Tilted Symmetry Axis Research Articles

Inversion for elastic parameters in weakly anisotropic media

Linearized inversion equations of quasi P waves (qP) and quasi S waves (qS) in arbitrary weakly anisotropic media are presented. The equations can be used for local determination of weak anisotropy (WA) parameters in inhomogeneous media. The equations for qS waves hold not only in regular directions but also in singularities. In the case of qS-wave coupling, the first arrival of qS waves can be used alone for inversion. We obtain three equations for each wave expressed in terms of WA parameters. One of the equations relates the WA parameters to slowness and the other two relate the WA parameters to polarization vectors. By eliminating one or two components of the slowness vector from the three equations, we get a new equation system which enables us to perform inversion when two (or one) components of the slowness vector are known. Since a coordinate system is selected to be identical to the polarization vectors, the inversion formulae for qS waves become linear ones. A synthetic multi-azimuthal multiple-source walkaway vertical seismic profile experiment for an orthorhombic inhomogeneous medium with tilted symmetry axes is carried out to illustrate the applicability and efficiency of the method.

Subduction zone anisotropy beneath Corvallis, Oregon: A serpentinite skid mark of trench‐parallel terrane migration?

We studied the back azimuth dependence of Ps converted phases at GSN station COR (Corvallis, Oregon) using broadband P receiver functions computed from 602 teleseismic earthquakes. The amplitudes and polarities of the transverse Ps phases are largely two‐lobed, which indicates anisotropy with a tilted symmetry axis. A double‐peaked Ps conversion at 4.5–6.5 s delay has the moveout of a dipping slab but is not consistent with simple deflection of the Ps converted phase by the dipping interface. A polarity flip on the transverse receiver function near north‐south back azimuth indicates an anisotropic symmetry axis aligned north‐south, far from the convergence direction (N68°E). The Ps phase is modeled using reflectivity synthetics with a highly anisotropic layer of depressed wave speed (VP ∼ 6.0–6.5 km/s) near 40 km depth, at the slab interface with the overriding North American plate. One‐dimensional modeling suggests 10% anisotropy in the supraslab layer, with a slow symmetry axis oriented N5°W at a 60° tilt from the vertical. Adjustments for the effect of slab interface dip on Ps amplitude suggest a somewhat lower 7% anisotropy. We infer a thin (∼7 km) anisotropic detachment zone for the northward slippage of the Siletz forearc terrane along the top of the descending slab, a motion consistent with GPS measurements and models of regional lithospheric dynamics. Serpentinite is a likely constituent for the deep anisotropic layer, owing to its reduced VP and high Poisson ratio. The ductile rheology and hydrated composition of serpentinite make it a plausible lithology for a mechanical detachment zone.

Born integral, stationary phase and linearized reflection coefficients in weak anisotropic media

SUMMARY We show that the linearized reflection coefficients for arbitrary anisotropic media embedded in an isotropic background can be derived directly from a Born formalism. Due to rapidly varying phases of the scattered waves from first-order perturbations in density and elastic parameters, the major contributions to the observed wavefield for any source–receiver pair far from the volume of scatterers arise from the stationary points of a scattering integral, called the Born integral. For simple interface models, such integrals can be evaluated analytically using the method of stationary phase. The resulting scattering function relates linearly to the approximate (linearized) reflection coefficient through a scaling factor determined by the angle of incidence and the properties of the background medium. We consider a homogeneous isotropic background to express the approximate reflection coefficients as a sum of an isotropic and an anisotropic reflection coefficient. The isotropic coefficient is a weighted sum of density and isotropic perturbations about the background, whereas the anisotropic coefficient is a weighted sum of anisotropic perturbations where the weights depend on the angles of incidence, the properties of the background medium as well as the azimuth of the plane of reflection with respect to some symmetry plane of the weakly anisotropic medium. We derived expressions for approximate PP and PS reflection coefficients of a weakly isotropic medium, a weakly orthorhombic medium and a weak but arbitrarily anisotropic medium underlying an isotropic medium. Our expressions for PP reflection coefficients are exactly the same as those obtained from first-order perturbation theory which were previously derived by linearization of the exact reflection coefficients. For converted PS waves, our expressions are valid only for small angles of incidence, but have much simpler forms than those obtained by linearization of exact reflection coefficients. We also derive the approximate PP reflection coefficient of a transversely isotropic medium with a tilted axis of symmetry in an explicit form and investigate the effects of dip of the symmetry axis on these reflection coefficients. Numerical results demonstrate that neglecting the dip of a moderately dipping (30°–60°) symmetry axis of a transversely isotropic medium yields significant errors in determining the weakly anisotropic parameters through an analysis of variations of amplitude with azimuth.

2.5D modelling, inversion and angle migration in anisotropic elastic media

ABSTRACT2.5D modelling approximates 3D wave propagation in the dip‐direction of a 2D geological model. Attention is restricted to raypaths for waves propagating in a plane. In this way, fast inversion or migration can be performed. For velocity analysis, this reduction of the problem is particularly useful.We review 2.5D modelling for Born volume scattering and Born–Helmholtz surface scattering. The amplitudes are corrected for 3D wave propagation, taking into account both in‐plane and out‐of‐plane geometrical spreading. We also derive some new inversion/migration results. An AVA‐compensated migration routine is presented that is simplified compared with earlier results. This formula can be used to create common‐image gathers for use in velocity analysis by studying the residual moveout. We also give a migration formula for the energy‐flux‐normalized plane‐wave reflection coefficient that models large contrast in the medium parameters not treated by the Born and the Born–Helmholtz equation results. All results are derived using the generalized Radon transform (GRT) directly in the natural coordinate system characterized by scattering angle and migration dip. Consequently, no Jacobians are needed in their calculation.Inversion and migration in an orthorhombic medium or a transversely isotropic (TI) medium with tilted symmetry axis are the lowest symmetries for practical purposes (symmetry axis is in the plane). We give an analysis, using derived methods, of the parameters for these two types of media used in velocity analysis, inversion and migration. The kinematics of the two media involve the same parameters, hence there is no distinction when carrying out velocity analysis. The in‐plane scattering coefficient, used in the inversion and migration, also depends on the same parameters for both media. The out‐of‐plane geometrical spreading, necessary for amplitude‐preserving computations, for the TI medium is dependent on the same parameters that govern in‐plane kinematics. For orthorhombic media, information on additional parameters is required that is not needed for in‐plane kinematics and the scattering coefficients.Resolution analysis of the scattering coefficient suggests that direct inversion by GRT yields unreliable parameter estimates. A more practical approach to inversion is amplitude‐preserving migration followed by AVA analysis.SYMBOLS AND NOTATIONA list of symbols and notation is given in Appendix D.

CDP mapping in tilted transversely isotropic (TTI) media. Part I: Method and effectiveness

ABSTRACTTransverse isotropy with a tilted axis of symmetry (TTI) causes image distortion if isotropic models are assumed during data processing. A simple anisotropic migration approach needs long computational times and is sensitive to the signal‐to‐noise ratio. This paper presents an efficient, general approach to common‐depth‐point (CDP) mapping to image the subsurface in TTI media from qP‐wave seismic data by adding anisotropic and dip parameters to the velocity model. The method consists of three steps: (i) calculating traveltimes and positions of the CDP points; (ii) determining CDP trajectories; (iii) CDP imaging. A crucial step is the rapid computation of traveltimes and raypaths in the TTI media, which is achieved by the Fermat method, specially adapted for anisotropic layered media. The algorithm can image the subsurface of a given model quickly and accurately, and is suitable for application to a bending reflector. The effectiveness of the method is demonstrated by comparing the raypaths, the traveltimes and the results of CDP mapping, when assuming isotropic media, transversely isotropic media with a vertical axis of symmetry (TIV), and TTI media.

Depth migration of shot records in heterogeneous, transversely isotropic media using optimum explicit operators

A space–frequency domain 2D depth‐migration scheme is generalized for imaging in the presence of anisotropy. The anisotropy model used is that of a transversely isotropic (TI) medium with a symmetry axis that can be either vertical or tilted. In the proposed scheme the anisotropy is described in terms of Thomsen parameters; however, the scheme can accommodate a wide range of anisotropy rather than only weak anisotropy. Short spatial convolution operators are used to extrapolate the wavefields recursively in the space–frequency domain for both qP‐ and qSV‐waves. The weighted least‐squares method for designing isotropic optimum operators is extended to asymmetric optimum explicit extrapolation operators in the presence of TI media with a tilted symmetry axis. Additionally, an efficient weighted quadratic‐programming design method is developed. The short spatial length of the derived operators makes it possible for the proposed scheme to handle lateral inhomogeneities. The performance of the operators, designed by combining the weighted least‐squares and weighted quadratic‐programming methods, is demonstrated by migration impulse responses of qP and qSV propagation modes for the weak and strong TI models with both vertical and tilted symmetry axes. Finally, a table‐driven shot‐record depth‐migration scheme is proposed, which is illustrated for finite‐difference modelled shot records in TI media.

Velocity analysis for tilted transversely isotropic media: A physical modeling example

Transverse isotropy with a tilted symmetry axis (TTI media) has been recognized as a common feature of shale formations in overthrust areas, such as the Canadian Foothills. Since TTI layers cause serious problems in conventional imaging, it is important to be able to reconstruct the velocity model suitable for anisotropic depth migration. Here, we discuss the results of anisotropic parameter estimation on a physical‐modeling data set. The model represents a simplified version of a typical overthrust section from the Alberta Foothills, with a horizontal reflector overlaid by a bending transversely isotropic layer. Assuming that the TTI layer is homogeneous and the symmetry axis stays perpendicular to its boundaries, we invert P-wave normal‐moveout (NMO) velocities and zero‐offset traveltimes for the symmetry‐direction velocity V0and the anisotropic parameters ε and δ. The coefficient ε is obtained using the traveltimes of a wave that crosses a dipping TTI block and reflects from the bottom of the model. The inversion for ε is based on analytic expressions for NMO velocity in media with intermediate dipping interfaces. Our estimates of both anisotropic coefficients are close to their actual values. The errors in the inversion, which are associated primarily with the uncertainties in picking the NMO velocities and traveltimes, can be reduced by a straighforward modification of the acquisition geometry. It should be emphasized that the moveout inversion also gives an accurate estimate of the thickness of the TTI layer, thus reconstructing the correct depth scale of the section. Although the physical model used here was relatively simple, our results demonstrate the principal feasibility of anisotropic velocity analysis and imaging in overthrust areas. The main problems in anisotropic processing for TTI models are likely to be caused by the lateral variation of the velocity field and overall structural complexity.

Inversion technique for transverse isotropy with a tilted symmetry axis

Computer simulation experiments and numerical modelling studies have been used to recover the P-wave anisotropic elastic parameters for transversely isotropic media with a tilted symmetry axis (TTI media). These parameters include P-wave velocity along the axis of symmetry α0, S-wave velocity along the axis of symmetryβ0, the P-wave anisotropy , and the near-vertical P-wave anisotropy δ. The angle of tilt in such media also has been recovered.The Levenberg-Marquardt method is utilised in carrying out the iterative inversion to recover the elastic parameters and angle of tilt ψ using a set of transmission data. This paper presents the 2-D case in which the vertical plane through the survey line contains the anisotropic axis of symmetry.The method was tested numerically on single layer models created with published anisotropic parameters having a range of tilt angles from zero to 90°. Traveltimes were computed using in-house software based on computed ray velocity (group velocity) functions. The traveltime data were inverted to recover the elastic parameters and the tilt angles of the symmetry axis. Results obtained are in good agreement with model parameters.In order to evaluate the performance of the inversion scheme in practice, random noise was added to the model data. The inversion algorithm proved to be robust and good results were obtained in the presence of noise. VSP field data were also successfully processed using the inversion program. The software includes options of constraining parameters to improve inversion accuracy.Elastic parameter recovery from field data may be used to improve the accuracy of pre-stack depth migration, depth estimation and reservoir volume estimations. Currently, VSP data must be recorded with dip direction for inversion with this 2-D software. Extension to the general 3-D case is currently being planned.

Influence of Anisotropic Elastic Parameters on Seismic Imaging: Numerical and Physical Modelling Studies

The wavefront curvature in an anisotropic medium is significantly influenced by the elastic parameters ε(near-vertical P-wave anisotropy), δ*. (P-wave anisotropy) and γ(SH-wave anisotropy). The effects of changes in these elastic parameters on the size of the Fresnel-zone have been studied using numerical modelling techniques and measured anisotropy of real sedimentary rocks. Ray paths for a P-wave gather above horizontal reflectors with a tilted axis of symmetry and contrasting velocity characteristics were also studied in order to determine how changes in elastic parameters affect the reflection point dispersal, and hence the smearing effect in stacked seismic data. Anisotropic migration of synthetic and physical modelling data was also carried out to demonstrate the effect of using inaccurate elastic parameters, and hence, wrong velocity functions in migrating seismic data.Numerical modelling results strongly indicate that the anisotropic elastic parameters significantly affect the size of the Fresnel zone, which determines the spatial resolving power for unmigrated seismic data. The magnitude of ray path distortion and reflection point dispersal is found to vary with the value and sign of the elastic parameter δ*. Ignoring anisotropy or using incorrect elastic parameters in migration will lead to wrong depth estimation and spatial mispositioning of reflection events.Experimental seismic data obtained using physical modelling demonstrate that anisotropy can significantly affect the spatial resolving power of seismic waves. The extent of these effects depends on the wavefront curvature which is influenced by the elastic parameters and orientation of the symmetry axis. These observations indicate that the imaging of reflection events from the base of thick shale sediments will be affected by anisotropy.

Inversion of azimuthally dependent NMO velocity in transversely isotropic media with a tilted axis of symmetry

Just as the transversely isotropic model with a vertical symmetry axis (VTI media) is typical for describing horizontally layered sediments, transverse isotropy with a tilted symmetry axis (TTI) describes dipping TI layers (such as tilted shale beds near salt domes) or crack systems. P-wave kinematic signatures in TTI media are controlled by the velocity [Formula: see text] in the symmetry direction, Thomsen’s anisotropic coefficients ε and δ, and the orientation (tilt ν and azimuth β) of the symmetry axis. Here, we show that all five parameters can be obtained from azimuthally varying P-wave NMO velocities measured for two reflectors with different dips and/or azimuths (one of the reflectors can be horizontal). The shear‐wave velocity [Formula: see text] in the symmetry direction, which has negligible influence on P-wave kinematic signatures, can be found only from the moveout of shear waves. Using the exact NMO equation, we examine the propagation of errors in observed moveout velocities into estimated values of the anisotropic parameters and establish the necessary conditions for a stable inversion procedure. Since the azimuthal variation of the NMO velocity is elliptical, each reflection event provides us with up to three constraints on the model parameters. Generally, the five parameters responsible for P-wave velocity can be obtained from two P-wave NMO ellipses, but the feasibility of the moveout inversion strongly depends on the tilt ν. If the symmetry axis is close to vertical (small ν), the P-wave NMO ellipse is largely governed by the NMO velocity from a horizontal reflector Vnmo(0) and the anellipticity coefficient η. Although for mild tilts the medium parameters cannot be determined separately, the NMO-velocity inversion provides enough information for building TTI models suitable for time processing (NMO, DMO, time migration). If the tilt of the symmetry axis exceeds 30°–40° (e.g., the symmetry axis can be horizontal), it is possible to find all P-wave kinematic parameters and construct the anisotropic model in depth. Another condition required for a stable parameter estimate is that the medium be sufficiently different from elliptical (i.e., ε cannot be close to δ). This limitation, however, can be overcome by including the SV-wave NMO ellipse from a horizontal reflector in the inversion procedure. While most of the analysis is carried out for a single layer, we also extend the inversion algorithm to vertically heterogeneous TTI media above a dipping reflector using the generalized Dix equation. A synthetic example for a strongly anisotropic, stratified TTI medium demonstrates a high accuracy of the inversion (subject to the above limitations).

Imaging structures below dipping TI media

Seismic anisotropy in dipping shales causes imaging and positioning problems for underlying structures. We developed an anisotropic depth‐migration approach for P-wave seismic data in transversely isotropic (TI) media with a tilted axis of symmetry normal to bedding. We added anisotropic and dip parameters to the depth‐imaging velocity model and used prestack depth‐migrated image gathers in a diagnostic manner to refine the anisotropic velocity model. The apparent position of structures below dipping anisotropic overburden changes considerably between isotropic and anisotropic migrations. The ray‐tracing algorithm used in a 2-D prestack Kirchhoff depth migration was modified to calculate traveltimes in the presence of TI media with a tilted symmetry axis. The resulting anisotropic depth‐migration algorithm was applied to physical‐model seismic data and field seismic data from the Canadian Rocky Mountain Thrust and Fold Belt. The anisotropic depth migrations offer significant improvements in positioning and reflector continuity over those obtained using isotropic algorithms.

P-SHconversions in a flat-layered medium with anisotropy of arbitrary orientation

SUMMARY P-SH conversion is commonly observed in teleseismic P waves, and is often attributed to dipping interfaces beneath the receiver. Our modelling suggests an alternative explanation in terms of flat-layered anisotropy. We use reflectivity techniques to compute three-component synthetic seismograms in a 1-D anisotropic layered medium. For each layer of the medium, we prescribe values of seismic velocities and hexagonally symmetric anisotropy about a common symmetry axis of arbitrary orientation. A compressional wave in an anisotropic velocity structure suffers conversion to both SV-and SH-polarized shear waves, unless the axis of symmetry is everywhere vertical or the wave travels parallel to all symmetry axes. The P-SV conversion forms the basis of the widely used ‘receiver function’ technique. The P-SH conversion occurs at interfaces where one or both layers are anisotropic. A tilted axis of symmetry and a dipping interface in isotropic media produce similar amplitudes of both direct (P) and converted (Ps) phases, leaving the backazimuth variation of the P-Ps delay as the main discriminant. Seismic anisotropy with a tilted symmetry axis leads to complex synthetic seismograms in velocity models composed of just a few flat homogeneous layers. It is possible therefore to model observations of P coda with prominent transverse components with relatively simple 1-D velocity structures. Successful retrieval of salient model characteristics appears possible using multiple realizations of a genetic-algorithm (GA) inversion of P coda from several backazimuths. Using GA inversion, we determine that six P coda recorded at station ARU in central Russia are consistent with models that possess strong (> 10 per cent) anisotropy in the top 5 km and between 30 and 43 km depth. The symmetry axes are tilted, and appear aligned with the seismic anisotropy orientation in the mantle under ARU suggested by SKS splitting.

Moveout analysis for tranversely isotropic media with a tilted symmetry axis

The transversely isotropic (TI) model with a tilted axis of symmetry may be typical, for instance, for sediments near the flanks of salt domes. This work is devoted to an analysis of reflection moveout from horizontal and dipping reflectors in the symmetry plane of TI media that contains the symmetry axis. While for vertical and horizontal transverse isotropy zero‐offset reflections exist for the full range of dips up to 90°, this is no longer the case for intermediate axis orientations. For typical homogeneous models with a symmetry axis tilted towards the reflector, wavefront distortions make it impossible to generate specular zero‐offset reflected rays from steep interfaces. The ‘missing’ dipping planes can be imaged only in vertically inhomogeneous media by using turning waves. These unusual phenomena may have serious implications in salt imaging. In non‐elliptical TI media, the tilt of the symmetry axis may have a drastic influence on normal‐moveout (NMO) velocity from horizontal reflectors, as well as on the dependence of NMO velocity on the ray parameter p (the ‘dip‐moveout (DMO) signature’). The DMO signature retains the same character as for vertical transverse isotropy only for near‐vertical and near‐horizontal orientation of the symmetry axis. The behaviour of NMO velocity rapidly changes if the symmetry axis is tilted away from the vertical, with a tilt of ±20° being almost sufficient to eliminate the influence of the anisotropy on the DMO signature. For larger tilt angles and typical positive values of the difference between the anisotropic parameters ε and δ, the NMO velocity increases with p more slowly than in homogeneous isotropic media; a dependence usually caused by a vertical velocity gradient. Dip‐moveout processing for a wide range of tilt angles requires application of anisotropic DMO algorithms. The strong influence of the tilt angle on P‐wave moveout can be used to constrain the tilt using P‐wave NMO velocity in the plane that includes the symmetry axis. However, if the azimuth of the axis is unknown, the inversion for the axis orientation cannot be performed without a 3D analysis of reflection traveltimes on lines with different azimuthal directions.

Free oscillations in an anisotropic earth: path-integral asymptotics

Summary We derive expressions for the influence of seismic anisotropy on the coupling of free oscillations using the Born approximation. By rotating to epicentral coordinates, the angular part of the coupling integrals can be approximated, in the limit of smooth structure, by an integral over the source-receiver great circle. To illustrate asymptotic coupling relationships, we examine a simplified case, in which the fourth-order elastic tensor Λαβγδ has an axis of symmetry ŵ of fixed orientation along the source-receiver great circle. In the path-integral formulation, mode-mode interaction satisfies a set of selection rules based on the angular degrees l1, l2 of the modes and the wavenumber s of elastic perturbation along the path. The familiar odd/even-parity selection rules for isotropic structure are a subset of these rules. If the elastic tensor is rotated to generalized spherical coordinates in an epicentral frame of reference, its components become complex-valued. Spheroidal-spheroidal and toroidal-toroidal coupling is dominated by Reαβγδ Spheroidal-toroidal coupling is dominated by ImΛαβγδ For simple cos 2η azimuthal velocity variations with a horizontal axis of symmetry ŵ, mixed-type coupling via ImΛαβγδ is maximal if ŵ is 45° to the propagation path. This agrees with full coupled-mode computations of long-period Love-Rayleigh scattering. Several terms in the elastic tensor Λαβγδ are non-zero only for a tilted axis of symmetry, and have distinct selection rules. Using the leading-order asymptotics for Legendre functions, coupling for smoothly varying structure is predicted to be small unless s= |l1–l2|. For coupling between like-type modes along a single dispersion branch, tomographic ‘source shifts’, focusing, and refraction effects can be identified.

Anisotropy in the inner core: relation between P-velocity and attenuation

We have measured both the differential travel times and the amplitude ratios of core phases PKP(DF) and PKP(BC), observed on digital broadband records worldwide, and analyzed them jointly, in order to try and distinguish between the effects of lateral heterogeneity and anisotropy in the inner core, and to test different hypotheses on the possible origin of such anisotropy. For the best quality data, we observe a correlation between stronger attenuation and faster velocities on a global scale, confirming earlier results obtained beneath Africa. Regional variations are observed in this relationship. In addition, for paths which have their turning point in equatorial regions, both the velocity and attenuation show clear dependence on the angle Θ of the ray path in the inner core with the Earth's rotation axis. By contrast, for paths with turning points at higher latitudes, this dependence is apparent only in the travel times. The ray azimuth at its turning point, computed in a reference system with a slightly tilted symmetry axis with respect to the Earth's rotation, appears to be a more successful parameter for analyzing anisotropy. The velocity-attenuation correlation suggests that crystal orientation is responsible for both the travel time and attenuation anisotropy, and that the heterogeneity level inside the inner core must be small. However, in order to investigate the pattern of anisotropy in more detail, simple models expressed in terms of Θ alone are no longer adequate, and more complex parametrization, possibly related to convective processes, will have to be considered.

P‐wave signatures and notation for transversely isotropic media: An overview

Progress in seismic inversion and processing in anisotropic media depends on our ability to relate different seismic signatures to the anisotropic parameters. While the conventional notation (stiffness coefficients) is suitable for forward modeling, it is inconvenient in developing analytic insight into the influence of anisotropy on wave propagation. Here, a consistent description of P‐wave signatures in transversely isotropic (TI) media with arbitrary strength of the anisotropy is given in terms of Thomsen notation. The influence of transverse isotropy on P‐wave propagation is shown to be practically independent of the vertical S‐wave velocity [Formula: see text], even in models with strong velocity variations. Therefore, the contribution of transverse isotropy to P‐wave kinematic and dynamic signatures is controlled by just two anisotropic parameters, ε and δ, with the vertical velocity [Formula: see text] being a scaling coefficient in homogeneous models. The distortions of reflection moveouts and amplitudes are not necessarily correlated with the magnitude of velocity anisotropy. The influence of transverse isotropy on P‐wave normal‐moveout (NMO) velocity in a horizontally layered medium, on small‐angle reflection coefficient, and on point‐force radiation in the symmetry direction is entirely determined by the parameter δ. Another group of signatures of interest in reflection seisimology—the dip‐dependence of NMO velocity, magnitude of nonhyperbolic moveout, time‐migration impulse response, and the radiation pattern near vertical—is dependent on both anisotropic parameters (ε and δ) and is primarily governed by the difference between ε and δ. Since P‐wave signatures are so sensitive to the value of ε − δ, application of the elliptical‐anisotropy approximation (ε = δ) in P‐wave processing may lead to significant errors. Many analytic expressions given in the paper remain valid in transversely isotropic media with a tilted symmetry axis. Moreover, the equation for NMO velocity from dipping reflectors, as well as the nonhyperbolic moveout equation, can be used in symmetry planes of any anisotropic media (e.g., orthorhombic).

Anisotropic wave propagation and zero-offset migration

In reflection seismology, migration may be defined as the transformation of apparent reflector positions to their true positions. In practice this usually means that diffractions are collapsed and reflector dip angles are changed. When a medium is anisotropic, and the axes of velocity symmetry are tilted with respect to the horizontal, horizontal reflection events migrate laterally as well. The lateral migration of reflections from horizontal layers was studied by constructing an anisotropic scale model representing a medium in which the bedding is inclined to the surface. This situation may be classified as one of transverse isotropy with a tilted axis of symmetry. The elastic parameters of the anisotropic model were recovered by P- and SH-wave transmission measurements carried out to simulate a walk-away VSP. Numerical modelling of a P-wave CMP gather above a horizontal reflector in such a medium indicated that there would be an asymmetrical distribution of reflection points. The zero-offset reflection point was displaced laterally by a distance equal to more than 20% of the depth, and the spread of reflection points was 15% of the depth. The NMO velocity was found to be a function of offset. A zero-offset reflection survey was carried out on the model. The P-wave data were migrated with an anisotropic migration algorithm which may be applied to wavefronts of any shape. On the resulting section, the horizontal reflection features are moved laterally by a distance equal to 20% of the depth back to their correct positions. The lateral displacement of reflections from a horizontal reflector beneath a medium with tilted anisotropic velocity characteristics has significance for the selection of drill locations when such a situation is encountered in the field.

Dip moveout in anisotropic media

When a common‐midpoint gather is collected above a dipping reflector, the point at which reflection occurs moves updip as the source‐receiver offset increases. Stacking velocity is constant, but it is a function of dip. Hence a stacked trace is not equivalent to a zero‐offset recorded trace. Dip moveout (DMO) is a processing step which converts traces to the equivalent of true zero‐offset records, making migration after stack (MAS) equivalent to migration before stack (MBS). The theory of velocity‐independent Gardner DMO is extended to triaxially elliptically anisotropic media in this paper. It is shown that the transformation is exact for homogeneous elliptically anisotropic media and that, after DMO, the stacking velocity is the horizontal component of the elliptically anisotropic velocity function. By taking three distinct seismic lines, three of the six constants of triaxial elliptical anisotropy may be determined. The remaining three cannot be obtained from surface seismic measurements. A simple numerical model of a point scatterer in a transversely isotropic medium with a tilted axis of symmetry is used to generate examples. The DMO process works when the velocity function is elliptical, but is not exact when the velocity function is nonelliptical.