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.     

Interval anisotropic parameter estimation from walkaway vertical seismic profiling data

This paper presents a new explicit method for the estimation of layered vertical transverse isotropic (VTI) anisotropic parameters from walkaway VSP data. This method is based on Dix‐type normal moveout (NMO) inversion. To estimate interval anisotropic parameters above a receiver array, the method uses time arrivals of surface‐related double‐reflected downgoing waves. A three‐term NMO approximation function is used to estimate NMO velocity and a non‐hyperbolic parameter. Assuming the vertical velocity is known from zero‐offset VSP data, Dix‐type inversion is applied to estimate the layered Thomsen anisotropic parameters ?, δ above the receivers array. Model results show reasonable accuracy for estimates through Dix‐type inversion. Results also show that in many cases we can neglect the influence of the velocity gradient on anisotropy estimates. First breaks are used to estimate anisotropic parameters within the walkaway receiver interval. Analytical uncertainty analysis is performed to NMO parameter estimates. Its conclusions are confirmed by modelling.     

Seismic characterization of vertical fractures described as general linear-slip interfaces

Fluid flow in many hydrocarbon reservoirs is controlled by aligned fractures which make the medium anisotropic on the scale of seismic wavelength. Applying the linear‐slip theory, we investigate seismic signatures of the effective medium produced by a single set of ‘general’ vertical fractures embedded in a purely isotropic host rock. The generality of our fracture model means the allowance for coupling between the normal (to the fracture plane) stress and the tangential jump in displacement (and vice versa). Despite its low (triclinic) symmetry, the medium is described by just nine independent effective parameters and possesses several distinct features which help to identify the physical model and estimate the fracture compliances and background velocities. For example, the polarization vector of the vertically propagating fast shear wave S₁ and the semi‐major axis of the S₁‐wave normal‐moveout (NMO) ellipse from a horizontal reflector always point in the direction of the fracture strike. Moreover, for the S₁‐wave both the vertical velocity and the NMO velocity along the fractures are equal to the shear‐wave velocity in the host rock. Analysis of seismic signatures in the limit of small fracture weaknesses allows us to select the input data needed for unambiguous fracture characterization. The fracture and background parameters can be estimated using the NMO ellipses from horizontal reflectors and vertical velocities of P‐waves and two split S‐waves, combined with a portion of the P‐wave slowness surface reconstructed from multi‐azimuth walkaway vertical seismic profiling (VSP) data. The stability of the parameter‐estimation procedure is verified by performing non‐linear inversion based on the exact equations.     

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.     

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.     

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.     

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.     

Migration velocity analysis for tilted transversely isotropic media

Tilted transversely isotropic formations cause serious imaging distortions in active tectonic areas (e.g., fold‐and‐thrust belts) and in subsalt exploration. Here, we introduce a methodology for P‐wave prestack depth imaging in tilted transversely isotropic media that properly accounts for the tilt of the symmetry axis as well as for spatial velocity variations. For purposes of migration velocity analysis, the model is divided into blocks with constant values of the anisotropy parameters ε and δ and linearly varying symmetry‐direction velocity V_P0 controlled by the vertical (k_z) and lateral (k_x) gradients. Since determination of tilt from P‐wave data is generally unstable, the symmetry axis is kept orthogonal to the reflectors in all trial velocity models. It is also assumed that the velocity V_P0 is either known at the top of each block or remains continuous in the vertical direction. The velocity analysis algorithm estimates the velocity gradients k_z and k_x and the anisotropy parameters ε and δ in the layer‐stripping mode using a generalized version of the method introduced by Sarkar and Tsvankin for factorized transverse isotropy with a vertical symmetry axis. Synthetic tests for several models typical in exploration (a syncline, uptilted shale layers near a salt dome and a bending shale layer) confirm that if the symmetry‐axis direction is fixed and V_P0 is known, the parameters k_z, k_x, ε and δ can be resolved from reflection data. It should be emphasized that estimation of ε in tilted transversely isotropic media requires using nonhyperbolic moveout for long offsets reaching at least twice the reflector depth. We also demonstrate that application of processing algorithms designed for a vertical symmetry axis to data from tilted transversely isotropic media may lead to significant misfocusing of reflectors and errors in parameter estimation, even when the tilt is moderate (30°). The ability of our velocity analysis algorithm to separate the anisotropy parameters from the velocity gradients can be also used in lithology discrimination and geologic interpretation of seismic data in complex areas.     

VTI介质长偏移距非双曲动校正公式优化

常规Alkhalifah动校正公式精度低,不能精确描述各向异性介质长偏移距地震反射同相轴的时距关系.本文以提高VTI介质长偏移距地震资料动校正公式的精度为目标,在分析VTI介质常规动校正方程的基础上,根据误差最小原理建立优化校正系数图版,实现对常规动校正公式大偏移距误差的修正,建立最优化校正Alkhalifah动校正方程,实现了对VTI介质长偏移距地震资料常规动校正方程的改进.之后由Fomel群速度公式导出高精度VTI模型长偏移距时距函数,提出了高精度VTI介质长偏移距地震资料动校正方程.将以上的动校正方程用于各向异性参数反演,模型计算表明最优化校正Alkhalifah动校正方程的反演精度是常规长偏移距动校正方程反演精度的2～4倍,高精度动校正方程的反演精度是常规动校正方程反演精度的2～8倍.     

Synthetic Experiments Dealing with Anisotropic Model Building

It has been shown in the past that the interval-NMO velocity and the non-ellipticity parameter largely control the P-wave reflection time moveout of VTI media. To invert for these two parameters, one needs either reasonably large offsets, or some structure in the subsurface in combination with relatively mild lateral velocity variation.This paper deals with a simulation of an inversion approach, building on the assumption that accurately measured V _NMO, as defined by small offset asymptotics for a particular reflector, were available. Instead of such measurements we take synthetically computed data. First, an isotropic model is constructed which explains these V _NMO. Subsequently, residual moveout in common image gathers is modelled by ray tracing (replacing real data), along with its sensitivity for changes in the interval-NMO velocity and the non-ellipticity parameter under the constraint that V _NMO is preserved. This enables iterative updating of the non-ellipticity parameter and the interval-NMO velocity in a layer that can be laterally inhomogeneous.This approach is successfully applied for a mildly dipping reflector at the bottom of a layer with laterally varying medium parameters. With the exact V _NMO assumed to be given, lateral inhomogeneity and anisotropy can be distinguished for such a situation. However, for another example with a homogeneous VTI layer overlying a curved reflector with dip up to 30°, there appears to be an ambiguity which can be understood by theoretical analysis. Consistently with existing theory using the NMO-ellipse, the presented approach is successfully applied to the latter example if V _NMO in the strike direction is combined with residual moveout in dip direction.     

Anisotropic migration velocity analysis: Application to a data set from West Africa

Although it is widely recognized that anisotropy can have a significant influence on the focusing and positioning of migrated reflection events, conventional depth imaging methods still operate with isotropic velocity fields. Here, we present an application of a 2D migration velocity analysis (MVA) algorithm, designed for factorized v(x, z) VTI (transversely isotropic with a vertical symmetry axis) media, to an offshore data set from West Africa. By approximating the subsurface with factorized VTI blocks, it is possible to decouple the spatial variations in the vertical velocity from the anisotropic parameters with minimal a priori information. Since our method accounts for lateral velocity variation, it produces more accurate estimates of the anisotropic parameters than those previously obtained with time‐domain techniques. The values of the anellipticity parameter η found for the massive shales exceed 0.2, which confirms that ignoring anisotropy in the study area can lead to substantial imaging distortions, such as mis‐stacking and mispositioning of dipping events. While some of these distortions can be removed by using anisotropic time processing, further marked improvement in image quality is achieved by prestack depth migration with the estimated factorized VTI model. In particular, many fault planes, including antithetic faults in the shallow part of the section, are better focused by the anisotropic depth‐migration algorithm and appear more continuous. Anisotropic depth migration facilitates structural interpretation by eliminating false dips at the bottom of the section and improving the images of a number of gently dipping features. One of the main difficulties in anisotropic MVA is the need to use a priori information for constraining the vertical velocity. In this case study, we successfully reconstructed the time–depth curve from reflection data by assuming that the vertical velocity is a continuous function of depth and estimating the vertical and lateral velocity gradients in each factorized block. If the subsurface contains strong boundaries with jumps in velocity, knowledge of the vertical velocity at a single point in a layer is sufficient for our algorithm to determine all relevant layer parameters.     

Traveltime parameters in a tilted elliptical anisotropic medium‡

In this paper, we derive analytical expressions for one‐way and two‐way kinematical parameters in elliptical tilted transverse isotropy media. We show that the homogeneous elliptical tilted transverse isotropy models result in hyperbolic moveout with a reflection point sideslip x₀, which can be considered as an additional traveltime parameter for one‐way wave propagation. For homogeneous elliptical tilted transverse isotropy models we show that the inversion of one‐way traveltime parameters suffers from the ambiguity for large tilts. It is shown that the accuracy of the inversion is sensitive to the error in x₀. We also derive and invert the traveltime parameters for a vertically heterogeneous elliptical tilted transverse isotropy model with a tilt gradually changing with depth. The a priori knowledge of parameter δ is very important for inversion. The wrong choise of this parameter results in significant errors in inverted model parameters.     

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.     

The analysis and application of simplified two-parameter moveout equation for C-waves in VTI anisotropy media

Several parameters are needed to describe the converted-wave （C-wave） moveout in processing multi-component seismic data, because of asymmetric raypaths and anisotropy. As the number of parameters increases, the converted wave data processing and analysis becomes more complex. This paper develops a new moveout equation with two parameters for C-waves in vertical transverse isotropy （VTI） media. The two parameters are the C-wave stacking velocity （Vc2） and the squared velocity ratio （7v,i） between the horizontal P-wave velocity and C-wave stacking velocity. The new equation has fewer parameters, but retains the same applicability as previous ones. The applicability of the new equation and the accuracy of the parameter estimation are checked using model and real data. The form of the new equation is the same as that for layered isotropic media. The new equation can simplify the procedure for C-wave processing and parameter estimation in VTI media, and can be applied to real C-wave processing and interpretation. Accurate Vc2 and Yvti can be deduced from C-wave data alone using the double-scanning method, and the velocity ratio model is suitable for event matching between P- and C-wave data.     

Piecewise‐continuous velocity inversion

We suggest a new method to determine the piecewise‐continuous vertical distribution of instantaneous velocities within sediment layers, using different order time‐domain effective velocities on their top and bottom points. We demonstrate our method using a synthetic model that consists of different compacted sediment layers characterized by monotonously increasing velocity, combined with hard rock layers, such as salt or basalt, characterized by constant fast velocities, and low velocity layers, such as gas pockets. We first show that, by using only the root‐mean‐square velocities and the corresponding vertical travel times (computed from the original instantaneous velocity in depth) as input for a Dix‐type inversion, many different vertical distributions of the instantaneous velocities can be obtained (inverted). Some geological constraints, such as limiting the values of the inverted vertical velocity gradients, should be applied in order to obtain more geologically plausible velocity profiles. In order to limit the non‐uniqueness of the inverted velocities, additional information should be added. We have derived three different inversion solutions that yield the correct instantaneous velocity, avoiding any a priori geological constraints. The additional data at the interface points contain either the average velocities (or depths) or the fourth‐order average velocities, or both. Practically, average velocities can be obtained from nearby wells, whereas the fourth‐order average velocity can be estimated from the quartic moveout term during velocity analysis. Along with the three different types of input, we consider two types of vertical velocity models within each interval: distribution with a constant velocity gradient and an exponential asymptotically bounded velocity model, which is in particular important for modelling thick layers. It has been shown that, in the case of thin intervals, both models lead to similar results. The method allows us to establish the instantaneous velocities at the top and bottom interfaces, where the velocity profile inside the intervals is given by either the linear or the exponential asymptotically bounded velocity models. Since the velocity parameters of each interval are independently inverted, discontinuities of the instantaneous velocity at the interfaces occur naturally. The improved accuracy of the inverted instantaneous velocities is particularly important for accurate time‐to‐depth conversion.     

Generalized Dix equation and analytic treatment of normal-moveout velocity for anisotropic media*

Despite the complexity of wave propagation in anisotropic media, reflection moveout on conventional common-midpoint (CMP) spreads is usually well described by the normal-moveout (NMO) velocity defined in the zero-offset limit. In their recent work, Grechka and Tsvankin showed that the azimuthal variation of NMO velocity around a fixed CMP location generally has an elliptical form (i.e. plotting the NMO velocity in each azimuthal direction produces an ellipse) and is determined by the spatial derivatives of the slowness vector evaluated at the CMP location. This formalism is used here to develop exact solutions for the NMO velocity in anisotropic media of arbitrary symmetry. For the model of a single homogeneous layer above a dipping reflector, we obtain an explicit NMO expression valid for all pure modes and any orientation of the CMP line with respect to the reflector strike. The contribution of anisotropy to NMO velocity is contained in the slowness components of the zero-offset ray (along with the derivatives of the vertical slowness with respect to the horizontal slownesses) — quantities that can be found in a straightforward way from the Christoffel equation. If the medium above a dipping reflector is horizontally stratified, the effective NMO velocity is determined through a Dix-type average of the matrices responsible for the ‘interval’ NMO ellipses in the individual layers. This generalized Dix equation provides an analytic basis for moveout inversion in vertically inhomogeneous, arbitrarily anisotropic media. For models with a throughgoing vertical symmetry plane (i.e. if the dip plane of the reflector coincides with a symmetry plane of the overburden), the semi-axes of the NMO ellipse are found by the more conventional rms averaging of the interval NMO velocities in the dip and strike directions. Modelling of normal moveout in general heterogeneous anisotropic media requires dynamic ray tracing of only one (zero-offset) ray. Remarkably, the expressions for geometrical spreading along the zero-offset ray contain all the components necessary to build the NMO ellipse. This method is orders of magnitude faster than multi-azimuth, multi-offset ray tracing and, therefore, can be used efficiently in traveltime inversion and in devising fast dip-moveout (DMO) processing algorithms for anisotropic media. This technique becomes especially efficient if the model consists of homogeneous layers or blocks separated by smooth interfaces. The high accuracy of our NMO expressions is illustrated by comparison with ray-traced reflection traveltimes in piecewise-homogeneous, azimuthally anisotropic models. We also apply the generalized Dix equation to field data collected over a fractured reservoir and show that P-wave moveout can be used to find the depth-dependent fracture orientation and to evaluate the magnitude of azimuthal anisotropy.     

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.     

Converted-wave imaging in anisotropic media: theory and case studies

Common‐conversion‐point binning associated with converted‐wave (C‐wave) processing complicates the task of parameter estimation, especially in anisotropic media. To overcome this problem, we derive new expressions for converted‐wave prestack time migration (PSTM) in anisotropic media and illustrate their applications using both 2D and 3D data examples. The converted‐wave kinematic response in inhomogeneous media with vertical transverse isotropy is separated into two parts: the response in horizontally layered vertical transverse isotrophy media and the response from a point‐scatterer. The former controls the stacking process and the latter controls the process of PSTM. The C‐wave traveltime in horizontally layered vertical transverse isotrophy media is determined by four parameters: the C‐wave stacking velocity V_C2, the vertical and effective velocity ratios γ₀ and γ_eff, and the C‐wave anisotropic parameter χ_eff. These four parameters are referred to as the C‐wave stacking velocity model. In contrast, the C‐wave diffraction time from a point‐scatterer is determined by five parameters: γ₀, V_P2, V_S2, η_eff and ζ_eff, where η_eff and ζ_eff are, respectively, the P‐ and S‐wave anisotropic parameters, and V_P2 and V_S2 are the corresponding stacking velocities. V_P2, V_S2, η_eff and ζ_eff are referred to as the C‐wave PSTM velocity model. There is a one‐to‐one analytical link between the stacking velocity model and the PSTM velocity model. There is also a simple analytical link between the C‐wave stacking velocities V_C2 and the migration velocity V_Cmig, which is in turn linked to V_P2 and V_S2. Based on the above, we have developed an interactive processing scheme to build the stacking and PSTM velocity models and to perform 2D and 3D C‐wave anisotropic PSTM. Real data applications show that the PSTM scheme substantially improves the quality of C‐wave imaging compared with the dip‐moveout scheme, and these improvements have been confirmed by drilling.     

Research Note: Insights into the data dependency on anisotropy: an inversion prospective

While velocity contrasts are responsible for most of the events recorded in our data, the long wavelength behavior of the velocity model is responsible for the geometrical shape of these events. For isotropic acoustic materials, the wave dependency on the long (wave propagation) and short (scattering) wavelength velocity components is stationary with the propagation angle. On the other hand, in representing a transversely isotropic with a vertical symmetry axis medium with the normal moveout velocity, the anellepticity parameter η, the vertical scaling parameter δ, and the sensitivity of waves vary with the polar angle for both the long and short wavelength features of the anisotropic dimensionless medium parameters (δ and η). For horizontal reflectors at reasonable depths, the long wavelength features of the η model is reasonably constrained by the long offsets, whereas the short wavelength features produce very week reflections at even reasonable offsets. Thus, for surface acquired seismic data, we could mainly invert for smooth η responsible for the geometrical shape of reflections. On the other hand, while the δ long wavelength components mildly affects the recorded data, its short wavelength variations can produce reflections at even zero offset, with a behavior pattern synonymous to density. The lack of the long wavelength δ information will mildly effect focusing but will cause misplacement of events in depth. With low enough frequencies (very low), we may be able to recover the long wavelength δ using full waveform inversion. However, unlike velocity, the frequencies needed for that should be ultra‐low to produce long‐wavelength scattering‐based model information as δ perturbations do not exert scattering at large offsets. For a combination given by the horizontal velocity, η, and ε, the diving wave influence of η is absorbed by the horizontal velocity, severely limiting the η influence on the data and full waveform inversion. As a result, with a good smooth η estimation, for example, from tomography, we can focus the full waveform inversion to invert for only the horizontal velocity and maybe ε as a parameter to fit the amplitude. This is possibly the most practical parametrization for inversion of surface seismic data in transversely isotropic with vertical symmetry axis media.