Seismic traveltime analysis for azimuthally anisotropic media: Theory and experiment

Geophysics ◽  
1997 ◽  
Vol 62 (5) ◽  
pp. 1570-1582 ◽  
Author(s):  
Colin M. Sayers ◽  
Daniel A. Ebrom
Keyword(s):  
Model Simulation ◽  
Symmetry Plane ◽  
Transversely Isotropic ◽  
Bedding Plane ◽  
Seismic Profile ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Principal Axes ◽  
Physical Model Simulation ◽  
Single Set

Natural fractures in reservoirs, and in the caprock overlying the reservoir, play an important role in determining fluid flow during production. The density and orientation of sets of fractures is therefore of great interest. Rocks possessing an anisotropic fabric and a preferred orientation of fractures display both polar and azimuthal anisotropy. Sedimentary rocks containing several sets of vertical fractures may be approximated as having monoclinic symmetry with symmetry plane parallel to the layers if, in the absence of fractures, the rock is transversely isotropic with symmetry axis perpendicular to the bedding plane. A nonhyperbolic traveltime equation, which can be used in the presence of azimuthally anisotropic layered media, can be obtained from an expansion of the inverse‐squared ray velocity in spherical harmonics. For a single set of aligned fractures, application of this equation to traveltime data acquired at a sufficient number of azimuths allows the strike of the fractures to be estimated. Analysis of the traveltimes measured in a physical model simulation of a reverse vertical seismic profile in an azimuthally anisotropic medium shows the medium to be orthorhombic with principal axes in agreement with those given by an independent shear‐wave experiment. In contrast to previous work, no knowledge of the orientation of the symmetry planes is required. The method is therefore applicable to P‐wave data collected at multiple azimuths using multiple offset vertical seismic profiling (VSP) techniques.

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.

Seismic critical-angle reflectometry: A method to characterize azimuthal anisotropy?

Geophysics ◽  
2007 ◽  
Vol 72 (3) ◽  
pp. D41-D50 ◽  
Author(s):  
Martin Landrø ◽  
Ilya Tsvankin
Keyword(s):  
Critical Angle ◽  
Transverse Isotropy ◽  
Symmetry Plane ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Head Wave ◽  
Orthorhombic Symmetry ◽  
Amplitude Curve ◽  
Diagnostic Features ◽  
Azimuthal Variation

Existing anisotropic parameter-estimation algorithms that operate with long-offset data are based on the inversion of either nonhyperbolic moveout or wide-angle amplitude-variation-with-offset (AVO) response. We show that valuable information about anisotropic reservoirs can also be obtained from the critical angle of reflected waves. To explain the behavior of the critical angle, we develop weak-anisotropy approximations for vertical transverse isotropy and then use Tsvankin’s notation to extend them to azimuthally anisotropic models of orthorhombic symmetry. The P-wave critical-angle reflection in orthorhombic media is particularly sensitive to the parameters [Formula: see text] and [Formula: see text] responsible for the symmetry-plane horizontal velocity in the high-velocity layer. The azimuthal variation of the critical angle for typical orthorhombic models can reach [Formula: see text], which translates into substantial changes in the critical offset of the reflected P-wave. The main diagnostic features of the critical-angle reflection employed in our method include the rapid amplitude increase at the critical angle and the subsequent separation of the head wave. Analysis of exact synthetic seismograms, generated with the reflectivity method, confirms that the azimuthal variation of the critical offset is detectable on wide-azimuth, long-spread data and can be qualitatively described by our linearized equations. Estimation of the critical offset from the amplitude curve of the reflected wave, however, is not straightforward. Additional complications may be caused by the overburden noise train and by the influence of errors in the overburden velocity model on the computation of the critical angle. Still, critical-angle reflectometry should help to constrain the dominant fracture directions and can be combined with other methods to reduce the uncertainty in the estimated anisotropy parameters.

3-D description of normal moveout in anisotropic inhomogeneous media

Geophysics ◽  
1998 ◽  
Vol 63 (3) ◽  
pp. 1079-1092 ◽  
Author(s):  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Fractured Reservoirs ◽  
Symmetry Plane ◽  
Transversely Isotropic ◽  
Inhomogeneous Media ◽  
P Wave ◽  
Orthorhombic Symmetry ◽  
Azimuthal Variation ◽  
Anisotropic Models ◽  
Transversely Isotropic Media ◽  
Isotropic Media

We present a new equation for normal‐moveout (NMO) velocity that describes azimuthally dependent reflection traveltimes of pure modes from both horizontal and dipping reflectors in arbitrary anisotropic inhomogeneous media. With the exception of anomalous areas such as those where common‐midpoint (CMP) reflection time decreases with offset, the azimuthal variation of NMO velocity represents an ellipse in the horizontal plane, with the orientation of the axes determined by the properties of the medium and the direction of the reflector normal. In general, a minimum of three azimuthal measurements is necessary to reconstruct the best‐fit ellipse and obtain NMO velocity in all azimuthal directions. This result provides a simple way to correct for the azimuthal variation in stacking velocity often observed in 3-D surveys. Even more importantly, analytic expressions for the parameters of the NMO ellipse can be used in the inversion of moveout data for the anisotropic coefficients of the medium. For homogeneous transversely isotropic media with a vertical axis of symmetry (VTI media), our equation for azimuthally dependent NMO velocity from dipping reflectors becomes a relatively simple function of phase velocity and its derivatives. We show that the zero‐dip NMO velocity Vnmo(0) and the anisotropic coefficient η are sufficient to describe the P-wave NMO velocity for any orientation of the CMP line with respect to the dip plane of the reflector. Using our formalism, Vnmo(0) and η (the only parameters needed for time processing) can be found from the dip‐dependent NMO velocity at any azimuth or, alternatively, from the azimuthally dependent NMO for a single dipping reflector. We also apply this theory to more complicated azimuthally anisotropic models with the orthorhombic symmetry used to describe fractured reservoirs. For reflections from horizontal interfaces in orthorhombic media, the axes of the normal moveout ellipse are aligned with the vertical symmetry planes. Therefore, azimuthal P-wave moveout measurements can be inverted for the orientation of the symmetry planes (typically determined by the fracture direction) and the NMO velocities within them. If the vertical velocity is known, symmetry‐plane NMO velocities make it possible to estimate two anisotropic parameters equivalent to Thomsen’s coefficient δ for transversely isotropic media.

Velocity-independent layer stripping of PP and PS reflection traveltimes

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. U59-U65 ◽  
Author(s):  
Pawan Dewangan ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Plane ◽  
Transversely Isotropic ◽  
P Wave ◽  
Isotropic Layer ◽  
Processing Parameter ◽  
Amplitude Variation With Offset ◽  
Interval Velocity ◽  
Velocity Models ◽  
Potential Applications ◽  
Stable Computation

Building accurate interval velocity models is critically important for seismic imaging and AVO (amplitude variation with offset) analysis. Here, we adapt the [Formula: see text] method to develop an exact technique for constructing the interval traveltime-offset function in a target zone beneath a horizontally layered overburden. All layers in the model can be anisotropic, with an essential assumption that the overburden has a horizontal symmetry plane (i.e., up-down symmetry). Our layer-stripping algorithm is entirely data-driven and, in contrast to the generalized Dix equations, does not require knowledge of the velocity field anywhere in the medium. Important advantages of our approach compared to the Dix-style formalism also include the ability to handle mode-converted waves, long-offset data, and laterally heterogeneous target layers with multiple, curved reflectors. Numerical tests confirm the high accuracy of the algorithm in computing the interval traveltimes of both PP- and PS-waves in a dipping, transversely isotropic layer with a tilted symmetry axis (TTI medium) beneath an anisotropic overburden. In combination with the inversion techniques developed for homogeneous TTI models, the proposed layer stripping of PP and PS data can be used to estimate the interval parameters of TTI formations in such important exploration areas as the Canadian Foothills. Potential applications of this methodology also include the dip-moveout inversion for the P-wave time-processing parameter [Formula: see text] and stable computation of the interval long-spread (nonhyperbolic) moveout for purposes of anisotropic velocity analysis.

Geometrical spreading of P-waves in horizontally layered, azimuthally anisotropic media

Geophysics ◽  
2005 ◽  
Vol 70 (5) ◽  
pp. D43-D53 ◽  
Author(s):  
Xiaoxia Xu ◽  
Ilya Tsvankin ◽  
Andrés Pech
Keyword(s):  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Azimuthal Velocity ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Geometrical Spreading ◽  
Weak Anisotropy ◽  
P Waves ◽  
Anisotropic Models ◽  
Wide Range

For processing and inverting reflection data, it is convenient to represent geometrical spreading through the reflection traveltime measured at the earth's surface. Such expressions are particularly important for azimuthally anisotropic models in which variations of geometrical spreading with both offset and azimuth can significantly distort the results of wide-azimuth amplitude-variation-with-offset (AVO) analysis. Here, we present an equation for relative geometrical spreading in laterally homogeneous, arbitrarily anisotropic media as a simple function of the spatial derivatives of reflection traveltimes. By employing the Tsvankin-Thomsen nonhyperbolic moveout equation, the spreading is represented through the moveout coefficients, which can be estimated from surface seismic data. This formulation is then applied to P-wave reflections in an orthorhombic layer to evaluate the distortions of the geometrical spreading caused by both polar and azimuthal anisotropy. The relative geometrical spreading of P-waves in homogeneous orthorhombic media is controlled by five parameters that are also responsible for time processing. The weak-anisotropy approximation, verified by numerical tests, shows that azimuthal velocity variations contribute significantly to geometrical spreading, and the existing equations for transversely isotropic media with a vertical symmetry axis (VTI) cannot be applied even in the vertical symmetry planes. The shape of the azimuthally varying spreading factor is close to an ellipse for offsets smaller than the reflector depth but becomes more complicated for larger offset-to-depth ratios. The overall magnitude of the azimuthal variation of the geometrical spreading for the moderately anisotropic model used in the tests exceeds 25% for a wide range of offsets. While the methodology developed here is helpful in modeling and analyzing anisotropic geometrical spreading, its main practical application is in correcting the wide-azimuth AVO signature for the influence of the anisotropic overburden.

Estimation of interval anisotropy parameters using velocity-independent layer stripping

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. WB117-WB127 ◽  
Author(s):  
Xiaoxiang Wang ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Plane ◽  
Synthetic Data ◽  
Single Layer ◽  
Transversely Isotropic ◽  
P Wave ◽  
Correlated Noise ◽  
Fractured Reservoir ◽  
Processing Parameter ◽  
Reflection Data ◽  
Error Amplification

Moveout analysis of long-spread P-wave data is widely used to estimate the key time-processing parameter [Formula: see text] in layered transversely isotropic media with a vertical symmetry axis (VTI). Inversion for interval [Formula: see text] values, however, suffers from instability caused by the trade-off between the effective moveout parameters and by subsequent error amplification during Dix-type layer stripping. We propose an alternative approach to nonhyperbolic moveout inversion based on the velocity-independent layer-stripping (VILS) method of Dewangan and Tsvankin. Also, we develop the 3D version of VILS and apply it to interval parameter estimation in orthorhombic media using wide-azimuth, long-spread data. If the overburden is laterally homogeneous and has a horizontal symmetry plane, VILS produces the exact interval traveltime-offset function in the target layer without knowledgeof the velocity field. Hence, Dix-type differentiation of moveout parameters used in existing techniques is replaced by the much more stable layer stripping of reflection traveltimes. The interval traveltimes are then inverted for the moveout parameters using the single-layer nonhyperbolic moveout equation. The superior accuracy and stability of the algorithm are illustrated on ray-traced synthetic data for typical VTI and orthorhombic models. Even small correlated noise in reflection traveltimes causes substantial distortions in the interval [Formula: see text] values computed by conventional Dix-type differentiation. In contrast, the output of VILS is insensitive to mild correlated traveltime errors. The algorithm is also tested on wide-azimuth P-wave reflection data recorded above a fractured reservoir at Rulison field in Colorado. The interval moveout parameters estimated by VILS in the shale layer above the reservoir are more plausible and less influenced by noise than those obtained by the Dix-type method.

Migration error in transversely isotropic media

Geophysics ◽  
1994 ◽  
Vol 59 (9) ◽  
pp. 1405-1418 ◽  
Author(s):  
Tariq Alkhalifah ◽  
Ken Larner
Keyword(s):  
Synthetic Data ◽  
Transversely Isotropic ◽  
Seismic Profile ◽  
Migration Velocity ◽  
Accurate Estimate ◽  
Position Error ◽  
P Wave ◽  
Position Errors ◽  
Isotropic Media ◽  
Error Behavior

Most migration algorithms today are based on the assumption that the earth is isotropic, an approximation that is often not valid and thus can lead to position errors on migrated images. Here, we compute curves of such position error as a function of reflector dip for transversely isotropic (TI) media characterized by Thomsen’s anisotropy parameters δ and ε. Depending on whether the migration velocity is derived from stacking velocity or vertical root‐mean‐square (rms) velocity, we find quite contrary sensitivities of the error behavior to the values of δ and ε. Likewise error‐versus‐dip behavior depends in a complicated way on vertical velocity gradient and vertical time, as well as orientation of the symmetry axis. Moreover, error behavior is dependent on just how δ and ε vary with depth. In addition to presenting such error curves, we show migrations of synthetic data that exemplify the mispositioning that results from ignoring anisotropy for P‐wave data. When migration is done using velocities derived from stacking velocity and when medium velocity increases with depth at rates typically encountered in practice, δ alone is sufficient to describe the position error. This is fortunate since the value of δ, unlike ε, can be obtained from combined vertical seismic profile (VSP) and surface seismic data. In contrast, when the migration velocity is obtained from the vertical rms velocity, the position errors depend strongly on ε, suggesting the importance of having an accurate estimate of ε when using an anisotropic migration algorithm.

Relationship of P-wave seismic attributes, azimuthal anisotropy, and commercial gas pay in 3-D P-wave multiazimuth data, Rulison Field, Piceance Basin, Colorado

Geophysics ◽  
1999 ◽  
Vol 64 (4) ◽  
pp. 1293-1311 ◽  
Author(s):  
Heloise B. Lynn ◽  
David Campagna ◽  
K. Michele Simon ◽  
Wallace E. Beckham
Keyword(s):  
Seismic Attributes ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Seismic Survey ◽  
Target Interval ◽  
Azimuthal Variation ◽  
Interval Velocity ◽  
Principal Axes ◽  
Naturally Fractured ◽  
Relationship Of

This case history is one of three field projects funded by the US Department of Energy as part of its ongoing research effort aimed to expand current levels of drilling and production efficiency in naturally‐fractured tight‐gas reservoirs. The original stated goal for the 3-D P-wave seismic survey was to evaluate and map fracture azimuth and relative fracture density throughout a naturally‐fractured gas reservoir interval. At Rulison field, this interval is the Cretaceous Mesaverde, approximately 2500 ft (760 m) of lenticular sands, silts, and shales. Three‐dimensional full‐azimuth P-wave data were acquired for the evaluation of azimuthal anisotropy and the relationship of the anisotropy to commercial pay in the target interval. The methodology is based on the evaluation of two restricted‐azimuth orthogonal (source‐receiver azimuth) 3-D P-wave volumes aligned with the natural principal axes of the azimuthal anisotropy, as estimated from velocity analysis of multiazimuth prestack gathers. The Dix interval velocity, as well as the interval amplitude variation with offset (AVO) gradient, was calculated for both azimuths for the gas‐saturated Mesaverde interval. The two seismic attributes best correlated with commercial gas pay (at a 21-well control set) were (1) values greater than 4% azimuthal variation in the interval velocity ratio (source‐receiver azimuth N60E/N30W) of the target interval (the gas‐saturated Mesaverde), and (2) the sum of the interval AVO gradients (N60E + N30W). The sum of the interval AVO gradients is an attribute sensitive to the presence of gas, but not diagnostic of an azimuthal variation in the amplitude. The two‐azimuth interval velocity anisotropy mapped over the survey area suggests spatial variations in the orientation of the maximum horizontal stress field and the open (to flow) fracture system.

An anelliptic approximation for geometric spreading in transversely isotropic and orthorhombic media

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. C37-C47 ◽  
Author(s):  
Shibo Xu ◽  
Alexey Stovas ◽  
Yanadet Sripanich
Keyword(s):  
Symmetry Plane ◽  
Transversely Isotropic ◽  
Seismic Wave Propagation ◽  
P Wave ◽  
Effective Parameters ◽  
Amplitude Variation With Offset ◽  
Acoustic Anisotropy ◽  
Geometric Spreading ◽  
Parametric Expression ◽  
Data Processing Methods

The relative geometric spreading along the raypath contributes to the amplitude decay of the seismic wave propagation that needs to be considered for amplitude variation with offset or other seismic data processing methods that require the true amplitude processing. Expressing the P-wave geometric spreading factor in terms of the offset-traveltime-based parameters is a more practical and convenient way because these parameters can be estimated from the nonhyperbolic velocity analysis. We have developed an anelliptic approximation for the relative geometric spreading of P-wave in a homogeneous transversely isotropic medium with vertical symmetry axis (VTI) and an orthorhombic (ORT) medium under the acoustic anisotropy assumption. The coefficients in our approximation are only defined within the symmetry planes and computed from fitting with the exact parametric expression. For an ORT model, due to the symmetric behavior in different symmetry planes, the other coefficients in the approximation can be easily obtained by making corresponding changes in indices from the computed coefficients in one symmetry plane. From the numerical examples, we found that for a homogeneous VTI model, the anelliptic approximation is more accurate than the generalized nonhyperbolic moveout approximation form for larger offset. For a homogeneous ORT model, our anelliptic approximation is more accurate than its traveltime-based counterparts. Using the Dix-type equations for the effective parameters, our anelliptic form approximation is extended to a multilayered VTI and ORT models and has accurate results in both models.

Estimation of interval anisotropic attenuation from reflection data

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. A69-A74 ◽  
Author(s):  
Jyoti Behura ◽  
Ilya Tsvankin
Keyword(s):  
Attenuation Coefficient ◽  
Reservoir Characterization ◽  
Symmetry Plane ◽  
Transversely Isotropic ◽  
Spectral Ratio ◽  
P Wave ◽  
Ratio Method ◽  
Reflection Seismic ◽  
Reflection Data ◽  
Target Layer

Knowledge of interval attenuation can be highly beneficial in reservoir characterization and lithology discrimination. We combine the spectral-ratio method with velocity-independent layer stripping to develop a technique for the estimation of the interval attenuation coefficient from reflection seismic data. The layer-stripping procedure is based on identifying the reflections from the top and bottom of the target layer that share the same ray segments in the overburden. The algorithm is designed for heterogeneous, arbitrarily anisotropic target layers, but the overburden is assumed to be laterally homogeneous with a horizontal symmetry plane. Although no velocity information about the overburden is needed, interpretation of the computed anisotropic attenuation coefficient involves the phase angle in the target layer. Tests on synthetic P-wave data from layered transversely isotropic and orthorhombic media confirm the high accuracy of 2D and 3D versions of the algorithm. We also demonstrate that the interval attenuation estimates are independent of the inhomogeneity angle of the incident and reflected waves.

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