PS-wave moveout inversion for tilted TI media: A physical-modeling study

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. D135-D143 ◽  
Author(s):  
Pawan Dewangan ◽  
Ilya Tsvankin ◽  
Mike Batzle ◽  
Kasper van Wijk ◽  
Matthew Haney
Keyword(s):  
Physical Modeling ◽  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Pure Mode ◽  
Velocity Analysis ◽  
S Wave ◽  
Reflection Data ◽  
Modeling Data ◽  
Modified Formula ◽  
Ti Media

Mode-converted PS-waves can provide critically important information for velocity analysis in transversely isotropic (TI) media. We demonstrate, with physical-modeling data, that the combination of long-spread reflection traveltimes of PP- and PS-waves can be inverted for the parameters of a horizontal TI layer with a tilted symmetry axis. The 2D multicomponent reflection data are acquired over a phenolic sample manufactured to simulate the effective medium formed by steeply dipping fracture sets or shale layers. The reflection moveout of PS-waves in this model is asymmetric with respect to the source and receiver positions, and the moveout-asymmetry attributes play a crucial role in constraining the TI parameters. Applying the modified [Formula: see text] method to the PP and PS traveltimes recorded in the symmetry-axis plane, we compute the time and offset asymmetry attributes of the PS-waves along with the traveltimes of the pure SS reflections. The algorithm of Dewangan and Tsvankin is then used to invert the combination of the moveout attributes of PP-, SS-, and PS-waves for the medium parameters and the thickness of the sample. It should be emphasized that the pure-mode (PP and SS) traveltimes alone are insufficient for the inversion, even if 3D wide-azimuth data are available. Our estimates of the symmetry axis tilt and layer thickness almost coincide with the actual values. The inverted model is also validated by reproducing the results of transmission experiments with both P- and S-wave sources. The transmitted SV wavefield exhibits a prominent cusp (triplication) accurately predicted by the parameter-estimation results.

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.

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.

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

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. D107-D121 ◽  
Author(s):  
Pawan Dewangan ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Elastic Anisotropy ◽  
Transversely Isotropic ◽  
Parameter Estimates ◽  
Pure Mode ◽  
Velocity Analysis ◽  
Nonlinear Inversion ◽  
Isotropic Layer ◽  
Weak Anisotropy ◽  
Time Asymmetry

One of the distinctive features of mode-converted waves is their asymmetric moveout (i.e., the PS-wave traveltime in general is different if the source and receiver are interchanged) caused by lateral heterogeneity or elastic anisotropy. If the medium is anisotropic, the PS-wave moveout asymmetry contains valuable information for parameter estimation that cannot be obtained from pure reflection modes. Here, we generalize the so-called [Formula: see text] method, which is designed to replace reflected PS modes in velocity analysis with pure (unconverted) SS-waves, by supplementing the output SS traces with the moveout-asymmetry attributes of PS-waves. The time-asymmetry attribute [Formula: see text] is computed in the slowness domain as the difference between the paired traveltimes of the PS arrivals corresponding to ray parameters (horizontal slownesses) of equal magnitude but opposite sign. Another useful asymmetry attribute is the offset [Formula: see text] of the PS-wave traveltime minimum on a common-midpoint (CMP) gather. We demonstrate the effectiveness of the developed algorithmand the importance of including the asymmetry attributes of PS-waves in anisotropic velocity analysis for a horizontal transversely isotropic layer with a tilted symmetry axis (or TTI) medium. Simple analytic expressions for the moveout asymmetry of PSV-waves, derived in the weak-anisotropy approximation, are verified by anisotropic ray tracing. The attribute [Formula: see text] is proportional to the anellipticity parameter [Formula: see text] and reaches its maximum when the symmetry axis deviates by 20°–30° from the vertical or horizontal direction. All relevant parameters of a TTI layer can be estimated by a nonlinear inversion of the NMO velocities and zero-offset traveltimes of PP- and SS- (SVSV) waves combined with the moveout-asymmetry attributes of the PSV-wave. The inversion of pure-mode (PP and SS) moveouts alone is nonunique, while the addition of the attributes [Formula: see text] and [Formula: see text] yields stable parameter estimates from 2D data acquired in the vertical symmetry-axis plane. If the TTI model is formed by obliquely dipping fractures, the anisotropic parameters can be inverted further for the fracture orientation and compliances.

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.

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.

Velocity analysis for tilted TI media: A physical‐modeling example

1999 ◽  
Author(s):  
Vladimir Grechka ◽  
Andres Pech ◽  
Ilya Tsvankin ◽  
Baoniu Han
Keyword(s):  
Physical Modeling ◽  
Velocity Analysis ◽  
Ti Media

Processing‐induced anisotropy

Geophysics ◽  
2002 ◽  
Vol 67 (6) ◽  
pp. 1920-1928 ◽  
Author(s):  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Seismic Data ◽  
Symmetry Axis ◽  
Homogeneous Medium ◽  
Transversely Isotropic ◽  
P Wave ◽  
Well Logs ◽  
Velocity Analysis ◽  
Induced Anisotropy ◽  
Vertical Heterogeneity ◽  
Vti Media

Processing of seismic data is often performed under the assumption that the velocity distribution in the subsurface can be approximated by a macromodel composed of isotropic homogeneous layers or blocks. Despite being physically unrealistic, such models are believed to be sufficient for describing the kinematics of reflection arrivals. In this paper, we examine the distortions in normal‐moveout (NMO) velocities caused by the intralayer vertical heterogeneity unaccounted for in velocity analysis. To match P‐wave moveout measurements from a horizontal or a dipping reflector overlaid by a vertically heterogeneous isotropic medium, the effective homogeneous overburden has to be anisotropic. This apparent anisotropy is caused not only by velocity monotonically increasing with depth, but also by random velocity variations similar to those routinely observed in well logs. Assuming that the effective homogeneous medium is transversely isotropic with a vertical symmetry axis (VTI), we express the VTI parameters through the actual depth‐dependent isotropic velocity function. If the reflector is horizontal, combining the NMO and vertical velocities always results in nonnegative values of Thomsen's coefficient δ. For a dipping reflector, the inversion of the P‐wave NMO ellipse yields a nonnegative Alkhalifah‐Tsvankin coefficient η that increases with dip. The values of η obtained by two other methods (2‐D dip‐moveout inversion and nonhyperbolic moveout analysis) are also nonnegative but generally differ from that needed to fit the NMO ellipse. For truly anisotropic (VTI) media, the influence of vertical heterogeneity above the reflector can lead to a bias toward positive δ and η estimates in velocity analysis.

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.

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