Stacking-velocity tomography for tilted orthorhombic media

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. C171-C180 ◽  
Author(s):  
Qifan Liu ◽  
Ilya Tsvankin
Keyword(s):  
Parameter Estimation ◽  
Subsurface Layer ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
P Waves ◽  
Additional Information ◽  
Reflection Data ◽  
Anisotropic Layers ◽  
Zero Offset ◽  
Velocity Tomography

Tilted orthorhombic (TOR) models are typical for dipping anisotropic layers, such as fractured shales, and can also be due to nonhydrostatic stress fields. Velocity analysis for TOR media, however, is complicated by the large number of independent parameters. Using multicomponent wide-azimuth reflection data, we develop stacking-velocity tomography to estimate the interval parameters of TOR media composed of homogeneous layers separated by plane dipping interfaces. The normal-moveout (NMO) ellipses, zero-offset traveltimes, and reflection time slopes of P-waves and split S-waves ([Formula: see text] and [Formula: see text]) are used to invert for the interval TOR parameters including the orientation of the symmetry planes. We show that the inversion can be facilitated by assuming that the reflector coincides with one of the symmetry planes, which is a common geologic constraint often employed for tilted transversely isotropic media. This constraint makes the inversion for a single TOR layer feasible even when the initial model is purely isotropic. If the dip plane is also aligned with one of the symmetry planes, we show that the inverse problem for [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-waves can be solved analytically. When only [Formula: see text]-wave data are available, parameter estimation requires combining NMO ellipses from a horizontal and dipping interface. Because of the increase in the number of independent measurements for layered TOR media, constraining the reflector orientation is required only for the subsurface layer. However, the inversion results generally deteriorate with depth because of error accumulation. Using tests on synthetic data, we demonstrate that additional information such as knowledge of the vertical velocities (which may be available from check shots or well logs) and the constraint on the reflector orientation can significantly improve the accuracy and stability of interval parameter estimation.

scholarly journals NMO‐velocity surfaces and Dix‐type formulas in anisotropic heterogeneous media

Geophysics ◽  
2002 ◽  
Vol 67 (3) ◽  
pp. 939-951 ◽  
Author(s):  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Cross Sections ◽  
Heterogeneous Media ◽  
Transversely Isotropic ◽  
Radius Vector ◽  
P Wave ◽  
Reflection Data ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Anisotropic Layers ◽  
Zero Offset

Reflection moveout of pure modes recorded on conventional‐length spreads is described by a normal‐moveout (NMO) velocity that depends on the orientation of the common‐midpoint (CMP) line. Here, we introduce the concept of NMO‐velocity surfaces, which are obtained by plotting the NMO velocity as the radius‐vector along all possible directions in 3‐D space, and use it to develop Dix‐type averaging and differentiation algorithms in anisotropic heterogeneous media. The intersection of the NMO‐velocity surface with the horizontal plane represents the NMO ellipse that can be estimated from wide‐azimuth reflection data. We demonstrate that the NMO ellipse and conventional‐spread moveout as a whole can be modeled by Dix‐type averaging of specifically oriented cross‐sections of the NMO‐velocity surfaces along the zero‐offset reflection raypath. This formalism is particularly simple to implement for a stack of homogeneous anisotropic layers separated by plane dipping boundaries. Since our method involves computing just a single (zero‐offset) ray for a given reflection event, it can be efficiently used in anisotropic stacking‐velocity tomography. Application of the Dix‐type averaging to layered transversely isotropic media with a vertical symmetry axis (VTI) shows that the presence of dipping interfaces above the reflector makes the P‐wave NMO ellipse dependent on the vertical velocity and anisotropic coefficients ε and δ. In contrast, P‐wave moveout in VTI models with a horizontally layered overburden is fully controlled by the NMO velocity of horizontal events and the Alkhalifah‐Tsvankin coefficient η ≈ ε − δ. Hence, in some laterally heterogeneous, layered VTI models P‐wave reflection data may provide enough information for anisotropic depth processing.

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.

The migrator’s equation for anisotropic media

Geophysics ◽  
1990 ◽  
Vol 55 (11) ◽  
pp. 1429-1434 ◽  
Author(s):  
N. F. Uren ◽  
G. H. F. Gardner ◽  
J. A. McDonald
Keyword(s):  
Physical Model ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Original Structure ◽  
Model Data ◽  
P Waves ◽  
Reflection Seismic ◽  
Isotropic Media ◽  
Zero Offset

The migrator’s equation, which gives the relationship between real and apparent dips on a reflector in zero‐offset reflection seismic sections, may be readily implemented in one step with a frequency‐domain migration algorithm for homogeneous media. Huygens’ principle is used to derive a similar relationship for anisotropic media where velocities are directionally dependent. The anisotropic form of the migrator’s equation is applicable to both elliptically and nonelliptically anisotropic media. Transversely isotropic media are used to demonstrate the performance of an f-k implementation of the migrator’s equation for anisotropic media. In such a medium SH-waves are elliptically anisotropic, while P-waves are nonelliptically anisotropic. Numerical model data and physical model data demonstrate the performance of the algorithm, in each case recovering the original structure. Isotropic and anisotropic migration of anisotropic physical model data are compared experimentally, where the anisotropic velocity function of the medium has a vertical axis of symmetry. Only when anisotropic migration is used is the original structure recovered.

Traveltime approximations for a layered transversely isotropic medium

Geophysics ◽  
2006 ◽  
Vol 71 (2) ◽  
pp. D23-D33 ◽  
Author(s):  
Bjørn Ursin ◽  
Alexey Stovas
Keyword(s):  
Taylor Series ◽  
Isotropic Medium ◽  
Continued Fraction ◽  
Transversely Isotropic ◽  
Horizontal Distance ◽  
Weak Anisotropy ◽  
Sh Waves ◽  
Numerical Studies ◽  
Anisotropic Layers ◽  
Zero Offset

We consider multiple transmitted, reflected, and converted qP-qSV-waves or multiple transmitted and reflected SH-waves in a horizontally layered medium that is transversely isotropic with a vertical symmetry axis (VTI). Traveltime and offset (horizontal distance) between a source and receiver, not necessarily in the same layer, are expressed as functions of horizontal slowness. These functions are given in terms of a Taylor series in slowness in exactly the same form as for a layered isotropic medium. The coefficients depend on the parameters of the anisotropic layers through which the wave has passed, and there is no weak anisotropy assumption. Using classical formulas, the traveltime or traveltime squared can then be expressed as a Taylor series in even powers of offset. These Taylor series give rise to a shifted hyperbola traveltime approximation and a new continued-fraction approximation, described by four parameters that match the Taylor series up to the sixth power in offset. Further approximations give several simplified continued-fraction approximations, all of which depend on three parameters: zero-offset traveltime, NMO velocity, and a heterogeneity coefficient. The approximations break down when there is a cusp in the group velocity for the qSV-wave. Numerical studies indicate that approximations of traveltime squared are generally better than those for traveltime. A new continued-fraction approximation that depends on three parameters is more accurate than the commonly used continued-fraction approximation and the shifted hyperbola.

AVO and AVA inversion for fractured reservoir characterization

Geophysics ◽  
2002 ◽  
Vol 67 (1) ◽  
pp. 300-306 ◽  
Author(s):  
Matteo Mario Beretta ◽  
Giancarlo Bernasconi ◽  
Giuseppe Drufuca
Keyword(s):  
Reservoir Characterization ◽  
Spectral Component ◽  
Plane Waves ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Fractured Reservoir ◽  
Fracture Density ◽  
P Waves ◽  
Data Set ◽  
Inversion Procedure

Seismic wave reflection amplitudes are used to detect fluids and fracture properties in reservoirs. This paper studies the characterization of a vertically fractured fluid‐filled reservoir by analyzing the reflection amplitudes of P‐waves with varying incident and azimuthal angles. The reservoir is modeled as a horizontal transversely isotropic medium embedded in an isotropic background, and the linearized P‐waves reflection coefficient are considered. The conditioning of the inverse problem is analyzed, and fracture density is found to be the best conditioned parameter. Using diffraction tomography under the Born approximation, an inversion procedure is proposed in the transformed k–ω domain to detect fracture density variations within the reservoir. Seismic data are rearranged in pairs of incident and reflected plane waves, enlightening only one spectral component of the fracture density field at a time. Only the observable spectral components are inverted. Moreover, working in the transformed domain, picking reflection amplitudes is not required. An example of the inversion applied to a synthetic data set is presented. The limitation of source and receiver numbers and the finite bandwidth of the wavelet produce a loss of resolution, but the overall fracture density variations are recovered correctly.

scholarly journals Theory of interval traveltime parameter estimation in layered anisotropic media

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. C253-C263 ◽  
Author(s):  
Yanadet Sripanich ◽  
Sergey Fomel
Keyword(s):  
Parameter Estimation ◽  
Fourth Order ◽  
Taylor Expansion ◽  
Symmetry Plane ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Interval Parameter ◽  
Effective Parameters ◽  
Zero Offset ◽  
Vti Media

Moveout approximations for reflection traveltimes are typically based on a truncated Taylor expansion of traveltime squared around the zero offset. The fourth-order Taylor expansion involves normal moveout velocities and quartic coefficients. We have derived general expressions for layer-stripping second- and fourth-order parameters in horizontally layered anisotropic strata and specified them for two important cases: horizontally stacked aligned orthorhombic layers and azimuthally rotated orthorhombic layers. In the first of these cases, the formula involving the out-of-symmetry-plane quartic coefficients has a simple functional form and possesses some similarity to the previously known formulas corresponding to the 2D in-symmetry-plane counterparts in vertically transversely isotropic (VTI) media. The error of approximating effective parameters by using approximate VTI formulas can be significant in comparison with the exact formulas that we have derived. We have proposed a framework for deriving Dix-type inversion formulas for interval parameter estimation from traveltime expansion coefficients in the general case and in the specific case of aligned orthorhombic layers. The averaging formulas for calculation of effective parameters and the layer-stripping formulas for interval parameter estimation are readily applicable to 3D seismic reflection processing in layered anisotropic media.

Estimation of interval anisotropy parameters using velocity-independent layer stripping

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. WB117-WB127 ◽  
Author(s):  
Xiaoxiang Wang ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Plane ◽  
Synthetic Data ◽  
Single Layer ◽  
Transversely Isotropic ◽  
P Wave ◽  
Correlated Noise ◽  
Fractured Reservoir ◽  
Processing Parameter ◽  
Reflection Data ◽  
Error Amplification

Moveout analysis of long-spread P-wave data is widely used to estimate the key time-processing parameter [Formula: see text] in layered transversely isotropic media with a vertical symmetry axis (VTI). Inversion for interval [Formula: see text] values, however, suffers from instability caused by the trade-off between the effective moveout parameters and by subsequent error amplification during Dix-type layer stripping. We propose an alternative approach to nonhyperbolic moveout inversion based on the velocity-independent layer-stripping (VILS) method of Dewangan and Tsvankin. Also, we develop the 3D version of VILS and apply it to interval parameter estimation in orthorhombic media using wide-azimuth, long-spread data. If the overburden is laterally homogeneous and has a horizontal symmetry plane, VILS produces the exact interval traveltime-offset function in the target layer without knowledgeof the velocity field. Hence, Dix-type differentiation of moveout parameters used in existing techniques is replaced by the much more stable layer stripping of reflection traveltimes. The interval traveltimes are then inverted for the moveout parameters using the single-layer nonhyperbolic moveout equation. The superior accuracy and stability of the algorithm are illustrated on ray-traced synthetic data for typical VTI and orthorhombic models. Even small correlated noise in reflection traveltimes causes substantial distortions in the interval [Formula: see text] values computed by conventional Dix-type differentiation. In contrast, the output of VILS is insensitive to mild correlated traveltime errors. The algorithm is also tested on wide-azimuth P-wave reflection data recorded above a fractured reservoir at Rulison field in Colorado. The interval moveout parameters estimated by VILS in the shale layer above the reservoir are more plausible and less influenced by noise than those obtained by the Dix-type method.

Parameter estimation in orthorhombic media using multicomponent wide-azimuth reflection data

Geophysics ◽  
2005 ◽  
Vol 70 (2) ◽  
pp. D1-D8 ◽  
Author(s):  
Vladimir Grechka ◽  
Andrés Pech ◽  
Ilya Tsvankin
Keyword(s):  
Parameter Estimation ◽  
Fractured Reservoirs ◽  
Symmetry Plane ◽  
Velocity Model ◽  
Naturally Fractured Reservoirs ◽  
Converted Wave ◽  
Reflection Data ◽  
Reflection Time ◽  
Horizontal Symmetry ◽  
Zero Offset

Orthorhombic models with a horizontal symmetry plane adequately describe seismic signatures recorded over many naturally fractured reservoirs. The inversion of wide-azimuth traveltimes of PP and SS (the fast [Formula: see text] and slow [Formula: see text]) reflections are discussed for Tsvankin's anisotropic parameters and the azimuths of the vertical symmetry planes of orthorhombic media. If shear waves are not excited, SS traveltimes can be found from PP and PS (converted-wave) data, which makes the method applicable to offshore surveys. The feasibility of parameter estimation is strongly dependent on reflector dip and orientation. For a horizontal reflector beneath a single orthorhombic layer, the vertical velocities and reflector depth cannot be found from conventional-spread reflection traveltimes alone. If the reflector is dipping, the inversion is ambiguous when the dip plane is close to one of the vertical symmetry planes of the orthorhombic layer above it. The parameter estimation becomes possible if the dip direction deviates by more than 10° from the nearest symmetry plane. We apply multicomponent stacking velocity tomography to perform velocity analysis for stratified orthorhombic models composed of homogeneous layers separated by plane or curved interfaces. The tomographic algorithm, which operates with the normal-moveout (NMO) ellipses, zero-offset traveltimes, and reflection time slopes of PP- and SS-waves, is designed to build the orthorhombic velocity model in the depth domain by estimating the anisotropic parameters and the shapes of the reflecting interfaces. Numerical tests show that for layered orthorhombic media, it is necessary to put constraints on the vertical velocities to avoid instability in the inversion of noise-contaminated reflection data.

Moveout inversion of wide-azimuth P-wave data for tilted TI media

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA23-WA29 ◽  
Author(s):  
Xiaoxiang Wang ◽  
Ilya Tsvankin
Keyword(s):  
Parameter Estimation ◽  
Input Data ◽  
Symmetry Axis ◽  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
Estimation Algorithm ◽  
Velocity Model ◽  
P Wave ◽  
Symmetry Direction ◽  
Zero Offset

Currently TTI (transversely isotropic with a tilted symmetry axis) models are widely used for velocity analysis and imaging in many exploration areas. We develop a 3D parameter-estimation algorithm for TTI media composed of homogeneous layers separated by plane dipping interfaces. The input data include P-wave NMO ellipses and time slopes (horizontal slownesses of the zero-offset rays) combined with borehole information. If the symmetry axis is perpendicular to the bottom of each layer, it is possible to estimate the interval symmetry-direction velocity VP0 , anisotropy parameter [Formula: see text], and the reflector orientation using a single constraint — the reflector depth. The algorithm can tolerate small [Formula: see text] deviation of the symmetry axis from the reflector normal. However, as is the case for the 2D problem, the parameter [Formula: see text] can seldom be obtained without nonhyperbolic moveout inversion. If the symmetry axis deviates from the reflector normal but is confined to the dip plane, stable parameter estimation requires specifying a relationship between the tilt and dip in each layer. When the tilt represents a free parameter, the input data have to be supplemented by wide-azimuth VSP traveltimes with the offset reaching at least 1/4 of the maximum reflector depth. Moreover, the additional angle coverage provided by VSP data may help resolve the parameter [Formula: see text] in the upper part of the model. The developed methodology can be used to build an accurate initial anisotropic velocity model for processing of wide-azimuth surveys.

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.

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