A new traveltime approximation for TI media

In a transversely isotropic (TI) medium, the trade-off between inhomogeneity and anisotropy can dramatically reduce our capability to estimate anisotropy parameters. By expanding the TI eikonal equation in power series in terms of the aneliptic parameter [Formula: see text], we derive an efficient tool to estimate (scan) for [Formula: see text] in a generally inhomogeneous, elliptically anisotropic background medium. For a homogeneous-tilted transversely isotropic medium, we obtain an analytic nonhyperbolic moveout equation that is accurate for large offsets. In the common case where we do not have well information and it is necessary to resolve the vertical velocity, the background medium can be assumed isotropic, and the traveltime equations becomes simpler. In all cases, the accuracy of this new TI traveltime equation exceeds previously published formulations and demonstrates how [Formula: see text] is better resolved at small offsets when the tilt is large.

Download Full-text

A complex point-source solution of the acoustic eikonal equation for Gaussian beams in transversely isotropic media

Geophysics ◽

10.1190/geo2019-0264.1 ◽

2020 ◽

Vol 85 (3) ◽

pp. T191-T207

Author(s):

Xingguo Huang ◽

Hui Sun ◽

Zhangqing Sun ◽

Nuno Vieira da Silva

Keyword(s):

Point Source ◽

Eikonal Equation ◽

Taylor Series Expansion ◽

Transversely Isotropic ◽

Complex Point ◽

Transversely Isotropic Media ◽

Isotropic Media ◽

Anisotropy Parameters ◽

Complex Point Source ◽

Complex Eikonal

The complex traveltime solutions of the complex eikonal equation are the basis of inhomogeneous plane-wave seismic imaging methods, such as Gaussian beam migration and tomography. We have developed analytic approximations for the complex traveltime in transversely isotropic media with a titled symmetry axis, which is defined by a Taylor series expansion over the anisotropy parameters. The formulation for the complex traveltime is developed using perturbation theory and the complex point-source method. The real part of the complex traveltime describes the wavefront, and the imaginary part of the complex traveltime describes the decay of the amplitude of waves away from the central ray. We derive the linearized ordinary differential equations for the coefficients of the Taylor-series expansion using perturbation theory. The analytical solutions for the complex traveltimes are determined by applying the complex point-source method to the background traveltime formula and subsequently obtaining the coefficients from the linearized ordinary differential equations. We investigate the influence of the anisotropy parameters and of the initial width of the ray tube on the accuracy of the computed traveltimes. The analytical formulas, as outlined, are efficient methods for the computation of complex traveltimes from the complex eikonal equation. In addition, those formulas are also effective methods for benchmarking approximated solutions.

Download Full-text

Scanning anisotropy parameters in complex media

Geophysics ◽

10.1190/1.3553015 ◽

2011 ◽

Vol 76 (2) ◽

pp. U13-U22 ◽

Cited By ~ 61

Author(s):

Tariq Alkhalifah

Keyword(s):

Anisotropic Medium ◽

Homogeneous Medium ◽

Transversely Isotropic ◽

Transient Behavior ◽

Complex Media ◽

Linear Partial Differential Equations ◽

Anisotropy Parameters ◽

Axis Of Symmetry ◽

Ti Media ◽

Taylor’S Series

Parameter estimation in an inhomogeneous anisotropic medium offers many challenges; chief among them is the trade-off between inhomogeneity and anisotropy. It is especially hard to estimate the anisotropy anellipticity parameter η in complex media. Using perturbation theory and Taylor’s series, I have expanded the solutions of the anisotropic eikonal equation for transversely isotropic (TI) media with a vertical symmetry axis (VTI) in terms of the independent parameter η from a generally inhomogeneous elliptically anisotropic medium background. This new VTI traveltime solution is based on a set of precomputed perturbations extracted from solving linear partial differential equations. The traveltimes obtained from these equations serve as the coefficients of a Taylor-type expansion of the total traveltime in terms of η. Shanks transform is used to predict the transient behavior of the expansion and improve its accuracy using fewer terms. A homogeneous medium simplification of the expansion provides classical nonhyperbolic moveout descriptions of the traveltime that are more accurate than other recently derived approximations. In addition, this formulation provides a tool to scan for anisotropic parameters in a generally inhomogeneous medium background. A Marmousi test demonstrates the accuracy of this approximation. For a tilted axis of symmetry, the equations are still applicable with a slightly more complicated framework because the vertical velocity and δ are not readily available from the data.

Download Full-text

Exact elastic impedance matrices for transversely isotropic medium

Geophysics ◽

10.1190/geo2015-0163.1 ◽

2016 ◽

Vol 81 (2) ◽

pp. C1-C15 ◽

Cited By ~ 12

Author(s):

Feng Zhang ◽

Xiang-Yang Li

Keyword(s):

Shale Gas ◽

Isotropic Medium ◽

Seismic Reflection ◽

Transversely Isotropic ◽

Seismic Inversion ◽

P Wave ◽

Reflection Coefficients ◽

Angle Of Incidence ◽

Transversely Isotropic Medium ◽

Anisotropy Parameters

Conventional elastic impedances are derived as scalars by means of the integration of reflectivity. In this sense, they are attributes of the seismic reflection but not the intrinsic physical property of the subsurface media. The derivation of these expressions shares the same assumptions as the reflectivity approximations, such as weak impedance contrast, small angle of incidence, or weak anisotropic media, and thus it limits the accuracy and interpretation capability. The exact P/SV impedance matrices relating the stress and strain represent the mechanical property of the subsurface media and yield reflection coefficients at an arbitrary angle of incidence. We have extended the impedance matrices to a transversely isotropic medium. The resulting elastic impedances (stress/velocity ratios) can be used to characterize those unconventional reservoir formations with strong seismic anisotropy, such as shale-gas and coal-bed methane. Our numerical analyses determined their variations with the phase angle and anisotropy parameters. The exact expressions of the P- and S-wave elastic impedances are used to model the seismic reflection coefficients, and thus they can be inverted simultaneously if all of the types of reflection waves are available. We then derive approximations of quasi-P-wave elastic impedances for seismic inversion of anisotropy parameters and further interpretation. Our applications on real logs and seismic data for a turbidite fan reservoir and a shale-gas reservoir determined the robust interpretation capability of derived elastic impedances in lithology characterizations.

Download Full-text

Small-angle AVO response of PS-waves in tilted transversely isotropic media

Geophysics ◽

10.1190/1.2329865 ◽

2006 ◽

Vol 71 (5) ◽

pp. C69-C79 ◽

Cited By ~ 13

Author(s):

Jyoti Behura ◽

Ilya Tsvankin

Keyword(s):

Small Angle ◽

Symmetry Axis ◽

Transversely Isotropic ◽

Normal Incidence ◽

Analytic Solutions ◽

Reflection Coefficients ◽

Axis Orientation ◽

Avo Inversion ◽

Anisotropy Parameters ◽

Ti Media

Field records for small source-receiver offsets often contain intensive converted PS-waves that may be caused by the influence of anisotropy on either side of the reflector. Here, we study the small-angle reflection coefficients of the split converted [Formula: see text]- and [Formula: see text]-waves ([Formula: see text] and [Formula: see text]) for a horizontal interface separating two transversely isotropic (TI) media with arbitrary orientations of the symmetry axis. The normal-incidence reflection coefficients [Formula: see text] and [Formula: see text] vanish when both half-spaces have a horizontal symmetry plane, which happens if the symmetry axis is vertical or horizontal (i.e., if the medium is VTI or HTI). For a tilted symmetry axis in either medium, however, the magnitude of the reflection coefficients can reach substantial values that exceed 0.1, even if the anisotropy strength is moderate. To study the influence exerted by the orientation of the symmetry axis and the anisotropy parameters, we develop concise weak-contrast, weak-anisotropyapproximations for the PS-wave reflection coefficients and com-pare them with exact numerical results. In particular, the analytic solutions show that the contributions made by the Thomsen parameters [Formula: see text] and [Formula: see text] and the symmetry-axis tilt [Formula: see text] to the coefficients [Formula: see text] and [Formula: see text] can be expressed through the first derivative of the P-wave phase velocity at normal incidence. If the symmetry-axis orientation and anisotropy parameters do not change across the interface, the normal-incidence reflection coefficients are insignificant, regardless of the strength of the velocity and density contrast. The AVO (amplitude variation with offset) gradients of the PS-waves are influenced primarily by the anisotropy of the incidence medium that causes shear-wave splitting and determines the partitioning of energy between the [Formula: see text] and [Formula: see text] modes. Because of their substantial amplitude, small-angle PS reflections in TI media contain valuable information for anisotropic AVO inversion of multicomponent data. Our analytic solutions provide a foundation for linear AVO-inversion algorithms and can be used to guide nonlinear inversion that is based on the exact reflection coefficients.

Download Full-text

Traveltime approximations and parameter estimation for orthorhombic media

Geophysics ◽

10.1190/geo2015-0367.1 ◽

2016 ◽

Vol 81 (4) ◽

pp. C127-C137 ◽

Cited By ~ 15

Author(s):

Nabil Masmoudi ◽

Tariq Alkhalifah

Keyword(s):

Eikonal Equation ◽

Homogeneous Medium ◽

Background Model ◽

Model Complexity ◽

Seismic Modeling ◽

Effective Parameters ◽

Computationally Efficient ◽

Efficient Tool ◽

Anisotropy Parameters ◽

Perturbation Parameters

Building anisotropy models is necessary for seismic modeling and imaging. However, anisotropy estimation is challenging due to the trade-off between inhomogeneity and anisotropy. Luckily, we can estimate the anisotropy parameters if we relate them analytically to traveltimes. Using perturbation theory, we have developed traveltime approximations for orthorhombic media as explicit functions of the anellipticity parameters [Formula: see text], [Formula: see text], and [Formula: see text] in inhomogeneous background media. The parameter [Formula: see text] is related to Tsvankin-Thomsen notation and ensures easier computation of traveltimes in the background model. Specifically, our expansion assumes an inhomogeneous ellipsoidal anisotropic background model, which can be obtained from well information and stacking velocity analysis. We have used the Shanks transform to enhance the accuracy of the formulas. A homogeneous medium simplification of the traveltime expansion provided a nonhyperbolic moveout description of the traveltime that was more accurate than other derived approximations. Moreover, the formulation provides a computationally efficient tool to solve the eikonal equation of an orthorhombic medium, without any constraints on the background model complexity. Although, the expansion is based on the factorized representation of the perturbation parameters, smooth variations of these parameters (represented as effective values) provides reasonable results. Thus, this formulation provides a mechanism to estimate the three effective parameters [Formula: see text], [Formula: see text], and [Formula: see text]. We have derived Dix-type formulas for orthorhombic medium to convert the effective parameters to their interval values.

Download Full-text

Plane-wave propagation in attenuative transversely isotropic media

Geophysics ◽

10.1190/1.2187792 ◽

2006 ◽

Vol 71 (2) ◽

pp. T17-T30 ◽

Cited By ~ 113

Author(s):

Yaping Zhu ◽

Ilya Tsvankin

Keyword(s):

Wave Propagation ◽

Plane Wave ◽

Wave Attenuation ◽

Transversely Isotropic ◽

Attenuation Coefficients ◽

The Real ◽

Velocity Anisotropy ◽

Anisotropy Parameters ◽

Attenuation Anisotropy ◽

Ti Media

Directionally dependent attenuation in transversely isotropic (TI) media can influence significantly the body-wave amplitudes and distort the results of the AVO (amplitude variation with offset) analysis. Here, we develop a consistent analytic treatment of plane-wave properties for TI media with attenuation anisotropy. We use the concept of homogeneous wave propagation, assuming that in weakly attenuative media the real and imaginary parts of the wave vector are parallel to one another. The anisotropic quality factor can be described by matrix elements [Formula: see text], defined as the ratios of the real and imaginary parts of the corresponding stiffness coefficients. To characterize TI attenuation, we follow the idea of the Thomsen notation for velocity anisotropy and replace the components [Formula: see text] by two reference isotropic quantities and three dimensionless anisotropy parameters [Formula: see text], and [Formula: see text]. The parameters [Formula: see text] and [Formula: see text] quantify the difference between the horizontal- and vertical-attenuation coefficients of P- and SH-waves, respectively, while [Formula: see text] is defined through the second derivative of the P-wave attenuation coefficient in the symmetry direction. Although the definitions of [Formula: see text], and [Formula: see text] are similar to those for the corresponding Thomsen parameters, the expression for [Formula: see text] reflects the coupling between the attenuation and velocity anisotropy. Assuming weak attenuation as well as weak velocity and attenuation anisotropy allows us to obtain simple attenuation coefficients linearized in the Thomsen-style parameters. The normalized attenuation coefficients for P- and SV-waves have the same form as the corresponding approximate phase-velocity functions, but both [Formula: see text] and the effective SV-wave attenuation-anisotropy parameter [Formula: see text] depend on the velocity-anisotropy parameters in addition to the elements [Formula: see text]. The linearized approximations not only provide valuable analytic insight, but they also remain accurate for the practically important range of small and moderate anisotropy parameters — in particular, for near-vertical and near-horizontal propagation directions.

Download Full-text

Diffraction traveltime approximation for TI media with an inhomogeneous background

Geophysics ◽

10.1190/geo2012-0413.1 ◽

2013 ◽

Vol 78 (5) ◽

pp. WC103-WC111 ◽

Cited By ~ 19

Author(s):

Umair bin Waheed ◽

Tariq Alkhalifah ◽

Alexey Stovas

Keyword(s):

Seismic Data ◽

Eikonal Equation ◽

Anisotropy Parameter ◽

Transversely Isotropic ◽

Transversely Isotropic Media ◽

Wide Range ◽

Isotropic Media ◽

Best Fitting ◽

Axis Of Symmetry ◽

Ti Media

Diffractions in seismic data contain valuable information that can help improve our modeling capability for better imaging of the subsurface. They are especially useful for anisotropic media because they inherently possess a wide range of dips necessary to resolve the angular dependence of velocity. We develop a scheme for diffraction traveltime computations based on perturbation of the anellipticity anisotropy parameter for transversely isotropic media with tilted axis of symmetry (TTI). The expansion, therefore, uses an elliptically anisotropic medium with tilt as the background model. This formulation has advantages on two fronts: first, it alleviates the computational complexity associated with solving the TTI eikonal equation, and second, it provides a mechanism to scan for the best-fitting anellipticity parameter [Formula: see text] without the need for repetitive modeling of traveltimes, because the traveltime coefficients of the expansion are independent of the perturbed parameter [Formula: see text]. The accuracy of such an expansion is further enhanced by the use of Shanks transform. We established the effectiveness of the proposed formulation with tests on a homogeneous TTI model and complex media such as the Marmousi and BP models.

Download Full-text

High-Contrast and Scattering-Type Transflective Liquid Crystal Displays Based on Polymer-Network Liquid Crystals

Polymers ◽

10.3390/polym12040739 ◽

2020 ◽

Vol 12 (4) ◽

pp. 739 ◽

Cited By ~ 1

Author(s):

Cheng-Kai Liu ◽

Wei-Hsuan Chen ◽

Chung-Yu Li ◽

Ko-Ting Cheng

Keyword(s):

Liquid Crystal ◽

Polymer Network ◽

Liquid Crystal Displays ◽

Quarter Wave Plate ◽

High Contrast ◽

Trade Off ◽

Twisted Nematic ◽

Quarter Wave ◽

The Common ◽

Linear Polarizer

The methods to enhance contrast ratios (CRs) in scattering-type transflective liquid crystal displays (ST-TRLCDs) based on polymer-network liquid crystal (PNLC) cells are investigated. Two configurations of ST-TRLCDs are studied and are compared with the common ST-TRLCDs. According to the comparisons, CRs are effectively enhanced by assembling a linear polarizer at the suitable position to achieve better dark states in the transmissive and reflective modes of the reported ST-TRLCDs with the optimized configuration, and its main trade-off is the loss of brightness in the reflective modes. The PNLC cell, which works as an electrically switchable polarizer herein, can be a PN-90° twisted nematic LC (PN-90° TNLC) cell or a homogeneous PNLC (H-PNLC) cell. The optoelectric properties of PN-90° TNLC and those of H-PNLC cells are compared in detail, and the results determine that the ST-TRLCD with the optimized configuration using an H-PNLC cell can achieve the highest CR. Moreover, no quarter-wave plate is used in the ST-TRLCD with the optimized configuration, so a parallax problem caused by QWPs can be solved. Other methods for enhancing the CRs of the ST-TRLCDs are also discussed.

Download Full-text