Wide-angle phase-slowness approximations in VTI media

Geophysics ◽  
2007 ◽  
Vol 72 (4) ◽  
pp. S177-S185 ◽  
Author(s):  
Ørjan Pedersen ◽  
Bjørn Ursin ◽  
Alexey Stovas
Keyword(s):  
Phase Shift ◽  
Taylor Series ◽  
Continued Fraction ◽  
Symmetry Axis ◽  
Wide Angle ◽  
Converted Wave ◽  
Numerical Tests ◽  
Series Representations ◽  
Vertical Propagation ◽  
Vti Media

An anisotropic medium with vertical symmetry axis (VTI) often presents a good model for describing real rocks. Propagation of quasi-P- and quasi-SV-waves in such media requires an expression of the vertical phase slowness, a complicated function of the horizontal phase slowness and the medium parameters. For converted-wave phase-shift migration methods, it is desired to have slowness expressions that are simple and accurate at wide angles of propagation. Taylor-series representations of the squared vertical slowness for quasi-P- and quasi-SV-waves result in new wide-angle phase-slowness approximations based on truncated series and continued-fraction representations. Slowness approximations that are exact for both vertical propagation and at a horizontal slowness corresponding to horizontally traveling qP-waves are derived. The approximation for quasi-SV-waves can be used in phase-shift migration in media where the quasi-SV wavefront contains triplications. These approximations are tested on several models and compared to previously published approximations. The numerical tests suggest that the new continued-fraction approximations are more accurate. They can be used in phase-shift migration algorithms, which are more efficient for large angles than the existing approximations.

Anisotropic full-waveform inversion of crosshole seismic data: A vertical symmetry axis field data application

Geophysics ◽  
2019 ◽  
Vol 84 (1) ◽  
pp. B15-B32 ◽  
Author(s):  
Shaun Hadden ◽  
R. Gerhard Pratt ◽  
Brendan Smithyman
Keyword(s):  
Symmetry Axis ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
S Waves ◽  
Traveltime Tomography ◽  
Full Waveform ◽  
Anisotropy Parameters ◽  
Vti Media ◽  
Ti Media ◽  
Acoustic Approximation

Anisotropic waveform tomography (AWT) uses anisotropic traveltime tomography followed by anisotropic full-waveform inversion (FWI). Such an approach is required for FWI in cases in which the geology is likely to exhibit anisotropy. An important anisotropy class is that of transverse isotropy (TI), and the special case of TI media with a vertical symmetry axis (VTI) media is often used to represent elasticity in undeformed sedimentary layering. We have developed an approach for AWT that uses an acoustic approximation to simulate waves in VTI media, and we apply this approach to crosshole data. In our approach, the best-fitting models of seismic velocity and Thomsen VTI anisotropy parameters are initially obtained using anisotropic traveltime tomography, and they are then used as the starting models for VTI FWI within the acoustic approximation. One common problem with the acoustic approach to TI media is the generation of late-arriving (spurious) S-waves as a by-product of the equation system. We used a Laplace-Fourier approach that effectively damps the spurious S-waves to suppress artifacts that might otherwise corrupt the final inversion results. The results of applying AWT to synthetic data illustrate the trade-offs in resolution between the two parameter classes of velocity and anisotropy, and they also verify anisotropic traveltime tomography as a valid method for generating starting models for FWI. The synthetic study further indicates the importance of smoothing the anisotropy parameters before proceeding to FWI inversions of the velocity parameter. The AWT technique is applied to real crosshole field gathers from a sedimentary environment in Western Canada, and the results are compared with the results from a simpler (elliptical) anisotropy model. The transversely isotropic approach yields an FWI image of the vertical velocity that (1) exhibits a superior resolution and (2) better predicts the field data than does the elliptical approach.

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.

Kirchhoff time migration for transversely isotropic media: An application to Trinidad data

Geophysics ◽  
2006 ◽  
Vol 71 (1) ◽  
pp. S29-S35 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Data Set ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Higher Order Terms ◽  
Isotropic Media ◽  
Migration Method ◽  
Vti Media ◽  
Vertical Inhomogeneity

Using a newly developed nonhyperbolic offset-mid-point traveltime equation for prestack Kirchhoff time migration, instead of the conventional double-square-root (DSR) equation, results in overall better images from anisotropic data. Specifically, prestack Kirchhoff time migration for transversely isotropic media with a vertical symmetry axis (VTI media) is implemented using an analytical offset-midpoint traveltime equation that represents the equivalent of Cheop's pyramid for VTI media. It includes higher-order terms necessary to better handle anisotropy as well as vertical inhomogeneity. Application of this enhanced Kirchhoff time-migration method to the anisotropic Marmousi data set demonstrates the effectiveness of the approach. Further application of the method to field data from Trinidad results in sharper reflectivity images of the subsurface, with the faults better focused and positioned than with images obtained using isotropic methods. The superiority of the anisotropic time migration is evident in the flatness of the image gathers.

Vector imaging of converted wave data in laterally uniform media with VTI anisotropy

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. S141-S145 ◽  
Author(s):  
Charlie Jing ◽  
Thomas A. Dickens ◽  
Graham A. Winbow
Keyword(s):  
Seismic Events ◽  
Imaging Method ◽  
Point Scatterer ◽  
Converted Wave ◽  
Prestack Migration ◽  
Wave Data ◽  
Enhanced Resolution ◽  
Wave Imaging ◽  
Transverse Isotropic ◽  
Vti Media

A vector imaging method has been developed for PS-converted waves in laterally homogeneous vertically transverse isotropic (VTI) media. It decomposes the converted-wave data into two upgoing quasi-shear waves ([Formula: see text] and [Formula: see text]) within the prestack migration algorithm according to subsurface image and surface receiver locations. Because the decomposition is performed as part of the migration, it is consistent with the dip and polarization of the seismic events, unlike traditional algorithms that use premigration rotations. Two shear-wave images with potentially enhanced resolution are formed simultaneously from the vector migration. The effects of VTI anisotropy on PS-converted wave imaging and the capability of the PS vector imaging algorithm to provide enhanced images are illustrated using a point-scatterer model.

Adsorption in Perfect Mixing Tank – Comparison of Exact and Approximate Kinetic Models

2014 ◽  
Vol 35 (3) ◽  
pp. 277-291 ◽  
Author(s):  
Krzysztof Kupiec ◽  
Monika Gwadera ◽  
Barbara Larwa
Keyword(s):  
Continued Fraction ◽  
Exact Analytical Solution ◽  
Approximate Model ◽  
Load Factor ◽  
Diffusion Resistance ◽  
Numerical Tests ◽  
Adsorbate Concentration ◽  
Perfect Mixing ◽  
Different Shapes ◽  
Good Agreement

Abstract Periodic adsorption in a perfect mixing tank of a limited volume was considered. It was assumed that the adsorption rate is limited by diffusion resistance in a pellet. The approximate model of diffusion kinetics based on a continued fraction approximation was compared with the exact analytical solution. For the approximate model an algorithm was developed to determine a temporal variation of the adsorbate concentration in the pellet. The comparison was made for different values of the adsorbent load factor. In the numerical tests different shapes of pellets were considered. Both the numerical tests as well as our own experimental results showed that the approximate model provides results that are in good agreement with the exact solution. In the experimental part of this work adsorption of p-nitrophenol and acetic acid from aqueous solutions on cylindrical pellets of activated carbon was conducted.

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.

Nonhyperbolic moveout analysis in VTI media using rational interpolation

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. D59-D71 ◽  
Author(s):  
Huub Douma ◽  
Alexander Calvert
Keyword(s):  
Field Data ◽  
Symmetry Axis ◽  
Interpolation Method ◽  
Transversely Isotropic ◽  
Rational Interpolation ◽  
Data Types ◽  
Computational Overhead ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Vti Media

Anisotropic velocity analysis using qP-waves in transversely isotropic media with a vertical symmetry axis (VTI) usually is done by inferring the anellipticity parameter [Formula: see text] and the normal moveout velocity [Formula: see text] from the nonhyperbolic character of the moveout. Several approximations explicit in these parameters exist with varying degrees of accuracy. Here, we present a rational interpolation approach to nonhyperbolic moveout analysis in the [Formula: see text] domain. This method has no additional computational overhead compared to using expressions explicit in [Formula: see text] and [Formula: see text]. The lack of such overhead stems from the observation that, for fixed [Formula: see text] and zero-offset two-way traveltime [Formula: see text], the moveout curve for different values of [Formula: see text] can be calculated by simple stretching of the offset axis. This observation is based on the assumptions that the traveltimes of qP-waves in transversely isotropic media mainly depend on [Formula: see text] and [Formula: see text], and that the shear-wave velocity along the symmetry axis has a negligibleinfluence on these traveltimes. The accuracy of the rational interpolation method is as good as that of these approximations. The method can be tuned accurately to any offset range of interest by increasing the order of the interpolation. We test the method using both synthetic and field data and compare it with the nonhyperbolic moveout equation of Alkhalifah and Tsvankin (1995) and the shifted hyperbola equation of Fomel (2004). Both data types confirm that for [Formula: see text], our method significantly outperforms the nonhyperbolic moveout equation in terms of combined unbiased parameter estimation with accurate moveout correction. Comparison with the shifted hyperbola equation of Fomel for Greenhorn-shale anisotropy establishes almost identical accuracy of the rational interpolation method and his equation. Even though the proposed method currently deals with homogeneous media only, results from application to synthetic and field data confirm the applicability of the proposed method to horizontally layered VTI media.

The Best Receiving Windows for Multi-Component Converted-Wave Seismic Data in VTI Media

2006 ◽  
Vol 49 (1) ◽  
pp. 184-195
Author(s):  
Jun LU ◽  
Yun WANG ◽  
Ying SHI
Keyword(s):  
Seismic Data ◽  
Converted Wave ◽  
Vti Media
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