Vertical Symmetry Axis Media Research Articles

Illumination compensation for the adjoint-state first-arrival traveltime tomography in VTI media

The sensitivity kernel of each parameter is angle-dependent in an anisotropic traveltime multiparameter inversion. To analyze the angle illumination in a transversely isotropic media with a vertical symmetry axis (VTI) by the adjoint state traveltime tomography (AST) method, a circular observation system is designed with receivers radially distributed at different propagated angles. Then, the nonuniform angular illumination in the gradient of different parameters in the VTI AST is revealed. The nonuniform angular illumination seriously hinders the simultaneous updates of multiple parameters toward the true solution and decreases the accuracy of the anisotropic multiparameter inversion. To overcome the illumination problem, a preconditioned AST method (PAST) is proposed by utilizing the diagonal elements of the approximate Hessian matrix as the angle illumination compensation for multiparameter gradients in the VTI medium. The PAST method not only compensates for angular illumination but also balances the ray density and eliminates gradient singular values, which improves the inversion accuracy for multiple parameters. The BP synthetic model test shows that the PAST method based on angular illumination compensation can invert the VTI multiparameter models more accurately. Therefore, the PAST inversion model can predict first-arrival traveltime information more accurately. Field data application in the East China Sea indicates that the inverted velocity of the proposed PAST fits well with the logging velocity, and the inverted anisotropic parameter agrees well with the Backus average values. Consequently, the image quality based on the PAST inverted models is slightly improved. It is proven that the proposed PAST method can better realize the near-surface multiparameter inversion of the VTI medium.

Wave reflection and transmission coefficients for layered transversely isotropic media with vertical symmetry axis under initial stress

SUMMARYThe stress-dependent wave reflection and transmission (R/T) coefficients for the layered transversely isotropic media with a vertical symmetry axis (VTI) are seldom investigated in the published literature. To fill this gap, we propose the exact formulas for the plane wave R/T coefficients on the welded and non-welded boundaries between two distinct VTI half-spaces under the effect of initial stress. The theory of acoustoelasticity is used to describe the influence of initial applied stress on the overall elastic properties of VTI media and to represent two different boundary conditions. The normal and tangential compliances are used to characterize the discontinuity of non-welded boundary based on the linear-slip model, and their stress dependences are ideally considered according to the effect of stress on fracture aperture. Then the plane-wave displacement equations are substituted into the boundary conditions to yield the analytic formulas for frequency-independent R/T coefficients for the welded interface and frequency-dependent R/T coefficients for the non-welded interface. The stress-dependent wave slowness vector and polarization vector embedded in R/T coefficients can be directly computed with the Christoffel equation given by the acoustoelastic equation. Modelling results graphically show the effects of initial stress on the angle-dependent wave velocities, Thomsen elastic anisotropy parameters, fracture compliances, the R/T coefficients and seismic reflection responses for welded and non-welded interfaces in detail. The R/T coefficients are more sensitive to initial stress at relatively large incidence angles for the designed two-layer model with welded or non-welded boundary. The proposed R/T coefficient formulas and modelling results are relevant to in situ stress detection, fracture characterization, and exploration for oil and gas in shale reservoirs in high-stress fields.

Perturbation‐based traveltime approximation for two acoustic assumptions in the transversely isotropic media with a vertical symmetry axis

ABSTRACTThe accuracy of an explicit traveltime‐offset approximation affects the results of the velocity analysis that plays a crucial role in the seismic data processing. Seismic anisotropy is important to account for large offset and azimuth since it can provide detailed information comparing with the isotropic assumption. For the perturbation‐based method, different traveltime approximation forms result in different accuracy. It is necessary to find the optimal traveltime approximation for specific signs and magnitudes of the anellipticity and different offset ranges. In this paper, a series of perturbation‐based traveltime approximations from different acoustic assumptions and parameterizations are specified. The perturbation coefficients are derived from the corresponding acoustic eikonal equations. We test the accuracy of the defined perturbation approximations in four homogeneous transversely isotropic medium with vertical symmetry axis (VTI) medium with a different set of anisotropy parameters and one multilayered transversely isotropic medium with vertical symmetry axis medium. The sensitivity in anellipticity for different approximations is also analysed. We find that different approximation achieves different accuracy under the circumstance of specific offset range and the value of anellipticity. Therefore, the optimal approximation form with higher accuracy can be selected based on different offset ranges and the magnitudes of anellipticity.

Exact and approximate reflection/transmission responses from a layer containing vertical fractures

SUMMARYAnalyses of vertical fractures are of great interest in characterizing the fluid flow and minimum in situ stress direction in reservoirs. Long-wavelength equivalent orthorhombic (ORT) media typically characterize the anisotropy induced by a set of vertical parallel fractures or two sets of vertical and mutually orthogonal fractures embedded into a transversely isotropic medium with a vertical symmetry axis (VTI). Reflection and transmission (R/T) responses quantify wave amplitude variations in 1-D media and help to reveal the model property enclosing the heterogeneity. Conventionally, the R/T responses are analysed for an interface bounded by two half-spaces. However, for a plane wave travelling through a subsurface layer, the wave scattering effects at the top and bottom of the layer interact with each other. For a continuous infinite ORT space cut in two halves along the horizontal symmetry plane, we focus on the plane wave R/T responses from an ORT layer that is placed between the two halves, where the azimuths of the vertical symmetry plane in the layer and in the upper and lower half-spaces are identical. The R/T coefficient modelling method can be found in many publications for the ORT layer with an arbitrary finite thickness. We decompose the exact R/T coefficients into series expansions that correspond to different orders of intrabed multiples in the ORT layer. Under the weak-contrast assumption for the ORT half-spaces and the ORT layer, we use the anisotropic background medium to obtain the first-order R/T coefficient approximations and second-order reflectivity approximations. There is no constraint for the middle layer thickness in the obtained first-order reflectivity approximations. In the proposed first-order transmissivity approximations and second-order reflectivity approximations, the layer thickness is assumed to be thin to obtain appropriate approximations for a few wave modes. The isotropic background medium is also considered for weakly anisotropic models to obtain simpler approximations that facilitate parametric analyses. For the ORT layer with its thickness much smaller than the propagating wave's wavelength, the influences of the layer thickness on R/T coefficients can be inspected conveniently from the derived approximations. Particularly, the R/T coefficients are analysed for the model which would be a homogeneous VTI medium, if the vertical parallel fractures were absent from the middle layer. Numerical tests demonstrate that the proposed R/T coefficient approximations perform well for the thin ORT layer. The approximation accuracy decreases when the thin layer thickness increases.

A Concise Anisotropic Slope Tomography Based on 2-D Quasi-Acoustic Eikonal Equation for VTI Media

Slope tomography is an appealing tomographic method for velocity and anisotropic model building. We present a concise anisotropic slope tomography based on the 2-D quasi-acoustic eikonal equation in transversely isotropic (TI) media with a vertical symmetry axis (VTI). First, we establish the Frechet derivatives of the 2-D qP-wave in VTI stereotomography by applying the ray perturbation theory to the 2-D quasi-acoustic eikonal equation for VTI media. Second, we devise an anisotropic slope tomography instead of an anisotropic stereotomography based on the derived Frechet derivatives because slope tomography is more concise, as well as stable for real applications. Besides, we adopt a similar strategy that is recently developed, parsimonious adjoint slope tomography, to calculate the scatterer coordinates after the parameters of the macro model are updated. With the help of a kinematic remigration of invariants observed at the surface, we outline a concise anisotropic slope tomography in VTI media. Compared with the recently developed anisotropic adjoint slope tomography, we still establish the linear system based on the ray perturbation theory and solve it using a conjugate gradient algorithm. The synthetic and real data examples show that the anisotropic parameters and the scatterer coordinates can be updated alternately and effectively by combining the kinematic remigration in VTI media with a 2-D qP-wave slope tomography.

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.

Attenuation compensation for time-reversal imaging in VTI media

Time reversal is a key component in reverse-time migration (RTM) and source localization using wavefield extrapolation. The successful implementation of time reversal depends on the time symmetry (reversibility) of the wave equation in acoustic and elastic media. This symmetry in time, however, is no longer valid in attenuative media, and attenuation is often anisotropic. Here, we employ a viscoelastic anisotropic wave equation that decouples the influence of energy dissipation and velocity dispersion. That equation helps compensate for anisotropic attenuation and restore the time symmetry by changing the signs of the dissipation-dominated terms in time-reversed propagation, while keeping the dispersion-related terms unchanged. We test the [Formula: see text]-compensated time-reversal imaging algorithm on synthetic microseismic data from a 2D transversely isotropic medium with a vertical symmetry axis (VTI). After back-propagating multicomponent data acquired in a vertical borehole, we image microseismic sources using wavefield focusing. The source excitation times are estimated by picking the maximum amplitude of the squared shear strain component [Formula: see text] at the source locations. Accounting for attenuation anisotropy produces superior source images and more accurate excitation times compared to those obtained without attenuation compensation or with purely isotropic attenuation coefficients. The algorithm is also applied to a modified BP TI model to investigate the influence of such factors as survey geometry, errors in velocity and attenuation, noise, and limited aperture.

Three-parameter normal moveout correction in layered anisotropic media: A stretch-free approach

In conventional normal moveout (NMO) correction, some parts of the recorded data at larger offsets are discarded because of NMO distortions. Deviation from the true traveltime of reflections due to the anisotropy and heterogeneity of the earth, and wavelet stretching are two reasons of these distortions. The magnitudes of both problems increase with increasing the offset to depth ratio. Therefore, to be able to keep larger offsets of shallower reflections, both problems should be obviated. Accordingly, first, we have studied different traveltime approximations being in use, alongside new parameterizations for two classical functional equations, to select suitable equations for NMO correction. We numerically quantify the fitting accuracy and uncertainty of known nonhyperbolic traveltime approximations for P-waves in transversely isotropic media with vertical symmetry axis (VTI). We select three suitable three-parameter approximations for NMO in layered VTI media as the VTI generalized moveout approximation, a double-square-root approximation, and a perturbation-based approximation. Second, we have developed an extension of the earlier proposed stretch-free NMO method, using the selected moveout approximations. This method involves an automatic modification of the input parameters in anisotropic NMO correction, for selected reflections. Our anisotropic stretch-free NMO method is tested on synthetic and three real data sets from Gulf of Mexico and Iranian oil fields. The results verify the success of the method in extending the usable offsets, by generating flat and stretch-free NMO corrected reflections.

Anisotropic full-waveform inversion of crosshole seismic data: A vertical symmetry axis field data application

Anisotropic waveform tomography (AWT) uses anisotropic traveltime tomography followed by anisotropic full-waveform inversion (FWI). Such an approach is required for FWI in cases in which the geology is likely to exhibit anisotropy. An important anisotropy class is that of transverse isotropy (TI), and the special case of TI media with a vertical symmetry axis (VTI) media is often used to represent elasticity in undeformed sedimentary layering. We have developed an approach for AWT that uses an acoustic approximation to simulate waves in VTI media, and we apply this approach to crosshole data. In our approach, the best-fitting models of seismic velocity and Thomsen VTI anisotropy parameters are initially obtained using anisotropic traveltime tomography, and they are then used as the starting models for VTI FWI within the acoustic approximation. One common problem with the acoustic approach to TI media is the generation of late-arriving (spurious) S-waves as a by-product of the equation system. We used a Laplace-Fourier approach that effectively damps the spurious S-waves to suppress artifacts that might otherwise corrupt the final inversion results. The results of applying AWT to synthetic data illustrate the trade-offs in resolution between the two parameter classes of velocity and anisotropy, and they also verify anisotropic traveltime tomography as a valid method for generating starting models for FWI. The synthetic study further indicates the importance of smoothing the anisotropy parameters before proceeding to FWI inversions of the velocity parameter. The AWT technique is applied to real crosshole field gathers from a sedimentary environment in Western Canada, and the results are compared with the results from a simpler (elliptical) anisotropy model. The transversely isotropic approach yields an FWI image of the vertical velocity that (1) exhibits a superior resolution and (2) better predicts the field data than does the elliptical approach.

Reflection reconstruction and imaging from seismic noise data for VTI media

The application of seismic interferometry provides a new approach to obtain useful information of subsurface structures using noise data. In this study we investigate the feasibility of passive seismic interferometry in anisotropic media. We simulate long duration passive noise measurements for a model containing a transversely isotropic layer with vertical symmetry axis (VTI) using the staggered-grid finite-difference method. Then we extract the Green’s functions using the cross-correlation method. Even for VTI media, various reflections and multiples can be reconstructed, and the arrival times of these reflections correspond well with the directly modelled responses. By changing the noise source configuration and anisotropic parameters, we explore the impacts of these factors on the interferometry record. The results show that noise source configuration has considerable influence on the quality of the retrieved response, especially for S-wave reflection events. Since long time ago, passive measurements recorded from a number of noise sources provide sufficient stationary phase points to retrieve the reflections; the effect of anisotropic parameters is small. The Hess VTI model demonstrates that passive seismic interferometry is feasible to wide range of anisotropic parameters and complex subsurface structures. Moreover, a migrated image is obtained from the retrieved virtual source gathers using reverse time migration, and the subsurface structures are imaged well. The synthetic results demonstrate that passive seismic interferometry is applicable to anisotropic media and that the interferometry results can be used to image subsurface structures.

Wavelength-dependent Fresnel beam propagator and migration in VTI media

Ray-based wave propagators are widely applied in seismic migration due to implementation flexibility and computational efficiency. The classic ray theory that under the high-frequency assumption requires sufficiently smooth velocity models, which limits the applications of ray-based methods because seismic waves are band-limited. Besides, it is desirable to extend seismic propagation and migration to general anisotropic case since the reality of subsurface. We adapt a wavelength-dependent smoothing (WDS) operator for transversely isotropic media with a vertical symmetry axis (VTI), which considers both characteristics of band-limited wave propagation and local anisotropic heterogeneity. Frequency-dependent traveltimes are computed with the WDS models by using an anisotropic dynamic programming approach. Then, a wavelength-dependent Fresnel beam propagator is constructed based on the frequency-dependent traveltimes. Analysis of traveltime fields demonstrates that wavelength-dependent Fresnel beam propagator can provide accurate wave propagating directions and traveltimes. We develop a wavelength-dependent Fresnel beam migration (WDFBM) method based on the wavelength-dependent Fresnel beam propagator for VTI media, which generates angle domain common imaging gathers (ADCIGs) efficiently. Numerical examples are included to demonstrate the effectiveness of our approach, which shows its potential application for subsequent velocity estimation via ADCIGs.

Skeletonized wave-equation inversion in vertical symmetry axis media without too much math

We have developed a tutorial for skeletonized inversion of pseudo-acoustic anisotropic vertical symmetry axis (VTI) data. We first invert for the anisotropic models using wave-equation traveltime inversion. Here, the skeletonized data are the traveltimes of transmitted and/or reflected arrivals that lead to simpler misfit functions and more robust convergence compared with full-waveform inversion. This provides a good starting model for waveform inversion. The effectiveness of this procedure is illustrated with synthetic data examples and a marine data set recorded in the Gulf of Mexico.

An acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis

Seismic-wave attenuation is an important component of describing wave propagation. Certain regions, such as gas clouds inside the earth, exert highly localized attenuation. In fact, the anisotropic nature of the earth induces anisotropic attenuation because the quasi P-wave dispersion effect should be profound along the symmetry direction. We have developed a 2D acoustic eikonal equation governing the complex-valued traveltime of quasi P-waves in attenuating, transversely isotropic media with a vertical-symmetry axis (VTI). This equation is derived under the assumption that the complex-valued traveltime of quasi P-waves in attenuating VTI media are independent of the S-wave velocity parameter [Formula: see text] in Thomsen’s notation and the S-wave attenuation coefficient [Formula: see text] in Zhu and Tsvankin’s notation. We combine perturbation theory and Shanks transform to develop practical approximations to the acoustic attenuating eikonal equation, capable of admitting an analytical description of the attenuation in homogeneous media. For a horizontal-attenuating VTI layer, we also derive the nonhyperbolic approximations for the real and imaginary parts of the complex-valued reflection traveltime. These equations reveal that (1) the quasi SV-wave velocity and the corresponding quasi SV-wave attenuation coefficient given as part of Thomsen-type notation barely affect the ray velocity and ray attenuation of quasi P-waves in attenuating VTI media; (2) combining the perturbation method and Shanks transform provides an accurate analytic eikonal solution for homogeneous attenuating VTI media; (3) for a horizontal attenuating VTI layer with weak attenuation, the real part of the complex-valued reflection traveltime may still be described by the existing nonhyperbolic approximations developed for nonattenuating VTI media, and the imaginary part of the complex-valued reflection traveltime still has the shape of nonhyperbolic curves. In addition, we have evaluated the possible extension of the proposed eikonal equation to realistic attenuating media, an alternative perturbation solution to the proposed eikonal equation, and the feasibility of applying the proposed nonhyperbolic equation for the imaginary part of the complex-valued traveltime to invert for interval attenuation parameters.

Separating quasi-P-wave in transversely isotropic media with a vertical symmetry axis by synthesized pressure applied to ocean-bottom seismic data elastic reverse time migration

Quasi-P (qP)-wavefield separation is a crucial step for elastic P-wave imaging in anisotropic media. It is, however, notoriously challenging to quickly and accurately obtain separated qP-wavefields. Based on the concepts of the trace of the stress tensor and the pressure fields defined in isotropic media, we have developed a new method to rapidly separate the qP-wave in a transversely isotropic medium with a vertical symmetry axis (VTI) by synthesized pressure from ocean-bottom seismic (OBS) data as a preprocessing step for elastic reverse time migration (ERTM). Another key aspect of OBS data elastic wave imaging is receiver-side 4C records back extrapolation. Recent studies have revealed that receiver-side tensorial extrapolation in isotropic media with ocean-bottom 4C records can sufficiently suppress nonphysical waves produced during receiver-side reverse time wavefield extrapolation. Similarly, the receiver-side 4C records tensorial extrapolation was extended to ERTM in VTI media in our studies. Combining a separated qP-wave by synthesizing pressure and receiver-side wavefield reverse time tensorial extrapolation with the crosscorrelation imaging condition, we have developed a robust, fast, flexible, and elastic imaging quality improved method in VTI media for OBS data.

Long-offset moveout for VTI using Padé approximation

The approximation of normal moveout is essential for estimating the anisotropy parameters of the transversally isotropic media with vertical symmetry axis (VTI). We have approximated the long-offset moveout using the Padé approximation based on the higher order Taylor series coefficients for VTI media. For a given anellipticity parameter, we have the best accuracy when the numerator is one order higher than the denominator (i.e., [[Formula: see text]]); thus, we suggest using [4/3] and [7/6] orders for practical applications. A [7/6] Padé approximation can handle a much larger offset and stronger anellipticity parameter. We have further compared the relative traveltime errors between the Padé approximation and several approximations. Our method shows great superiority to most existing methods over a wide range of offset (normalized offset up to 2 or offset-to-depth ratio up to 4) and anellipticity parameter (0–0.5). The Padé approximation provides us with an attractive high-accuracy scheme with an error that is negligible within its convergence domain. This is important for reducing the error accumulation especially for deeper substructures.

Research Note: Insights into the data dependency on anisotropy: an inversion prospective

ABSTRACTWhile velocity contrasts are responsible for most of the events recorded in our data, the long wavelength behavior of the velocity model is responsible for the geometrical shape of these events. For isotropic acoustic materials, the wave dependency on the long (wave propagation) and short (scattering) wavelength velocity components is stationary with the propagation angle. On the other hand, in representing a transversely isotropic with a vertical symmetry axis medium with the normal moveout velocity, the anellepticity parameter η, the vertical scaling parameter δ, and the sensitivity of waves vary with the polar angle for both the long and short wavelength features of the anisotropic dimensionless medium parameters (δ and η). For horizontal reflectors at reasonable depths, the long wavelength features of the η model is reasonably constrained by the long offsets, whereas the short wavelength features produce very week reflections at even reasonable offsets. Thus, for surface acquired seismic data, we could mainly invert for smooth η responsible for the geometrical shape of reflections. On the other hand, while the δ long wavelength components mildly affects the recorded data, its short wavelength variations can produce reflections at even zero offset, with a behavior pattern synonymous to density. The lack of the long wavelength δ information will mildly effect focusing but will cause misplacement of events in depth. With low enough frequencies (very low), we may be able to recover the long wavelength δ using full waveform inversion. However, unlike velocity, the frequencies needed for that should be ultra‐low to produce long‐wavelength scattering‐based model information as δ perturbations do not exert scattering at large offsets. For a combination given by the horizontal velocity, η, and ε, the diving wave influence of η is absorbed by the horizontal velocity, severely limiting the η influence on the data and full waveform inversion. As a result, with a good smooth η estimation, for example, from tomography, we can focus the full waveform inversion to invert for only the horizontal velocity and maybe ε as a parameter to fit the amplitude. This is possibly the most practical parametrization for inversion of surface seismic data in transversely isotropic with vertical symmetry axis media.

An efficient symplectic reverse time migration method using a local nearly analytic discrete operator in acoustic transversely isotropic media with a vertical symmetry axis

Reverse time migration (RTM) has become an extremely important imaging method due to its ability of handling complicated velocity models. In recent years, there has been an increasing need to perform RTM with anisotropic velocity models such as transversely isotropic media with a vertical symmetry axis (VTI). Finite difference (FD) is one of the most widely used numerical methods in RTM. However, it often suffers from serious numerical dispersion, resulting in significant loss of imaging resolution. Over the years, a nearly analytic discrete operator has been developed to approximate the high-order spatial partial differential operators, which is effective in suppressing numerical dispersion. Using this operator, we developed a nearly analytic symplectic (NAS) method to numerically solve acoustic VTI wave equations, which preserves the symplectic structure. We have investigated its properties including the stability condition, numerical dispersion relation, and computational efficiency. Also, we applied the NAS method to solve wave equations in RTM in acoustic VTI media. This new method not only reduced numerical dispersion and improved computational efficiency, compared to the conventional numerical methods such as the Lax-Wendroff correction (LWC) method, but it also was symplectic. Our forward-modeling results showed that the computational efficiency of NAS is about four times and its overall memory requirement is about 75% of those of LWC. The RTM results of the 2D prestack Hess acoustic VTI model showed that RTM by using NAS obtained better image quality than that by using LWC when coarse grids were used, which demonstrated that the NAS method can be promising in large-scale anisotropic RTM.

Low-frequency layer-induced anisotropy

It is often useful to compare velocities estimated from seismic experiments with those measured by well logs at much smaller scales and higher frequencies. In the presence of fine layering, seismic velocities and anisotropy are scale- and frequency-dependent and upscaling of well logs is necessary to allow comparison. When upscaling well-log data measured at sonic frequencies, we assumed a layered medium with layer thickness given by logging step. Standard upscaling gives the exact solution for effective properties of a layered medium assuming that all constituents of the medium are linearly elastic and there are no anelastic energy losses. This method is static, meaning that the solution is obtained in the zero frequency limit. We expand upscaling to low frequencies and propose to use a seismic wavelet estimate for weighted averaging of effective properties. By expanding the logarithm of the propagator matrix computed from the stack of horizontal transversely isotropic layers with a vertical symmetry axis (VTI) in series with frequency and keeping the first three terms in this series, we obtain the low-frequency extension of Backus upscaling. The effective medium properties computed for individual nonzero frequencies correspond to a nonphysical medium with a vertical symmetry axis. To preserve the VTI symmetry, the effective slowness surface obtained for each individual frequency is approximated by a VTI medium by fitting the coefficients of the Taylor series derived from corresponding eigenvalues of the effective system matrix and similar series obtained for the vertical slownesses of different wave modes. The frequency-dependent anisotropy parameters are obtained from Taylor series coefficients for individual wave modes. Afterwards, these coefficients are averaged with frequencies weighted by the spectrum of a seismic wavelet extracted at the corresponding depth interval. The proposed averaging technique is data-driven and takes into account the low-frequency behavior of seismic waves with near-vertical propagation.

Accuracy comparison of nonhyperbolic moveout approximations for qP-waves in VTI media

There are many non-hyperbolic moveout approximations for VTI media with various accuracy claims. We perform an analytical and numerical comparison of the 11 moveout approximations developed for a homogeneous transversely isotropic medium with vertical symmetry axis (VTI). The comparison is based on evaluating their long-offset asymptotes and higher-order Taylor series coefficients computed for a VTI medium. To make the different approximations comparable we select approximation coefficients to preserve the fourth-order coefficient in Taylor series for traveltime squared. To make the comparison simpler, we define the traveltime as a function of normalized offset and one parameter which is the anelliptical parameter. We select the six best approximations and perform comparative analysis by plotting the asymptotic traveltime parameters and high-order Taylor coefficients versus anelliptical parameter. We also compute the relative error in traveltime for selected approximations for homogeneous and horizontally layered VTI models.

Wave Equation Prestack Depth Migration Method in 3D VTI Media

AbstractBased on the 3‐D dispersion equation for the transversely isotropic media with a vertical symmetry axis (VTI), we proposed one‐way operators in the wavenumber‐space domain such that the terms that contain spatial coordinates and the terms that contain wavenumbers are separated. The coefficients in one‐way operators are determined using an optimization scheme for more accurately matching to the exact phase‐shift operator with a wide‐angle. The resulting 3‐D VTI wave equation prestack migration algorithm can accommodate a wide range of anisotropy rather than the weak anisotropy. For its practical aspects, we also present a frequency‐dependent varying‐step depth extrapolation scheme for reducing the computational cost of wavefield depth extrapolation and an anti‐alias one‐way propagator aiming at imaging the steep dipping structures with 3‐D sparsely sampling dataset in 3‐D VTI media. The proposed 3‐D VTI wave equation prestack migration scheme is demonstrated by SEG 2‐D Hess VTI dataset and a 3‐D migration impulse response with its layout resulting from the 3‐D field data.