Inversion of azimuthally dependent NMO velocity in transversely isotropic media with a tilted axis of symmetry

Geophysics ◽  
2000 ◽  
Vol 65 (1) ◽  
pp. 232-246 ◽  
Author(s):  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Wave Velocity ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Model Parameters ◽  
Symmetry Direction ◽  
Time Migration ◽  
Inversion Procedure ◽  
Tti Media

Just as the transversely isotropic model with a vertical symmetry axis (VTI media) is typical for describing horizontally layered sediments, transverse isotropy with a tilted symmetry axis (TTI) describes dipping TI layers (such as tilted shale beds near salt domes) or crack systems. P-wave kinematic signatures in TTI media are controlled by the velocity [Formula: see text] in the symmetry direction, Thomsen’s anisotropic coefficients ε and δ, and the orientation (tilt ν and azimuth β) of the symmetry axis. Here, we show that all five parameters can be obtained from azimuthally varying P-wave NMO velocities measured for two reflectors with different dips and/or azimuths (one of the reflectors can be horizontal). The shear‐wave velocity [Formula: see text] in the symmetry direction, which has negligible influence on P-wave kinematic signatures, can be found only from the moveout of shear waves. Using the exact NMO equation, we examine the propagation of errors in observed moveout velocities into estimated values of the anisotropic parameters and establish the necessary conditions for a stable inversion procedure. Since the azimuthal variation of the NMO velocity is elliptical, each reflection event provides us with up to three constraints on the model parameters. Generally, the five parameters responsible for P-wave velocity can be obtained from two P-wave NMO ellipses, but the feasibility of the moveout inversion strongly depends on the tilt ν. If the symmetry axis is close to vertical (small ν), the P-wave NMO ellipse is largely governed by the NMO velocity from a horizontal reflector Vnmo(0) and the anellipticity coefficient η. Although for mild tilts the medium parameters cannot be determined separately, the NMO-velocity inversion provides enough information for building TTI models suitable for time processing (NMO, DMO, time migration). If the tilt of the symmetry axis exceeds 30°–40° (e.g., the symmetry axis can be horizontal), it is possible to find all P-wave kinematic parameters and construct the anisotropic model in depth. Another condition required for a stable parameter estimate is that the medium be sufficiently different from elliptical (i.e., ε cannot be close to δ). This limitation, however, can be overcome by including the SV-wave NMO ellipse from a horizontal reflector in the inversion procedure. While most of the analysis is carried out for a single layer, we also extend the inversion algorithm to vertically heterogeneous TTI media above a dipping reflector using the generalized Dix equation. A synthetic example for a strongly anisotropic, stratified TTI medium demonstrates a high accuracy of the inversion (subject to the above limitations).

Stacking-velocity inversion with borehole constraints for tilted TI media

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. D69-D77 ◽  
Author(s):  
Xiaoxiang Wang ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
P Wave ◽  
Simultaneous Estimation ◽  
Borehole Data ◽  
Symmetry Direction ◽  
Velocity Inversion ◽  
Interval Parameters ◽  
Tti Media ◽  
Zero Offset

Transversely isotropic models with a tilted symmetry axis (TTI) play an increasingly important role in seismic imaging, especially near salt bodies and in active tectonic areas. Here, we present a 2D parameter-estimation methodology for TTI media based on combining P-wave normal-moveout (NMO) velocities, zero-offset traveltimes, and reflection time slopes with borehole data that include check-shot traveltimes as well as the reflector depths and dips. For a dipping TTI layer with the symmetry axis confined to the dip plane of the reflector, simultaneous estimation of the symmetry-direction velocity [Formula: see text], the anisotropy parameters [Formula: see text] and [Formula: see text], and the tilt [Formula: see text] of the symmetry axis proves to be ambiguous despite the borehole constraints. If the symmetry axis is orthogonal to the reflector, [Formula: see text] and [Formula: see text] can be recovered with high accuracy, even when the symmetry axis deviates by [Formula: see text] from the reflector normal. The parameter [Formula: see text], however, cannot be constrained for dips smaller than 60° without using nonhyperbolic moveout. To invert for the interval parameters of layered TTI media, we apply 2D stacking-velocity inversion supplemented with the same borehole constraints. The dip planes in all layers are assumed to be aligned, and the symmetry axis is set orthogonal to the reflector in each layer. Information about reflector dips can be replaced with near-offset walkaway vertical seismic profiling (VSP) traveltimes. Tests on noise-contaminated data demonstrate that the algorithm produces stable estimates of the interval parameters [Formula: see text] and [Formula: see text], if the range of dips does not exceed 30°. Our method can be used to build an accurate initial TTI model for post-migration reflection tomography and other techniques that employ migration velocity analysis.

Velocity analysis for tilted transversely isotropic media: A physical modeling example

Geophysics ◽  
2001 ◽  
Vol 66 (3) ◽  
pp. 904-910 ◽  
Author(s):  
Vladimir Grechka ◽  
Andres Pech ◽  
Ilya Tsvankin ◽  
Baoniu Han
Keyword(s):  
Physical Modeling ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Velocity Analysis ◽  
Isotropic Layer ◽  
Anisotropic Parameter ◽  
Data Set ◽  
Symmetry Direction

Transverse isotropy with a tilted symmetry axis (TTI media) has been recognized as a common feature of shale formations in overthrust areas, such as the Canadian Foothills. Since TTI layers cause serious problems in conventional imaging, it is important to be able to reconstruct the velocity model suitable for anisotropic depth migration. Here, we discuss the results of anisotropic parameter estimation on a physical‐modeling data set. The model represents a simplified version of a typical overthrust section from the Alberta Foothills, with a horizontal reflector overlaid by a bending transversely isotropic layer. Assuming that the TTI layer is homogeneous and the symmetry axis stays perpendicular to its boundaries, we invert P-wave normal‐moveout (NMO) velocities and zero‐offset traveltimes for the symmetry‐direction velocity V0 and the anisotropic parameters ε and δ. The coefficient ε is obtained using the traveltimes of a wave that crosses a dipping TTI block and reflects from the bottom of the model. The inversion for ε is based on analytic expressions for NMO velocity in media with intermediate dipping interfaces. Our estimates of both anisotropic coefficients are close to their actual values. The errors in the inversion, which are associated primarily with the uncertainties in picking the NMO velocities and traveltimes, can be reduced by a straighforward modification of the acquisition geometry. It should be emphasized that the moveout inversion also gives an accurate estimate of the thickness of the TTI layer, thus reconstructing the correct depth scale of the section. Although the physical model used here was relatively simple, our results demonstrate the principal feasibility of anisotropic velocity analysis and imaging in overthrust areas. The main problems in anisotropic processing for TTI models are likely to be caused by the lateral variation of the velocity field and overall structural complexity.

P‐wave signatures and notation for transversely isotropic media: An overview

Geophysics ◽  
1996 ◽  
Vol 61 (2) ◽  
pp. 467-483 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Wave Propagation ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Seismic Inversion ◽  
P Wave ◽  
Symmetry Direction ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Progress in seismic inversion and processing in anisotropic media depends on our ability to relate different seismic signatures to the anisotropic parameters. While the conventional notation (stiffness coefficients) is suitable for forward modeling, it is inconvenient in developing analytic insight into the influence of anisotropy on wave propagation. Here, a consistent description of P‐wave signatures in transversely isotropic (TI) media with arbitrary strength of the anisotropy is given in terms of Thomsen notation. The influence of transverse isotropy on P‐wave propagation is shown to be practically independent of the vertical S‐wave velocity [Formula: see text], even in models with strong velocity variations. Therefore, the contribution of transverse isotropy to P‐wave kinematic and dynamic signatures is controlled by just two anisotropic parameters, ε and δ, with the vertical velocity [Formula: see text] being a scaling coefficient in homogeneous models. The distortions of reflection moveouts and amplitudes are not necessarily correlated with the magnitude of velocity anisotropy. The influence of transverse isotropy on P‐wave normal‐moveout (NMO) velocity in a horizontally layered medium, on small‐angle reflection coefficient, and on point‐force radiation in the symmetry direction is entirely determined by the parameter δ. Another group of signatures of interest in reflection seisimology—the dip‐dependence of NMO velocity, magnitude of nonhyperbolic moveout, time‐migration impulse response, and the radiation pattern near vertical—is dependent on both anisotropic parameters (ε and δ) and is primarily governed by the difference between ε and δ. Since P‐wave signatures are so sensitive to the value of ε − δ, application of the elliptical‐anisotropy approximation (ε = δ) in P‐wave processing may lead to significant errors. Many analytic expressions given in the paper remain valid in transversely isotropic media with a tilted symmetry axis. Moreover, the equation for NMO velocity from dipping reflectors, as well as the nonhyperbolic moveout equation, can be used in symmetry planes of any anisotropic media (e.g., orthorhombic).

Ray-based gridded tomography for tilted transversely isotropic media

Geophysics ◽  
2013 ◽  
Vol 78 (1) ◽  
pp. C11-C23 ◽  
Author(s):  
Xiaoxiang Wang ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
A Priori ◽  
Transversely Isotropic ◽  
P Wave ◽  
Model Parameters ◽  
Velocity Analysis ◽  
Symmetry Direction ◽  
Reflection Data ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Reflection tomography in the migrated domain can help reconstruct heterogeneous, anisotropic velocity fields needed for accurate depth imaging of complex geologic structures. The presence of anisotropy, however, increases the uncertainty in velocity analysis and typically requires a priori constraints on the model parameters. Here, we develop a 2D P-wave tomographic algorithm for heterogeneous transversely isotropic media with a tilted symmetry axis (TTI) and investigate the conditions necessary for stable estimation of the symmetry-direction velocity [Formula: see text] and the anisotropy parameters [Formula: see text] and [Formula: see text]. The model is divided into rectangular cells, and the parameters [Formula: see text], [Formula: see text], [Formula: see text], and the tilt [Formula: see text] of the symmetry axis are defined at the grid points. To increase the stability of the inversion, the symmetry axis is set orthogonal to the imaged reflectors, with the tilt interpolated inside each layer. The iterative migration velocity analysis involves efficient linearized parameter updating designed to minimize the residual moveout in image gathers for all available reflection events. The moveout equation in the depth-migrated domain includes a nonhyperbolic term that describes long-offset data, which are particularly sensitive to [Formula: see text]. Synthetic tests for models with a “quasi-factorized” TTI syncline (i.e., [Formula: see text] and [Formula: see text] are constant inside the anisotropic layer) and a TTI thrust sheet demonstrate that stable parameter estimation requires either strong smoothness constraints or additional information from walkaway VSP (vertical seismic profiling) traveltimes. If the model is quasi-factorized with a linear spatial variation of [Formula: see text], it may be possible to obtain the interval TTI parameters just from long-spread reflection data.

Analytic study of the geometrical spreading of P-waves in a layered transversely isotropic medium with a vertical symmetry axis

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1305-1315 ◽  
Author(s):  
Hongbo Zhou ◽  
George A. McMechan
Keyword(s):  
Isotropic Medium ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Analytical Formula ◽  
Transversely Isotropic ◽  
P Wave ◽  
Geometrical Spreading ◽  
Transversely Isotropic Medium ◽  
Weak Anisotropy ◽  
Special Cases

An analytical formula for geometrical spreading is derived for a horizontally layered transversely isotropic medium with a vertical symmetry axis (VTI). With this expression, geometrical spreading can be determined using only the anisotropy parameters in the first layer, the traveltime derivatives, and the source‐receiver offset. Explicit, numerically feasible expressions for geometrical spreading are obtained for special cases of transverse isotropy (weak anisotropy and elliptic anisotropy). Geometrical spreading can be calculated for transversly isotropic (TI) media by using picked traveltimes of primary nonhyperbolic P-wave reflections without having to know the actual parameters in the deeper subsurface; no ray tracing is needed. Synthetic examples verify the algorithm and show that it is numerically feasible for calculation of geometrical spreading. For media with a few (4–5) layers, relative errors in the computed geometrical spreading remain less than 0.5% for offset/depth ratios less than 1.0. Errors that change with offset are attributed to inaccuracy in the expression used for nonhyberbolic moveout. Geometrical spreading is most sensitive to errors in NMO velocity, followed by errors in zero‐offset reflection time, followed by errors in anisotropy of the surface layer. New relations between group and phase velocities and between group and phase angles are shown in appendices.

Inversion of reflection traveltimes for transverse isotropy

Geophysics ◽  
1995 ◽  
Vol 60 (4) ◽  
pp. 1095-1107 ◽  
Author(s):  
Ilya Tsvankin ◽  
Leon Thomsen
Keyword(s):  
Wave Velocity ◽  
Vertical Velocity ◽  
Joint Inversion ◽  
Transverse Isotropy ◽  
Taylor Series Expansion ◽  
High Sensitivity ◽  
Transversely Isotropic ◽  
P Wave ◽  
S Wave ◽  
Quartic Term

In anisotropic media, the short‐spread stacking velocity is generally different from the root‐mean‐square vertical velocity. The influence of anisotropy makes it impossible to recover the vertical velocity (or the reflector depth) using hyperbolic moveout analysis on short‐spread, common‐midpoint (CMP) gathers, even if both P‐ and S‐waves are recorded. Hence, we examine the feasibility of inverting long‐spread (nonhyperbolic) reflection moveouts for parameters of transversely isotropic media with a vertical symmetry axis. One possible solution is to recover the quartic term of the Taylor series expansion for [Formula: see text] curves for P‐ and SV‐waves, and to use it to determine the anisotropy. However, this procedure turns out to be unstable because of the ambiguity in the joint inversion of intermediate‐spread (i.e., spreads of about 1.5 times the reflector depth) P and SV moveouts. The nonuniqueness cannot be overcome by using long spreads (twice as large as the reflector depth) if only P‐wave data are included. A general analysis of the P‐wave inverse problem proves the existence of a broad set of models with different vertical velocities, all of which provide a satisfactory fit to the exact traveltimes. This strong ambiguity is explained by a trade‐off between vertical velocity and the parameters of anisotropy on gathers with a limited angle coverage. The accuracy of the inversion procedure may be significantly increased by combining both long‐spread P and SV moveouts. The high sensitivity of the long‐spread SV moveout to the reflector depth permits a less ambiguous inversion. In some cases, the SV moveout alone may be used to recover the vertical S‐wave velocity, and hence the depth. Success of this inversion depends on the spreadlength and degree of SV‐wave velocity anisotropy, as well as on the constraints on the P‐wave vertical velocity.

Multiparameter TTI tomography of P-wave reflection and VSP data

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. WC51-WC63 ◽  
Author(s):  
Xiaoxiang Wang ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Model Updating ◽  
Transversely Isotropic ◽  
P Wave ◽  
Ocean Bottom ◽  
Symmetry Direction ◽  
Direction Parallel ◽  
The North ◽  
Vsp Data ◽  
Anisotropy Parameters

Transversely isotropic models with a tilted symmetry axis (TTI media) are widely used in depth imaging of complex geologic structures. Here, we present a modification of a previously developed 2D P-wave tomographic algorithm for estimating heterogeneous TTI velocity fields and apply it to synthetic and field data. The symmetry-direction velocity [Formula: see text], anisotropy parameters [Formula: see text] and [Formula: see text], and symmetry-axis tilt [Formula: see text] are defined on a rectangular grid. To ensure stable reconstruction of the TTI parameters, reflection data are combined with walkaway vertical seismic profiling (VSP) traveltimes in joint tomographic inversion. To improve the convergence of the algorithm, we develop a three-stage model-updating procedure that gradually relaxes the constraints on the spatial variations of the anisotropy parameters, while the symmetry axis is kept orthogonal to the reflectors. Only at the final stage of the inversion are the parameters [Formula: see text], [Formula: see text], and [Formula: see text] updated on the same grid. We also incorporate geologic constraints into tomography by designing regularization terms that penalize parameter variations in the direction parallel to the interfaces. First, we examine the performance of the regularized joint tomography of reflection and VSP data for two sections of the BP TTI model that contain an anticline and a salt dome. All three TTI parameters in the shallow part of both sections (down to 5 km) are well resolved by the proposed model-updating process. Then, the algorithm is applied to a 2D section from 3D ocean-bottom seismic data acquired at Volve field in the North Sea. The inverted TTI model produces well-focused reflectors throughout the section and accurately positions the key horizons, which is confirmed by the available well markers.

The effects of transverse isotropy on reflection amplitude versus offset

Geophysics ◽  
1987 ◽  
Vol 52 (4) ◽  
pp. 564-567 ◽  
Author(s):  
J. Wright
Keyword(s):  
Wave Velocity ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
S Wave ◽  
Reflection Amplitude ◽  
Amplitude Versus Offset ◽  
Transversely Isotropic Media ◽  
P Wave Velocity ◽  
Isotropic Media

Studies have shown that elastic properties of materials such as shale and chalk are anisotropic. With the increasing emphasis on extraction of lithology and fluid content from changes in reflection amplitude with shot‐to‐group offset, one needs to know the effects of anisotropy on reflectivity. Since anisotropy means that velocity depends upon the direction of propagation, this angular dependence of velocity is expected to influence reflectivity changes with offset. These effects might be particularly evident in deltaic sand‐shale sequences since measurements have shown that the P-wave velocity of shales in the horizontal direction can be 20 percent higher than the vertical P-wave velocity. To investigate this behavior, a computer program was written to find the P- and S-wave reflectivities at an interface between two transversely isotropic media with the axis of symmetry perpendicular to the interface. Models for shale‐chalk and shale‐sand P-wave reflectivities were analyzed.

Modeling and inversion of PS-wave moveout asymmetry for tilted TI media: Part 2 — Dipping TTI layer

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. D123-D134 ◽  
Author(s):  
Pawan Dewangan ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Vertical Plane ◽  
Transversely Isotropic ◽  
Estimation Algorithm ◽  
Velocity Model ◽  
Inversion Algorithm ◽  
S Wave ◽  
Weak Anisotropy ◽  
Symmetry Direction ◽  
Tti Media

Dipping transversely isotropic layers with a tilted symmetry axis (TTI media) cause serious imaging problems in fold-and-thrust belts and near salt domes. Here, we apply the modified [Formula: see text] method introduced in Part 1 to the inversion of long-offset PP and PS reflection data for the parameters of a TTI layer with the symmetry axis orthogonal to the bedding. The inversion algorithm combines the time- and offset-asymmetry attributes of the PSV-wave with the hyperbolic PP- and SS-wave moveout in the symmetry-axis plane (i.e., the vertical plane that contains the symmetry axis). The weak-anisotropy approximations for the moveout-asymmetry attributes, verified by numerical analysis, indicate that small-offset (leading) terms do not contain independent information for the inversion. Therefore, the parameter-estimation algorithm relies on PS data recorded at large offsets (the offset-to-depth ratio has to reach at least two), which makes the results generally less stable than those for a horizontal TTI layer in Part1. The least-resolved parameter is Thomsen’s coefficient [Formula: see text]that does not directly influence the moveout of either pure or converted modes. Still, the contribution of the PS-wave asymmetry attributes helps to constrain the TTI model for large tilts [Formula: see text] of the symmetry axis [Formula: see text]. The accuracy of the inversion for large tilts can be improved further by using wide-azimuth PP and PS reflections. With high-quality PS data, the inversion remains feasible for moderate tilts [Formula: see text], but it breaks down for models with smaller values of [Formula: see text] in which the moveout asymmetry is too weak. The tilt itself and several combinations of the medium parameters (e.g., the ratio of the P- and S-wave velocities in the symmetry direction), however, are well constrained for all symmetry-axis orientations. The results of Parts 1 and 2 show that 2D measurements of the PS-wave asymmetry attributes can be used effectively in anisotropic velocity analysis for TTI media. In addition to providing an improved velocity model for imaging beneath TTI beds, our algorithms yield information for lithology discrimination and structural interpretation.

Decoupled equations for reverse time migration in tilted transversely isotropic media

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. T37-T45 ◽  
Author(s):  
Ge Zhan ◽  
Reynam C. Pestana ◽  
Paul L. Stoffa
Keyword(s):  
Broad Band ◽  
Transversely Isotropic ◽  
Rapid Expansion ◽  
P Wave ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Coupled Equations ◽  
Time Migration ◽  
Tti Media ◽  
Wave Modes

Conventional modeling and migration for tilted transversely isotropic (TTI) media may suffer from numerical instabilities and shear wave artifacts due to the coupling of the P-wave and SV-wave modes in the TTI coupled equations. Starting with the separated P- and SV-phase velocity expressions for vertical transversely isotropic (VTI) media, we extend these decoupled equations for modeling and reverse time migration (RTM) in acoustic TTI media. Compared with the TTI coupled equations published in the geophysical literature, the new TTI decoupled equations provide a more stable solution due to the complete separation of the P-wave and SV-wave modes. The pseudospectral method is the most convenient method to implement these equations due to the form of wavenumber expressions and has the added benefit of being highly accurate and thus avoiding numerical dispersion. The rapid expansion method (REM) in time is employed to produce a broad band numerically stable time evolution of the wavefields. Synthetic results validate the proposed TTI decoupled equations and show that modeling and RTM in TTI media with the decoupled equations remain numerically stable even for models with strong anisotropy and sharp contrasts.

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