3D description and inversion of reflection moveout of PS‐waves in anisotropic media

Common‐midpoint moveout of converted waves is generally asymmetric with respect to zero offset and cannot be described by the traveltime series t²(x²) conventionally used for pure modes. Here, we present concise parametric expressions for both common‐midpoint (CMP) and common‐conversion‐point (CCP) gathers of PS‐waves for arbitrary anisotropic, horizontally layered media above a plane dipping reflector. This analytic representation can be used to model 3D (multi‐azimuth) CMP gathers without time‐consuming two‐point ray tracing and to compute attributes of PS moveout such as the slope of the traveltime surface at zero offset and the coordinates of the moveout minimum. In addition to providing an efficient tool for forward modelling, our formalism helps to carry out joint inversion of P and PS data for transverse isotropy with a vertical symmetry axis (VTI media). If the medium above the reflector is laterally homogeneous, P‐wave reflection moveout cannot constrain the depth scale of the model needed for depth migration. Extending our previous results for a single VTI layer, we show that the interval vertical velocities of the P‐ and S‐waves (V_P0 and V_S0) and the Thomsen parameters ε and δ can be found from surface data alone by combining P‐wave moveout with the traveltimes of the converted PS(PSV)‐wave. If the data are acquired only on the dip line (i.e. in 2D), stable parameter estimation requires including the moveout of P‐ and PS‐waves from both a horizontal and a dipping interface. At the first stage of the velocity‐analysis procedure, we build an initial anisotropic model by applying a layer‐stripping algorithm to CMP moveout of P‐ and PS‐waves. To overcome the distorting influence of conversion‐point dispersal on CMP gathers, the interval VTI parameters are refined by collecting the PS data into CCP gathers and repeating the inversion. For 3D surveys with a sufficiently wide range of source–receiver azimuths, it is possible to estimate all four relevant parameters (V_P0, V_S0, ε and δ) using reflections from a single mildly dipping interface. In this case, the P‐wave NMO ellipse determined by 3D (azimuthal) velocity analysis is combined with azimuthally dependent traveltimes of the PS‐wave. On the whole, the joint inversion of P and PS data yields a VTI model suitable for depth migration of P‐waves, as well as processing (e.g. transformation to zero offset) of converted waves.     

Preserved travel‐time smoothing in orthorhombic media

Certain degree of smoothness of velocity models is required for most ray‐based migration and tomography. Applying conventional smoothing in model parameters results in offset‐dependent travel‐time errors for reflected events, which can be large even for small contrasts in model parameters between the layers. This causes the shift in both the depth and residual moveout of the migrated images. To overcome this problem in transversely isotropic medium with a vertical symmetry axis, the preserved travel‐time smoothing method was proposed earlier. We extend this method for orthorhombic media with and without azimuthal variation between the layers. We illustrate this method for a single interface between two orthorhombic layers and show that the smoothing‐driven errors in travel time are very small for practical application.     

The kinematics of S waves in acoustic orthorhombic media

Orthorhombic models are often used in the seismic industry nowadays to describe azimuthal and polar anisotropy and reasonably realistic in capturing the features of the earth interior. It is challenging to handle so many model parameters in the seismic data processing. In order to reduce the number of the parameters for P wave, the acoustic orthorhombic medium is proposed by setting all on-axis S wave velocities to zero. However, due to the coupled behaviour for P and S waves in the orthorhombic model, the ‘S wave artefacts’ are still remained in the acoustic orthorhombic model, which kinematics needs to be defined and analysed. In this paper, we analyse the behaviour of S wave in acoustic orthorhombic media. By analysis of the slowness surface in acoustic orthorhombic media, we define the S waves (or S wave artefacts) that are more complicated in shape comparing to the one propagating in an acoustic transversely isotropic medium with a vertical symmetry axis. The kinematic properties of these waves are defined and analysed in both phase and group domain. The caustics, amplitude and the multi-layered case for S wave in acoustic orthorhombic model are also discussed. It is shown that there are two waves propagating in this acoustic orthorhombic medium. One of these waves is similar to the one propagating in acoustic vertical symmetry axis media, whereas another one has a very complicated shape consisting of two crossing surfaces.     

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.     

Triplications for the converted wave in transversely isotropic media with a tilted symmetry axis

In seismic data processing, serious problems could be caused by the existence of triplication and need to be treated properly for tomography and other inversion methods. The triplication in transversely isotropic medium with a vertical symmetry axis has been well studied and concluded that the triplicated traveltime only occurs for S wave and there is no triplication for P and converted PS waves since the P wave convexity slowness always compensates the S wave slowness concavity. Compared with the vertical symmetry axis model, the research of the triplication in transversely isotropic medium with a tilted symmetry axis is still keeping blank. In order to analyse the triplication for the converted wave in the tilted symmetry axis model, we examine the traveltime of the triplication from the curvature of averaged P and S wave slowness. Three models are defined and tested in the numerical examples to illustrate the behaviour of the tilted symmetry axis model for the triplicated traveltime with the change of the rotation angle. Since the orientation of an interface is related to the orientation of the symmetry axis, the triplicated traveltime is encountered for the converted wave in the tilted symmetry axis model assuming interfaces to be planar and horizontal. The triplicated region is influenced by the place and level of the concave curvature of the P and S wave slowness.     

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.     

Non-hyperbolic moveout inversion of wide-azimuth P-wave data for orthorhombic media

The azimuthally varying non‐hyperbolic moveout of P‐waves in orthorhombic media can provide valuable information for characterization of fractured reservoirs and seismic processing. Here, we present a technique to invert long‐spread, wide‐azimuth P‐wave data for the orientation of the vertical symmetry planes and five key moveout parameters: the symmetry‐plane NMO velocities, V⁽¹⁾_nmo and V⁽²⁾_nmo , and the anellipticity parameters, η⁽¹⁾, η⁽²⁾ and η⁽³⁾ . The inversion algorithm is based on a coherence operator that computes the semblance for the full range of offsets and azimuths using a generalized version of the Alkhalifah–Tsvankin non‐hyperbolic moveout equation. The moveout equation provides a close approximation to the reflection traveltimes in layered anisotropic media with a uniform orientation of the vertical symmetry planes. Numerical tests on noise‐contaminated data for a single orthorhombic layer show that the best‐constrained parameters are the azimuth ? of one of the symmetry planes and the velocities V⁽¹⁾_nmo and V⁽²⁾_nmo , while the resolution in η⁽¹⁾ and η⁽²⁾ is somewhat compromised by the trade‐off between the quadratic and quartic moveout terms. The largest uncertainty is observed in the parameter η⁽³⁾ , which influences only long‐spread moveout in off‐symmetry directions. For stratified orthorhombic models with depth‐dependent symmetry‐plane azimuths, the moveout equation has to be modified by allowing the orientation of the effective NMO ellipse to differ from the principal azimuthal direction of the effective quartic moveout term. The algorithm was successfully tested on wide‐azimuth P‐wave reflections recorded at the Weyburn Field in Canada. Taking azimuthal anisotropy into account increased the semblance values for most long‐offset reflection events in the overburden, which indicates that fracturing is not limited to the reservoir level. The inverted symmetry‐plane directions are close to the azimuths of the off‐trend fracture sets determined from borehole data and shear‐wave splitting analysis. The effective moveout parameters estimated by our algorithm provide input for P‐wave time imaging and geometrical‐spreading correction in layered orthorhombic media.     

Dense 3D residual moveout analysis as a tool for HTI parameter estimation

Three‐dimensional residual moveout analysis is the basic step in velocity model refinement. The analysis is generally carried out using horizontal and/or vertical semblances defined on a sparse set of in‐lines or cross‐lines with densely sampled source–receiver offsets. An alternative approach, which we call dense residual moveout analysis (DRMA), is to use all the bins of a three‐dimensional survey but sparsely sampled offsets. The proposed technique is very fast and provides unbiased and statistically efficient estimates of the residual moveout. Indeed, for the sparsest possible offset distribution, when only near‐ and far‐angle stacks are used, the variance of the residual moveout estimate is only 1.4 times larger than the variance of the least‐squares estimate obtained using all offsets. The high performance of DRMA makes it a useful tool for many applications, of which azimuthal velocity analysis is considered here. For a horizontal transverse isotropy (HTI) model, a deterministic procedure is proposed to define, at every point of residual moveout estimation, the azimuthal angle of the HTI axis of symmetry, the Thomsen anisotropy coefficients, and the interval (or root‐mean‐square) velocities in both the HTI isotropy and symmetry planes. The procedure is not restricted by DRMA assumptions; for example, it is also applicable to semblance‐based residual moveout estimates. The high resolution of the technique is illustrated by azimuthal velocity analysis over an oilfield in West Siberia.     

Elastic full waveform inversion of multi‐component OBC seismic data

Elastic full waveform inversion of seismic reflection data represents a data‐driven form of analysis leading to quantification of sub‐surface parameters in depth. In previous studies attention has been given to P‐wave data recorded in the marine environment, using either acoustic or elastic inversion schemes. In this paper we exploit both P‐waves and mode‐converted S‐waves in the marine environment in the inversion for both P‐ and S‐wave velocities by using wide‐angle, multi‐component, ocean‐bottom cable seismic data. An elastic waveform inversion scheme operating in the time domain was used, allowing accurate modelling of the full wavefield, including the elastic amplitude variation with offset response of reflected arrivals and mode‐converted events. A series of one‐ and two‐dimensional synthetic examples are presented, demonstrating the ability to invert for and thereby to quantify both P‐ and S‐wave velocities for different velocity models. In particular, for more realistic low velocity models, including a typically soft seabed, an effective strategy for inversion is proposed to exploit both P‐ and mode‐converted PS‐waves. Whilst P‐wave events are exploited for inversion for P‐wave velocity, examples show the contribution of both P‐ and PS‐waves to the successful recovery of S‐wave velocity.     

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.     

P‐wave stacking‐velocity tomography for VTI media

A major complication caused by anisotropy in velocity analysis and imaging is the uncertainty in estimating the vertical velocity and depth scale of the model from surface data. For laterally homogeneous VTI (transversely isotropic with a vertical symmetry axis) media above the target reflector, P‐wave moveout has to be combined with other information (e.g. borehole data or converted waves) to build velocity models for depth imaging. The presence of lateral heterogeneity in the overburden creates the dependence of P‐wave reflection data on all three relevant parameters (the vertical velocity V_P0 and the Thomsen coefficients ε and δ) and, therefore, may help to determine the depth scale of the velocity field. Here, we propose a tomographic algorithm designed to invert NMO ellipses (obtained from azimuthally varying stacking velocities) and zero‐offset traveltimes of P‐waves for the parameters of homogeneous VTI layers separated by either plane dipping or curved interfaces. For plane non‐intersecting layer boundaries, the interval parameters cannot be recovered from P‐wave moveout in a unique way. Nonetheless, if the reflectors have sufficiently different azimuths, a priori knowledge of any single interval parameter makes it possible to reconstruct the whole model in depth. For example, the parameter estimation becomes unique if the subsurface layer is known to be isotropic. In the case of 2D inversion on the dip line of co‐orientated reflectors, it is necessary to specify one parameter (e.g. the vertical velocity) per layer. Despite the higher complexity of models with curved interfaces, the increased angle coverage of reflected rays helps to resolve the trade‐offs between the medium parameters. Singular value decomposition (SVD) shows that in the presence of sufficient interface curvature all parameters needed for anisotropic depth processing can be obtained solely from conventional‐spread P‐wave moveout. By performing tests on noise‐contaminated data we demonstrate that the tomographic inversion procedure reconstructs both the interfaces and the VTI parameters with high accuracy. Both SVD analysis and moveout inversion are implemented using an efficient modelling technique based on the theory of NMO‐velocity surfaces generalized for wave propagation through curved interfaces.     

Velocity model updating in prestack Kirchhoff time migration for PS converted waves: Part I – Theory

A velocity model updating approach is developed based on moveout analysis of the diffraction curve of PS converted waves in prestack Kirchhoff time migration. The diffraction curve can be expressed as a product of two factors: one factor depending on the PS converted‐wave velocity only, and the other factor depending on all parameters. The velocity‐dependent factor represents the hyperbolic behaviour of the moveout and the other is a scale factor that represents the non‐hyperbolic behaviour of the moveout. This non‐hyperbolic behaviour of the moveout can be corrected in prestack Kirchhoff time migration to form an inverse normal‐moveout common‐image‐point gather in which only the hyperbolic moveout is retained. This hyperbolic moveout is the moveout that would be obtained in an isotropic equivalent medium. A hyperbolic velocity is then estimated from this gather by applying hyperbolic moveout analysis. Theoretical analysis shows that for any given initial velocity, the estimated hyperbolic velocity converges by an iterative procedure to the optimal velocity if the velocity ratio is optimal or to a value closer to the optimal velocity if the velocity ratio is not optimal. The velocity ratio (V_P/V_S) has little effect on the estimation of the velocity. Applying this technique to a synthetic seismic data set confirms the theoretical findings. This work provides a practical method to obtain the velocity model for prestack Kirchhoff time migration.     

Characterization of dipping fractures in a transverselyisotropic background

Although 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.     

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.     

Out‐of‐plane geometrical spreading in anisotropic media

Two-dimensional seismic processing is successful in media with little structural and velocity variation in the direction perpendicular to the plane defined by the acquisition direction and the vertical axis. If the subsurface is anisotropic, an additional limitation is that this plane is a plane of symmetry. Kinematic ray propagation can be considered as a two-dimensional process in this type of medium. However, two-dimensional processing in a true-amplitude sense requires out-of-plane amplitude corrections in addition to compensation for in-plane amplitude variation. We provide formulae for the out-of-plane geometrical spreading for P- and S-waves in transversely isotropic and orthorhombic media. These are extensions of well-known isotropic formulae.
For isotropic and transversely isotropic media, the ray propagation is independent of the azimuthal angle. The azimuthal direction is defined with respect to a possibly tilted axis of symmetry. The out-of-plane spreading correction can then be calculated by integrating quantities which describe in-plane kinematics along in-plane rays. If, in addition, the medium varies only along the vertical direction and has a vertical axis of symmetry, no ray tracing need be carried out. All quantities affecting the out-of-plane geometrical spreading can be derived from traveltime information available at the observation surface.
Orthorhombic media possess no rotational symmetry and the out-of-plane geometrical spreading includes parameters which, even in principle, are not invertible from in-plane experiments. The exact and approximate formulae derived for P- and S-waves are nevertheless useful for modelling purposes.     

Research Note: Overpressure prediction from C‐wave seismic data

Recently, mode converted shear waves (C‐waves) have been shown to enable overpressure prediction in media where primary wave acquisition is inhibited by gas and fluid effects – C‐wave moveout is analysed and a long standing relationship between differential stress and primary‐wave (P‐wave) velocity is modified and employed. Though pore‐pressure prediction based on C‐waves is supported by empirical evidence from laboratory and field experiments, a theoretical justification has yet to be developed. In this research note, we provide a supporting algebra for the original relationship between pore pressure and C‐wave velocity.