A pure quasi-P-wave equation in acoustic transversely isotropic media with the weak-anisotropy decomposition

Transversely isotropic media with a vertical symmetry axis (VTI) is the model of the subsurface suitable for processing seismic reflection data surface of sedimentary basins formed by shale. The propagation of P waves in VTI media is characterized by four independent elastic parameters and complex algebraic equations for the phase and group velocities. Therefore, there is a need to obtain approximations accurate to the phase and group velocities in VTI homogeneous media. Several authors have described approaches to the phase and group velocities with only three parameters, in homogeneous horizontally layered VTI media through hypotheses as weak anisotropy of the medium and anellipticity wavefront. In this work, we have used rational approximants in shifted-hiperbola approximation and obtained rational anelliptitic approximations of the phase and group velocities in homogeneous horizontally stratified VTI media. We have verified the accuracy of the approximations, compared with other approximations in the literature. As an application, we converted the group velocity approximations in nonhyperbolic moveout approximations and performed parameter estimation by means of semblancebased velocity analysis. The results show the validity of anelliptic rational approximations in inverse processes. Introduction Due to the limitations of isotropic models in more complex lithologies, such as sedimentary basins formed by shales, the seismic reflection survey is considered as a model of subsurface anisotropic media, especially the VTI media. In homogeneous media TIV, the wavefront of the SH phase velocity is elliptical and has exact equation that depends only two elastic parameters. However, the Pand SV-waves have: strongly anelliptical wavefront for both phase velocity and group velocity; algebraically complicated exact equations for the phase velocity; and are characterized by five independent elastic moduli tensors ( ). Moreover, even in TIV media, it is difficult to explain exact equations for the group velocity. Another remarkable feature of anisotropic media is the nonhyperbolic behavior of moveout curve. Thus, it is necessary to obtain approximations for the phase and group velocities and thus for moveout curves, which have precision and are practical to perform the steps of seismic data processing. Thomsen (1986), using the physical characteristics of the elastic parameters and the properties in the vertical direction, introduced a parameter of elastic moduli which facilitates the study of the effects of wave propagation in homogeneous anisotropic media VTI. Alkhalifah and Tsvankin (1995) found that only three of these parameters influence the propagation of P-waves in TIV media. Authors such as Muir and Dellinger (1985), Thomsen (1986), Dellinger (1993), Alkhalifah and Tsvankin (1995), Alkhalifah (1998), Fomel (2004), Psensic (2013), among others, have shown approaches the phase velocity in homogeneous VTI media, which depend explicitly only three elastic parameters. Dellinger, and Muir (1985) and Dellinger (1993) showed the anelliptical approximation of the phase velocities, using the properties of the elliptical anisotropy. Assuming analogy in the form of approximations obtained elliptical approximation of group velocity and consequently the moveout approximations for TI media. Fowler (2003) defined the elliptical component of the phase velocity, and through convenient parameterization of elastic parameters obtained anelliptical approximation for phase velocity in VTI media, equivalent to those obtained by other authors. However, using heuristics pure, converted them to approaches: by dispersion relations, group velocities and time equations. Fomel (2004) inspired by the anelliptical approximation (Dellinger et al,1993) used the shifted-hyperbola approximation (Malovichko 1978; Sword 1987; de Bazelaire 1988; Castle 1994) and Stoltstretch correction (Stolt 1978; Fomel and Vaillant 2001 ) to obtain, separately, the acoustic phase velocity approximation (Alkhalifah, 1998) of the P-wave in VTI media. However, by analogy form, obtained group velocity approximation and non-hyperbolic traveltime, very accurate. In this work, we obtained anelliptical approximation for the phase velocity of the compressional wave in vertical media such as rational approximating the shiftedhyperbola approximation (Fomel, 2004). Using the conversion technique by similarity of form (Dellinger, 1993), we obtained anelliptical approximations to group velocity towards these; we obtain new nonhyperbolic moveout approximations. To prove the accuracy of such approximations, we calculated the relative errors of these compared to other approximations. We also conducted semblance-based velocity analysis, to show the robustness of rational approximations of traveltime in estimating parameters. Phase velocity in VTI media The wave propagation in VTI media is characterized by the independent elastic parameters, density-normalized:

An approximation is of practical interest whenever an exact approach is not available or is too complicated to be used. Kinematic properties of wave propagation in orthorhombic media are generally more complicated than in transversely isotropic media — an issue that emphasizes the necessity of proper approximate equations that keep a balance between accuracy and simplicity. Exact phase velocity equation in orthorhombic media is algebraically too complicated for some practical purposes, even after acoustic assumption. Although the exact phase velocity equation is readily calculated, there is not an explicit equation for the exact group velocity as a function of group angle nor for the traveltime as a function of offset. Accordingly, we have developed new approximate phase velocity, group velocity, and moveout equations for acoustic orthorhombic media in a simple and uniform functional form. They fit to their corresponding exact kinematic properties, within and outside the orthorhombic symmetry planes. We find a higher accuracy of our approximations compared with other existing approximations in a variety of orthorhombic models. As an example, we convert our phase velocity approximation to a dispersion relation in the frequency domain and use it for wavefield modeling in a heterogeneous orthorhombic model. Our dispersion relation is simpler and more accurate than the original equation being in use in the wave extrapolation modeling by low-rank approximation.

We present approximate formulas for the qP-wave phase velocity, polarization vector, and normal moveout velocity in an arbitrary weakly anisotropic medium obtained with first-order perturbation theory. All these quantities are expressed in terms of weak anisotropy (WA) parameters, which represent a natural generalization of parameters introduced by Thomsen. The formulas presented and the WA parameters have properties of Thomsen's formulas and parameters: (1) the approximate equations are considerably simpler than exact equations for qP waves, (2) the WA parameters are nondimensional quantities, and (3) in isotropic media, the WA parameters are zero and the corresponding equations reduce to equations for isotropic media. In contrast to Thomsen's parameters, the WA parameters are related linearly to the density normalized elastic parameters. For the transversely isotropic media with vertical axis of symmetry, the equations presented and the WA parameters reduce to the equations and linearized parameters of Thomsen. The accuracy of the formulas presented is tested on two examples of anisotropic media with relatively strong anisotropy: on a transversely isotropic medium with the horizontal axis of symmetry and on a medium with triclinic anisotropy. Although anisotropy is rather strong, the approximate formulas presented yield satisfactory results.

Anelliptic approximations for phase and group velocities of qP waves in transversely isotropic (TI) media have been widely applied in various seismic data processing and imaging tasks. We have revisited previously proposed approximations and suggested two improvements. The first improvement involves finding an empirical connection between anelliptic parameters along different fitting axes based on laboratory measurements of anisotropy of rock samples of different types. The relationship between anelliptic parameters observed was strongly linear suggesting a novel set of anisotropic parameters suitable for the study of qP-wave signatures. The second improvement involves suggesting a new functional form for the anelliptic parameter term to achieve better fitting along the horizontal axis. These modifications led to improved three-parameter and four-parameter approximations for phase and group velocities of qP-waves in TI media. In several model comparisons, the new three-parameter approximations appeared to be more accurate than previous approximations with the same number of parameters. These modifications also served as a foundation for an extension to orthorhombic media, where qP velocities involved nine independent elastic parameters. As determined by previous researchers, qP-wave propagation in orthorhombic media could be adequately approximated using just six combinations of those nine parameters. We have developed novel six-parameter approximations for phase and group velocities for qP-waves in orthorhombic media. The orthorhombic phase-velocity approximation provides a more accurate alternative to previously known approximations and can find applications in full-wave modeling, imaging, and inversion. The group-velocity approximation is also highly accurate and can find applications in ray tracing and velocity analysis.

ABSTRACTSeismic anisotropy in geological media is now widely accepted. Parametrizations and explicit approximations for the velocities in such media, considered as purely elastic and moderately anisotropic, are now standards and have even been extended to arbitrary types of anisotropy. In the case of attenuating media, some authors have also recently published different parametrizations and velocity and attenuation approximations in viscoelastic anisotropic media of particular symmetry type (e.g., transversely isotropic or orthorhombic). This paper extends such work to media of arbitrary anisotropy type, that is to say to triclinic media. In the case of homogeneous waves and using the so‐called ‘correspondence principle’, it is shown that the viscoelastic equations (for the phase velocities, phase slownesses, moduli, wavenumbers, etc.) are formally identical to the corresponding purely elastic equations available in the literature provided that all the corresponding quantities are complex (except the unit vector in the propagation direction that remains real). In contrast to previous work, the new parametrization uses complex anisotropy parameters and constitutes a simple extension to viscoelastic media of previous work dealing with non‐attenuating elastic media of arbitrary anisotropy type. We make the link between these new complex anisotropy parameters and measurable parameters, as well as with previously published anisotropy parameters, demonstrating the usefulness of the new parametrization. We compute the explicit complete directional dependence of the exact and of the approximate (first and higher‐order perturbation) complex phase velocities of the three body waves (qP, qS1 and qS2). The exact equations are successfully compared with the ultrasonic phase velocities and phase attenuations of the three body waves measured in a strongly attenuating water‐saturated sample of Vosges sandstone exhibiting moderate velocity anisotropy but very strong attenuation anisotropy. The approximate formulas are checked on experimental data. Compared to the exact solutions, the errors observed on the first‐order approximate velocities are small (<1%) for qP‐waves and moderate (<10%) for qS‐waves. The corresponding errors on the quality factorQare moderate (<6%) for qP‐waves but critically large (up to 160%) for the qS‐waves. The use of higher‐order approximations substantially improves the accuracy, for instance typical maximum relative errors do not exceed 0.06% on all the velocities and 0.6% on all the quality factorsQ, for third‐order approximations. All the results obtained on other rock samples confirm the results obtained on this rock. The simplicity of the derivations and the generality of the results are striking and particularly convenient for practical applications.

Effects of anisotropy on time-depth relation in transversely isotropic medium with a vertical axis of symmetry

Prestack Kirchhoff time migration for transversely isotropic media with a vertical symmetry axis (VTI media) is implemented using an offset‐midpoint traveltime equation, Cheop’s pyramid equivalent equation for VTI media. The derivation of such an equation for VTI media requires approximations that pertain to high frequency and weak anisotropy. Yet the resultant offset‐midpoint traveltime equation for VTI media is highly accurate for even strong anisotropy. It is also strictly dependent on two parameters: NMO velocity and the anisotropy parameter, η. It reduces to the exact offset‐midpoint traveltime equation for isotropic media when η = 0. In vertically inhomogeneous media, the NMO velocity and η parameters in the offset‐midpoint traveltime equation are replaced by their effective values: the velocity is replaced by the rms velocity and η is given by a more complicated equation that includes summation of the fourth power of velocity.

Sensitivity analysis of an inverse procedure for determination of elastic coefficients for strong anisotropy

Most techniques used to estimate anisotropy from multiple‐source offset VSP data assume angles measured from particle motion as an incidence angle. However, the difference between P‐wave polarization and the propagation direction for an anisotropic medium can be higher than 8 degrees. This difference provides a nonnegligible error in the estimation of anisotropy parameters from phase velocities. An exact model, proposed to describe P‐ and SV‐phase velocity variations for a transversely isotropic medium (TIM), takes into account the polarization angles. This model is a function of two anisotropy parameters (η and τ), of the vertical P‐ and SV‐wave phase velocities and of the polarization angle γ. However, η and τ can be used to express the polarization angle equation in a much simpler way. To quantify the error in estimated anisotropy parameters due to the assumption that the polarization angle is equal to the incidence angle, I study five TIMs. Each medium has an anisotropy that is representative of those observed in seismic surveying. The anisotropy parameters are recovered by inverting the P‐ and SV‐wave phase velocities for different incidence angles, and these incidence angles are assumed to be equal to the corresponding polarization angles. The mean error in estimated parameters is about 10 percent. This error is about the same as the one that would be obtained for velocities with uncertainties in their measurements. Unfortunately, the inversion of phase velocities measured from a real multiple‐source offset VSP to estimate anisotropy parameters needs, for calculating the misfit function, to add both errors in velocities due to hypothesis for angles and errors in velocity measurements due to uncertainties in data. In this case an exact model eliminates errors due to the assumption for the model and provides a more accurate estimation of anisotropy parameters.

With the incident P-wave, we derive approximate formulas for amplitudes and polarizations of waves reflected from and transmitted through a planar, horizontal boundary between an overlying isotropic medium and an underlying tilted transversely isotropic (TTI) medium assuming that the directions of the phase and group velocities are consistent. Provided that the velocities in the isotropic medium are equal to the velocities along the symmetry axis direction, we derive the relational expression between the propagation angle in the TTI medium and the propagation angle in the hypothetical isotropic medium, under the condition that the horizontal slowness is the same, and then we update the approximate formula of the polarization in the TTI medium. Provided that the slow and fast transverse waves (qS and SH) are generated simultaneously in the anisotropic interface, we linearize for a six-order Zoeppritz equation, derive the azimuthal formula of longitudinal and S-waves, and determine their detailed expressions within the symmetry axis plane. According to the derived azimuthal AVO formula, we establish medium models, compare the derived AVO with the precision, and obtain the following conclusions: (1) The dip angle for the symmetry axis with respect to the vertical may have a sufficiently large impact on AVO, and the vertical longitudinal wave can generate an S-wave. (2) For the derived AVO formula, within the symmetry axis plane, the fitting effect of the approximate and exact formulas is good; however, within the other incident planes, taking the azimuth angle 45° as an example, the approximation is suitable for the large impedance contrast if the anisotropic parameters are set properly. (3) The error between the approximation and precision is mainly caused by the difference between the reflected and transmitted angles, the velocities’ derivation with respect to azimuth, and the division of approximation into isotropic and anisotropic parts.

To compute the phase velocities in the weakly anisotropic media, we propose to transform the Christoffel matrix K into an adapted coordinate system, and, then, apply the perturbation theory to the resulting matrix X. For a weakly anisotropic medium, the off-diagonal elements of the matrix X are small compared to the diagonal ones, and two of them are equal to 0. The diagonal elements of the matrix X are initial approximations of the phase velocities squared. To refine them, it is proposed to use either iterative schemes or Taylor series expansions. The initial terms of the series and the formulas of iterative schemes are expressed through the elements of the matrix X and have a compact analytical representation. The odd-order terms in the series are equal to 0. To approximate the phase velocities of the S1 and S2 waves, a stable method is proposed based on solving a quadratic equation with the coefficients being expressed in terms of the matrix elements and the precomputed value of the qP wave phase velocity squared. For all iterative schemes and series, the convergence conditions are derived. The polarization vector of the wave with the square of the phase velocity is defined as the column with maximum modulus of cofactor of the matrix K-I. The group velocities vectors are computed based on the known components of the polarization vector, the directional vector, and the density-normalized stiffness coefficients. The computational accuracy is demonstrated for the standard orthorhombic model. It is shown how the perturbation theory can be applied to media with strong anisotropy. To do this, first we need to apply several QR transforms or Jacobi rotations of the Christoffel matrix, and then use the perturbation theory. This method with four Jacobi rotations is applied to the calculation of the phase velocities squared for a triclinic medium with a maximum number (32) of singularity points. In this case, the phase velocities are computed with a relative error less than 0,004 %.

To improve the computational efficiency of reverse-time migration (RTM) for vertically transverse isotropic (VTI) media, various acoustic approximations of the elastic wave equations have been presented. Among these, the pseudo-acoustic wave equation, which combines differential and scalar operators, has the advantage that it does not produce shear wave artefacts. In this study, we investigate the feasibility of pseudo-acoustic anisotropic RTM (PA-RTM) for the analysis of synthetic and observed ocean-bottom cable (OBC) datasets of the Volve oil field in the North Sea. To analyse the influence of anisotropic parameters on RTM images and the sensitivity of data components to seismic anisotropy, we perform PA-RTM using synthetic data by incorporating various background models, as well as pressure wavefields and vertical particle acceleration. The synthetic experiments demonstrate that the anisotropic parameter ε plays an important role, whereas δ is negligible in PA-RTM for VTI media, and that vertical particle acceleration is less affected by seismic anisotropy than the pressure wavefields. Our experiments using observed data show that reflectors are better imaged by PA-RTM than by isotropic-RTM, particularly near the reservoir below the strongly anisotropic region, which is supported by angle-domain common image gathers. These results indicate that PA-RTM using only the vertical component is a suitable option for VTI media when insufficient seismic anisotropy data are available.

An accurate pure qP-wave equation in transverse isotropic (TI) media and its efficient and stable implementation are valuable for seismic imaging and inversion. Owing to the complexity of the qP-wave phase velocity expression in anisotropic media, it is difficult to construct such a pure qP-wave equation. In this paper, we combine the Taylor expansion and scalar operator methods to formulate an efficient and stable pure qP-wave equation in TI media. First, the Taylor expansion method is used to convert the square-root term into a fractional term in the qP-wave phase velocity expression. We further improve the approximation accuracy of the resulting equation by a correction technique. Then, the scalar operator is applied to scalarize the equivalent form of the fractional term in the approximated dispersion equation, deriving a simple and easy-to-implement pure qP-wave equation. We use the optical flow method to compute the direction of wave propagation, which improves the calculation accuracy of the scalar operators. Numerical experiments with representative models demonstrate that the new method has higher accuracy and better adaptability to models with strong anisotropy, complex structure, and rapid variation of the tilt angle than previous methods.

Coal seams have beddings and fissures and are typically anisotropic media. Current channel wave theories are mainly based on isotropic media, and few studies exist on the dispersion characteristics of Rayleigh channel waves in anisotropic models, such as transversely isotropic (TI) media. We chose the generalized reflection-transmission coefficient method to solve the dispersion curves of Rayleigh channel waves in TI media. However, it is difficult to solve the associated dispersion equations of Rayleigh channel waves using this method directly. Therefore, we have extended the generalized reflection-transmission coefficient method and determined the improved accuracy through numerical simulation. We analyzed the dispersion characteristics of Rayleigh channel waves of several typical coal seam models in TI media. The results showed that in the three-layer model, the difference in fundamental-mode dispersion curves between vertically transversely isotropic (VTI) media and isotropic media was relatively small; however, the differences in the higher order dispersion curves were slightly larger. The difference in the Airy phase velocity between horizontal transversely isotropic (HTI) and isotropic media was relatively large. When the coefficient of variation in the qP waves (δV) was greater than 0, the fundamental-mode and first-order phase velocity curves of HTI media exhibited an evident intersection at the head end. In the dirt-band-containing coal seam model, within the 350 and 550 Hz band, the high-frequency velocity of fundamental-mode phase velocity curve of isotropy and HTI media was slightly higher than the low-frequency velocity, which is a notable phenomenon.

In two-phase anisotropic media, fast P-waves, slow P-waves and SV-waves are coupled with each other. Fast P-wavefronts show an elliptical anisotropic feature. Although the wavelet phase of the fast P-wavefront in a solid and that in a fluid are the same, the wavelet phases of the slow P-waves are opposite. In cases where there is an obvious slow P-wave in the wavefield, the slow P-wave in fluid is stronger than that in the solid. The attenuation mechanism introduced in Biot's theory is of secondary importance for the fast P-wave and the SV-wave, but dissipation coefficients have a significant effect on the slow P-wave. If there is a large dissipation in the media, the slow P-wave will be attenuated very quickly. When the dissipation coefficients have strong anisotropy, the slow P-wave is attenuated not just along the direction that has a big dissipation coefficient, but for the entire slow P-wavefield. Hence the slow P-wave is hardly observable in practice on surface seismic records. It is clearly a limitation of conventional seismic analysis which we should be aware of in reservoir geophysics.