Simulation of improved pure P-wave equation in transversely isotropic media with a horizontal symmetry axis

Characterizing the expressions of seismic waves in elastic anisotropic media depends on multiparameters. To reduce the complexity, decomposing the P-mode wave from elastic seismic data is an effective way to describe the considerably accurate kinematics with fewer parameters. The acoustic approximation for transversely isotropic media is widely used to obtain P-mode wave by setting the axial S-wave phase velocity to zero. However, the separated pure P-wave of this approach is coupled with undesired S-wave in anisotropic media called S-wave artefacts. To eliminate the S-wave artefacts in acoustic waves for anisotropic media, we set the vertical S-wave phase velocity as a function related to propagation directions. Then, we derive a pure P-wave equation in transversely isotropic media with a horizontal symmetry axis by introducing the expression of vertical S-wave phase velocity. The differential form of new expression for pure P-wave is reduced to second-order by inserting the expression of S-wave phase velocity as an auxiliary operator. The results of numerical simulation examples by finite difference illustrate the stability and accuracy of the derived pure P-wave equation.     

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.     

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.     

Kinematic parameters of pure‐ and converted‐mode waves in elastic orthorhombic media

I derive the kinematic properties of single‐mode P, S1, and S2 waves as well as converted PS1, PS2, and S1S2 waves in elastic orthorhombic media including vertical velocity, two normal moveout velocities defined in vertical symmetry planes, and three anelliptic parameters (two of them are defined in vertical symmetry plane and one parameter is the cross‐term one). I show that the azimuthal dependence of normal moveout velocity and anellipticity is different in phase and group domains. The effects on‐vertical‐axis singularity and on‐vertical‐axis triplication are considered for pure‐mode S1 and S2 waves and converted‐mode S1S2 waves. The conditions and properties of on‐vertical‐axis triplication are defined in terms of kinematic parameters. The results are illustrated in four homogeneous orthorhombic models and one multilayered orthorhombic model with no variation in azimuthal orientation for all the layers.     

利用随钻正交偶极子声波测井评价地层各向异性的数值研究

在随钻测井条件下,由于钻铤占据了井孔内的大部分空间,充液井孔中沿着井轴方向传播的模式波的特性与电缆测井非常不同.本文建立了随钻正交偶极子测井声学模型,采用三维有限差分方法模拟了偶极子声源在随钻条件下各向异性地层井孔内激发的声场,研究了地层的声学各向异性在随钻正交偶极子声波测井中的响应特征.数值模拟结果表明,在随钻测井条件下,对于井轴同TI地层对称轴垂直的情况,弯曲波分裂现象仍然存在,通过正交偶极子测量方式和合适的反演算法能够准确有效地确定地层的快横波方位角,可以考虑采用同正演理论相结合的反演算法来获得地层的快、慢横波速度及声学各向异性信息;对于井轴同介质对称轴呈一定夹角的TI地层井孔,情况变得非常复杂,不同井斜倾角下弯曲波的速度的变化趋势并非同对应的地层横波速度的变化趋势完全一致,不过在一定的频段内,地层横波速度仍然是弯曲波的最主要控制因素.对于本文研究的模型,当井轴同介质对称轴的夹角大于大于60°时,此时获得的弯曲波的各向异性值基本能够反映对应角度下地层横波速度的各向异性信息.     

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.     

VTI介质中地震波反射波合成记录的方法研究

在向向异性介质中，由于地震波的相速度和群速度有较大差异，相角和群角不同，相速度、群速度及群角相与角之间的关系比较复杂，因此与各同性相比较，计算地震波相速度，群速度及群角就更加困难，其合成地震记录的难度也就之增大，本文根据VTI（具有垂直对称轴的横向各向同性）介质中地震波的运动学特征，应用射线追踪方法，计算了VTI介质中的地震波的旅行时，并合成了反 地震记录，这为反射地震波的旅行时及速度分析提供了基础。     

S‐wave kinematics in acoustic transversely isotropic media with a vertical symmetry axis

Acoustic transversely isotropic models are widely used in seismic exploration for P‐wave processing and analysis. In isotropic acoustic media only P‐wave can propagate, while in an acoustic transversely isotropic medium both P and S waves propagate. In this paper, we focus on kinematic properties of S‐wave in acoustic transversely isotropic media. We define new parameters better suited for S‐wave kinematics analysis. We also establish the travel time and relative geometrical spreading equations and analyse their properties. To illustrate the behaviour of the S‐wave in multi‐layered acoustic transversely isotropic media, we define the Dix‐type equations that are different from the ones widely used for the P‐wave propagation.     

Analysis of different parameterisations of waveform inversion of compressional body waves in an elastic transverse isotropic Earth with a vertical axis of symmetry

In a multi‐parameter waveform inversion, the choice of the parameterisation influences the results and their interpretations because leakages and the tradeoff between parameters can cause artefacts. We review the parameterisation selection when the inversion focuses on the recovery of the intermediate‐to‐long wavenumbers of the compressional velocities from the compressional body (P) waves. Assuming a transverse isotropic medium with a vertical axis of symmetry and weak anisotropy, analytical formulas for the radiation patterns are developed to quantify the tradeoff between the shear velocity and the anisotropic parameters and the effects of setting to zero the shear velocity in the acoustic approach. Because, in an anisotropic medium, the radiation patterns depend on the angle of the incident wave with respect to the vertical axis, two particular patterns are discussed: a transmission pattern when the ingoing and outgoing slowness vectors are parallel and a reflection pattern when the ingoing and outgoing slowness vectors satisfy Snell's law. When the inversion aims at recovering the long‐to‐intermediate wavenumbers of the compressional velocities from the P‐waves, we propose to base the parameterisation choice on the transmission patterns. Since the P‐wave events in surface seismic data do not constrain the background (smooth) vertical velocity due to the velocity/depth ambiguity, the preferred parameterisation contains a parameter that has a transmission pattern concentrated along the vertical axis. This parameter can be fixed during the inversion which reduces the size of the model space. The review of several parameterisations shows that the vertical velocity, the Thomsen parameter δ, or the Thomsen parameter ε have a transmission pattern along the vertical axis depending on the parameterisation choice. The review of the reflection patterns of those selected parameterisations should be done in the elastic context. Indeed, when reflection data are also inverted, there are potential leakages of the shear parameter at intermediate angles when we carry out acoustic inversion.     

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.     

裂缝诱导双相HTI介质地震波场错格伪谱法模拟与波场特征分析

裂缝诱导的双相具有水平对称轴的横向各向同性(HTI)介质模型是由一组平行排列的垂直裂缝嵌入到统计各向同性的流体饱和多孔隙岩石中而组成的,它综合考虑了裂缝型储层岩石的各向异性和孔隙性.高精度的地震波场数值模拟技术是研究该介质中地震波传播规律的主要方法.本文结合错格伪谱法和时间分裂法,求解描述该介质中地震波传播的一阶速度-应力方程.模拟了单层和双层模型中的地震波场,并对其进行了特征分析.研究结果表明:错格伪谱法能有效消除标准网格伪谱法波场模拟结果中出现的数值伪影现象,与时间分裂法结合能够获得稳定的、高精度的模拟结果;裂缝诱导双相HTI介质中的地震波场兼具裂缝各向异性介质和双相介质中传播的地震波的波场特征.     

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.     

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.     

Algorithms for staggered-grid computations for poroelastic,elastic, acoustic,and scalar wave equations

Heterogeneous wave equations are more complicated numerically than homogeneous wave equations, but are necessary for physical validity. A wide variety of numerical solutions of seismic wave equations is available, but most produce strong numerical artefacts and local instabilities where model parameters change rapidly. Accuracy and stability of heterogeneous equations is achieved through staggered-grid formulations. A new pseudospectral staggered-grid algorithm is developed for the poroelastic (Biot) equations. The algorithm may be reduced to handle the elastic and acoustic limits of the Biot equations. Comparisons of results from poroelastic, elastic, acoustic and scalar computations for a 2D model show that porous medium parameters may affect amplitudes significantly. The use of homogeneous wave equations for modelling of a heterogeneous medium, or of a centred rather than a staggered grid, or of simplified (e.g. acoustic) wave equations when elastic or poroelastic media are synthesized, may produce erroneous or ambiguous interpretations.     

Reflected waves in finely layered tilted orthorhombic media

Upscaling in seismics is a homogenization of finely layered media in the zero-frequency limit. An upscaling technique for arbitrary anisotropic layers has been developed by Schoenberg and Muir. Applying this technique to a stack of layers of orthorhombic (ORT) symmetry whose vertical symmetry planes are aligned, results in an effective homogeneous layer with orthorhombic symmetry. If the symmetry planes in a horizontal orthorhombic layer are rotated with respect to vertical, the medium is referred to as tilted orthorhombic (TOR) medium, and the stack composed of TOR layers in zero-frequency limit will produce an effective medium of a lower symmetry than orthorhombic. We consider a P-wave that propagates through a stack of thin TOR layers, then it is reflected (preserving the mode) at some interface below the stack, and then propagates back through the same stack. We propose to use a special modified medium for the upscaling in case of this sequential down- and up-propagation: each TOR layer in the stack is replaced by two identical TOR layers whose tilt angles have the opposite algebraic sign. In this modified medium, one-way propagation of a seismic wave (any wave mode) is equivalent to propagation of a pure-mode reflection in the original medium. We apply this idea to study the contribution from an individual layer from the stack and show how the approach can be applied to a stack of TOR layers. To demonstrate the applicability of the model, we use well log data for the upscaling. The model we propose for the upscaling can be used in well-seismic ties to correct the effective parameters obtained from well log data for the presence of tilt, if latter is confirmed by additional measurements (for example, borehole imaging).     

THE INSENSITIVITY OF REFLECTED SH WAVES TO ANISOTROPY IN AN UNDERLYING LAYERED MEDIUM1

Propagation in the plane of mirror symmetry of a monoclinic medium, with displacement normal to the plane, is the most general circumstance in anisotropic media for which pure shear-wave propagation can occur at all angles. Because the pure shear mode is uncoupled from the other two modes, its slowness surface in the plane is an ellipse. When the mirror symmetry plane is vertical the pure shear waves in this plane are SH waves and the elliptical SH sheet of the slowness surface is, in general, tilted with respect to the vertical axis. Consider a half-space of such a monoclinic medium, called medium M, overlain by a half-space of isotropic medium I with plane SH waves incident on medium M propagating in the vertical symmetry plane of M. Contrary to the appearance of a lack of symmetry about the vertical axis due to the tilt of the SH-wave slowness ellipse, the reflection and transmission coefficients are symmetrical functions of the angle of incidence, and further, there exists an isotropic medium E with uniquely determined density and shear speed which gives exactly the same reflection and transmission coefficients underlying medium J as does monoclinic medium M. This means that the underlying monoclinic medium M can be replaced by isotropic medium E without changing the reflection and transmission coefficients for all values of the angle of incidence. Thus no set of SH seismic experiments performed in the isotropic medium in the symmetry plane of the underlying half-space can reveal anything about the monoclinic anisotropy of that underlying half-space. Moreover, even when the underlying monoclinic half-space is stratified, there exists a stratified isotropic half-space that gives the identical reflection coefficient as the stratified monoclinic half-space for all angles of incidence and all frequencies.     

On shear‐wave triplications in a multilayered transeversely isotropic medium with vertical symmetry axis

The presence of triplications (caustics) can be a serious problem in seismic data processing and analysis. The traveltime curve becomes multi‐valued and the geometrical spreading correction factor tends to zero due to energy focusing. We analyse the conditions for the qSV‐wave triplications in a homogeneous transversely isotropic medium with vertical symmetry axis. The proposed technique can easily be extended to the case of horizontally layered vertical symmetry axis medium. We show that the triplications of the qSV‐wave in a multilayered medium imply certain algebra. We illustrate this algebra on a two‐layer vertical symmetry axis model.     

Numerical analysis of one-dimensional nonlinear acoustic wave

Numerical investigations on one-dimensional nonlinear acoustic wave with third and fourth order nonlinearities are presented using high-order finite-difference (HFD) operators with a simple flux-limiter (SFL) algorithm. As shown by our numerical tests, the HFDSFL method is able to produce more stable, accurate and conservative solutions to the nonlinear acoustic waves than those computed by finite-difference combined with the flux-corrected-transport algorithm. Unlike the linear acoustic waves, the nonlinear acoustic waves have variable phase velocity and waveform both in time-space (t-x) domain and frequency-wavenumber (f-k) domain; of our special interest is the behaviour during the propagation of nonlinear acoustic waves: the waveforms are strongly linked to the type of medium nonlinearities, generation of harmonics, frequency and wavenumber peak shifts. In seismic sense, these characteristics of nonlinear wave will introduce new issues during such seismic processing as Normal Moveout and f-k filter. Moreover, as shown by our numerical experiment for a four-layer model, the nonlinearities of media will introduce extra velocity errors in seismic velocity inversion.     

Effective ellipsoidal models for wavefield extrapolation in tilted orthorhombic media

Wavefield computations using the ellipsoidally anisotropic extrapolation operator offer significant cost reduction compared to that for the orthorhombic case, especially when the symmetry planes are tilted and/or rotated. However, ellipsoidal anisotropy does not provide accurate wavefield representation or imaging for media of orthorhombic symmetry. Therefore, we propose the use of ‘effective ellipsoidally anisotropic’ models that correctly capture the kinematic behaviour of wavefields for tilted orthorhombic (TOR) media. We compute effective velocities for the ellipsoidally anisotropic medium using kinematic high-frequency representation of the TOR wavefield, obtained by solving the TOR eikonal equation. The effective model allows us to use the cheaper ellipsoidally anisotropic wave extrapolation operators. Although the effective models are obtained by kinematic matching using high-frequency asymptotic, the resulting wavefield contains most of the critical wavefield components, including frequency dependency and caustics, if present, with reasonable accuracy. The proposed methodology offers a much better cost versus accuracy trade-off for wavefield computations in TOR media, particularly for media of low to moderate anisotropic strength. Furthermore, the computed wavefield solution is free from shear-wave artefacts as opposed to the conventional finite-difference based TOR wave extrapolation scheme. We demonstrate applicability and usefulness of our formulation through numerical tests on synthetic TOR models.