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.

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 Transformation to zero offset in transversely isotropic media

Geophysics ◽  
1996 ◽  
Vol 61 (4) ◽  
pp. 947-963 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Ray Tracing ◽  
Impulse Response ◽  
Homogeneous Medium ◽  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Amplitude Distribution ◽  
Isotropic Media ◽  
Zero Offset ◽  
Vertical Inhomogeneity

Nearly all dip‐moveout correction (DMO) implementations to date assume isotropic homogeneous media. Usually, this has been acceptable considering the tremendous cost savings of homogeneous isotropic DMO and considering the difficulty of obtaining the anisotropy parameters required for effective implementation. In the presence of typical anisotropy, however, ignoring the anisotropy can yield inadequate results. Since anisotropy may introduce large deviations from hyperbolic moveout, accurate transformation to zero‐offset in anisotropic media should address such nonhyperbolic moveout behavior of reflections. Artley and Hale’s v(z) ray‐tracing‐based DMO, developed for isotropic media, provides an attractive approach to treating such problems. By using a ray‐tracing procedure crafted for anisotropic media, I modify some aspects of their DMO so that it can work for v(z) anisotropic media. DMO impulse responses in typical transversely isotropic (TI) models (such as those associated with shales) deviate substantially from the familiar elliptical shape associated with responses in homogeneous isotropic media (to the extent that triplications arise even where the medium is homogeneous). Such deviations can exceed those caused by vertical inhomogeneity, thus emphasizing the importance of taking anisotropy into account in DMO processing. For isotropic or elliptically anisotropic media, the impulse response is an ellipse; but as the key anisotropy parameter η varies, the shape of the response differs substantially from elliptical. For typical η > 0, the impulse response in TI media tends to broaden compared to the response in an isotropic homogeneous medium, a behavior opposite to that encountered in typical v(z) isotropic media, where the response tends to be squeezed. Furthermore, the amplitude distribution along the DMO operator differs significantly from that for isotropic media. Application of this anisotropic DMO to data from offshore Africa resulted in a considerably better alignment of reflections from horizontal and dipping reflectors in common‐midpoint gather than that obtained using an isotropic DMO. Even the presence of vertical inhomogeneity in this medium could not eliminate the importance of considering the shale‐induced anisotropy.

Theory of traveltime shifts around compacting reservoirs: 3D solutions for heterogeneous anisotropic media

Geophysics ◽  
2009 ◽  
Vol 74 (1) ◽  
pp. D25-D36 ◽  
Author(s):  
Rodrigo Felício Fuck ◽  
Andrey Bakulin ◽  
Ilya Tsvankin
Keyword(s):  
Isotropic Medium ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Time Lapse ◽  
Theory Of Elasticity ◽  
Closed Form Expression ◽  
Reservoir Compaction ◽  
Velocity Changes ◽  
Zero Offset ◽  
Induced Velocity

Time-lapse traveltime shifts of reflection events recorded above hydrocarbon reservoirs can be used to monitor production-related compaction and pore-pressure changes. Existing methodology, however, is limited to zero-offset rays and cannot be applied to traveltime shifts measured on prestack seismic data. We give an analytic 3D description of stress-related traveltime shifts for rays propagating along arbitrary trajectories in heterogeneous anisotropic media. The nonlinear theory of elasticity helps to express the velocity changes in and around the reservoir through the excess stresses associated with reservoir compaction. Because this stress-induced velocity field is both heterogeneous and anisotropic, it should be studied using prestack traveltimes or amplitudes. Then we obtain the traveltime shifts by first-order perturbation of traveltimes that accounts not only for the velocity changes but also for 3D deformation of reflectors. The resulting closed-form expression can be used efficiently for numerical modeling of traveltime shifts and, ultimately, for reconstructing the stress distribution around compacting reservoirs. The analytic results are applied to a 2D model of a compacting rectangular reservoir embedded in an initially homogeneous and isotropic medium. The computed velocity changes around the reservoir are caused primarily by deviatoric stresses and produce a transversely isotropic medium with a variable orientation of the symmetry axis and substantial values of the Thomsen parameters [Formula: see text] and [Formula: see text]. The offset dependence of the traveltime shifts should play a crucial role in estimating the anisotropy parameters and compaction-related deviatoric stress components.

Dip moveout in anisotropic media

Geophysics ◽  
1990 ◽  
Vol 55 (7) ◽  
pp. 863-867 ◽  
Author(s):  
N. F. Uren ◽  
G. H. F. Gardner ◽  
J. A. McDonald
Keyword(s):  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Point Scatterer ◽  
Velocity Function ◽  
Processing Step ◽  
Axis Of Symmetry ◽  
Zero Offset ◽  
Seismic Lines ◽  
Seismic Measurements ◽  
Simple Numerical Model

When a common‐midpoint gather is collected above a dipping reflector, the point at which reflection occurs moves updip as the source‐receiver offset increases. Stacking velocity is constant, but it is a function of dip. Hence a stacked trace is not equivalent to a zero‐offset recorded trace. Dip moveout (DMO) is a processing step which converts traces to the equivalent of true zero‐offset records, making migration after stack (MAS) equivalent to migration before stack (MBS). The theory of velocity‐independent Gardner DMO is extended to triaxially elliptically anisotropic media in this paper. It is shown that the transformation is exact for homogeneous elliptically anisotropic media and that, after DMO, the stacking velocity is the horizontal component of the elliptically anisotropic velocity function. By taking three distinct seismic lines, three of the six constants of triaxial elliptical anisotropy may be determined. The remaining three cannot be obtained from surface seismic measurements. A simple numerical model of a point scatterer in a transversely isotropic medium with a tilted axis of symmetry is used to generate examples. The DMO process works when the velocity function is elliptical, but is not exact when the velocity function is nonelliptical.

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.

Forbidden directions for inhomogeneous pure shear waves in dissipative anisotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (2) ◽  
pp. 522-530 ◽  
Author(s):  
José M. Carcione ◽  
Fabio Cavallini
Keyword(s):  
Wave Propagation ◽  
Pure Shear ◽  
Plane Waves ◽  
Symmetry Plane ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Material Interfaces ◽  
Inhomogeneous Waves ◽  
Propagation Properties ◽  
The Waves

In this work we investigate the wave‐propagation properties of pure shear, inhomogeneous, viscoelastic plane waves in the symmetry plane of a monoclinic medium. In terms of seismic propagation, the problem is to describe SH‐waves traveling through a fractured transversely isotropic formation where we assume that the waves are inhomogeneous with amplitudes varying across surfaces of constant phase. This assumption is widely supported by theoretical and experimental evidence. The results are presented in terms of polar diagrams of the quality factor, attenuation, slowness, and energy velocity curves. Inhomogeneous waves are more anisotropic and dissipative than homogeneous viscoelastic plane waves, for which the wavenumber and attenuation directions coincide. Moreover, the theory predicts, beyond a given degree of inhomogeneity, the existence of “stop bands” where there is no wave propagation. This phenomenon does not occur in dissipative isotropic and elastic anisotropic media. The combination of anelasticity and anisotropy activates these bands. They exist even in very weakly anisotropic and quasi‐elastic materials; only a finite value of Q is required. Weaker anisotropy does not affect the width of the bands, but increases the threshold of inhomogeneity above which they appear; moreover, near the threshold, lower attenuation implies narrower bands. A numerical simulation suggests that, in the absence of material interfaces or heterogeneities, the wavefield is mainly composed of homogeneous waves.

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.

scholarly journals Dip‐moveout processing by Fourier transform in anisotropic media

Geophysics ◽  
1997 ◽  
Vol 62 (4) ◽  
pp. 1260-1269 ◽  
Author(s):  
John E. Anderson ◽  
Ilya Tsvankin
Keyword(s):  
Transverse Isotropy ◽  
Computing Time ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
P Wave ◽  
Surface Reflection ◽  
Anisotropic Models ◽  
Reflection Data ◽  
Isotropic Media ◽  
Vti Media

Conventional dip‐moveout (DMO) processing is designed for isotropic media and cannot handle angle‐dependent velocity. We show that Hale's isotropic DMO algorithm remains valid for elliptical anisotropy but may lead to serious errors for nonelliptical models, even if velocity anisotropy is moderate. Here, Hale's constant‐velocity DMO method is extended to anisotropic media. The DMO operator, to be applied to common‐offset data corrected for normal moveout (NMO), is based on the analytic expression for dip‐dependent NMO velocity given by Tsvankin. Since DMO correction in anisotropic media requires knowledge of the velocity field, it should be preceded by an inversion procedure designed to obtain the normal‐moveout velocity as a function of ray parameter. For transversely isotropic models with a vertical symmetry axis (VTI media), P‐wave NMO velocity depends on a single anisotropic coefficient (η) that can be determined from surface reflection data. Impulse responses and synthetic examples for typical VTI media demonstrate the accuracy and efficiency of this DMO technique. Once the inversion step has been completed, the NMO-DMO sequence does not take any more computing time than the genetic Hale method in isotropic media. Our DMO operator is not limited to vertical transverse isotropy as it can be applied in the same fashion in symmetry planes of more complicated anisotropic models such as orthorhombic.

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.

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