Anellipticity Parameter Research Articles

On the singularity point in acoustic orthorhombic media

Abstract The acoustic orthorhombic model is widely used in seismic modeling and processing of P-wave data. However, the anisotropic acoustic models have so called S-wave artifacts (1 artifact in transversely isotropic acoustic medium and two artifacts in orthorhombic acoustic medium). I show that S-wave artifacts can have one singularity point that results in complications in polarization field and the group velocity surface. The conditions of the existence of this point are defined in terms of anellipticity parameters. This singularity point and its group velocity image are the objects of my analysis.

The accuracy of approximating P-wave traveltimes in orthotropic media using VTI equations

SUMMARY Analysis of azimuthally varying non-hyperbolic moveout of P-wave traveltimes in orthotropic media can provide valuable information for reservoir fracture characterization and seismic data processing. Moveout velocities and azimuthal orientation of the two vertical symmetry planes plus anellipticity parameters defined for the three symmetry planes of an orthotropic medium can be estimated from the inversion of P-wave traveltimes measured in long-spread, wide-azimuth seismic surveys. Several orthotropic moveout equations have been developed to this date that can be used in the traveltime data inversion. The accuracy of the inversion results is primarily determined by the complexity of the equation and the assumptions (i.e. elastic or acoustic) made to derive it. Here, we show that often simpler moveout equations derived for transverse isotropy media with vertical axis of symmetry (VTI) can be used to estimate anellipticity parameters of the orthotropic media with high accuracy. We make an accuracy comparison analysis between non-hyperbolic moveout equations that are widely used in modelling P-wave traveltimes in VTI media. Our comparison analysis is based on the accuracy of estimating anellipticity parameters defined for two vertical [η(1) and η(2)] and one horizontal [η(3)] symmetry planes of a wide range of orthotropic media. The anellipticity parameters are estimated from the inversion of numerical traveltimes that are generated for a single and horizontal orthotropic layer. We show that for the survey geometry used in this analysis, most of the moveout equations can produce reliable estimates of the anellipticity parameters η(1) and η(2). Despite the availability of traveltime data from wide-azimuths and long source–receiver offsets, none of the moveout equations can estimate η(3) as accurate as η(1) and η(2) estimations. We show that while all the equations produce mostly accurate estimates of η(1) and η(2), generalized moveout approximation (GMA) provides the greatest accuracy. We also assess the validity of using GMA to invert P-wave traveltimes for three anellipticity parameters in terms of anisotropy in the background medium, fracture weakness, source–receiver offset (group angle) and traveltime data coverage in both the vertical (dip) and horizontal (azimuthal) planes. We show that, for a reasonable offset range, GMA can be reliably used to estimate anellipticity parameters η(1) and η(2) for an orthotropic medium that is constructed by intensive fracturing of a host medium with strong VTI anisotropy.

Multiparameter Acoustic Inversion for Variable-Tilt Transversely Isotropic Media With Generalized Radon Transform

The pseudoacoustic approximations are commonly used for migration and inversion in transversely isotropic (TI) media, as they accurately characterize the P-wave propagation and are simpler than their elastic counterparts, resulting in computational savings. This paper is devoted to presenting an approach for generalized Radon transform (GRT) migration and inversion in acoustic TI media with a tilted symmetry axis (TTI). In parameterizing an acoustic TTI medium with the P-wave normal moveout velocity (NMO) <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v_n</i> , Thomsen’s parameter δ, and anelliptic parameter η, a concise single-scattering integral for NMO pressure is obtained by perturbing the TTI medium from a background non-elliptically anisotropic medium. It results in explicitly representing the perturbation scattering patterns of each parameter ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v_n</i> , δ, η), helping us understand the scattering angular influence of the perturbed parameters. The application of GRT on this scattering integral allows a direct construction of the acoustic TTI inversion operator. Numerical examples verify the effectiveness of the proposed acoustic TTI GRT inversion method and show its considerably good performance in the presence of steeply dipping anisotropic layering.

P-, SV- and SH-wave eikonal approximations for HTI media based on perturbation theory

SUMMARY The eikonal equation for transversely isotropic media is a complex quartic partial differential equation that causes instability in traveltime inversion. The phenomenon of shear wave splitting in a transversely isotropic model with a horizontal symmetry axis (HTI) provides information for inversion. However, the involvement of shear waves makes the computation of forward modelling and inversion more time-consuming than when only using Pwaves. Most studies of traveltime approximation based on the eikonal equation are based on an acoustic assumption, which does not work for shear waves. Therefore, based on perturbation theory, we derive approximate solutions of the eikonal equation in a heterogeneous elastic HTI model for P, SV and SH waves to improve the traveltime computation efficiency. The approximate solutions of linear, quadratic and Shanks forms are based on the linearization of the traveltime function in terms of fracture weaknesses in an isotropic background, which are suitable for joint P-, SV- and SH-wave traveltime inversion and tomography for fracture parameters. When applied for a homogenous model, the proposed formulation results in a simple and accurate traveltime approximation. The first- and second-order perturbation coefficients are used to analyse the sensitivity of the traveltime to fracture weakness. To improve the accuracy of our approximation, we also expand the approximation of the P-wave traveltime with respect to the anellipticity parameter η in an elliptic background, which is evaluated by performing an error test. Based on the accuracy and stability of the perturbation, we select a set of linear equations with respect to the fracture weaknesses and a set of more accurate nonlinear equations to serve different needs.

Perturbation-based moveout approximation in the elastic transversely isotropic media with a vertical symmetry axis

Explicit approximations in traveltime are usually used in seismic data processing such as simulation through ray tracing, velocity analysis for model building, and imaging from time migration. Accurate approximation is key to achieving high-resolution imaging of the structure. We have proposed a perturbation-based approximation for P- and S-waves traveltime in a transversely isotropic medium with a vertical symmetry axis. The perturbation coefficients in this approximation are derived directly from the implicit parametric offset-traveltime equation. The comparison with the generalized moveout approximation (GMA) for P and S waves is applied in numerical examples. We find that our proposed approximation indicates higher accuracy than the GMA counterpart for P and S waves. The error sensitivity of P and S waves in the depth, vertical velocity ratio, and the anellipticity parameter also is analyzed to examine the influence of these factors. In addition, we investigate the anisotropy estimation through two approximations for P and S waves and find that our proposed perturbation method results in high accuracy, which can significantly improve the data processing quality. For the S wave, our approximation also is more accurate than the GMA counterpart and could be implemented for joint tomographic inversions.

Oriented NMO correction of VTI data in the absence of near-offset traces

Calculating an accurate seismic velocity model serves an important role in many seismic imaging techniques. The process of velocity model building is often time-consuming, specifically for anisotropic areas, where more than a single parameter is involved in the process. In the past few years, more time-efficient approaches have been considered to estimate seismic velocity as well as anellipticity parameter or heterogeneity factor using local event slopes. Nevertheless, some of these techniques are not practical due to curvature-dependency, or due to the lack of near-offset data. To address such limitations, we have used a curvature-independent approach for normal-moveout correction as well as parameter estimation in vertical transverse isotropic media, which is based on local estimation of vertical traveltime using a shifted-hyperbola approximation in the absence of near-offset data. The performance of our approach is tested on synthetic and field common-midpoint gathers. It is also assessed in different signal-to-noise ratios and different missing-near-offset situations. Our findings are consistent with the results achieved by the previous methods that were not developed for sparse data.

Perturbation of phase velocities in elastic orthorhombic media

The parameterization of anisotropic models is very important when focusing on specific signatures of seismic waves and reducing the parameter crosstalk involved in inverting seismic data. The parameterization is strongly dependent on the problem at hand. We have developed a new parameterization for an elastic orthorhombic (ORT) model with on-axes compressional wave (P-wave) and shear wave (S-wave) velocities and new symmetric anelliptic parameters. The perturbation approach is well defined for P-waves in acoustic ORT media. In the elastic ORT media, the P-wave perturbation coefficients are very similar to their acoustic counterparts. However, the S-wave perturbation coefficients are still unknown. The perturbation coefficients can be interpreted as sensitivity coefficients, and they are important in many applications. We apply the second-order perturbation in anelliptic parameters for P-, S1-, and S2-wave phase velocities in the elastic ORT model. We find that when using the conventional method some perturbation coefficients for S-waves are not defined in the vicinity of the singularity point in an elliptical background model. Thus, we adopt an alternative perturbation approach that overcomes this problem. We compute the first- and second-order perturbation coefficients for P- and S-waves. The perturbation-based approximations are very accurate for P- and S-waves compared with the exact solutions, based on a numerical example. The reductions to transversely isotropic and acoustic ORT models are also considered for analysis. We also determine how perturbations in anelliptic parameters affect S-wave triplications in an elastic ORT model.

Linearized scattering problem in acoustic transversely isotropic media with a vertical symmetry axis

The ray-based generalized Radon transform (GRT) inversion/migration strategy is studied in this paper for acoustic transversely isotropic media with a vertical symmetry axis (VTI media) represented by the P-wave normal moveout velocity, Thomsen parameter δ, anelliptic parameter η, and density. The multi-parameter GRT inversion solutions for multi-shot/multi-offset acquisition geometries are derived from the single-scattering approximation of the wavefields and an elliptically anisotropic background. Results from classical two-dimensional numerical tests show the true amplitude properties of the GRT backprojection operators.

Wave characteristics in elliptical orthorhombic media

An elliptical anisotropic medium is defined as a simplified representation of anisotropy in which the anelliptic parameters are set to zero in all symmetry planes. Despite the fact that this model is rather seldom observed for real rocks, it is often used as a reference model. The P-wave equations for an elliptical anisotropic medium are well known. However, the shear wave (S-wave) equations have not been derived. Thus, we have defined all wave modes in elliptical orthorhombic models focusing mostly on the S-wave properties. We determine that all wave modes in elliptical orthorhombic model are generally coupled and analyze the effect of the additive coupling term. As a result, there is an S-wave fundamental singularity point located in one of the symmetry planes depending on the relative magnitude of S-wave stiffness coefficients.

Rational approximation of P-wave kinematics — Part 2: Orhorhombic media

Orthorhombic anisotropy is a modern standard for 3D seismic studies in complex geologic settings. Several seismic data processing methods and wave propagation modeling algorithms in orthorhombic media rely on phase-velocity, group-velocity, and traveltime approximations. The algebraic simplicity of an approximate equation is an important factor in these media because the governing equations are more complicated than transversely isotropic media. To approximate the P-wave kinematics in acoustic orthorhombic media, we have developed a new 3D general functional equation that has a simple rational form. Using the general form, we adopt two versions of rational approximations for the phase velocity, group velocity, and traveltime. The first version uses a simpler functional form and parameter definition within the orthorhombic symmetry planes. The second version is more accurate, using one parameter that is defined out of the symmetry planes. For the phase velocity, we obtain another approximation that is no longer rational but is still algebraically simple, exact for 3D transversely isotropic media, and it is exact within the symmetry planes of orthorhombic media. We find superior accuracy in our approximations compared with previous ones, using numerical studies on multiple moderately anisotropic orthorhombic models. We investigate the effect of the negative anellipticity parameters on the accuracy and find that, in models in which the error of the existing most accurate approximations exceeds 2%, the error of the new approximations remains below 0.2%. The adopted approximations are algebraically simpler and stably more accurate than existing approximations; therefore, they may be considered as attractive alternatives for the existing approximations in many practical applications. We extend the applicability of our approximations by using them to obtain the equations of group direction as a function of phase direction and vice versa, which are useful in wave propagation modeling methods.

Rational approximation of P-wave kinematics — Part 1: Transversely isotropic media

In seismic data processing and several wave propagation modeling algorithms, the phase velocity, group velocity, and traveltime equations are essential. To have these equations in explicit form, or to reduce algebraic complexity, approximation methods are used. For the approximation of P-wave kinematics in acoustic transversely isotropic media, we have developed a new flexible 2D functional equation in a continued fraction form. Using different orders of the continued fraction, we obtain different approximations for (1) phase velocity as a function of phase direction, (2) group velocity as a function of group direction, and (3) traveltime as a function of offset. Then, we use them in the approximation of the group direction as a function of phase direction, and phase direction as a function of group direction. The proposed approximations have a rational form, which is considered algebraically simple and computationally efficient. The used continued fraction form rapidly converges to exact kinematics. By introducing the optimal ray into our approximations and using it for parameter definition, the convergence becomes faster, so the accuracy of the existing most accurate approximations is available by the third order, and new most accurate approximations are obtained by the fourth order of the proposed general form. The error of the most accurate version of the proposed approximations is below 0.001% for moderate anisotropic models with an anellipticity parameter up to 0.3. This high accuracy is considered to be attractive in practical implementations that use the kinematic equations and their derivatives.

Improving the accuracy of the expanded anisotropic eikonal equation at larger offsets using Levin T-transformation

In an anisotropic model, traveltime can be determined approximately by numerical solution of the eikonal equation in terms of an anellipticity parameter η, using perturbation theory. However, its accuracy decreases under the effect of strong anisotropy at larger offsets. It becomes invalid for determining normal moveout velocity and anellipticity parameter in seismic processing. We propose a new approach using Levin T-transformation to transform the expanded traveltime in the transversely isotropic medium with vertical axis of symmetry (VTI) into rational form. The objective of this study is to provide a new traveltime approximation that is more accurate at larger offsets. In this study, we derive Levin algorithm and determine the optimal value of Levin parameters, which is a key step in achieving better accuracy. In a numerical experiment, we compare the accuracy between Levin T-transformation and second sequence of Shanks transformation in a homogeneous VTI medium. We also implement both approximations in a velocity analysis and stacking traces using synthetic common midpoint gathers on a multilayer earth model. The proposed method shows a superiority in accuracy to existing methods over a range of offsets with offset-to-depth ratio up to 6 and anellipticity parameter 0–0.5.

Reducing residual normal moveout by globally optimized generalized moveout approximation in vertically transverse isotropy (VTI) media

The anisotropic phenomenon in wave propagation has been widely recognized on various scales. Knowledge of anisotropic effects is essential for interpretation and processing of seismic data. The existence of anisotropy leads to the moveout of nonhyperbolic even in a homogenous layer. A transversely isotropic medium with a vertical symmetry axis (VTI) is a reasonable approximation of horizontally layered anisotropic medium. The approximation of travel time is important for reducing the residual normal moveout of layered VTI media. The aim of this study is to develop globally optimized generalized moveout approximation in reducing residual normal moveout in VTI media. A comparative analysis was carried out to recent method for given an ellipticity parameter (0 ≤ η ≥ 0.5 ) and wide offset depth to ratio (0 to 4). The result shows that the globally optimized generalized moveout approximation is better in reducing residual moveout at large offsets with a stronger anellipticity parameter than existing methods. This is essential for reducing the accumulation of error especially for deeper substructures.

Scattering pattern analysis and true‐amplitude generalized Radon transform migration for acoustic transversely isotropic media with a vertical axis of symmetry

ABSTRACTIn multi‐parameter ray‐based anisotropic migration/inversion, it is essential that we have an understanding of the scattering mechanism corresponding to parameter perturbations. Because the complex nonlinearity in the anisotropic inversion problem is intractable, the construction of true‐amplitude linearized migration/inversion procedures is needed and important. By using the acoustic medium assumption for transversely isotropic media with a vertical axis of symmetry and representing the anisotropy with P‐wave normal moveout velocity, Thomsen parameter δ and anelliptic parameter η, we formalize the linearized inverse scattering problem for three‐dimensional pseudo‐acoustic equations. Deploying the single‐scattering approximation and an elliptically anisotropic background introduces a new linear integral operator that connects the discontinuous perturbation parameters with the multi‐shot/multi‐offset P‐wave scattered data. We further apply the high‐frequency asymptotic Green's function and its derivatives to the integral operator, and then the scattering pattern of each perturbation parameter can be explicitly presented. By naturally establishing a connection to generalized Radon transform, the pseudo‐inverse of the integral operator can be solved by the generalized Radon transform inversion. In consideration of the structure of this pseudo‐inverse operator, the computational implementation is done pointwise by shooting a fan of rays from the target imaging area towards the acquisition system. Results from two‐dimensional numerical tests show amplitude‐preserving images with high quality.

Characterization of a shale-gas reservoir based on a seismic amplitude variation with offset inversion for transverse isotropy with vertical axis of symmetry media and quantitative seismic interpretation

The Lower Silurian shale-gas formation in the south of the Sichuan Basin represents a strong transverse isotropy with vertical axis of symmetry (VTI) feature. Successful characterization of shale-gas formation requires handling the great influence of anisotropy in the seismic wave propagation. Seismic amplitude variation with offset (AVO) inversion for VTI media using PP-waves only is a difficult issue because more than three parameters need to be estimated and such an inverse problem is highly ill posed. We have applied an AVO inversion method for VTI media based on a modified approximation of the PP-wave reflection coefficient. This approximation consists of only three model parameters: the acoustic impedance (attribute [Formula: see text]), shear modulus proportional to the anellipticity parameter (attribute [Formula: see text]), and the approximated horizontal P-wave velocity (attribute [Formula: see text]), which can be well-inverted and have great interpretation capability in shale-gas reservoir characterization. A statistical-rock-physics method was then applied to the inverted attributes for quantitative interpretation of the shale-gas reservoir. A Markov random field is combined with Bayesian rule to improve the continuity and accuracy of the interpretation results. Shales can be successfully discriminated from surrounding formations by using the attribute pair [Formula: see text]-[Formula: see text], and the organic-rich gas-bearing shale can be successfully identified by using the attribute pair [Formula: see text]-[Formula: see text]. Comparison between the prediction results and well logs demonstrates the feasibility of the inversion and quantitative interpretation approaches.

Low-frequency layer-induced kinematic parameters in transversely isotropic media and orthorhombic media

SUMMARY We develop a method to compute frequency-dependent kinematic parameters for an effective orthorhombic (ORT) medium. In order to investigate the influence of fracture weaknesses on the kinematic parameters, the effective ORT medium is composed based on the linear slip theory and derived by applying the limited Baker–Campbell–Hausdorff series. The frequency-dependent kinematic parameters 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). We also investigate the influence of volume fraction, frequency, velocity ratio and fracture weaknesses on the effective kinematic parameters.

P‐wave complex‐valued traveltimes in homogeneous attenuating transversely isotropic media

ABSTRACTComputation of complex‐valued traveltimes provides an efficient approach to describe the seismic wave attenuation for applications like attenuation tomography, inverse Q filtering and Kirchhoff migration with absorption compensation. Attenuating acoustic transverse isotropy can be used to describe the directional variation of velocity and attenuation of P‐waves in thin‐bedding geological structures. We present an approximate method to solve the acoustic eikonal equation for an attenuating transversely isotropic medium with a vertical symmetry axis. We take into account two similar parameterizations of an attenuating vertical symmetry axis medium. The first parameterization uses the normal moveout velocity, whereas the second parameterization uses the horizontal velocity. For each parameterization, we combine perturbation theory and the Shanks transform in different ways to derive analytic solutions. Numerical examples show that the analytic solutions derived from the second parameterization yield better accuracy. The Shanks transform solution with respect to only the anellipticity parameter from the second parameterization is demonstrated numerically to be the most accurate among all the analytic solutions.

Seismic amplitude inversion for the transversely isotropic media with vertical axis of symmetry

ABSTRACTTransverse isotropy with a vertical axis of symmetry is a common form of anisotropy in sedimentary basins, and it has a significant influence on the seismic amplitude variation with offset. Although exact solutions and approximations of the PP‐wave reflection coefficient for the transversely isotropic media with vertical axis of symmetry have been explicitly studied, it is difficult to apply these equations to amplitude inversion, because more than three parameters need to be estimated, and such an inverse problem is highly ill‐posed. In this paper, we propose a seismic amplitude inversion method for the transversely isotropic media with a vertical axis of symmetry based on a modified approximation of the reflection coefficient. This new approximation consists of only three model parameters: attribute A, the impedance (vertical phase velocity multiplied by bulk density); attribute B, shear modulus proportional to an anellipticity parameter (Thomsen's parameter ε−δ); and attribute C, the approximate horizontal P‐wave phase velocity, which can be well estimated by using a Bayesian‐framework‐based inversion method. Using numerical tests we show that the derived approximation has similar accuracy to the existing linear approximation and much higher accuracy than isotropic approximations, especially at large angles of incidence and for strong anisotropy. The new inversion method is validated by using both synthetic data and field seismic data. We show that the inverted attributes are robust for shale‐gas reservoir characterization: the shale formation can be discriminated from surrounding formations by using the crossplot of the attributes A and C, and then the gas‐bearing shale can be identified through the combination of the attributes A and B. We then propose a rock‐physics‐based method and a stepwise‐inversion‐based method to estimate the P‐wave anisotropy parameter (Thomsen's parameter ε). The latter is more suitable when subsurface media are strongly heterogeneous. The stepwise inversion produces a stable and accurate Thomsen's parameter ε, which is proved by using both synthetic and field data.

Rock elasticity as a function of the uniaxial stress: laboratory measurements and theoretical modelling of vertical transversely isotropic and orthorhombic shales

ABSTRACTWe apply a rock‐physics model that describes the relationship between the effective stress and rock elasticity. We experimentally obtain and analyse a data set containing one vertical transversely isotropic and one orthorhombic shale sample. The vertical transversely isotropic symmetry of the first sample is caused by the layered structure of the rock. The seismic orthorhombicity of the second sample could be explained after microscopic analysis of thin section, which demonstrates an imperfect disorder of inhomogeneities. Both samples were loaded uniaxially in a quasi‐static regime. During the loading, we measured stress‐dependent seismic velocities and sample deformations. For the analysis of the stress‐dependent velocities and stiffnesses, we modelled the measured data set using a recent generalization of the porosity deformation approach. Comparison of the experimentally determined and numerically modelled data supports the applicability of the theory and helps in the interpretation of experimentally obtained data. In agreement with the theory, uniaxial stress increases the elliptic component of the seismic anisotropy and does not impact the anellipticity parameter. We demonstrate the distinct influence of the stiff and compliant porosities on the stress sensitivity of the elastic properties.

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

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