configuração do meio geológico pode ser corretamente tratada, em especial na presença de anisotropia. Por outro lado, a RTM é dispendiosa computacionalmente e requer o uso de clusters de computadores por parte da indústria. Este artigo apresenta em detalhes uma implementação da RTM para meios transversalmente isotrópicos inclinados (TTI), abordando as principais dificuldades na sua implementação, além dos recursos computacionais exigidos. O algoritmo desenvolvido é aplicado a casos simples e a um benchmark padrão, conhecido como BP TTI 2007.Palavras-chave: diferenças finitas, modelagem numérica de ondas, anisotropia sísmica.

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.

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