Migration velocity analysis for TI media in the presence of quadratic lateral velocity variation

Geophysics ◽  
2012 ◽  
Vol 77 (6) ◽  
pp. U87-U96 ◽  
Author(s):  
Mamoru Takanashi ◽  
Ilya Tsvankin
Keyword(s):  
Migration Velocity ◽  
P Wave ◽  
Velocity Variation ◽  
Small Scale ◽  
Velocity Analysis ◽  
Lateral Heterogeneity ◽  
Lateral Velocity ◽  
Symmetry Direction ◽  
Migration Velocity Analysis ◽  
Ti Media

One of the most serious problems in anisotropic velocity analysis is the trade-off between anisotropy and lateral heterogeneity, especially if velocity varies on a scale smaller than the maximum offset. We have developed a P-wave MVA (migration velocity analysis) algorithm for transversely isotropic (TI) models that include layers with small-scale lateral heterogeneity. Each layer is described by constant Thomsen parameters [Formula: see text] and [Formula: see text] and the symmetry-direction velocity [Formula: see text] that varies as a quadratic function of the distance along the layer boundaries. For tilted TI media (TTI), the symmetry axis is taken orthogonal to the reflectors. We analyzed the influence of lateral heterogeneity on image gathers obtained after prestack depth migration and found that quadratic lateral velocity variation in the overburden can significantly distort the moveout of the target reflection. Consequently, medium parameters beneath the heterogeneous layer(s) are estimated with substantial error, even when borehole information (e.g., check shots or sonic logs) is available. Because residual moveout in the image gathers is highly sensitive to lateral heterogeneity in the overburden, our algorithm simultaneously inverts for the interval parameters of all layers. Synthetic tests for models with a gently dipping overburden demonstrate that if the vertical profile of the symmetry-direction velocity [Formula: see text] is known at one location, the algorithm can reconstruct the other relevant parameters of TI models. The proposed approach helps increase the robustness of anisotropic velocity model-building and enhance image quality in the presence of small-scale lateral heterogeneity in the overburden.

Analysis of image gathers in factorized VTI media

Geophysics ◽  
2003 ◽  
Vol 68 (6) ◽  
pp. 2016-2025 ◽  
Author(s):  
Debashish Sarkar ◽  
Ilya Tsvankin
Keyword(s):  
Vertical Velocity ◽  
Velocity Gradient ◽  
Migration Velocity ◽  
P Wave ◽  
Velocity Variation ◽  
Velocity Analysis ◽  
Lateral Heterogeneity ◽  
Velocity Models ◽  
Isotropic Media ◽  
Vti Media

Because events in image gathers generated after prestack depth migration are sensitive to the velocity field, they are often used in migration velocity analysis for isotropic media. Here, we present an analytic and numerical study of P‐wave image gathers in transversely isotropic media with a vertical symmetry axis (VTI) and establish the conditions for flattening such events and positioning them at the true reflector depth. Application of the weak‐anisotropy approximation leads to concise expressions for reflections in image gathers from homogeneous and factorized v(z) media in terms of the VTI parameters and the vertical velocity gradient kz. Flattening events in image gathers for any reflector dip requires accurate values of the zero‐dip NMO velocity at the surface [Vnmo (z = 0)], the gradient kz, and the anellipticity coefficient η. For a fixed error in Vnmo and kz, the magnitude of residual moveout of events in image gathers decreases with dip, while the moveout caused by an error in η initially increases for moderate dips but then decreases as dips approach 90°. Flat events in image gathers in VTI media, however, do not guarantee the correct depth scale of the model because reflector depth depends on the vertical migration velocity. For factorized v(x, z) media with a linear velocity variation in both the x‐ and z‐directions, the moveout on image gathers is controlled by Vnmo (x = z = 0), kz, η, and a combination of the horizontal velocity gradient kx and the Thomsen parameter δ (specifically, kx[Formula: see text]). If too large a value of any of these four quantities is used in migration, reflections in the image gathers curve downward (i.e., they are undercorrected; the inferred depth increases with offset), while a negative error results in overcorrection. Lateral heterogeneity tends to increase the sensitivity of moveout of events in image gathers to the parameter η, and errors in η may lead to measurable residual moveout of horizontal events in v(x, z) media even for offset‐to‐depth ratios close to unity. These results provide a basis for extending to VTI media conventional velocity analysis methods operating with image gathers. Although P‐wave traveltimes alone cannot be used to separate anisotropy from lateral heterogeneity (i.e., kx is coupled to δ), moveout of events in image gathers does constrain the vertical gradient kz. Hence, it may be possible to build VTI velocity models in depth by supplementing reflection data with minimal a priori information, such as the vertical velocity at the top of the factorized VTI layer.

Image gathers of SV-waves in homogeneous and factorized VTI media

Geophysics ◽  
2005 ◽  
Vol 70 (5) ◽  
pp. D55-D64 ◽  
Author(s):  
Ramzy M. Al-Zayer ◽  
Ilya Tsvankin
Keyword(s):  
Shear Wave ◽  
Vertical Velocity ◽  
Migration Velocity ◽  
P Wave ◽  
Model Parameters ◽  
Velocity Analysis ◽  
Wave Data ◽  
Migration Velocity Analysis ◽  
Wave Migration ◽  
Vti Media

Reflection moveout of SV-waves in transversely isotropic media with a vertical symmetry axis (VTI media) can provide valuable information about the model parameters and help to overcome the ambiguities in the inversion of P-wave data. Here, to develop a foundation for shear-wave migration velocity analysis, we study SV-wave image gathers obtained after prestack depth migration. The key issue, addressed using both approximate analytic results and Kirchhoff migration of synthetic data, is whether long-spread SV data can constrain the shear-wave vertical velocity [Formula: see text] and the depth scale of VTI models. For homogeneous media, the residual moveout of horizontal SV events on image gathers is close to hyperbolic and depends just on the NMO velocity [Formula: see text] out to offset-to-depth ratios of about 1.7. Because [Formula: see text] differs from [Formula: see text], flattening moderate-spread gathers of SV-waves does not ensure the correct depth of the migrated events. The residual moveout rapidly becomes nonhyperbolic as the offset-to-depth ratio approaches two, with the migrated depths at long offsets strongly influenced by the SV-wave anisotropy parameter σ. Although the combination of [Formula: see text] and σ is sufficient to constrain the vertical velocity [Formula: see text] and reflector depth, the tradeoff between σ and the Thomsen parameter ε on long-spread gathers causes errors in time-to-depth conversion. The residual moveout of dipping SV events is also controlled by the parameters [Formula: see text], σ, and ε, but in the presence of dip, the contributions of both σ and ε are significant even at small offsets. For factorized v(z) VTI media with a constant SV-wave vertical-velocity gradient [Formula: see text], flattening of horizontal events for a range of depths requires the correct NMO velocity at the surface, the gradient [Formula: see text], and, for long offsets, the parameters σ and ε. On the whole, the nonnegligible uncertainty in the estimation of reflector depth from SV-wave moveout highlights the need to combine P- and SV-wave data in migration velocity analysis for VTI media.

Migration Velocity Analysis in TI Media

2010 ◽  
Author(s):  
J. Panizzardi ◽  
C. Andreoletti ◽  
N. Bienati ◽  
L. Casasanta
Keyword(s):  
Migration Velocity ◽  
Velocity Analysis ◽  
Migration Velocity Analysis ◽  
Ti Media

Migration velocity analysis: Accounting for the effects of lateral velocity variations

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 164-175 ◽  
Author(s):  
Hans J. Tieman
Keyword(s):  
Migration Velocity ◽  
Velocity Analysis ◽  
Lateral Velocity ◽  
Traveltime Inversion ◽  
Long Wavelength ◽  
Migration Velocity Analysis ◽  
Unstable Process ◽  
Velocity Variations ◽  
Iterative Procedures ◽  
Migration Analysis

The most common migration velocity analysis algorithms, iterative profile migration, focusing analysis, and stack power analysis are based on restrictive subsurface assumptions that may cause the methods to break down in the presence of lateral velocity variations. Typically, the subsurface is assumed to have either a constant velocity, or at most, a depth variable velocity. Recent innovations include the use of traveltime inversion philosophy to invert migration measurements of the curvature as a function of offset exhibited by an event following migration. Traveltime inversion makes few assumptions regarding the subsurface, but is a rather unstable process. Thus, an important question is, “Under what conditions do the traditional methods break down and make the use of traveltime inversion methods mandatory?” Marrying the common migration analysis with tomography results in a set of equations that, while useful for generating updates from migration measurements, are too complex for answering the above question. However, by restricting the subsurface to low relief structures and assuming small angle wave propagation, these updating equations can be approximated by forms that are identical to the traditional updating equations, except for a factor that is dependent upon the magnitude, position, and spatial wavelength of potential lateral velocity variations. These simpler equations indicate that even relatively long wavelength anomalies can cause updates to point in the wrong direction and iterative procedures to diverge, and under certain conditions, cause the accuracy of these updates to decrease dramatically.

An analytical approach to migration velocity analysis

Geophysics ◽  
1997 ◽  
Vol 62 (4) ◽  
pp. 1238-1249 ◽  
Author(s):  
Zhenyue Liu
Keyword(s):  
Synthetic Data ◽  
Migration Velocity ◽  
Velocity Estimation ◽  
Velocity Analysis ◽  
Lateral Velocity ◽  
Depth Migration ◽  
Curvature Analysis ◽  
Derivative Function ◽  
Migration Velocity Analysis ◽  
Residual Curvature

Prestack depth migration provides a powerful tool for velocity analysis in complex media. Both prominent approaches to velocity analysis—depth‐focusing analysis and residual‐curvature analysis, rely on approximate formulas to update velocity. Generally, these formulas are derived under the assumptions of horizontal reflector, lateral velocity homogeneity, or small offset. Therefore, the conventional methods for updating velocity lack accuracy and computational efficiency when velocity has large, lateral variations. Here, based on ray theory, I find the analytic representation for the derivative of imaged depths with respect to migration velocity. This derivative function characterizes a general relationship between residual moveout and residual velocity. Using the derivative function and the perturbation method, I derive a new formula to update velocity from residual moveout. In the derivation, I impose no limitation on offset, dip, or velocity distribution. Consequently, I revise the residual‐curvature‐analysis method for velocity estimation in the postmigrated domain. Furthermore, my formula provides sensitivity and error estimation for migration‐based velocity analysis, which is helpful in quantifying the reliability of the estimated velocity. The theory and methodology in this paper have been tested on synthetic data (including the Marmousi data).

Common-reflection-point migration velocity analysis of 2D P-wave data from TTI media

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. C65-C79 ◽  
Author(s):  
Ernesto V. Oropeza ◽  
George A. McMechan
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Migration Velocity ◽  
P Wave ◽  
Velocity Analysis ◽  
Reflection Point ◽  
Axis Orientation ◽  
Velocity Models ◽  
Migration Velocity Analysis ◽  
Free Data

We have developed a common-reflection-point (CRP)-based kinematic migration velocity analysis for 2D P-wave reflection data to estimate the four transversely isotropic (TI) parameters [Formula: see text], [Formula: see text], and [Formula: see text], and the tilt angle [Formula: see text] of the symmetry axis in a TI medium. In each iteration, the tomographic parameter was updated alternately with prestack anisotropic ray-based migration. Iterations initially used layer stripping to reduce the number of degrees of freedom; after convergence was reached, a couple of more iterations over all parameters and all CRPs ensured global interlayer coupling and parameter interaction. The TI symmetry axis orientation was constrained to be locally perpendicular to the reflectors. The [Formula: see text] dominated the inversion, and so it was weighted less than [Formula: see text] and [Formula: see text] in the parameter updates. Estimates of [Formula: see text] and [Formula: see text] were influenced if the error in [Formula: see text] was [Formula: see text]; estimates of [Formula: see text] were also influenced if the error in [Formula: see text] was [Formula: see text]. Examples included data for a simple model with a homogeneous TI layer whose dips allowed recovery of all anisotropy parameters from noise-free data, and a more realistic model (the BP tilted transversely isotropic (TTI) model) for which only [Formula: see text], [Formula: see text], and [Formula: see text] were recoverable. The adequacy of the traveltimes predicted by the inverted anisotropic models was tested by comparing migrated images and common image gathers, with those produced using the known velocity models.

Migration velocity analysis in factorized VTI media

Geophysics ◽  
2004 ◽  
Vol 69 (3) ◽  
pp. 708-718 ◽  
Author(s):  
Debashish Sarkar ◽  
Ilya Tsvankin
Keyword(s):  
Vertical Velocity ◽  
Velocity Gradient ◽  
Migration Velocity ◽  
P Wave ◽  
Velocity Analysis ◽  
Prestack Depth Migration ◽  
Depth Migration ◽  
Migration Velocity Analysis ◽  
Wave Migration ◽  
Vti Media

One of the main challenges in anisotropic velocity analysis and imaging is simultaneous estimation of velocity gradients and anisotropic parameters from reflection data. Approximating the subsurface by a factorized VTI (transversely isotropic with a vertical symmetry axis) medium provides a convenient way of building vertically and laterally heterogeneous anisotropic models for prestack depthmigration. The algorithm for P‐wave migration velocity analysis (MVA) introduced here is designed for models composed of factorized VTI layers or blocks with constant vertical and lateral gradients in the vertical velocity VP0. The anisotropic MVA method is implemented as an iterative two‐step procedure that includes prestack depth migration (imaging step) followed by an update of the medium parameters (velocity‐analysis step). The residual moveout of the migrated events, which is minimized during the parameter updates, is described by a nonhyperbolic equation whose coefficients are determined by 2D semblance scanning. For piecewise‐factorized VTI media without significant dips in the overburden, the residual moveout of P‐wave events in image gathers is governed by four effective quantities in each block: (1) the normal‐moveout velocity Vnmo at a certain point within the block, (2) the vertical velocity gradient kz, (3) the combination kx[Formula: see text] of the lateral velocity gradient kx and the anisotropic parameter δ, and (4) the anellipticity parameter η. We show that all four parameters can be estimated from the residual moveout for at least two reflectors within a block sufficiently separated in depth. Inversion for the parameter η also requires using either long‐spread data (with the maximum offset‐to‐depth ratio no less than two) from horizontal interfaces or reflections from dipping interfaces. To find the depth scale of the section and build a model for prestack depth migration using the MVA results, the vertical velocity VP0 needs to be specified for at least a single point in each block. When no borehole information about VP0 is available, a well‐focused image can often be obtained by assuming that the vertical‐velocity field is continuous across layer boundaries. A synthetic test for a three‐layer model with a syncline structure confirms the accuracy of our MVA algorithm in estimating the interval parameters Vnmo, kz, kx, and η and illustrates the influence of errors in the vertical velocity on the image quality.

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