Moveout approximation for horizontal transversely isotropic and vertical transversely isotropic layered medium. Part I: 1D ray propagation‡

Anisotropy in subsurface geological models is primarily caused by two factors: sedimentation in shale/sand layers and fractures. The sedimentation factor is mainly modelled by vertical transverse isotropy (VTI), whereas the fractures are modelled by a horizontal transversely isotropic medium (HTI). In this paper we study hyperbolic and non‐hyperbolic normal reflection moveout for a package of HTI/VTI layers, considering arbitrary azimuthal orientation of the symmetry axis at each HTI layer. We consider a local 1D medium, whose properties change vertically, with flat interfaces between the layers. In this case, the horizontal slowness is preserved; thus, the azimuth of the phase velocity is the same for all layers of the package. In general, however, the azimuth of the ray velocity differs from the azimuth of the phase velocity. The ray azimuth depends on the layer properties and may be different for each layer. In this case, the use of the Dix equation requires projection of the moveout velocity of each layer on the phase plane. We derive an accurate equation for hyperbolic and high‐order terms of the normal moveout, relating the traveltime to the surface offset, or alternatively, to the subsurface reflection angle. We relate the azimuth of the surface offset to its magnitude (or to the reflection angle), considering short and long offsets. We compare the derived approximations with analytical ray tracing.     

Traveltime approximation in vertical transversely isotropic layered media

The well‐known asymptotic fractional four‐parameter traveltime approximation and the five‐parameter generalised traveltime approximation in stratified multi‐layer transversely isotropic elastic media with a vertical axis of symmetry have been widely used for pure‐mode and converted waves. The first three parameters of these traveltime expansions are zero‐offset traveltime, normal moveout velocity, and quartic coefficient, ensuring high accuracy of traveltimes at short offsets. The additional parameter within the four‐parameter approximation is an effective horizontal velocity accounting for large offsets, which is important to avoid traveltime divergence at large offsets. The two additional parameters in the above‐mentioned five‐parameter approximation ensure higher accuracy up to a given large finite offset with an exact match at this offset. In this paper, we propose two alternative five‐parameter traveltime approximations, which can be considered extensions of the four‐parameter approximation and an alternative to the five‐parameter approximation previously mentioned. The first three short‐offset parameters are the same as before, but the two additional long‐offset parameters are different and have specific physical meaning. One of them describes the propagation in the high‐velocity layer of the overburden (nearly horizontal propagation in the case of very large offsets), and the other characterises the intercept time corresponding to the critical slowness that includes contributions of the lower velocity layers only. Unlike the above‐mentioned approximations, both of the proposed traveltime approximations converge to the theoretical (asymptotic) linear traveltime at the limit case of very large (“infinite”) offsets. Their accuracy for moderate to very large offsets, for quasi‐compressional waves, converted waves, and shear waves polarised in the horizontal plane, is extremely high in cases where the overburden model contains at least one layer with a dominant higher velocity compared with the other layers. We consider the implementation of the proposed traveltime approximations in all classes of problems in which the above‐mentioned approximations are used, such as reflection and diffraction analysis and imaging.     

Slowness-domain kinematical characteristics for horizontally layered orthorhombic media. Part I: Critical slowness match

Kinematical characteristics of reflected waves in anisotropic elastic media play an important role in the seismic imaging workflow. Considering compressional and converted waves, we derive new, azimuthally dependent, slowness-domain approximations for the kinematical characteristics of reflected waves (radial and transverse offsets, intercept time and traveltime) for layered orthorhombic media with varying azimuth of the vertical symmetry planes. The proposed method can be considered an extension of the well-known ‘generalized moveout approximation’ in the slowness domain, from azimuthally isotropic to azimuthally anisotropic models. For each slowness azimuth, the approximations hold for a wide angle range, combining power series coefficients in the vicinity of both the normal-incidence ray and an additional wide-angle ray. We consider two cases for the wide-angle ray: a ‘critical slowness match’ and a ‘pre-critical slowness match’ studied in Parts I and II of this work, respectively. For the critical slowness match, the approximations are valid within the entire slowness range, up to the critical slowness. For the ‘pre-critical slowness match’, the approximations are valid only within the bounded slowness range; however, the accuracy within the defined range is higher. The critical slowness match is particularly effective when the subsurface model includes a dominant high-velocity layer where, for nearly critical slowness values, the propagation in this layer is almost horizontal. Comparing the approximated kinematical characteristics with those computed by numerical ray tracing, we demonstrate high accuracy.     

Perturbation‐based moveout approximations in anisotropic media

The moveout approximations play an important role in seismic data processing. The standard hyperbolic moveout approximation is based on an elliptical background model with two velocities: vertical and normal moveout. We propose a new set of moveout approximations based on a perturbation series in terms of anellipticity parameters using the alternative elliptical background model defined by vertical and horizontal velocities. We start with a transversely isotropic medium with a vertical symmetry axis. Then, we extend this approach to a homogeneous orthorhombic medium. To define the perturbation coefficients for a new background, we solve the eikonal equation with horizontal velocities in transversely isotropic medium with a vertical symmetry axis and orthorhombic media. To stabilise the perturbation series and improve the accuracy, the Shanks transform is applied for all the cases. We select different parameterisations for both velocities and anellipticity parameters for an orthorhombic model. From the comparison in traveltime error, the new moveout approximations result in better accuracy comparing with the standard perturbation‐based methods and other approximations.     

Normal moveout velocity for pure‐mode and converted waves in layered orthorhombic medium

We study the azimuthally dependent hyperbolic moveout approximation for small angles (or offsets) for quasi‐compressional, quasi‐shear, and converted waves in one‐dimensional multi‐layer orthorhombic media. The vertical orthorhombic axis is the same for all layers, but the azimuthal orientation of the horizontal orthorhombic axes at each layer may be different. By starting with the known equation for normal moveout velocity with respect to the surface‐offset azimuth and applying our derived relationship between the surface‐offset azimuth and phase‐velocity azimuth, we obtain the normal moveout velocity versus the phase‐velocity azimuth. As the surface offset/azimuth moveout dependence is required for analysing azimuthally dependent moveout parameters directly from time‐domain rich azimuth gathers, our phase angle/azimuth formulas are required for analysing azimuthally dependent residual moveout along the migrated local‐angle‐domain common image gathers. The angle and azimuth parameters of the local‐angle‐domain gathers represent the opening angle between the incidence and reflection slowness vectors and the azimuth of the phase velocity ψ_phs at the image points in the specular direction. Our derivation of the effective velocity parameters for a multi‐layer structure is based on the fact that, for a one‐dimensional model assumption, the horizontal slowness and the azimuth of the phase velocity ψ_phs remain constant along the entire ray (wave) path. We introduce a special set of auxiliary parameters that allow us to establish equivalent effective model parameters in a simple summation manner. We then transform this set of parameters into three widely used effective parameters: fast and slow normal moveout velocities and azimuth of the slow one. For completeness, we show that these three effective normal moveout velocity parameters can be equivalently obtained in both surface‐offset azimuth and phase‐velocity azimuth domains.     

Slowness domain offset and traveltime approximations in layered vertical transversely isotropic media

Considering horizontally layered transversely isotropic media with vertical symmetry axis and all types of pure‐mode and converted waves we present a new wide‐angle series approximation for the kinematical characteristics of reflected waves: horizontal offset, intercept time, and total reflection traveltime as functions of horizontal slowness. The method is based on combining (gluing) both zero‐offset and (large) finite‐offset series coefficients. The horizontal slowness is bounded by the critical value, characterised by nearly horizontal propagation within the layer with the highest horizontal velocity. The suggested approximation uses five parameters to approximate the offset, six parameters to approximate the intercept time or the traveltime, and seven parameters to approximate any two or all three kinematical characteristics. Overall, the method is very accurate for pure‐mode compressional waves and shear waves polarised in the horizontal plane and for converted waves. The application of the method to pure‐mode shear waves polarised in the vertical plane is limited due to cusps and triplications. To demonstrate the high accuracy of the method, we consider a synthetic, multi‐layer model, and we plot the normalised errors with respect to numerical ray tracing.     

Generalized non‐hyperbolic approximation for qP‐wave relative geometrical spreading in a layered transversely isotropic medium with a vertical symmetry axis

Compensation for geometrical spreading along the ray‐path is important in amplitude variation with offset analysis especially for not strongly attenuative media since it contributes to the seismic amplitude preservation. The P‐wave geometrical spreading factor is described by a non‐hyperbolic moveout approximation using the traveltime parameters that can be estimated from the velocity analysis. We extend the P‐wave relative geometrical spreading approximation from the rational form to the generalized non‐hyperbolic form in a transversely isotropic medium with a vertical symmetry axis. The acoustic approximation is used to reduce the number of parameters. The proposed generalized non‐hyperbolic approximation is developed with parameters defined by two rays: vertical and a reference rays. For numerical examples, we consider two choices for parameter selection by using two specific orientations for reference ray. We observe from the numerical tests that the proposed generalized non‐hyperbolic approximation gives more accurate results in both homogeneous and multi‐layered models than the rational counterpart.     

A new parameterization for generalized moveout approximation,based on three rays

A simple and accurate traveltime approximation is important in many applications in seismic data processing, inversion and modelling stages. Generalized moveout approximation is an explicit equation that approximates reflection traveltimes in general two-dimensional models. Definition of its five parameters can be done from properties of finite offset rays, for general models, or by explicit calculation from model properties, for specific models. Two versions of classical finite-offset parameterization for this approximation use traveltime and traveltime derivatives of two rays to define five parameters, which makes them asymmetrical. Using a third ray, we propose a balance between the number of rays and the order of traveltime derivatives. Our tests using different models also show the higher accuracy of the proposed method. For acoustic transversely isotropic media with a vertical symmetry axis, we calculate a new moveout approximation in the generalized moveout approximation functional form, which is explicitly defined by three independent parameters of zero-offset two-way time, normal moveout velocity and anellipticity parameter. Our test shows that the maximum error of the proposed transversely isotropic moveout approximation is about 1/6 to 1/8 of that of the moveout approximation that had been reported as the most accurate approximation in these media. The higher accuracy is the result of a novel parameterization that do not add any computational complexity. We show a simple example of its application on synthetic seismic data.     

Mapping of moveout attributes using local slopes

Decomposing seismic data in local slopes is the basic idea behind velocity‐independent imaging. Using accurate moveout approximations enables computing moveout attributes such as normal moveout velocity and nonhyperbolic parameters as functions of zero‐offset travel time. Mapping of moveout attributes is performed from the pre‐stack seismic data domain into the time‐migrated image domain. The different moveout attributes have different accuracy for a given moveout approximation that depends on the corresponding order of travel‐time derivative. The most accurate attribute is the zero‐offset travel time, and the nonhyperbolic parameter has the worst accuracy, regardless of the moveout approximation. Typically, the mapping of moveout attributes is performed using a point‐to‐point procedure, whereas the generalized moveout approximation requires two point‐to‐point mappings. Testing the attribute mapping on the different models shows that the accuracy of mapped attributes is model dependent, whereas the generalized moveout approximation gives practically exact results.     

Moveout approximation for horizontal transversely isotropic and vertical transversely isotropic layered medium. Part II: effective model‡

We use residual moveouts measured along continuous full azimuth reflection angle gathers, in order to obtain effective horizontal transversely isotropic model parameters. The angle gathers are generated through a special angle domain imaging system, for a wide range of reflection angles and full range of phase velocity azimuths. The estimation of the effective model parameters is performed in two stages. First, the background horizontal transversely isotropic (HTI)/vertical transversely isotropic (VTI) layered model is used, along with the values of reflection angles, for converting the measured residual moveouts (or traveltime errors) into azimuthally dependent normal moveout (NMO) velocities. Then we apply a digital Fourier transform to convert the NMO velocities into azimuthal wavenumber domain, in order to obtain the effective HTI model parameters: vertical time, vertical compression velocity, Thomsen parameter delta and the azimuth of the medium axis of symmetry. The method also provides a reliability criterion of the HTI assumption. The criterion shows whether the medium possesses the HTI type of symmetry, or whether the azimuthal dependence of the residual traveltime indicates to a more complex azimuthal anisotropy. The effective model used in this approach is defined for a 1D structure with a set of HTI, VTI and isotropic layers (with at least one HTI layer). We describe and analyse the reduction of a multi‐layer structure into an equivalent effective HTI model. The equivalent model yields the same NMO velocity and the same offset azimuth on the Earth's surface as the original layered structure, for any azimuth of the phase velocity. The effective model approximates the kinematics of an HTI/VTI layered structure using only a few parameters. Under the hyperbolic approximation, the proposed effective model is exact.     

S-wave in 2D acoustic transversely isotropic media with a tilted symmetry axis

In an acoustic transversely isotropic medium, there are two waves that propagate. One is the P-wave and another one is the S-wave (also known as S-wave artefact). This paper is devoted to analyse the S-wave in two-dimensional acoustic transversely isotropic media with a tilted symmetry axis. We derive the S-wave slowness surface and traveltime function in a homogeneous acoustic transversely isotropic medium with a tilted symmetry axis. The S-wave traveltime approximations in acoustic transversely isotropic media with a tilted symmetry axis can be mapped from the counterparts for acoustic transversely isotropic media with a vertical symmetry axis. We consider a layered two-dimensional acoustic transversely isotropic medium with a tilted symmetry axis to analyse the S-wave moveout. We also illustrate the behaviour of the moveout for reflected S-wave and converted waves.     

Weak‐anisotropy approximation for P‐wave reflection coefficient at the boundary between two tilted transversely isotropic media

Existing and commonly used in industry nowadays, closed‐form approximations for a P‐wave reflection coefficient in transversely isotropic media are restricted to cases of a vertical and a horizontal transverse isotropy. However, field observations confirm the widespread presence of rock beds and fracture sets tilted with respect to a reflection boundary. These situations can be described by means of the transverse isotropy with an arbitrary orientation of the symmetry axis, known as tilted transversely isotropic media. In order to study the influence of the anisotropy parameters and the orientation of the symmetry axis on P‐wave reflection amplitudes, a linearised 3D P‐wave reflection coefficient at a planar weak‐contrast interface separating two weakly anisotropic tilted tranversely isotropic half‐spaces is derived. The approximation is a function of the incidence phase angle, the anisotropy parameters, and symmetry axes tilt and azimuth angles in both media above and below the interface. The expression takes the form of the well‐known amplitude‐versus‐offset “Shuey‐type” equation and confirms that the influence of the tilt and the azimuth of the symmetry axis on the P‐wave reflection coefficient even for a weakly anisotropic medium is strong and cannot be neglected. There are no assumptions made on the symmetry‐axis orientation angles in both half‐spaces above and below the interface. The proposed approximation can be used for inversion for the model parameters, including the orientation of the symmetry axes. Obtained amplitude‐versus‐offset attributes converge to well‐known approximations for vertical and horizontal transverse isotropic media derived by Rüger in corresponding limits. Comparison with numerical solution demonstrates good accuracy.     

The modified generalized moveout approximation: a new parameter selection

Non‐hyperbolic generalised moveout approximation is a powerful tool to approximate the travel‐time function by using information obtained from two rays. The standard approach for parameter selection is using three parameters defined from zero‐offset ray and two parameters obtained from a reference ray. These parameters include the travel time and travel‐time derivatives of different order. The original parameter selection implies more fit at zero offset compared with offset from a reference ray. We propose an alternative approach for parameter selection within the frame of generalised moveout approximation by transferring more fit from the zero offset to a reference ray by changing in parameter selection. The modified approximation is tested against the original one in few analytical model examples, including the multi‐layered model.     

Anisotropic inversion of refracted waves in vertical cable data in the presence of dip

Refracted arrivals are analysed to estimate the near‐surface anisotropy of marine sediments using a vertical‐cable (VC) configuration. In the presence of dip, the horizontal and vertical ray‐slownesses are obtained from the observed apparent slownesses in the up‐ and downdip directions using a sum or difference at each azimuth. The multiple azimuths generated by a VC geometry permit the ray‐slowness distribution of the marine sediments to be determined. An inversion procedure is developed to provide dip and anisotropy parameters for refractive layers from the measured refraction traveltimes in multilayered azimuthally isotropic and anisotropic media. Two sets of transversely isotropic models are used to analyse the azimuthal variations of apparent and ray slownesses. In the first set, we fix the anisotropic parameters of the models but vary the dip (0°, 5° and 10°) to test the effects of the presence of dip. In the second set, we vary the P‐wave anisotropy strength (5.2%, 10.3%, 15.8% and 22.0%) to examine the sensitivity and accuracy of ray‐slowness approximations which are independent of dip. We test this inversion procedure on synthetic P‐wave VC data calculated for six different models by a finite‐difference method. The results of applications to real VC data acquired from the North Sea are also presented.     

On anelliptic approximations for qP velocities in VTI media

A unified approach to approximating phase and group velocities of qP seismic waves in a transversely isotropic medium with vertical axis of symmetry (VTI) is developed. While the exact phase‐velocity expressions involve four independent parameters to characterize the elastic medium, the proposed approximate expressions use only three parameters. This makes them more convenient for use in surface seismic experiments, where the estimation of all four parameters is problematic. The three‐parameter phase‐velocity approximation coincides with the previously published ‘acoustic’ approximation of Alkhalifah. The group‐velocity approximation is new and noticeably more accurate than some of the previously published approximations. An application of the group‐velocity approximation for finite‐difference computation of traveltimes is shown.     

Reflection and transmission responses of layered transversely isotropic viscoelastic media

We consider a layered heterogeneous viscoelastic transversely isotropic medium with a vertical symmetry axis (a viscoelastic TIV medium) and parameters that depend on depth only. This takes into account intrinsic attenuation, anisotropy and thin layering. The seismic wavefield is decomposed into up- and downgoing waves scaled by the vertical energy flux. This scaling gives important symmetry relationships for both reflection and transmission (R/T) responses. For a stack of homogeneous layers, the exact reflection response can be computed in a numerically stable way by a simple layer-recursive algorithm. We derive exact plane-wave R/T coefficients and several linear and quadratic approximations between two viscoelastic TIV media, as functions of the real-valued horizontal slowness. The approximations are valid for pre- and post-critical values of horizontal slowness provided that the proper complex square roots are used when computing the vertical slowness. Numerical examples demonstrate that the quadratic approximations can be used for large differences in medium parameters, while the linear approximations can be used for small differences. For weak anisotropy it is sufficient to use an isotropic background medium, while for strong anisotropy it is necessary to use a weak TIV or TIV background medium. We also extend the O'Doherty–Anstey formula to the P- and SV-wave transmission responses of a stack of viscoelastic TIV layers, taking into account intrinsic attenuation, anisotropy and thin layering.     

Anisotropic velocity analysis1

Results from walkaway VSP and shale laboratory experiments show that shale anisotropy can be significantly anelliptic. Heterogeneity and anellipticity both lead to non-hyperbolic moveout curves and the resulting ambiguity in velocity analysis is investigated for the case of a factorizable anisotropic medium with a linear dependence of velocity on depth. More information can be obtained if there are several reflectors. The method of Dellinger et al. for anisotropic velocity analysis in layered transversely isotropic media is examined and is shown to be restricted to media having relatively small anellipticity. A new scheme, based on an expansion of the inverse-squared group velocity in spherical harmonics, is presented. This scheme can be used for larger anellipticity, and is applicable for horizontal layers having monoclinic symmetry with the symmetry plane parallel to the layers. The method is applied to invert the results of anisotropic ray tracing on a model Sand/shale sequence. For transversely isotropic media with small anisotropy, the scheme reduces to the method of Byun et al. and Byun and Corrigan. The expansion in spherical harmonics allows the P-phase slowness surface of each layer to be determined in analytic form from the layer parameters obtained by inversion without the need to assume that the anisotropy is weak.