scholarly journals A transversely isotropic medium with a tilted symmetry axis normal to the reflector

Geophysics ◽  
2010 ◽  
Vol 75 (3) ◽  
pp. A19-A24 ◽  
Author(s):  
Tariq Alkhalifah ◽  
Paul Sava
Keyword(s):  
Closed Form ◽  
Model Building ◽  
Incidence Angle ◽  
Transversely Isotropic ◽  
Velocity Model ◽  
Reflection Angle ◽  
Velocity Model Building ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Zero Offset

The computational tools for imaging in transversely isotropic media with tilted axes of symmetry (TTI) are complex and in most cases do not have an explicit closed-form representation. Developing such tools for a TTI medium with tilt constrained to be normal to the reflector dip (DTI) reduces their complexity and allows for closed-form representations. The homogeneous-case zero-offset migration in such a medium can be performed using an isotropic operator scaled by the velocity of the medium in the tilt direction. For the nonzero-offset case, the reflection angle is always equal to the incidence angle, and thus, the velocities for the source and receiver waves at the reflection point are equal and explicitly dependent on the reflection angle. This fact allows for the development of explicit representations for angle decomposition as well as moveout formulas for analysis of extended images obtained by wave-equation migration. Although setting the tilt normal to the reflector dip may not be valid everywhere (i.e., on salt flanks), it can be used in the process of velocity model building, in which such constrains are useful and typically are used.

Correction for the influence of velocity lenses on nonhyperbolic moveout inversion for VTI media

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA13-WA21 ◽  
Author(s):  
Mamoru Takanashi ◽  
Ilya Tsvankin
Keyword(s):  
Model Building ◽  
Incidence Angle ◽  
Transversely Isotropic ◽  
Velocity Model ◽  
Low Velocity ◽  
Anisotropic Parameter ◽  
Velocity Model Building ◽  
Offset Time ◽  
Zero Offset ◽  
Push Down

Nonhyperbolic moveout analysis plays an increasingly important role in velocity model building because it provides valuable information for anisotropic parameter estimation. However, lateral heterogeneity associated with stratigraphic lenses such as channels and reefs can significantly distort the moveout parameters, even when the structure is relatively simple. We analyze the influence of a low-velocity isotropic lens on nonhyperbolic moveout inversion for horizontally layered VTI (transversely isotropic with a vertical symmetry axis) models. Synthetic tests demonstrate that a lens can cause substantial, laterally varying errors in the normal-moveout velocity [Formula: see text] and the anellipticity parameter [Formula: see text]. The area influenced by the lens can be identified using the residual moveout after the nonhyperbolic moveout correction as well as the dependence of errors in [Formula: see text] and [Formula: see text] on spreadlength. To remove such errors in [Formula: see text] and [Formula: see text], we propose a correction algorithm designed for a lens embedded in a horizontally layered overburden. This algorithm involves estimation of the incidence angle of the ray passing through the lens for each recorded trace. With the assumption that lens-related perturbation of the raypath is negligible, the lens-induced traveltime shifts are computed from the corresponding zero-offset time distortion (i.e., from “pull-up” or “push-down” anomalies). Synthetic tests demonstrate that this algorithm substantially reduces the errors in the effective and interval parameters [Formula: see text] and [Formula: see text]. The corrected traces and reconstructed “background” values of [Formula: see text] and [Formula: see text] are suitable for anisotropic time imaging and producing a high-quality stack.

scholarly journals Large-offset P-wave traveltime in layered transversely isotropic media

Geophysics ◽  
2021 ◽  
pp. 1-49
Author(s):  
Mohammad Mahdi Abedi ◽  
David Pardo
Keyword(s):  
Asymptotic Series ◽  
Transversely Isotropic ◽  
Velocity Estimation ◽  
Horizontal Velocity ◽  
P Wave ◽  
Transversely Isotropic Media ◽  
Wide Range ◽  
Isotropic Media ◽  
Heterogeneity Parameter ◽  
Zero Offset

Large-offset seismic data processing, imaging, and velocity estimation require an accurate traveltime approximation over a wide range of offsets. In layered transversely isotropic media with vertical symmetry axis (VTI), the accuracy of traditional traveltime approximations is limited to near offsets. Herein, we propose a new traveltime approximation that maintains the accuracy of the classical equations around zero offset, and exhibits the correct curvilinear asymptote at infinitely large offsets. Our approximation is based on the conventional acoustic assumption. Its equation incorporates six parameters. To define them, we use the Taylor series expansion of the exact traveltime around zero offset, and a new asymptotic series for infinite offset. Our asymptotic equation shows that the traveltime behavior at infinitely large offsets is dominated by the properties of the layer with the maximum horizontal velocity in the sequence. The parameters of our approximation depend on: the effective zero offset traveltime, the normal moveout velocity, the anellipticity, a new large-offset heterogeneity parameter, and the properties of the layer with the maximum horizontal velocity in the sequence. We apply our traveltime approximation: (1) to directly calculate traveltime and ray parameter at given offsets, as analytical forward modeling; and (2) to estimate the first four of the aforementioned parameters for the layers beneath a known high-velocity layer. Our large-offset heterogeneity parameter includes the layering effect on the reflections traveltime.

Oriented NMO correction of VTI data in the absence of near-offset traces

Geophysics ◽  
2021 ◽  
pp. 1-41
Author(s):  
Ali Raeisdana ◽  
M. Javad Khoshnavaz ◽  
Hamid Reza Siahkoohi
Keyword(s):  
Model Building ◽  
Seismic Velocity ◽  
Imaging Techniques ◽  
Velocity Model ◽  
The Past ◽  
Velocity Model Building ◽  
Isotropic Media ◽  
Nmo Correction ◽  
Transverse Isotropic ◽  
Normal Moveout Correction

Calculating an accurate seismic velocity model serves an important role in many seismic imaging techniques. The process of velocity model building is often time-consuming, specifically for anisotropic areas, where more than a single parameter is involved in the process. In the past few years, more time-efficient approaches have been considered to estimate seismic velocity as well as anellipticity parameters or heterogeneity factor using local event slopes. Nevertheless, some of these techniques are not practical due to curvature-dependency, or due to the lack of near-offset data. To address such limitations, we use a curvature-independent approach for normal-moveout correction as well as parameter estimation in vertical transverse isotropic media, which is based on local estimation of vertical traveltime using a shifted hyperbola approximation in the absence of near-offset data. The performance of the proposed approach is tested on synthetic and field common-midpoint gathers. It is also assessed in different signal-to-noise ratios and different missing-near-offset situations. Our findings are consistent with the results achieved by the previous methods that were not developed for sparse data.

scholarly journals Migration Using a Transversely Isotropic Medium with Symmetry Normal to the Reflector Dip

2011 ◽  
Vol 2011 ◽  
pp. 1-5 ◽  
Author(s):  
Tariq Alkhalifah ◽  
Paul Sava
Keyword(s):  
Model Building ◽  
Symmetry Axis ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Velocity Model ◽  
Downward Continuation ◽  
Velocity Model Building ◽  
Lateral Inhomogeneity ◽  
Velocity Information ◽  
Model Phase

A transversely isotropic (TI) model in which the tilt is constrained to be normal to the dip (DTI model) allows for simplifications in the imaging and velocity model building efforts as compared to a general TI (TTI) model. Although this model cannot be represented physically in all situations, for example, in the case of conflicting dips, it handles arbitrary reflector orientations under the assumption of symmetry axis normal to the dip. Using this assumption, we obtain efficient downward continuation algorithms compared to the general TTI ones, by utilizing the reflection features of such a model. Phase-shift migration can be easily extended to approximately handle lateral inhomogeneity using, for example, the split-step approach. This is possible because, unlike the general TTI case, the DTI model reduces to VTI for zero dip. These features enable a process in which we can extract velocity information by including tools that expose inaccuracies in the velocity model in the downward continuation process. We test this model on synthetic data corresponding to a general TTI medium and show its resilience.

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.

Fowler DMO and time migration for transversely isotropic media

Geophysics ◽  
1996 ◽  
Vol 61 (3) ◽  
pp. 835-845 ◽  
Author(s):  
John Anderson ◽  
Tariq Alkhalifah ◽  
Ilya Tsvankin
Keyword(s):  
Transverse Isotropy ◽  
Computing Time ◽  
Transversely Isotropic ◽  
P Wave ◽  
Effective Parameters ◽  
Weak Anisotropy ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Zero Offset

The main advantage of Fowler’s dip‐moveout (DMO) method is the ability to perform velocity analysis along with the DMO removal. This feature of Fowler DMO becomes even more attractive in anisotropic media, where imaging methods are hampered by the difficulty in reconstructing the velocity field from surface data. We have devised a Fowler‐type DMO algorithm for transversely isotropic media using the analytic expression for normal‐moveout velocity. The parameter‐estimation procedure is based on the results of Alkhalifah and Tsvankin showing that in transversely isotropic media with a vertical axis of symmetry (VTI) the P‐wave normal‐moveout (NMO) velocity as a function of ray parameter can be described fully by just two coefficients: the zero‐dip NMO velocity [Formula: see text] and the anisotropic parameter η (η reduces to the difference between Thomsen parameters ε and δ in the limit of weak anisotropy). In this extension of Fowler DMO, resampling in the frequency‐wavenumber domain makes it possible to obtain the values of [Formula: see text] and η by inspecting zero‐offset (stacked) panels for different pairs of the two parameters. Since most of the computing time is spent on generating constant‐velocity stacks, the added computational effort caused by the presence of anisotropy is relatively minor. Synthetic and field‐data examples demonstrate that the isotropic Fowler DMO technique fails to generate an accurate zero‐offset section and to obtain the zero‐dip NMO velocity for nonelliptical VTI models. In contrast, this anisotropic algorithm allows one to find the values of the parameters [Formula: see text] and η (sufficient to perform time migration as well) and to correct for the influence of transverse isotropy in the DMO processing. When combined with poststack F-K Stolt migration, this method represents a complete inversion‐processing sequence capable of recovering the effective parameters of transversely isotropic media and producing migrated images for the best‐fit homogeneous anisotropic model.

Multicomponent stacking‐velocity tomography for transversely isotropic media

Geophysics ◽  
2002 ◽  
Vol 67 (5) ◽  
pp. 1564-1574 ◽  
Author(s):  
Vladimir Grechka ◽  
Andres Pech ◽  
Ilya Tsvankin
Keyword(s):  
Velocity Field ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Accurate Estimation ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Inversion Procedure ◽  
Important Processing ◽  
Zero Offset

Accurate estimation of the velocity field is the most difficult step in imaging of seismic data for anisotropic media. Here, the velocity‐analysis problem is examined for the most common anisotropic model of sedimentary formations—transverse isotropy (TI) with arbitrary orientation of the symmetry axis. We show that supplementing wide‐azimuth reflected PP data with mode‐converted (PS) waves yields more stable estimates of the anisotropic coefficients and, in many cases, helps to constrain the model in depth. An important processing step preceding the inversion is computation of the traveltimes of the pure SS‐waves from those of the PP‐, and PS‐waves based on a technique recently developed by Grechka and Tsvankin. This procedure allows us to replace PS‐wave moveout, which is generally asymmetric with respect to zero offset, with the symmetric (hyperbolic on short spreads) moveout of the pure SS reflections. Then, generalizing the algorithm previously suggested for PP data, we develop a joint tomographic inversion of the normal‐moveout (NMO) ellipses and zero‐offset traveltimes of PP‐ and SS‐waves. Application of the method to wide‐azimuth PP and PS reflections from a dipping interface beneath a homogeneous TI layer shows that for a range of reflector dips and tilt angles of the symmetry axis, it is possible to build the anisotropic velocity field in the depth domain. We also extend our inversion procedure to layered TI media with curved interfaces and study its stability in the presence of noise and heterogeneity.

Regularization in slope tomography

Geophysics ◽  
2008 ◽  
Vol 73 (5) ◽  
pp. VE39-VE47 ◽  
Author(s):  
Jessé C. Costa ◽  
Francisco J. da Silva ◽  
Ellen N. Gomes ◽  
Jörg Schleicher ◽  
Luiz André Melo ◽  
...  
Keyword(s):  
Wave Equation ◽  
Model Building ◽  
Synthetic Data ◽  
Velocity Model ◽  
Model Estimation ◽  
Distinct Effect ◽  
Additional Information ◽  
Numerical Tests ◽  
Reflection Angle ◽  
Velocity Model Building

Seismic imaging in depth is limited by the accuracy of velocity model estimation. Slope tomography uses the slowness components and traveltimes of picked reflection or diffraction events for velocity model building. The unavoidable data incompleteness requires additional information to assure stability to inversion. One natural constraint for ray-based tomography is a smooth velocity model. We propose a new, reflection-angle-based kind of smoothness constraint as regularization in slope tomography and have compared its effects to three other, more conventional constraints. The effects of these constraints were evaluated through angle-domain common-image gathers, computed with wave-equation migration using the estimated velocity model. We found that the smoothness constraints have a distinct effect on the velocity model but a weaker effect on the migrated data. In numerical tests on synthetic data, the new constraint leads to geologically more consistent models.

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