Polarization, phase velocity, and NMO velocity of qP-waves in arbitrary weakly anisotropic media

Geophysics ◽  
1998 ◽  
Vol 63 (5) ◽  
pp. 1754-1766 ◽  
Author(s):  
Ivan Pšenčk ◽  
Dirk Gajewski
Keyword(s):  
Phase Velocity ◽  
Polarization Vector ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Natural Generalization ◽  
Weak Anisotropy ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Approximate Formulas

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.

Normal moveout from dipping reflectors in anisotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 268-284 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Weak Anisotropy ◽  
Anisotropic Models ◽  
Transversely Isotropic Media ◽  
Wide Range ◽  
Isotropic Media ◽  
The Difference

Description of reflection moveout from dipping interfaces is important in developing seismic processing methods for anisotropic media, as well as in the inversion of reflection data. Here, I present a concise analytic expression for normal‐moveout (NMO) velocities valid for a wide range of homogeneous anisotropic models including transverse isotropy with a tilted in‐plane symmetry axis and symmetry planes in orthorhombic media. In transversely isotropic media, NMO velocity for quasi‐P‐waves may deviate substantially from the isotropic cosine‐of‐dip dependence used in conventional constant‐velocity dip‐moveout (DMO) algorithms. However, numerical studies of NMO velocities have revealed no apparent correlation between the conventional measures of anisotropy and errors in the cosine‐of‐dip DMO correction (“DMO errors”). The analytic treatment developed here shows that for transverse isotropy with a vertical symmetry axis, the magnitude of DMO errors is dependent primarily on the difference between Thomsen parameters ε and δ. For the most common case, ε − δ > 0, the cosine‐of‐dip–corrected moveout velocity remains significantly larger than the moveout velocity for a horizontal reflector. DMO errors at a dip of 45 degrees may exceed 20–25 percent, even for weak anisotropy. By comparing analytically derived NMO velocities with moveout velocities calculated on finite spreads, I analyze anisotropy‐induced deviations from hyperbolic moveout for dipping reflectors. For transversely isotropic media with a vertical velocity gradient and typical (positive) values of the difference ε − δ, inhomogeneity tends to reduce (sometimes significantly) the influence of anisotropy on the dip dependence of moveout velocity.

scholarly journals An efficient wavefield inversion for transversely isotropic media with a vertical axis of symmetry

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. R195-R206 ◽  
Author(s):  
Chao Song ◽  
Tariq Alkhalifah
Keyword(s):  
High Resolution ◽  
Optimization Problem ◽  
Source Function ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Trade Off ◽  
Transversely Isotropic Media ◽  
Highly Nonlinear ◽  
Isotropic Media ◽  
Axis Of Symmetry

Conventional full-waveform inversion (FWI) aims at retrieving a high-resolution velocity model directly from the wavefields measured at the sensor locations resulting in a highly nonlinear optimization problem. Due to the high nonlinearity of FWI (manifested in one form in the cycle-skipping problem), it is easy to fall into local minima. Considering that the earth is truly anisotropic, a multiparameter inversion imposes additional challenges in exacerbating the null-space problem and the parameter trade-off issue. We have formulated an optimization problem to reconstruct the wavefield in an efficient matter with background models by using an enhanced source function (which includes secondary sources) in combination with fitting the data. In this two-term optimization problem to fit the wavefield to the data and to the background wave equation, the inversion for the wavefield is linear. Because we keep the modeling operator stationary within each frequency, we only need one matrix inversion per frequency. The inversion for the anisotropic parameters is handled in a separate optimization using the wavefield and the enhanced source function. Because the velocity is the dominant parameter controlling the wave propagation, it is updated first. Thus, this reduces undesired updates for anisotropic parameters due to the velocity update leakage. We find the effectiveness of this approach in reducing parameter trade-off with a distinct Gaussian anomaly model. We find that in using the parameterization [Formula: see text], and [Formula: see text] to describe the transversely isotropic media with a vertical axis of symmetry model in the inversion, we end up with high resolution and minimal trade-off compared to conventional parameterizations for the anisotropic Marmousi model. Application on 2D real data also indicates the validity of our method.

scholarly journals Frequency domain multiparameter acoustic inversion for transversely isotropic media with a vertical axis of symmetry

Geophysics ◽  
2019 ◽  
Vol 84 (1) ◽  
pp. C1-C14 ◽  
Author(s):  
Ramzi Djebbi ◽  
Tariq Alkhalifah
Keyword(s):  
Frequency Domain ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
The Other ◽  
Sensitivity Analyses ◽  
Data Set ◽  
Trade Off ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Axis Of Symmetry

Multiparameter full-waveform inversion for transversely isotropic media with a vertical axis of symmetry (VTI) suffers from the trade-off between the parameters. The trade-off results in the leakage of one parameter’s update into the other. It affects the accuracy and convergence of the inversion. The sensitivity analyses suggested a parameterization using the horizontal velocity [Formula: see text], Thomsen’s parameter [Formula: see text], and the anelliptic parameter [Formula: see text] to reduce the trade-off for surface recorded seismic data. We aim to invert for this parameterization using the scattering integral (SI) method. The available Born sensitivity kernels, within this approach, can be used to calculate additional inversion information. We mainly compute the diagonal of the approximate Hessian, used as a conjugate-gradient preconditioner, and the gradients’ step lengths. We consider modeling in the frequency domain. The large computational cost of the SI method can be avoided with direct Helmholtz equation solvers. We applied our method to the VTI Marmousi II model for various inversion strategies. We found that we can invert the [Formula: see text] accurately. For the [Formula: see text] parameter, only the short wavelengths are well-recovered. On the other hand, the [Formula: see text] parameter impact is weak on the inversion results and can be fixed. However, a good background [Formula: see text], with accurate long wavelengths, is needed to correctly invert for [Formula: see text]. Furthermore, we invert a real data set acquired by CGG from offshore Australia. We simultaneously invert all three parameters using our inversion approach. The velocity model is improved, and additional layers are recovered. We confirm the accuracy of the results by comparing them with well-log information, as well as looking at the data and angle gathers.

The effect of transverse isotropy on isotropic traveltime inversion of vertical seismic profile data—A modeling study

Geophysics ◽  
1990 ◽  
Vol 55 (9) ◽  
pp. 1235-1241 ◽  
Author(s):  
Jan Douma
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Seismic Profile ◽  
Isotropic Layer ◽  
Low Velocity ◽  
Traveltime Inversion ◽  
Axis Of Symmetry ◽  
Transversely Isotropic Layer

Traveltime inversion of multioffset VSP data can be used to determine the depths of the interfaces in layered media. Many inversion schemes, however, assume isotropy and consequently may introduce erroneous structures for anisotropic media. Synthetic traveltime data are computed for layered anisotropic media and inverted assuming isotropic layers. Only the interfaces between these layers are inverted. For a medium consisting of a horizontal isotropic low‐velocity layer on top of a transversely isotropic layer with a horizontal axis of symmetry (e.g., anisotropy due to aligned vertical cracks), 2-D isotropic inversion results in an anticline. For a given axis of symmetry the form of this anticline depends on the azimuth of the source‐borehole direction. The inversion result is a syncline (in 3-D a “bowl” structure), regardless of the azimuth of the source‐borehole direction for a vertical axis of symmetry of the transversely isotropic layer (e.g., anisotropy due to horizontal bedding).

Moveout approximations for P- and SV-waves in dip-constrained transversely isotropic media

Geophysics ◽  
2013 ◽  
Vol 78 (6) ◽  
pp. C53-C59 ◽  
Author(s):  
Véronique Farra ◽  
Ivan Pšenčík
Keyword(s):  
Transversely Isotropic ◽  
Arbitrary Orientation ◽  
Transversely Isotropic Medium ◽  
Weak Anisotropy ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Dip Angle ◽  
P And Sv Waves ◽  
Axes Of Symmetry

We generalize the P- and SV-wave moveout formulas obtained for transversely isotropic media with vertical axes of symmetry (VTI), based on the weak-anisotropy approximation. We generalize them for 3D dip-constrained transversely isotropic (DTI) media. A DTI medium is a transversely isotropic medium whose axis of symmetry is perpendicular to a dipping reflector. The formulas are derived in the plane defined by the source-receiver line and the normal to the reflector. In this configuration, they can be easily obtained from the corresponding VTI formulas. It is only necessary to replace the expression for the normalized offset by the expression containing the apparent dip angle. The final results apply to general 3D situations, in which the plane reflector may have arbitrary orientation, and the source and the receiver may be situated arbitrarily in the DTI medium. The accuracy of the proposed formulas is tested on models with varying dip of the reflector, and for several orientations of the horizontal source-receiver line with respect to the dipping reflector.

scholarly journals Dip‐moveout error in transversely isotropic media with linear velocity variation in depth

Geophysics ◽  
1993 ◽  
Vol 58 (10) ◽  
pp. 1442-1453 ◽  
Author(s):  
Ken L. Larner
Keyword(s):  
Homogeneous Model ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Velocity Variation ◽  
Transversely Isotropic Medium ◽  
Velocity Gradients ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Homogeneous Media

Levin modeled the moveout, within common‐midpoint (CMP) gathers, of reflections from plane‐dipping reflectors beneath homogeneous, transversely isotropic media. For some media, when the axis of symmetry for the anisotropy was vertical, he found departures in stacking velocity from predictions based upon the familiar cosine‐of‐dip correction for isotropic media. Here, I do similar tests, again with transversely isotropic models with vertical axis of symmetry, but now allowing the medium velocity to vary linearly with depth. Results for the same four anisotropic media studied by Levin show behavior of dip‐corrected stacking velocity with reflector dip that, for all velocity gradients considered, differs little from that for the counterpart homogeneous media. As with isotropic media, traveltimes in an inhomogeneous, transversely isotropic medium can be modeled adequately with a homogeneous model with vertical velocity equal to the vertical rms velocity of the inhomogeneous medium. In practice, dip‐moveout (DMO) is based on the assumption that either the medium is homogeneous or its velocity varies with depth, but in both cases isotropy is assumed. It turns out that for only one of the transversely isotropic media considered here—shale‐limestone—would v(z) DMO fail to give an adequate correction within CMP gathers. For the shale‐limestone, fortuitously the constant‐velocity DMO gives a better moveout correction than does the v(z) DMO.

scholarly journals Traveltime approximations for transversely isotropic media with an inhomogeneous background

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA31-WA42 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Transversely Isotropic Media ◽  
Higher Order Terms ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Series Type ◽  
Ti Media ◽  
Taylor’S Series

A transversely isotropic (TI) model with a tilted symmetry axis is regarded as one of the most effective approximations to the Earth subsurface, especially for imaging purposes. However, we commonly utilize this model by setting the axis of symmetry normal to the reflector. This assumption may be accurate in many places, but deviations from this assumption will cause errors in the wavefield description. Using perturbation theory and Taylor’s series, I expand the solutions of the eikonal equation for 2D TI media with respect to the independent parameter [Formula: see text], the angle the tilt of the axis of symmetry makes with the vertical, in a generally inhomogeneous TI background with a vertical axis of symmetry. I do an additional expansion in terms of the independent (anellipticity) parameter [Formula: see text] in a generally inhomogeneous elliptically anisotropic background medium. These new TI traveltime solutions are given by expansions in [Formula: see text] and [Formula: see text] with coefficients extracted from solving linear first-order partial differential equations. Pade approximations are used to enhance the accuracy of the representation by predicting the behavior of the higher-order terms of the expansion. A simplification of the expansion for homogenous media provides nonhyperbolic moveout descriptions of the traveltime for TI models that are more accurate than other recently derived approximations. In addition, for 3D media, I develop traveltime approximations using Taylor’s series type of expansions in the azimuth of the axis of symmetry. The coefficients of all these expansions can also provide us with the medium sensitivity gradients (Jacobian) for nonlinear tomographic-based inversion for the tilt in the symmetry axis.

scholarly journals Mapping moveout approximations in TI media

Geophysics ◽  
2014 ◽  
Vol 79 (1) ◽  
pp. C19-C26 ◽  
Author(s):  
Alexey Stovas ◽  
Tariq Alkhalifah
Keyword(s):  
Transversely Isotropic ◽  
Vertical Axis ◽  
Accurate Approximation ◽  
Seismic Modeling ◽  
Effective Parameters ◽  
Transversely Isotropic Media ◽  
Vertical Heterogeneity ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Ti Media

Moveout approximations play a very important role in seismic modeling, inversion, and scanning for parameters in complex media. We developed a scheme to map one-way moveout approximations for transversely isotropic media with a vertical axis of symmetry (VTI), which is widely available, to the tilted case (TTI) by introducing the effective tilt angle. As a result, we obtained highly accurate TTI moveout equations analogous with their VTI counterparts. Our analysis showed that the most accurate approximation is obtained from the mapping of generalized approximation. The new moveout approximations allow for, as the examples demonstrate, accurate description of moveout in the TTI case even for vertical heterogeneity. The proposed moveout approximations can be easily used for inversion in a layered TTI medium because the parameters of these approximations explicitly depend on corresponding effective parameters in a layered VTI medium.

The migrator’s equation for anisotropic media

Geophysics ◽  
1990 ◽  
Vol 55 (11) ◽  
pp. 1429-1434 ◽  
Author(s):  
N. F. Uren ◽  
G. H. F. Gardner ◽  
J. A. McDonald
Keyword(s):  
Physical Model ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Original Structure ◽  
Model Data ◽  
P Waves ◽  
Reflection Seismic ◽  
Isotropic Media ◽  
Zero Offset

The migrator’s equation, which gives the relationship between real and apparent dips on a reflector in zero‐offset reflection seismic sections, may be readily implemented in one step with a frequency‐domain migration algorithm for homogeneous media. Huygens’ principle is used to derive a similar relationship for anisotropic media where velocities are directionally dependent. The anisotropic form of the migrator’s equation is applicable to both elliptically and nonelliptically anisotropic media. Transversely isotropic media are used to demonstrate the performance of an f-k implementation of the migrator’s equation for anisotropic media. In such a medium SH-waves are elliptically anisotropic, while P-waves are nonelliptically anisotropic. Numerical model data and physical model data demonstrate the performance of the algorithm, in each case recovering the original structure. Isotropic and anisotropic migration of anisotropic physical model data are compared experimentally, where the anisotropic velocity function of the medium has a vertical axis of symmetry. Only when anisotropic migration is used is the original structure recovered.

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