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.

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.

Geometrical spreading in a layered transversely isotropic medium with vertical symmetry axis

Geophysics ◽  
2003 ◽  
Vol 68 (6) ◽  
pp. 2082-2091 ◽  
Author(s):  
Bjørn Ursin ◽  
Ketil Hokstad
Keyword(s):  
Ray Tracing ◽  
Seismic Data ◽  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Analytical Approximation ◽  
Geometrical Spreading ◽  
S Wave ◽  
Weak Anisotropy ◽  
P Waves ◽  
Amplitude Versus

Compensation for geometrical spreading is important in prestack Kirchhoff migration and in amplitude versus offset/amplitude versus angle (AVO/AVA) analysis of seismic data. We present equations for the relative geometrical spreading of reflected and transmitted P‐ and S‐wave in horizontally layered transversely isotropic media with vertical symmetry axis (VTI). We show that relatively simple expressions are obtained when the geometrical spreading is expressed in terms of group velocities. In weakly anisotropic media, we obtain simple expressions also in terms of phase velocities. Also, we derive analytical equations for geometrical spreading based on the nonhyperbolic traveltime formula of Tsvankin and Thomsen, such that the geometrical spreading can be expressed in terms of the parameters used in time processing of seismic data. Comparison with numerical ray tracing demonstrates that the weak anisotropy approximation to geometrical spreading is accurate for P‐waves. It is less accurate for SV‐waves, but has qualitatively the correct form. For P waves, the nonhyperbolic equation for geometrical spreading compares favorably with ray‐tracing results for offset‐depth ratios less than five. For SV‐waves, the analytical approximation is accurate only at small offsets, and breaks down at offset‐depth ratios less than unity. The numerical results are in agreement with the range of validity for the nonhyperbolic traveltime equations.

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.

Normal moveout from dipping reflectors in anisotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 268-284 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Weak Anisotropy ◽  
Anisotropic Models ◽  
Transversely Isotropic Media ◽  
Wide Range ◽  
Isotropic Media ◽  
The Difference

Description of reflection moveout from dipping interfaces is important in developing seismic processing methods for anisotropic media, as well as in the inversion of reflection data. Here, I present a concise analytic expression for normal‐moveout (NMO) velocities valid for a wide range of homogeneous anisotropic models including transverse isotropy with a tilted in‐plane symmetry axis and symmetry planes in orthorhombic media. In transversely isotropic media, NMO velocity for quasi‐P‐waves may deviate substantially from the isotropic cosine‐of‐dip dependence used in conventional constant‐velocity dip‐moveout (DMO) algorithms. However, numerical studies of NMO velocities have revealed no apparent correlation between the conventional measures of anisotropy and errors in the cosine‐of‐dip DMO correction (“DMO errors”). The analytic treatment developed here shows that for transverse isotropy with a vertical symmetry axis, the magnitude of DMO errors is dependent primarily on the difference between Thomsen parameters ε and δ. For the most common case, ε − δ > 0, the cosine‐of‐dip–corrected moveout velocity remains significantly larger than the moveout velocity for a horizontal reflector. DMO errors at a dip of 45 degrees may exceed 20–25 percent, even for weak anisotropy. By comparing analytically derived NMO velocities with moveout velocities calculated on finite spreads, I analyze anisotropy‐induced deviations from hyperbolic moveout for dipping reflectors. For transversely isotropic media with a vertical velocity gradient and typical (positive) values of the difference ε − δ, inhomogeneity tends to reduce (sometimes significantly) the influence of anisotropy on the dip dependence of moveout velocity.

Quartic moveout coefficient: 3D description and application to tilted TI media

Geophysics ◽  
2003 ◽  
Vol 68 (5) ◽  
pp. 1600-1610 ◽  
Author(s):  
Andres Pech ◽  
Ilya Tsvankin ◽  
Vladimir Grechka
Keyword(s):  
Heterogeneous Media ◽  
Symmetry Axis ◽  
High Sensitivity ◽  
Transversely Isotropic ◽  
Seismic Inversion ◽  
P Wave ◽  
Individual Values ◽  
Weak Anisotropy ◽  
Essential Information ◽  
Ti Media

Nonhyperbolic (long‐spread) moveout provides essential information for a number of seismic inversion/processing applications, particularly for parameter estimation in anisotropic media. Here, we present an analytic expression for the quartic moveout coefficient A4 that controls the magnitude of nonhyperbolic moveout of pure (nonconverted) modes. Our result takes into account reflection‐point dispersal on irregular interfaces and is valid for arbitrarily anisotropic, heterogeneous media. All quantities needed to compute A4 can be evaluated during the tracing of the zero‐offset ray, so long‐spread moveout can be modeled without time‐consuming multioffset, multiazimuth ray tracing. The general equation for the quartic coefficient is then used to study azimuthally varying nonhyperbolic moveout of P‐waves in a dipping transversely isotropic (TI) layer with an arbitrary tilt ν of the symmetry axis. Assuming that the symmetry axis is confined to the dip plane, we employed the weak‐anisotropy approximation to analyze the dependence of A4 on the anisotropic parameters. The linearized expression for A4 is proportional to the anellipticity coefficient η ≈ ε − δ and does not depend on the individual values of the Thomsen parameters. Typically, the magnitude of nonhyperbolic moveout in tilted TI media above a dipping reflector is highest near the reflector strike, whereas deviations from hyperbolic moveout on the dip line are substantial only for mild dips. The azimuthal variation of the quartic coefficient is governed by the tilt ν and reflector dip φ and has a much more complicated character than the NMO–velocity ellipse. For example, if the symmetry axis is vertical (VTI media, ν = 0) and the dip φ < 30°, A4 goes to zero on two lines with different azimuths where it changes sign. If the symmetry axis is orthogonal to the reflector (this model is typical for thrust‐and‐fold belts), the strike‐line quartic coefficient is defined by the well‐known expression for a horizontal VTI layer (i.e., it is independent of dip), while the dip‐line A4 is proportional to cos4 φ and rapidly decreases with dip. The high sensitivity of the quartic moveout coefficient to the parameter η and the tilt of the symmetry axis can be exploited in the inversion of wide‐azimuth, long‐spread P‐wave data for the parameters of TI media.

Reflection and transmission responses in a layered transversely isotropic medium with horizontal symmetry axis

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. C143-C157 ◽  
Author(s):  
Song Jin ◽  
Alexey Stovas
Keyword(s):  
Good Accuracy ◽  
Symmetry Axis ◽  
Local Property ◽  
Transversely Isotropic ◽  
Weak Anisotropy ◽  
Reflection And Transmission ◽  
Numerical Tests ◽  
Transversely Isotropic Media ◽  
Horizontal Symmetry ◽  
Isotropic Media

Seismic wave reflection and transmission (R/T) responses characterize the subsurface local property, and the widely spread anisotropy has considerable influences even at small incident angles. We have considered layered transversely isotropic media with horizontal symmetry axes (HTI), and the symmetry axes were not restricted to be aligned. With the assumption of weak contrast across the interface, linear approximations for R/T coefficients normalized by vertical energy flux are derived based on a simple layered HTI model. We also obtain the approximation with the isotropic background medium under an additional weak anisotropy assumption. Numerical tests illustrate the good accuracy of the approximations compared with the exact results.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document