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.

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.

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.

Velocity analysis using nonhyperbolic moveout in transversely isotropic media

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1839-1854 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
P Wave ◽  
Transversely Isotropic Media ◽  
Vertical Heterogeneity ◽  
Isotropic Media ◽  
D Values ◽  
Two Parameters ◽  
Vti Media ◽  
Ti Media

P‐wave reflections from horizontal interfaces in transversely isotropic (TI) media have nonhyperbolic moveout. It has been shown that such moveout as well as all time‐related processing in TI media with a vertical symmetry axis (VTI media) depends on only two parameters, [Formula: see text] and η. These two parameters can be estimated from the dip‐moveout behavior of P‐wave surface seismic data. Alternatively, one could use the nonhyperbolic moveout for parameter estimation. The quality of resulting estimates depends largely on the departure of the moveout from hyperbolic and its sensitivity to the estimated parameters. The size of the nonhyperbolic moveout in TI media is dependent primarily on the anisotropy parameter η. An “effective” version of this parameter provides a useful measure of the nonhyperbolic moveout even in v(z) isotropic media. Moreover, effective η, [Formula: see text], is used to show that the nonhyperbolic moveout associated with typical TI media (e.g., shales, with η ≃ 0.1) is larger than that associated with typical v(z) isotropic media. The departure of the moveout from hyperbolic is increased when typical anisotropy is combined with vertical heterogeneity. Larger offset‐to‐depth ratios (X/D) provide more nonhyperbolic information and, therefore, increased stability and resolution in the inversion for [Formula: see text]. The X/D values (e.g., X/D > 1.5) needed for obtaining stability and resolution are within conventional acquisition limits, especially for shallow targets. Although estimation of η using nonhyperbolic moveouts is not as stable as using the dip‐moveout method of Alkhalifah and Tsvankin, particularly in the absence of large offsets, it does offer some flexibility. It can be applied in the absence of dipping reflectors and also may be used to estimate lateral η variations. Application of the nonhyperbolic inversion to data from offshore Africa demonstrates its usefulness, especially in estimating lateral and vertical variations in η.

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.

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.

Migration of P‐wave reflection data in transversely isotropic media

Geophysics ◽  
1994 ◽  
Vol 59 (4) ◽  
pp. 591-596 ◽  
Author(s):  
Suhas Phadke ◽  
S. Kapotas ◽  
N. Dai ◽  
Ernest R. Kanasewich
Keyword(s):  
Wave Equation ◽  
Dispersion Relation ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Limited Range ◽  
Horizontal Velocity ◽  
P Wave ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Wave propagation in transversely isotropic media is governed by the horizontal and vertical wave velocities. The quasi‐P(qP) wavefront is not an ellipse; therefore, the propagation cannot be described by the wave equation appropriate for elliptically anisotropic media. However, for a limited range of angles from the vertical, the dispersion relation for qP‐waves can be approximated by an ellipse. The horizontal velocity necessary for this approximation is different from the true horizontal velocity and depends upon the physical properties of the media. In the method described here, seismic data is migrated using a 45-degree wave equation for elliptically anisotropic media with the horizontal velocity determined by comparing the 45-degree elliptical dispersion relation and the quasi‐P‐dispersion relation. The method is demonstrated for some synthetic data sets.

Anisotropic complex Padé hybrid finite-difference depth migration

Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. S51-S59 ◽  
Author(s):  
Daniela Amazonas ◽  
Rafael Aleixo ◽  
Jörg Schleicher ◽  
Jessé C. Costa
Keyword(s):  
Finite Difference ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Evanescent Waves ◽  
Acoustic Wave Equation ◽  
Depth Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Numerical Instabilities ◽  
Branch Cut

Standard real-valued finite-difference (FD) and Fourier finite-difference (FFD) migrations cannot handle evanescent waves correctly, which can lead to numerical instabilities in the presence of strong velocity variations. A possible solution to these problems is the complex Padé approximation, which avoids problems with evanescent waves by rotating the branch cut of the complex square root. We have applied this approximation to the acoustic wave equation for vertical transversely isotropic media to derive more stable FD and hybrid FD/FFD migrations for such media. Our analysis of the dispersion relation of the new method indicates that it should provide more stable migration results with fewer artifacts and higher accuracy at steep dips. Our studies lead to the conclusion that the rotation angle of the branch cut that should yield the most stable image is 60° for FD migration, as confirmed by numerical impulse responses and work with synthetic data.

Analytic study of the effective parameters for determination of the NMO velocity function in transversely isotropic media

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1855-1866 ◽  
Author(s):  
Jack K. Cohen
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Velocity Function ◽  
Transversely Isotropic Media ◽  
Wave Phase Velocity ◽  
Isotropic Media ◽  
Anisotropy Parameters ◽  
Vti Media ◽  
Ray Parameter

In their studies of transversely isotropic media with a vertical symmetry axis (VTI media), Alkhalifah and Tsvankin observed that, to a high numerical accuracy, the normal moveout (NMO) velocity for dipping reflectors as a function of ray parameter p depends mainly on just two parameters, each of which can be determined from surface P‐wave observations. They substantiated this result by using the weak‐anisotropy approximation and exploited it to develop a time‐domain processing sequence that takes into account vertical transverse isotropy. In this study, the two‐parameter Alkhalifah‐Tsvankin result was further examined analytically. It was found that although there is (as these authors already observed) some dependence on the remaining parameters of the problem, this dependence is weak, especially in the practically important regimes of weak to moderately strong transverse isotropy and small ray parameter. In each of these regimes, an analytic solution is derived for the anisotropy parameter η required for time‐domain P‐wave imaging in VTI media. In the case of elliptical anisotropy (η = 0), NMO velocity expressed through p is fully controlled just by the zero‐dip NMO velocity—one of the Alkhalifah‐ Tsvankin parameters. The two‐parameter representation of NMO velocity also was shown to be exact in another limit—that of the zero shear‐wave vertical velociy. The analytic results derived here are based on new representations for both the P‐wave phase velocity and normal moveout velocity in terms of the ray parameter, with explicit expressions given for the cases of vanishing onaxis shear speed, weak to moderate transverse isotropy, and small to moderate ray parameter. Using these formulas, I have rederived and, in some cases, extended in a uniform manner various results of Tsvankin, Alkhalifah, and others. Examples include second‐order expansions in the anisotropy parameters for both the P‐wave phase‐velocity function and NMO‐velocity function, as well as expansions in powers of the ray parameter for both of these functions. I have checked these expansions against the corresponding exact functions for several choices of the anisotropy parameters.

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