3-D moveout inversion in azimuthally anisotropic media with lateral velocity variation: Theory and a case study

Geophysics ◽  
1999 ◽  
Vol 64 (4) ◽  
pp. 1202-1218 ◽  
Author(s):  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Correction Term ◽  
Anisotropic Media ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Velocity Variation ◽  
Variation Theory ◽  
Fractured Reservoir ◽  
Lateral Velocity ◽  
Data Set ◽  
Zero Offset

Reflection moveout recorded over an azimuthally anisotropic medium (e.g., caused by vertical or dipping fractures) varies with the azimuth of the source‐receiver line. Normal‐moveout (NMO) velocity, responsible for the reflection traveltimes on conventional‐length spreads, forms an elliptical curve in the horizontal plane. While this result remains valid in the presence of arbitrary anisotropy and heterogeneity, the inversion of the NMO ellipse for the medium parameters has been discussed so far only for horizontally homogeneous models above a horizontal or dipping reflector. Here, we develop an analytic moveout correction for weak lateral velocity variation in horizontally layered azimuthally anisotropic media. The correction term is proportional to the curvature of the zero‐offset traveltime surface at the common midpoint and, therefore, can be estimated from surface seismic data. After the influence of lateral velocity variation on the effective NMO ellipses has been stripped, the generalized Dix equation can be used to compute the interval ellipses and evaluate the magnitude of azimuthal anisotropy (measured by P-wave NMO velocity) within the layer of interest. This methodology was applied to a 3-D “wide‐azimuth” data set acquired over a fractured reservoir in the Powder River Basin, Wyoming. The processing sequence included 3-D semblance analysis (based on the elliptical NMO equation) for a grid of common‐midpoint “supergathers,” spatial smoothing of the effective NMO ellipses and zero‐offset traveltimes, correction for lateral velocity variation, and generalized Dix differentiation. Our estimates of depth‐varying fracture trends in the survey area, based on the interval P-wave NMO ellipses, are in good agreement with the results of outcrop and borehole measurements and the rotational analysis of four‐ component S-wave data.

Analysis of multiazimuthal VSP data for anisotropy and AVO

Geophysics ◽  
1999 ◽  
Vol 64 (4) ◽  
pp. 1172-1180 ◽  
Author(s):  
W. Scott Leaney ◽  
Colin M. Sayers ◽  
Douglas E. Miller
Keyword(s):  
Fractured Reservoirs ◽  
Vertical Axis ◽  
Horizontal Stress ◽  
Azimuthal Anisotropy ◽  
Carbonate Reservoir ◽  
P Wave ◽  
Fractured Reservoir ◽  
Data Set ◽  
Vsp Data ◽  
The Impact

Multioffset vertical seismic profile (VSP) experiments, commonly referred to as walkaways, enable anisotropy to be measured reliably in the field. The results can be fed into modeling programs to study the impact of anisotropy on velocity analysis, migration, and amplitude versus offset (AVO). Properly designed multioffset VSPs can also provide the target AVO response measured under optimum conditions, since the wavelet is recorded just above the reflectors of interest with minimal reflection point dispersal. In this paper, the multioffset VSP technique is extended to include multioffset azimuths, and a multiazimuthal multiple VSP data set acquired over a carbonate reservoir is analyzed for P-wave anisotropy and AVO. Direct arrival times down to the overlying shale and reflection times and amplitudes from the carbonate are analyzed. Data analysis involves a three‐term fit to account for nonhyperbolic moveout, dip, and azimuthal anisotropy. Results indicate that the overlying shale is transversely isotropic with a vertical axis of symmetry (VTI), while the carbonate shows 4–5% azimuthal anisotropy in traveltimes. The fast direction is consistent with the maximum horizontal stress orientation determined from break‐out logs and is also consistent with the strike of major faults. AVO analysis of the reflection from the top of the carbonate layer shows a critical angle reduction in the fast direction and maximum gradient in the slow direction. This agrees with modeling and indicates a greater amplitude sensitivity in the slow direction—the direction perpendicular to fracture strike. In principle, 3-D surveys should have wide azimuthal coverage to characterize fractured reservoirs. If this is not possible, it is important to have azimuthal line coverage in the minimum horizontal stress direction to optimize the use of AVO for fractured reservoir characterization. This direction can be obtained from multiazimuthal walkaways using the azimuthal P-wave analysis techniques presented.

Anisotropic geometrical-spreading correction for wide-azimuth P-wave reflections

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. D161-D170 ◽  
Author(s):  
Xiaoxia Xu ◽  
Ilya Tsvankin
Keyword(s):  
Transversely Isotropic ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Fractured Reservoir ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Velocity Anisotropy ◽  
Reflection Data ◽  
Fitting Procedure ◽  
Avo Attributes

Compensation for geometrical spreading along a raypath is one of the key steps in AVO (amplitude-variation-with-offset) analysis, in particular, for wide-azimuth surveys. Here, we propose an efficient methodology to correct long-spread, wide-azimuth reflection data for geometrical spreading in stratified azimuthally anisotropic media. The P-wave geometrical-spreading factor is expressed through the reflection traveltime described by a nonhyperbolic moveout equation that has the same form as in VTI (transversely isotropic with a vertical symmetry axis) media. The adapted VTI equation is parameterized by the normal-moveout (NMO) ellipse and the azimuthally varying anellipticity parameter [Formula: see text]. To estimate the moveout parameters, we apply a 3D nonhyperbolic semblance algorithm of Vasconcelos and Tsvankin that operates simultaneously with traces at all offsets andazimuths. The estimated moveout parameters are used as the input in our geometrical-spreading computation. Numerical tests for models composed of orthorhombic layers with strong, depth-varying velocity anisotropy confirm the high accuracy of our travetime-fitting procedure and, therefore, of the geometrical-spreading correction. Because our algorithm is based entirely on the kinematics of reflection arrivals, it can be incorporated readily into the processing flow of azimuthal AVO analysis. In combination with the nonhyperbolic moveout inversion, we apply our method to wide-azimuth P-wave data collected at the Weyburn field in Canada. The geometrical-spreading factor for the reflection from the top of the fractured reservoir is clearly influenced by azimuthal anisotropy in the overburden, which should cause distortions in the azimuthal AVO attributes. This case study confirms that the azimuthal variation of the geometrical-spreading factor often is comparable to or exceeds that of the reflection coefficient.

Survey design for coal-scale 3D-PS seismic reflection

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. P57-P70 ◽  
Author(s):  
Shaun Strong ◽  
Steve Hearn
Keyword(s):  
Survey Design ◽  
High Amplitude ◽  
Azimuthal Anisotropy ◽  
Mine Planning ◽  
P Wave ◽  
Depth Range ◽  
Low Velocity ◽  
Converted Wave ◽  
Data Set ◽  
Economic Considerations

Survey design for converted-wave (PS) reflection is more complicated than for standard P-wave surveys, due to raypath asymmetry and increased possibility of phase distortion. Coal-scale PS surveys (depth [Formula: see text]) require particular consideration, partly due to the particular physical properties of the target (low density and low velocity). Finite-difference modeling provides a pragmatic evaluation of the likely distortion due to inclusion of postcritical reflections. If the offset range is carefully chosen, then it may be possible to incorporate high-amplitude postcritical reflections without seriously degrading the resolution in the stack. Offsets of up to three times target depth may in some cases be usable, with appropriate quality control at the data-processing stage. This means that the PS survey design may need to handle raypaths that are highly asymmetrical and that are very sensitive to assumed velocities. A 3D-PS design was used for a particular coal survey with the target in the depth range of 85–140 m. The objectives were acceptable fold balance between bins and relatively smooth distribution of offset and azimuth within bins. These parameters are relatively robust for the P-wave design, but much more sensitive for the case of PS. Reduction of the source density is more acceptable than reduction of the receiver density, particularly in terms of the offset-azimuth distribution. This is a fortuitous observation in that it improves the economics of a dynamite source, which is desirable for high-resolution coal-mine planning. The final-survey design necessarily allows for logistical and economic considerations, which implies some technical compromise. However, good fold, offset, and azimuth distributions are achieved across the survey area, yielding a data set suitable for meaningful analysis of P and S azimuthal anisotropy.

Velocity estimation in laterally varying media

Geophysics ◽  
1982 ◽  
Vol 47 (6) ◽  
pp. 884-897 ◽  
Author(s):  
Walter S. Lynn ◽  
Jon F. Claerbout
Keyword(s):  
Taylor Series Expansion ◽  
Velocity Estimation ◽  
Lateral Resolution ◽  
Velocity Variation ◽  
Lateral Velocity ◽  
Depth Migration ◽  
Derivative Term ◽  
Fourth Derivative ◽  
Zero Offset ◽  
Lateral Variations

In areas of large lateral variations in velocity, stacking velocities computed on the basis of hyperbolic moveout can differ substantially from the actual root mean square (rms) velocities. This paper addresses the problem of obtaining rms or migration velocities from stacking velocities in such areas. The first‐order difference between the stacking and the vertical rms velocities due to lateral variations in velocity are shown to be related to the second lateral derivative of the rms slowness [Formula: see text]. Approximations leading to this relation are straight raypaths and that the vertical rms slowness to a given interface can be expressed as a second‐order Taylor series expansion in the midpoint direction. Under these approximations, the effect of the first lateral derivative of the slowness on the traveltime is negligible. The linearization of the equation relating the stacking and true velocities results in a set of equations whose inversion is unstable. Stability is achieved, however, by adding a nonphysical fourth derivative term which affects only the higher spatial wavenumbers, those beyond the lateral resolution of the lateral derivative method (LDM). Thus, given the stacking velocities and the zero‐offset traveltime to a given event as a function of midpoint, the LDM provides an estimate of the true vertical rms velocity to that event with a lateral resolution of about two mute zones or cable lengths. The LDM is applicable when lateral variations of velocity greater than 2 percent occur over the mute zone. At variations of 30 percent or greater, the internal assumptions of the LDM begin to break down. Synthetic models designed to test the LDM when the different assumptions are violated show that, in all cases, the results are not seriously affected. A test of the LDM on field data having a lateral velocity variation caused by sea floor topography gives a result which is supported by depth migration.

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.

Plumes: Response of time migration to lateral velocity variation

Geophysics ◽  
1995 ◽  
Vol 60 (4) ◽  
pp. 1118-1127 ◽  
Author(s):  
Dimitri Bevc ◽  
James L. Black ◽  
Gopal Palacharla
Keyword(s):  
Impulse Response ◽  
Synthetic Data ◽  
Migration Velocity ◽  
Velocity Variation ◽  
Velocity Dependence ◽  
Geometrical Construction ◽  
Lateral Velocity ◽  
Time Migration ◽  
Offset Time ◽  
Zero Offset

We analyze how time migration mispositions events in the presence of lateral velocity variation by examining the impulse response of depth modeling followed by time migration. By examining this impulse response, we lay the groundwork for the development of a remedial migration operator that links time and depth migration. A simple theory by Black and Brzostowski predicted that the response of zero‐offset time migration to a point diffractor in a v(x, z) medium would be a distinctive, cusp‐shaped curve called a plume. We have constructed these plumes by migrating synthetic data using several time‐migration methods. We have also computed the shape of the plumes by two geometrical construction methods. These two geometrical methods compare well and explain the observed migration results. The plume response is strongly influenced by migration velocity. We have studied this dependency by migrating synthetic data with different velocities. The observed velocity dependence is confirmed by geometrical construction. A simple first‐order theory qualitatively explains the behavior of zero‐offset time migration, but a more complete understanding of migration velocity dependence in a v(x, z) medium requires a higher order finite‐offset theory.

scholarly journals Physical and numerical modeling of tuning and diffraction in azimuthally anisotropic media

Geophysics ◽  
2000 ◽  
Vol 65 (5) ◽  
pp. 1613-1621 ◽  
Author(s):  
Richard L. Gibson ◽  
Stephen Theophanis ◽  
M. Nafi Toksöz
Keyword(s):  
Numerical Modeling ◽  
Fractured Reservoirs ◽  
Homogeneous Model ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Orthorhombic Symmetry ◽  
Fractured Reservoir ◽  
Sh Wave ◽  
Wave Data ◽  
Ray Methods

Fractured reservoirs are an important target for exploration and production geophysics, and the azimuthal anisotropy often associated with these reservoirs can strongly influence seismic wave propagation. We created a physical model of a fractured reservoir to simulate some of these propagation effects. The reservoir is represented by a phenolite disk that is thin with respect to the elastic wavelengths in the experiment, creating model dimensions that are representative of realistic reservoirs. Phenolite is strongly anisotropic with orthorhombic symmetry, which suggests that azimuthal amplitude versus offset (AVO) effects should be obvious in data. We acquired both SH- and P-wave data in common‐offset gathers with a near offset and a far offset and found that although the SH-wave data show clear azimuthal variations in AVO, the P-wave signals show no apparent changes with azimuth. We then applied numerical modeling to analyze the data. Because ray methods cannot model diffractions from the disk edge, we first used a ray‐Born technique to simulate variations in waveforms associated with such scattering. The synthetic seismograms reproduced variations in the SH-wave waveforms accurately, though the amplitude contrast between acquisition azimuths was overestimated. Assuming a laterally homogeneous model, we then applied ray methods to simulate tuning effects in SH- and P-wave data and confirmed that in spite of the large contrasts in elastic properties, the tuning of the P-wave reflections from the thin disk changed so there was negligible contrast in AVO with azimuth. Models of field scale reservoirs showed that the same effects could be expected for field applications.

scholarly journals Stable wide‐angle Fourier finite‐difference downward extrapolation of 3‐D wavefields

Geophysics ◽  
2002 ◽  
Vol 67 (3) ◽  
pp. 872-882 ◽  
Author(s):  
Biondo Biondi
Keyword(s):  
Finite Difference ◽  
Continuation Method ◽  
Phase Error ◽  
Azimuthal Anisotropy ◽  
Laplacian Operator ◽  
Lateral Velocity ◽  
Data Set ◽  
Frequency Dependent ◽  
Reference Velocity ◽  
Velocity Variations

I present an unconditionally stable, implicit finite‐difference operator that corrects the constant‐velocity phase‐shift operator for lateral velocity variations. The method is based on the Fourier finite‐difference (FFD) method. Contrary to previous results, my correction operator is stable even when the medium velocity has sharp discontinuities, and the reference velocity is higher than the medium velocity. The stability of the new correction enables the definition of a new downward‐continuation method based on the interpolation of two wavefields: the first wavefield is obtained by applying the FFD correction starting from a reference velocity lower than the medium velocity; the second wavefield is obtained by applying the FFD correction starting from a reference velocity higher than the medium velocity. The proposed Fourier finite‐difference plus interpolation (FFDPI) method combines the advantages of the FFD technique with the advantages of interpolation. A simple and economical procedure for defining frequency‐dependent interpolation weight is presented. When the interpolation step is performed using these frequency‐dependent interpolation weights, it significantly reduces the residual phase error after interpolation, the frequency dispersion caused by the discretization of the Laplacian operator, and the azimuthal anisotropy caused by splitting. Tests on zero‐offset data from the SEG‐EAGE salt data set show that the FFDPI method improves the imaging of a fault reflection with respect to a similar interpolation scheme that uses a split‐step correction for adapting to lateral velocity variations.

Velocity model building for a single-offset 3C VSP data via deformable-layer tomography: a Texas salt dome example

Geophysics ◽  
2021 ◽  
pp. 1-44
Author(s):  
Yukai Wo ◽  
Jingjing Zong ◽  
Hao Hu ◽  
Hua-Wei Zhou ◽  
Robert R. Stewart
Keyword(s):  
Velocity Model ◽  
Salt Dome ◽  
P Wave ◽  
Velocity Variation ◽  
Shear Mode ◽  
P Waves ◽  
Data Set ◽  
Image Domain ◽  
Vsp Data ◽  
Geometric Variation

We have applied multiscale deformable-layer tomography (DLT) to build a laterally varying velocity model, using a single-offset vertical seismic profile (VSP) data set acquired for a salt proximity survey in southern Texas. The purpose of the VSP survey is to delineate the 2D salt flank using the P-wave reflections. Previous study has identified an anhydrate layer as the cap rock of the salt dome. The large impedance contrasts of this anhydrite layer generate strong downgoing P (sediment)-S (anhydrite)-P (salt) waves recorded by downhole geophones. Incidentally, the P-S-P-waves have similar traveltimes as those of the P-wave salt flank reflections, thus contaminating the imaging of the salt flank. Identifying shear-mode contamination requires an accurate velocity model of anhydrite. However, the extremely poor coverage of the single-offset VSP greatly challenges tomographic techniques to determine the lateral velocity variation. We tackle this problem using multiscale DLT, which characterizes the velocity field by a set of deformable layers. We constrain the layer velocities using the check-shot data and invert for the geometric variation. The inverted model indicates that the anhydrite layer has a “thick-thin-thick” lateral variation with offset, and the S-wave in the anhydrite layer helps in imaging the P-S-P-waves along the well track. The estimated anhydrite layer geometry is validated by the kinematic accuracies of P-waves in the data domain and P-S-P-waves in the image domain. Some in-salt dipping structures are determined by multiscale DLT as well. This field data example indicates that multiscale DLT is feasible for estimating velocities using VSP data of the single-offset situation. An accurate velocity model is the key for modeling and adaptive subtraction of the shear-mode contamination related to the salt geometry.

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