Thomsen Parameters Research Articles

Aligned vertical fractures, HTI reservoir symmetry and Thomsen seismic anisotropy parameters for polar media

ABSTRACTCertain crack‐influence parameters of Sayers and Kachanov are shown to be directly related to Thomsen's weak‐anisotropy seismic parameters for fractured reservoirs when the crack/fracture density is small enough. These results are then applied to the problem of seismic wave propagation in polar reservoirs, i.e., those anisotropic reservoirs having two axes that are equivalent but distinct from the third axis), especially for horizontal transversely isotropic seismic wave symmetry due to the presence of aligned vertical fractures and resulting in azimuthal seismic wave symmetry at the Earth's surface. The approach presented suggests one method of inverting for fracture density from wave speed data. A significant fraction of the technical effort in the paper is devoted to showing how to predict the angular location of the true peak (or trough) of the quasi‐SV‐wave for polar media and especially how this peak is related to another angle that is very easy to compute. The axis of symmetry is always treated here as the x3‐axis for either vertical transversely isotropic symmetry (due, for example, to horizontal cracks), or horizontal transversely isotropic symmetry (due to aligned vertical cracks). Then, the meaning of the stiffnesses is derived from the fracture analysis in the same way for vertical transversely isotropic and horizontal transversely isotropic media, but for horizontal transverse isotropy the wave speeds relative to the Earth's surface are shifted by 90o in the plane perpendicular to the aligned vertical fractures. Skempton's poroelastic coefficient B is used as a general means of quantifying the effects of fluids inside the fractures. Explicit Biot‐Gassmann‐consistent formulas for Thomsen's parameters are also obtained for either drained or undrained fractures resulting in either vertical transversely isotropic or horizontal transversely isotropic symmetry of the reservoir.

Theory of traveltime shifts around compacting reservoirs: 3D solutions for heterogeneous anisotropic media

Time-lapse traveltime shifts of reflection events recorded above hydrocarbon reservoirs can be used to monitor production-related compaction and pore-pressure changes. Existing methodology, however, is limited to zero-offset rays and cannot be applied to traveltime shifts measured on prestack seismic data. We give an analytic 3D description of stress-related traveltime shifts for rays propagating along arbitrary trajectories in heterogeneous anisotropic media. The nonlinear theory of elasticity helps to express the velocity changes in and around the reservoir through the excess stresses associated with reservoir compaction. Because this stress-induced velocity field is both heterogeneous and anisotropic, it should be studied using prestack traveltimes or amplitudes. Then we obtain the traveltime shifts by first-order perturbation of traveltimes that accounts not only for the velocity changes but also for 3D deformation of reflectors. The resulting closed-form expression can be used efficiently for numerical modeling of traveltime shifts and, ultimately, for reconstructing the stress distribution around compacting reservoirs. The analytic results are applied to a 2D model of a compacting rectangular reservoir embedded in an initially homogeneous and isotropic medium. The computed velocity changes around the reservoir are caused primarily by deviatoric stresses and produce a transversely isotropic medium with a variable orientation of the symmetry axis and substantial values of the Thomsen parameters [Formula: see text] and [Formula: see text]. The offset dependence of the traveltime shifts should play a crucial role in estimating the anisotropy parameters and compaction-related deviatoric stress components.

Nonlinear traveltime inversion scheme for crosshole seismic tomography in tilted transversely isotropic media

Crosshole seismic tomography often is applied to image the velocity structure of an interwell medium. If the rocks are anisotropic, the tomographic technique must be adapted to the complex situation; otherwise, it leads to a false interpretation. We propose a nonlinear kinematic inversion method for crosshole seismic tomography in composite transversely isotropic media with known dipping symmetry axes. This method is based on a new version of the first-order traveltime perturbation equation. It directly uses the derivative of the phase velocity rather than the eigenvectors of the body-wave modes to overcome the singularity problem for application to the two quasi-shear waves. We applied an iterative nonlinear solver incorporating our kinematic ray-tracing scheme and directly compute the Jacobian matrix in an arbitrary reference medium. This reconstructs the five elastic moduli or Thomsen parameters from the first-arrival traveltimes of the three seismic body waves (qP, qSV, qSH) in strongly and weakly anisotropic media. We conducted three synthetic experiments that involve determining anisotropic parameters for a homogeneous rock, reconstructing a fault embedded in a strongly anisotropic background, and imaging a complicated four-layer model containing a small channel and a buried dipping interface. We compared results of our nonlinear inversion method with isotropic tomography and the traditional linear anisotropic inversion scheme, which showed the capability and superiority of the new scheme for crosshole tomographic imaging.

Velocity sensitivity of seismic body waves to the anisotropic parameters of a TTI-medium

We formulate the derivatives of the phase and group velocities for each of the anisotropic parameters in a tilted transversely isotropic medium (TTI-medium). This is a common geological model in seismic exploration and has five elastic moduli or related Thomsen parameters and two orientation angles defining the axis of symmetry of the rock. We present two independent methods to compute the derivatives and examine the formulae with real anisotropic rocks. The formulations and numerical computations do not encounter any singularity problem when applied to the two quasi shear waves, which is a problem with other approaches. The two methods yield the same results, which show in a quantitative way the sensitivity behaviour of the phase and the group velocities to all of the elastic moduli or Thomsen's anisotropic parameters as well as the orientation angles in the 2D and 3D cases. One can recognize the dominant (strong effect) and weak (or 'dummy') parameters for the three seismic body-wave modes (qP, qSV, qSH) and their effective domains over the whole range of phase-slowness directions. These sensitivity patterns indicate the possibility of nonlinear kinematic inversion with the three wave modes for determining the anisotropic parameters and imaging an anisotropic medium.

Non-linear traveltime inversion for 3-D seismic tomography in strongly anisotropic media

We have developed two, new non-linear traveltime inversion schemes for 3-D seismic tomography in anisotropic media. They differ from the traditional linearized inversion approach and offer five significant improvements: (1) they are based on an alternative form of the first-order traveltime perturbation equation, derived so as to simplify the inversion formulae and overcome the quasi shear wave singularity problem; (2) robust 3-D ray tracing is employed which enables the simultaneous computation of the first-arrival traveltimes and ray paths for the three body waves (qP, qS1 and qS2) in arbitrary anisotropic media; (3) the Jacobian matrix used in the update is based on an efficient computation for a 3-D anisotropic model, so that the inversion is applicable to both weakly and strongly anisotropic situations, unlike most previous approaches which assume weak anisotropy; (4) a local-search, constrained minimization is applied to the non-linear inversion which makes anisotropic tomographic imaging an iterative procedure; (5) there is an option to invert for the elastic moduli directly or the Thomsen parameters directly in heterogenous, tilted transversely isotropic media, using any source–receiver recording geometry. We have examined the imaging capability of the non-linear solver with individual body-wave modes using a 3-D synthetic anisotropic model incorporating two targets, a ‘high velocity’ and a ‘low velocity’ anomaly, embedded in a titled transversely isotropic medium. The model is illuminated by means of azimuthal VSP and crosshole measurements. The experimental results show that the two non-linear inversion schemes successfully image the ‘targets’ and yield satisfactory 3-D tomograms of the elastic moduli and the Thomsen parameters.

Sensitivity of Dix-type inverse formulae

The problem to be addressed here is how accurately we can estimate layer thickness and layer parameters for a transverse isotropic medium with a vertical symmetry axis (VTI media) and for an isotropic medium with a velocity function, for both P- and S-waves, which varies linearly with depth. Layer parameters for a VTI medium are the vertical P- and S-wave velocities and the well-known Thomsen parameters e and δ. For an isotropic medium with velocity functions that vary linearly with depth the layer parameters are the P- and S-wave velocities at the top of each layer and the P- and S-wave velocity gradients. Layer thickness and layer parameters are inverted from estimated P- and S-wave (qP and qSV for VTI media) traveltime parameters using Dix-type inversion formulae. The sensitivity of these inverse formulae has been investigated. By combining qP- and qSV-wave traveltime data from the VTI model, we obtain good estimates for the layer thickness and vertical qP- and qSV-wave velocities. A maximum offset-to-depth ratio of more than 2 is recommended. It is shown that the constraint in the heterogeneity factor gets poorer with increasing noise. The estimated e and δ have larger errors, and cannot be regarded as valid for further processing. The inversion formulae for estimating layer thickness and P- and S-wave velocity functions are very sensitive to errors in the estimated traveltime parameters.

Elastic moduli of anisotropic clay

Clay minerals are important components in shales, controlling their elastic properties and anisotropy. The elasticity of crystalline clay minerals differs significantly from that of clay in situ because of the ability of clay particles to bind water. In the ma-jority of published works, only isotropic moduli for in situ clays are reported. However, anisotropy is inherent in the clay elas-ticity. We develop an inversion technique for determination of the stiffness tensor of in situ clay from the shale’s stiffness tensor. As an example, we obtain the stiffness tensor of a “water-clay” composite from the data on the water-saturated Greenhorn shale sample, whose clay composition consists of almost equal amounts of illite and smectite and comparable amounts of kaolinite and chlorite. The stiffness tensor of the water-clay composite is found for the Greenhorn shale with step-by-step inversion based upon an effective medium theory. The inversion usesa nonlinear optimization technique with bounds imposed on the estimated parameters. In the inversion, we apply different approaches of the effective medium theory using a published method referred to here as the generalized singular approxi-mation (GSA). The GSA method makes it possible to take into account the microstructure of shales. The resulting elasticity constants of the anisotropic (transversely isotropic) in situ clay composite are [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] (in GPa); and the density equals [Formula: see text]. The Thomsen parameters for the clay composite are [Formula: see text], [Formula: see text], and [Formula: see text]. The elasticity constants found for this clay composite can be used in the theoretical analysis of shales that have a similar composition of clay but with different mineral compositions. The inversion technique developed can be used for general shale water-clay composites when the mineral composition and orientation of the clay platelets are known.

Stress induced elastic anisotropy of the Etnean basalt: Theoretical and laboratory examination

A theory, the stress‐sensitivity approach, has been developed, which relates the elastic moduli of anisotropic rocks to the stress tensor and pore pressure for an arbitrary symmetry of the applied load. The theory explains the stress‐induced changes of seismic velocities in terms of stress‐induced changes of the pore space geometry. The stress dependent anisotropy is described in terms of Thomsen's anisotropy parameters ε, γ and δ. To test the theory we analyze the laboratory (high frequency) results of deformation of an isotropically crack damaged dry lava flow basalt from Mt. Etna volcano. The theory states that, under an anisotropic (i.e. axisymmetric triaxial) load and in the case of an initially isotropic rock, (1) the anisotropy parameters are linear functions of the stress exponentials (i. e. exponential functions of principal stress components) and (2) the ratio of these anisotropy parameters as a function of the stress is constant. In order to verify these relationships, the stress exponentials and the anisotropy parameters based on the measured velocities are computed as well as the expected ratio of the Thomsen's parameters. Our experimental results are in very good agreement with the theoretically predicted relations.

Modelagem e migração em profundidade 2D em meios com simetria polar local

This paper shows a technique based on the phase-shift method (PSM) to implement pre-stack depth migration on locally transverse isotropic media (LTI), with the symmetry axis direction varies continually along the layers. For seismic modeling, a generalization of the finite differences method for the solution of the elastic wave equation was used. With this procedure, it was possible to accommodate seismic modeling on LTI media defined by six parameters at each grid point, i.e. , density, P and S wave propagation velocities along the local symmetry axis, Thomsen parameters and the direction of the local symmetry axis itself. In order to separate from the seismograms the qP and qSV wavefields, an algorithm based on the Christoffel equation was implemented. The migration for each common shot gather is implemented solely by phase-shift based algorithms, which means that not only the depropagation of the registered wavefield, but also the generation of the time matrices involved in the imaging condition were obtained in this manner for each set of parameters at each depth level. The migration results using qP-qP and qP-qSV reflections show that the horizons were located precisely, and that the process is stable in relation to the symmetry axis variations. The proposed method is for multicomponent seismic acquisitions and might be applied to marine seismic data using streamers, or Ocean Bottom Cables or vertical cables. Since the proposed method uses phase-shift algorithms, its parallel implementation can be highly efficient.

Weak elastic anisotropy in a cracked rock

Abstract The elastic properties of a cracked rock are transversely isotropic if its crack density distribution is axisymmetric. The three Thomsen parameters ε, δ and γ are used to characterize the seismic properties of a weakly anisotropic rock. In this study we calculated these anisotropy parameters in such a cracked rock under dry and saturated conditions. We derived analytic expressions for a dry rock based on an approximate model proposed by earlier workers, which shows elliptically anisotropic behaviour. Guided by microstructural observations we adopted a two-parameter axisymmetric distribution to characterize the crack density, which predicts that in a fluid-saturated rock the parameters δ and γ are related to ε in a nonlinear manner. All three parameters considered are sensitively dependent on the difference in the densities of cracks with normals parallel and perpendicular to the symmetry axis. Theoretical predictions of the model are compared with laboratory and field measurements of seismic anisotropy.

Small-angle AVO response of PS-waves in tilted transversely isotropic media

Field records for small source-receiver offsets often contain intensive converted PS-waves that may be caused by the influence of anisotropy on either side of the reflector. Here, we study the small-angle reflection coefficients of the split converted [Formula: see text]- and [Formula: see text]-waves ([Formula: see text] and [Formula: see text]) for a horizontal interface separating two transversely isotropic (TI) media with arbitrary orientations of the symmetry axis. The normal-incidence reflection coefficients [Formula: see text] and [Formula: see text] vanish when both half-spaces have a horizontal symmetry plane, which happens if the symmetry axis is vertical or horizontal (i.e., if the medium is VTI or HTI). For a tilted symmetry axis in either medium, however, the magnitude of the reflection coefficients can reach substantial values that exceed 0.1, even if the anisotropy strength is moderate. To study the influence exerted by the orientation of the symmetry axis and the anisotropy parameters, we develop concise weak-contrast, weak-anisotropyapproximations for the PS-wave reflection coefficients and com-pare them with exact numerical results. In particular, the analytic solutions show that the contributions made by the Thomsen parameters [Formula: see text] and [Formula: see text] and the symmetry-axis tilt [Formula: see text] to the coefficients [Formula: see text] and [Formula: see text] can be expressed through the first derivative of the P-wave phase velocity at normal incidence. If the symmetry-axis orientation and anisotropy parameters do not change across the interface, the normal-incidence reflection coefficients are insignificant, regardless of the strength of the velocity and density contrast. The AVO (amplitude variation with offset) gradients of the PS-waves are influenced primarily by the anisotropy of the incidence medium that causes shear-wave splitting and determines the partitioning of energy between the [Formula: see text] and [Formula: see text] modes. Because of their substantial amplitude, small-angle PS reflections in TI media contain valuable information for anisotropic AVO inversion of multicomponent data. Our analytic solutions provide a foundation for linear AVO-inversion algorithms and can be used to guide nonlinear inversion that is based on the exact reflection coefficients.

Comparing P-S conversion point location in isotropic and anisotropic media

The determination of the exact position of the P-S conversion point (CP) is an important issue in converted-wave seismology. Frequently, this issue is approached considering that the rock formation is isotropic. This simple approach may lead to significant errors due to the anisotropic features of the subsurface. This paper presents a comparison between the position of the CP obtained from the isotropic solution and the position of the CP considering two types of anisotropy: the transversely isotropic medium with vertical axis of symmetry, the so-called VTI, and an orthorhombic medium. In order to determine the location of the CP, the modeling program ANRAY, based on ray tracing method, was used. It is shown that for the VTI case the discrepancy between the true position of the CP and the one found by assuming isotropy can be in the order of hundreds of meters, even for offset/depth ratio, (X/Z), less than 2. On the other hand, for some particular values of Thomsen's parameters <FONT FACE=Symbol>Î</FONT> and delta, it can be demonstrated that, the position of the CP practically coincides with the isotropic case. For orthorhombic symmetry (azimuthal anisotropy), the CP, in general, is no longer located in the vertical plane defined by the source-receiver positions, i.e., the discrepancy between CP locations is expressed by an inline and an offline components. The offline component value depends on the anisotropic nature of the medium. This observation is not valid only for orthorhombic models since similar discrepancies can also be observed in cases where the anisotropy is induced by cross-dipping structures or in models with lower anisotropic symmetries.

Plane-wave propagation in attenuative transversely isotropic media

Directionally dependent attenuation in transversely isotropic (TI) media can influence significantly the body-wave amplitudes and distort the results of the AVO (amplitude variation with offset) analysis. Here, we develop a consistent analytic treatment of plane-wave properties for TI media with attenuation anisotropy. We use the concept of homogeneous wave propagation, assuming that in weakly attenuative media the real and imaginary parts of the wave vector are parallel to one another. The anisotropic quality factor can be described by matrix elements [Formula: see text], defined as the ratios of the real and imaginary parts of the corresponding stiffness coefficients. To characterize TI attenuation, we follow the idea of the Thomsen notation for velocity anisotropy and replace the components [Formula: see text] by two reference isotropic quantities and three dimensionless anisotropy parameters [Formula: see text], and [Formula: see text]. The parameters [Formula: see text] and [Formula: see text] quantify the difference between the horizontal- and vertical-attenuation coefficients of P- and SH-waves, respectively, while [Formula: see text] is defined through the second derivative of the P-wave attenuation coefficient in the symmetry direction. Although the definitions of [Formula: see text], and [Formula: see text] are similar to those for the corresponding Thomsen parameters, the expression for [Formula: see text] reflects the coupling between the attenuation and velocity anisotropy. Assuming weak attenuation as well as weak velocity and attenuation anisotropy allows us to obtain simple attenuation coefficients linearized in the Thomsen-style parameters. The normalized attenuation coefficients for P- and SV-waves have the same form as the corresponding approximate phase-velocity functions, but both [Formula: see text] and the effective SV-wave attenuation-anisotropy parameter [Formula: see text] depend on the velocity-anisotropy parameters in addition to the elements [Formula: see text]. The linearized approximations not only provide valuable analytic insight, but they also remain accurate for the practically important range of small and moderate anisotropy parameters — in particular, for near-vertical and near-horizontal propagation directions.

Image gathers of SV-waves in homogeneous and factorized 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.

Wave Equation Depth Migration with the Optimum Split-Step Fourier Method In 3-D VTI Media

A dispersion equation for transversely isotropic media with a vertical symmetry axis (VTI) is derived from the phase velocity expressed in terms of Thomsen parameters. Based on the vertical wavenumber solved from the dispersion equation, the optimum split-step Fourier 3-D wave equation depth migration method in isotropic media is extended to 3-D VTI media. The macro velocity model used for describing VTI media is consistent with the current velocity estimation method and can simultaneously accommodate the isotropy, weak anisotropy and strong anisotropy. The higher order correction terms added to the phase-shift method makes the proposed algorithm be able to model wider-angle wavefield with high accuracy in the presence of severe lateral velocity variations in 3-D VTI media. The accuracy of the proposed algorithm is demonstrated by 2-D and 3-D migration impulse responses.

Nonlinear rock physics model for estimation of 3D subsurface stress in anisotropic formations: Theory and laboratory verification

We develop a rock physics model based on nonlinear elasticity that describes the dependence of the effective stiffness tensor as a function of a 3D stress field in intrinsically anisotropic formations. This model predicts the seismic velocity of both P‐ and S‐waves in any direction for an arbitrary 3D stress state. Therefore, the model overcomes the limitations of existing empirical velocity‐stress models that link P‐wave velocity in isotropic rocks to uniaxial or hydrostatic stress. To validate this model, we analyze ultrasonic velocity measurements on stressed anisotropic samples of shale and sandstone. With only three nonlinear constants, we are able to predict the stress dependence of all five elastic medium parameters comprising the transversely isotropic stiffness tensor. We also show that the horizontal stress affects vertical S‐wave velocity with the same order of magnitude as vertical stress does. We develop a weak‐anisotropy approximation that directly links commonly measured anisotropic Thomsen parameters to the principal stresses. Each Thomsen parameter is simply a sum of corresponding background intrinsic anisotropy and stress‐induced contribution. The stress‐induced part is controlled by the difference between horizontal and vertical stresses and coefficients depending on nonlinear constants. Thus, isotropic rock stays isotropic under varying but hydrostatic load, whereas transversely isotropic rock retains the same values of dimensionless Thomsen parameters. Only unequal horizontal and vertical stresses alter anisotropy. Since Thomsen parameters conveniently describe seismic signatures, such as normal‐moveout velocities and amplitude‐variation‐with‐offset gradients, this approximation is suitable for designing new methods for the estimation of 3D subsurface stress from multicomponent seismic data.

Characterization of dipping fractures in a transverselyisotropic background

ABSTRACTAlthough it is believed that natural fracture sets predominantly have near‐vertical orientation, oblique stresses and some other mechanisms may tilt fractures away from the vertical. Here, we examine an effective medium produced by a single system of obliquely dipping rotationally invariant fractures embedded in a transversely isotropic with a vertical symmetry axis (VTI) background rock. This model is monoclinic with a vertical symmetry plane that coincides with the dip plane of the fractures.Multicomponent seismic data acquired over such a medium possess several distinct features that make it possible to estimate the fracture orientation. For example, the vertically propagating fast shear wave (and the fast converted PS‐wave) is typically polarized in the direction of the fracture strike. The normal‐moveout (NMO) ellipses of horizontal reflection events are co‐orientated with the dip and strike directions of the fractures, which provides an independent estimate of the fracture azimuth. However, the polarization vector of the slow shear wave at vertical incidence does not lie in the horizontal plane – an unusual phenomenon that can be used to evaluate fracture dip. Also, for oblique fractures the shear‐wave splitting coefficient at vertical incidence becomes dependent on fracture infill (saturation).A complete medium‐characterization procedure includes estimating the fracture compliances and orientation (dip and azimuth), as well as the Thomsen parameters of the VTI background. We demonstrate that both the fracture and background parameters can be obtained from multicomponent wide‐azimuth data using the vertical velocities and NMO ellipses of PP‐waves and two split SS‐waves (or the traveltimes of PS‐waves) reflected from horizontal interfaces. Numerical tests corroborate the accuracy and stability of the inversion algorithm based on the exact expressions for the vertical and NMO velocities.

Analysis of image gathers in factorized 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.

Quartic moveout coefficient: 3D description and application to tilted TI media

Nonhyperbolic (long‐spread) moveout provides essential information for a number of seismic inversion/processing applications, particularly for parameter estimation in anisotropic media. Here, we present an analytic expression for the quartic moveout coefficient A4that controls the magnitude of nonhyperbolic moveout of pure (nonconverted) modes. Our result takes into account reflection‐point dispersal on irregular interfaces and is valid for arbitrarily anisotropic, heterogeneous media. All quantities needed to compute A4can be evaluated during the tracing of the zero‐offset ray, so long‐spread moveout can be modeled without time‐consuming multioffset, multiazimuth ray tracing.The general equation for the quartic coefficient is then used to study azimuthally varying nonhyperbolic moveout of P‐waves in a dipping transversely isotropic (TI) layer with an arbitrary tilt ν of the symmetry axis. Assuming that the symmetry axis is confined to the dip plane, we employed the weak‐anisotropy approximation to analyze the dependence of A4on the anisotropic parameters. The linearized expression for A4is proportional to the anellipticity coefficient η ≈ ε − δ and does not depend on the individual values of the Thomsen parameters. Typically, the magnitude of nonhyperbolic moveout in tilted TI media above a dipping reflector is highest near the reflector strike, whereas deviations from hyperbolic moveout on the dip line are substantial only for mild dips.The azimuthal variation of the quartic coefficient is governed by the tilt ν and reflector dip φ and has a much more complicated character than the NMO–velocity ellipse. For example, if the symmetry axis is vertical (VTI media, ν = 0) and the dip φ &lt; 30°, A4goes to zero on two lines with different azimuths where it changes sign. If the symmetry axis is orthogonal to the reflector (this model is typical for thrust‐and‐fold belts), the strike‐line quartic coefficient is defined by the well‐known expression for a horizontal VTI layer (i.e., it is independent of dip), while the dip‐line A4is proportional to cos4φ and rapidly decreases with dip. The high sensitivity of the quartic moveout coefficient to the parameter η and the tilt of the symmetry axis can be exploited in the inversion of wide‐azimuth, long‐spread P‐wave data for the parameters of TI media.

Seismic anisotropy in gas‐hydrate‐ and gas‐bearing sediments on the Blake Ridge, from a walkaway vertical seismic profile

We present results from an analysis of anisotropy in marine sediments using walkaway vertical seismic profiles from the Blake Ridge, offshore South Carolina. We encountered transverse isotropy (TI) with a vertical symmetry axis in a gas‐hydrate‐bearing unit of clay and claystone with Thomsen parameters ε = 0.05 ± 0.02 and δ = 0.04 ± 0.06. TI increased to ε = 0.16 ± 0.04 and δ = 0.19 ± 0.12 in the underlying gas zone. Rock physics modeling suggests that the observed TI is caused by a partial alignment of clay particles rather than high‐velocity gas‐hydrate veins. Similarly, the increase of TI in the gas zone is not caused by thin low‐velocity gas layers but rather, we speculate, by the sharp contrast between seismic properties of an anisotropic sediment frame and elongated gas‐bearing pore voids. Our results underscore the significance of anisotropy for integrating near‐vertical and wide‐angle seismic data.