Article

Prestack Gaussian-beam depth migration in anisotropic media

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Abstract

Gaussian-beam depth migration is a useful alternative to Kirchhoff and wave-equation migrations. It overcomes the limitations of Kirchhoff migration in imaging multipathing arrivals, while retaining its efficiency and its capability of imaging steep dips with turning waves. Extension of this migration method to anisotropic media has, however, been hampered by the difficulties in traditional kinematic and dynamic ray-tracing systems in inhomogeneous, anisotropic media. Formulated in terms of elastic parameters, the traditional anisotropic ray-tracing systems aredifficult to implement and inefficient for computation, especially for the dynamic ray-tracing system. They may also result inambiguity in specifying elastic parameters for a given medium.To overcome these difficulties, we have reformulated the ray-tracing systems in terms of phase velocity.These reformulated systems are simple and especially useful for general transversely isotropic and weak orthorhombic media, because the phase velocities for these two types of media can be computed with simple analytic expressions. These two types of media also represent the majority of anisotropy observed in sedimentary rocks. Based on these newly developed ray-tracing systems, we have extended prestack Gaussian-beam depth migration to general transversely isotropic media. Test results with synthetic data show that our anisotropic, prestack Gaussian-beam migration is accurate and efficient. It produces images superior to those generated by anisotropic, prestack Kirchhoff migration.

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... Asterisk * denotes complex conjugate. These quantities are computed by Gaussian beam ray tracing (Hill 2001) modified for P-and S-wave anisotropy (Zhu, Gray, and Wang 2007). Hill (2001) has given prescriptions for forming slant stacks D h and specifying beam centre locations L. ...
... In addition, the full elastic version of the eikonal equation is cumbersome for ray tracing, requiring evaluation of complicated right-hand-side functions and solution of an eigenvalue problem at each ray step. Zhu et al. (2007) overcome these difficulties for P-wave ray tracing by basing the eikonal equation on phase velocity, for which there exist simple analytical expressions involving Thomsen parameters for transverse isotropic (TI) and orthorhombic media. We follow this approach to obtain the governing ray equations for SV-waves in inhomogeneous TI media. ...
Article
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Gaussian Beam depth (GBM) migration overcomes the single-wavefront limitation of the majority of Kirchhoff implementations, and constitutes a cost-effective alternative to full-wavefield imaging methods such as reverse time migration. Common-offset (CO) beam migration was originally derived to exploit the symmetries available in marine towed-streamer acquisition. However, sparse acquisition geometries, such as cross-spread and ocean bottom, do not easily accommodate the requirements for common-azimuth migration. Seismic data interpolation or regularization can be used to mitigate this problem and to form well-populated common offset-vector (COV) tiles. Unfortunately, this procedure is computationally intensive and can, in the case of converted-wave imaging with sparse receivers, compromise the final image resolution. For all these reasons, we introduce a common-shot (or common-receiver) controlled-beam migration (CBM) implementation which allows the migration of datasets particularly rich in azimuth, without any regularization pre-processing required. CBM is a specialized version of GBM aimed at signal-to-noise ratio enhancement. Using some examples, we demonstrate that PS-imaging of ocean bottom-node (OBN) data benefits from this formulation, particularly in the shallow subsurface where regularization is both most necessary and most challenging.
... eat significance to study the migration algorithm in TI media, which can eliminate the influence of anisotropy and realise accurate images of subsurface structures. Gaussian beam migration for anisotropy has been developed in the past few years. Alkhalifah (1995) first proposed a Gaussian beam poststack depth migration method for anisotropic media. Zhu et al. (2007) presented a prestack Gaussian beam depth migration method in anisotropic media and applied it with encouraging results to synthetic data. Based on the study of Gaussian beam poststack depth migration for anisotropic media (Alkhalifah, 1995), an anisotropic Gaussian beam prestack depth migration (GB- PSDM) method is presented in this pap ...
... Based on the study of Gaussian beam poststack depth migration for anisotropic media (Alkhalifah, 1995), an anisotropic Gaussian beam prestack depth migration (GB- PSDM) method is presented in this paper. Unlike the method of Zhu et al. (2007), the anisotropic kinematic and dynamic ray tracing in our migration method are not formulated in terms of phase velocity and group velocity. The purpose of this study is to provide an accurate migration method for P-wave data in TI media. ...
Article
Full-text available
The transversely isotropic (TI) media approximation is commonly applied to assist in the processing of seismic data acquired in sedimentary environments. Based on anisotropic kinematic and dynamic ray tracing systems, a P-wave Gaussian beam prestack depth migration (GB-PSDM) method for TI media is introduced in this paper. The imaging principle of anisotropic GB-PSDM and the corresponding migration parameters are presented on the basis of the GB-PSDM method in isotropic media. Tests of synthetic and field seismic data show that the method is an accurate and efficient anisotropic prestack depth migration method in TI media.
... Asterisk * denotes complex conjugate. These quantities are computed by Gaussian beam ray tracing (Hill 2001) modified for P-and S-wave anisotropy (Zhu, Gray, and Wang 2007). Hill (2001) has given prescriptions for forming slant stacks D h and specifying beam centre locations L. ...
... In addition, the full elastic version of the eikonal equation is cumbersome for ray tracing, requiring evaluation of complicated right-hand-side functions and solution of an eigenvalue problem at each ray step. Zhu et al. (2007) overcome these difficulties for P-wave ray tracing by basing the eikonal equation on phase velocity, for which there exist simple analytical expressions involving Thomsen parameters for transverse isotropic (TI) and orthorhombic media. We follow this approach to obtain the governing ray equations for SV-waves in inhomogeneous TI media. ...
Article
Gaussian beam depth migration overcomes the single‐wavefront limitation of most implementations of Kirchhoff migration and provides a cost‐effective alternative to full‐wavefield imaging methods such as reverse‐time migration. Common‐offset beam migration was originally derived to exploit symmetries available in marine towed‐streamer acquisition. However, sparse acquisition geometries, such as cross‐spread and ocean bottom, do not easily accommodate requirements for common‐offset, common‐azimuth (or common‐offset‐vector) migration. Seismic data interpolation or regularization can be used to mitigate this problem by forming well‐populated common‐offset‐vector volumes. This procedure is computationally intensive and can, in the case of converted‐wave imaging with sparse receivers, compromise the final image resolution. As an alternative, we introduce a common‐shot (or common‐receiver) beam migration implementation, which allows migration of datasets rich in azimuth, without any regularization pre‐processing required. Using analytic, synthetic, and field data examples, we demonstrate that converted‐wave imaging of ocean‐bottom‐node data benefits from this formulation, particularly in the shallow subsurface where regularization for common‐offset‐vector migration is both necessary and difficult.
... In P-wave (acoustic) imaging, isotropic imaging algorithms have been mostly developed for TTI and vertical transverse isotropy (VTI) media (Alkhalifah & Fomel, 2009;Behera, Khare, & Sarkar, 2011;Koren, Ravve, & Levy, 2010;Zhu, Gray, & Wang, 2007). Transverse isotropy (TI), the simplest form of anisotropy, exists when thin bed sequences, perpendicular to the symmetry axis, are isotropic. ...
Book
Seismic Imaging Methods and Application for Oil and Gas Exploration connects the legacy of field data processing and imaging with new research methods using diffractions and anisotropy in the field of geophysics. Topics covered include seismic data acquisition, seismic data processing, seismic wave modeling, high-resolution imaging, and anisotropic modeling and imaging. This book is a necessary resource for geophysicist working in the oil and gas and mineral exploration industries, as well as for students and academics in exploration geophysics.
... Ray tracing system in isotropic media has a simple form. With the phase velocity in TI media, it could be extended into TI media (Cerveny, 2001;Zhu et al., 2007). But it computes traveltime along the ray path which needs to be interpolated into the computing grid for depth imaging. ...
... ation (GBM) (Hill, 1990Hill, ,2001 Hale, 1992;) is developed as a ray-based depth imaging method whose accuracy rivals that of wave equation migration method. Besides, the anisotropy is inherently a scale-dependent property in seismic exploration, which should be considered because the long offset and wide azimuth acquisition system is widely used. Zhu et al. (2007) extend the GBM to general transversely isotropic (TI) media based on a developed ray tracing system. However, the choice of initial beam width is quasi-empirical, which limits the accuracy of GBM. Alternatively, the beam width of GBM has no definitely physical meaning. From the physical point of view, the spatial vicinity of the ray nam ...
Conference Paper
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In order to overcome the shortcomings of conventional Kirchhoff prestack depth migration (PSDM) and Gaussian beam migration (GBM), a new Fresnel beam migration (FBM) method is proposed by introducing the traveltime field produced by dynamic programming approach in anisotropic media. The beam width (Fresnel zone) of our method has a more definite physical meaning than GBM, which achieve the higher accuracy because the traveltime calculation method has no limitation on large velocity contrast. The major steps of FBM are traveltime computation, the central ray trajectory tracing, Fresnel zone calculation and wavefield back-propagation. The proposed method resolves spatial ambiguities caused by elliptical traveltime isochrones summation with limited aperture, and it is an economic way to generate angle gathers for velocity model building. The numerical examples demonstrate the accuracy and effectiveness of this method.
... There are many researches for traveltime calculation in anisotropic media (Cerveny, 1972;Shearer and Chapman, 1988;Qian and Symes, 2002;Kumar et al., 2004;. Zhu et al. (2007) extended the GBM to general transversely isotropic (TI) media based on a developed ray tracing system. Vanelle and Gajewski (2013) introduce the trueamplitude Kirchhoff migration based on the traveltime in anisotropic media. ...
... Besides, the anisotropy as an inherently scale-dependent property of subsurface strata should be considered because the long offset and wide azimuth acquisition system is widely used. Zhu et al. (2007) extended the GBM to general transversely isotropic (TI) media based on a developed ray tracing system. Vanelle and Gajewski (2013) introduce the true-amplitude KPSDM based on the traveltime in anisotropic media. In fact, to adapt the ray-based depth migration methods to anisotropic media is not very difficult. In this paper, the charact ...
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A characteristic wave decomposition (CWD) method is presented in this paper, which can express the seismic data sparsely. We perform CWD by double beam forming for a compressed seismic wave-field and its index. Based on the beam-formed wave-field in characteristic domain, a beam-based characteristic wave imaging method (CWI) is put forward. Due to the flexibility and efficiency to generate angle gathers for velocity model building, the CWI is a useful alternative to Kirchhoff and wave-equation migrations. Extension of CWI to anisotropic media by introducing ray tracing system in transverse isotropic with a vertical symmetry (VTI) media achieve a better imaging result. The CWI has a theoretical speedup of 1~2 orders of magnitude over the conventional Kirchhoff migration methods. Besides, it can handle low signal to noise ratio data and target oriented imaging conveniently, and angle gathers can be produced naturally by CWI. Consequently, the CWI is an efficient technique for large scale seismic imaging and 3D velocity model building.
... Compensating attenuation-associated effects can be accomplished by replacing the real-valued velocity with a complex velocity (Traynin et al. 2008). To extend to anisotropic media, fast kinematic and dynamic ray-tracing are the key steps for Gaussian beam forward and adjoint operators (Alkhalifah 1995;Červený 2003;Zhu et al. 2007;Casasanta and Gray 2015b). In an anisotropic media, both halfopening angle and azimuth angle should be considered because of varying velocities at different azimuths, and the linearized modeling and the adjoint migration operator becomes and † , respectively. ...
Article
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Compared with traditional adjoint-based migration, least-squares migration (LSM) can reduce finite-frequency effects, remove acquisition footprints and improve spatial resolution by solving a linear inverse problem for subsurface reflectivity. One important requirement for the success of LSM is having an accurate migration velocity model. Because of low signal-to-noise ratio (SNR), inaccurate traveltime picking, lack of low-frequency signals and limited acquisition aperture, it is still challenging to build an accurate velocity model using ray-based tomography or full waveform inversion. LSM with large velocity errors results in erroneous reflector locations, strong swing artifact and even non-convergence. To mitigate these issues, we develop a novel least-squares imaging framework in the subsurface half-opening angle domain. Instead of using high-wavenumber velocity perturbations as the reflectivity model as in traditional LSM, we parameterize the wave equation with an angle-dependent reflectivity, and derive the corresponding linearized forward modeling and adjoint migration operators. Because Gaussian Beam migration naturally incorporates propagation directions in wavefield extrapolation, we compute the Green’s function using the Gaussian beam summation method. To improve the common-image gather (CIG) quality for low-fold and low-SNR data, a shaping regularization over the half-opening angles is introduced in the conjugate gradient scheme to iteratively update the angle-dependent reflectivity model. A flattening-enhanced summation is used to compute the stacked images by accounting for the depth moveout of CIGs caused by velocity errors, and produces constructive stacking results. Numerical experiments for benchmark models and a land survey demonstrate that the proposed method can improve LSM convergence and produce high-quality angle-dependent and stacked images even with inaccurate migration velocity models.
... The first extension to the anisotropic model was developed by Alkhalifah (1995), where beam migration is performed for post-stack data. Subsequently Zhu et al. (2007) created an anisotropic beam migration for pre-stack data. In addition, Neal et al. (2009) investigated anisotropic beam depth migration in the deep water in the Gulf of Mexico. ...
Article
An approach for true-amplitude seismic beam imaging of multicomponent seismic data in 2-D anisotropic elastic media is presented and discussed. Here, the recovered true-amplitude function is a scattering potential. This approach is amigration procedure based on the weighted summation of pre-stack data. The true-amplitude weights are computed by applying Gaussian beams (GBs). We shoot a pair of properly chosen GBs with a fixed dip and opening angles from the current imaging point towards an acquisition system. This pair of beams is used to compute a true-amplitude selective image of a rapid velocity variation. The total trueamplitude image is constructed by superimposing selective images computed for a range of available dip angles. The global regularity of the GBs allows one to disregard whether a ray field is regular or irregular. P- and S-wave GBs can be used to handle raw multicomponent data without separating the waves. The use of anisotropic GBs allows one to take into account the anisotropy of the background model.
... Most isotropic time-and depth-migration algorithms ͑Kirchhoff, Stolt, phase-shift, phase-shift-plus-interpolation ͑PSPI͒, Gaussian beam, finite-difference, etc.͒ have been generalized for VTI and, in many cases, for TTI media ͑e.g., Sena and Toksöz, 1993;Anderson et al., 1996;Alkhalifah, 1997;Ren et al., 2005;Zhu et al., 2007a͒. The key issue in anisotropic processing, however, is reliable estimation of the velocity model from reflection data combined with borehole and other information. ...
... Vestrum et al. (1999) and Grech et al. (2002) showed that the positioning and continuity of image structures below TTI strata can be improved, i.e. anisotropic images can be better focused, by anisotropic depth migration. Zhu et al. (2007) proposed a reformulated prestack Gaussian-beam depth migration (GBM) to restore the images of a step structure overlain by a TTI medium model and a dipping anisotropic thrust sheet. The algorithm of the modified prestack GBM was reformulated based on raytracing systems in terms of phase velocity. ...
Article
Full-text available
In active tectonic regions, the primary formations are often tilted and subjected to the processes of folding and/or faulting. Dipping formations may be categorised as tilted transverse isotropy (TTI). While carrying out hydrocarbon exploration in areas of orogenic structures, mispositioning and defocusing effects in apparent reflections are often caused by the tilted transverse isotropy of the overburden. In this study, scaled physical modelling was carried out to demonstrate the behaviours of seismic wave propagation and imaging problems incurred by transverse isotropic (TI) overburdens that possess different orientations of the symmetry axis. To facilitate our objectives, zero-offset reflections were acquired from four stratum-fault models to image the same structures that were overlain by a TI (phenolite) slab. The symmetry axis of the TI slab was vertical, tilted or horizontal. In response to the symmetry axis orientations, spatial shifts and asymmetrical diffraction patterns in apparent reflections were observed in the acquired profiles. Given the different orientations of the symmetry axis, numerical manipulations showed that the imaged events could be well described by theoretical ray paths computed by the trial-and-error ray method and Fermat's principle (TERF) method. In addition, outputs of image restoration show that the imaging problems, i.e. spatial shift in the apparent reflections, can be properly handled by the ray-based anisotropic 2D Kirchhoff time migration (RAKTM) method.
... There is a rich literature on the use of Gaussian beam summation in computing high-frequency seismic wavefields in smoothly varying inhomogeneous media and on the successful applications of Gaussian beams to seismic depth migration (Červený, Popov, and Pšenčík 1982;Popov 1982;Nowack and Aki 1984;Červený 1985;Babich and Popov 1989;Hill 1990Hill , 2001Nowack 2003;Gray 2005;Červený, Klimeš, and Pšenčík 2007;Zhu, Gray, and Wang 2007;Gray and Bleistein 2009;Popov et al. 2010). The beam solution is usually formed in the frequency domain, and it is used to decompose the space-frequency domain seismic data into Gaussian beams that are localized both in position and direction. ...
Article
We present a Gaussian packet migration method based on Gabor frame decomposition and asymptotic propagation of Gaussian packets. A Gaussian packet has both Gaussian-shaped time–frequency localization and space–direction localization. Its evolution can be obtained by ray tracing and dynamic ray tracing. In this paper, we first briefly review the concept of Gaussian packets. After discussing how initial parameters affect the shape of a Gaussian packet, we then propose two Gabor-frame-based Gaussian packet decomposition methods that can sparsely and accurately represent seismic data. One method is the dreamlet–Gaussian packet method. Dreamlets are physical wavelets defined on an observation plane and can represent seismic data efficiently in the local time–frequency space–wavenumber domain. After decomposition, dreamlet coefficients can be easily converted to the corresponding Gaussian packet coefficients. The other method is the Gabor-frame Gaussian beam method. In this method, a local slant stack, which is widely used in Gaussian beam migration, is combined with the Gabor frame decomposition to obtain uniform sampled horizontal slowness for each local frequency. Based on these decomposition methods, we derive a poststack depth migration method through the summation of the backpropagated Gaussian packets and the application of the imaging condition. To demonstrate the Gaussian packet evolution and migration/imaging in complex models, we show several numerical examples. We first use the evolution of a single Gaussian packet in media with different complexities to show the accuracy of Gaussian packet propagation. Then we test the point source responses in smoothed varying velocity models to show the accuracy of Gaussian packet summation. Finally, using poststack synthetic data sets of a four-layer model and the two-dimensional SEG/EAGE model, we demonstrate the validity and accuracy of the migration method. Compared with the more accurate but more time-consuming one-way wave-equation-based migration, such as beamlet migration, the Gaussian packet method proposed in this paper can correctly image the major structures of the complex model, especially in subsalt areas, with much higher efficiency. This shows the application potential of Gaussian packet migration in complicated areas.
... The Gaussian beam method for the research of wave-propagation phenomena in rather complicated geophysical models are broadly applied in seismic depth imaging owing to its effectiveness and not so time-consuming numerical procedures with low memory requirements (Hill, 1990(Hill, , 2001Hale, 1992;Gray et al., 2005Gray et al., , 2009. To date, it has been effectively extended to complex media, including elastic and anisotropic models (Protasov et al., 2012;Alkhalifah, 1995;Zhu et al., 2007), which are used to obtain a better final pre-stack depth image based on the algorithm in frequency domain. Adhering to the basic framework of Gaussian beam migration (GBM), in addition, numerous novel seismic beam techniques have been investigated (Nowack, 2008(Nowack, , 2011Huang et al. 2015), which provide several excellent alternatives to enhance the imaging accuracy or computational efficiency of beam migration algorithms in frequency domain. ...
... otropic media have been published (Gray 2005; Gray andBleistein 2009;Popov et al. 2010;Yue et al. 2010;Han et al. 2013Han et al. , 2014a). Meanwhile, Gaussian beam migration in anisotropic media was studied. The first extension to the anisotropic media was developed byAlkhalifah (1995), where Gaussian beam migration is performed for poststack data.Zhu et al. (2007)proposed a prestack Gaussian beam depth migration method in anisotropic media.Han et al. (2014b)presented a converted wave Gaussian beam migration method for TI media and applied it with encouraging results to synthetic data. Protasov (2015) presented a true-amplitude Gaussian beam imaging method of multicomponent seismic data in anisotr ...
Article
Full-text available
An approach for extracting angle-domain common-image gathers (ADCIGs) from anisotropic Gaussian beam prestack depth migration (GB-PSDM) is presented in this paper. The propagation angle is calculated in the process of migration using the real-value traveltime information of Gaussian beam. Based on the above, we further investigate the effects of anisotropy on GB-PSDM, where the corresponding ADCIGs are extracted to assess the quality of migration images. The test results of the VTI syncline model and the TTI thrust sheet model show that anisotropic parameters ε, δ, and tilt angle 𝜃, have a great influence on the accuracy of the migrated image in anisotropic media, and ignoring any one of them will cause obvious imaging errors. The anisotropic GB-PSDM with the true anisotropic parameters can obtain more accurate seismic images of subsurface structures in anisotropic media.
... Alkhalifah (1995) first proposed a Gaussian beam poststack depth migration method for anisotropic media. Zhu et al. (2007) presented a prestack Gaussian beam depth migration method in anisotropic media and applied it with encouraging results to synthetic data. However, these studies of Gaussian beam migration are only applied to acoustic wave migration. ...
Article
Full-text available
Increasing amounts of multi-component seismic data are being acquired on land and offshore because more complete seismic wavefield information is beneficial for structural imaging, fluid detection, and reservoir monitoring. S-waves are typically influenced more by anisotropy in a medium than are P-waves; as a result, the anisotropy cannot be ignored during the converted PS-wave imaging. Gaussian beam migration, an elegant and efficient depth migration method, is becoming a new topic in the study of PS-wave migration; its accuracy is comparable to that of wave-equation migration, and its flexibility is comparable to that of Kirchhoff migration. In this paper, we introduce an anisotropic Gaussian beam prestack depth migration (GB-PSDM) method for the converted PS-wave, in which the anisotropic media can be a transversely isotropic (TI) medium with a vertical or tilted symmetry axis. We present the PS-wave common shot gathers GB-PSDM imaging condition and derive the ray tracing of P- and SV-waves in two-dimensional TI media. The migration impulse responses of P- and SV-propagation modes in TI media with both vertical and tilted symmetry axes are presented. The results of numerical examples indicate that the method introduced here offers significant improvements in the quality of converted PS-wave imaging compared with an isotropic algorithm.
... Most isotropic time-and depth-migration algorithms ͑Kirchhoff, Stolt, phase-shift, phase-shift-plus-interpolation ͑PSPI͒, Gaussian beam, finite-difference, etc.͒ have been generalized for VTI and, in many cases, for TTI media ͑e.g., Sena and Toksöz, 1993;Anderson et al., 1996;Alkhalifah, 1997;Ren et al., 2005;Zhu et al., 2007a͒. The key issue in anisotropic processing, however, is reliable estimation of the velocity model from reflection data combined with borehole and other information. ...
Article
Full-text available
Recent advances in parameter estimation and seismic pro-cessing have allowed incorporation of anisotropic models into a wide range of seismic methods. In particular, vertical and tilted transverse isotropy are currently treated as an inte-gral part of velocity fields employed in prestack depth migra-tion algorithms, especially those based on the wave equation. We briefly review the state of the art in modeling, processing, and inversion of seismic data for anisotropic media. Topics include optimal parameterization, body-wave modeling methods, P-wave velocity analysis and imaging, processing in the -p domain, anisotropy estimation from vertical-seis-mic-profiling VSP surveys, moveout inversion of wide-azi-muth data, amplitude-variation-with-offset AVO analysis, processing and applications of shear and mode-converted waves, and fracture characterization. When outlining future trends in anisotropy studies, we emphasize that continued progress in data-acquisition technology is likely to spur tran-sition from transverse isotropy to lower anisotropic symme-tries e.g., orthorhombic. Further development of inversion and processing methods for such realistic anisotropic models should facilitate effective application of anisotropy parame-ters in lithology discrimination, fracture detection, and time-lapse seismology.
... Combining the advantages of both Kirchhoff and wave-equation migrations, ray-based beam migrations such as Gaussian beam (e.g. Hill 2001; Albertin et al., 2001; Gray, 2005; Zhu et al., 2007) provide a powerful tool for seismic imaging of the Canadian Foothills. They overcome the limitations of Kirchhoff migration in imaging multipathing arrivals while retaining its flexibility with input geometries and its capability of imaging steep dips and overturned structures with turning waves. ...
Article
Full-text available
We have developed a beam method for shot-domain prestack depth migration. Based on a complex-ray Maslov formulation, this method overcomes the limitations of Kirchhoff migration in imaging multipathing arrivals, while retaining its flexibility with input geometry and its capability of imaging steep dips and overturned structures with turning waves. It is especially useful for seismic imaging of the Canadian Foothills, where subsurface structures are complex and data-acquisition geometries are often irregular. We demonstrate in this study the application of this method to 2D and 3D datasets from the Foothills. We show that the beam method produces better images than the Kirchhoff method on both shallow and deeper parts of the migrated sections, and when compared with the Kirchhoff images, the beam migration results are often cleaner and have much less migration swinging artifacts.
... The pioneering work of GBM was proposed by Hill (1990Hill ( , 2001. Later, GBM was implemented with a more flexible acquisition geometry (Gray 2005;Nowack et al. 2005;Gray & Bleistein 2009;Protasov & Tcheverda 2011;Yang et al. 2014Yang et al. , 2015a) and extended to elastic as well as anisotropic media (Alkhalifah 1995;Zhu et al. 2007;Protasov & Tcheverda 2012;Casasanta & Gray 2015;Protasov 2015;Yang et al. 2015b. Popov et al. (2007Popov et al. ( , 2010) develop a novel reverse-time migration workflow using GBS, which has a good performance for imaging complex faulted blocks and subsalt reflectors. ...
Article
Full-text available
With a finite recording aperture, a limited source spectrum and unbalanced illumination, traditional imaging methods are insufficient to generate satisfactory depth profiles with high resolution and high amplitude fidelity. This is because traditional migration uses the adjoint operator of the forward modelling rather than the inverse operator.We propose a least-squares migration approach based on the time-domain Gaussian beam summation, which helps to balance subsurface illumination and improve image resolution. Based on the Born approximation for the isotropic acoustic wave equation, we derive a linear time-domain Gaussian beam modelling operator, which significantly reduces computational costs in comparison with the spectral method. Then, we formulate the corresponding adjoint Gaussian beam migration, as the gradient of an L2-norm waveform misfit function. An L1-norm regularization is introduced to the inversion to enhance the robustness of least-squares migration, and an approximated diagonal Hessian is used as a pre-conditioner to speed convergence. Synthetic and field data examples demonstrate that the proposed approach improves imaging resolution and amplitude fidelity in comparison with traditional Gaussian beam migration. © The Author(s) 2018. Published by Oxford University Press on behalf of The Royal Astronomical Society.
... Moreover, ray-based depth migration methods are popular in generating offset/angle gathers for velocity model building due to its flexibility and efficiency. These migration methods can be extended to anisotropic case by adapting the wave propagators to anisotropy (Bleistein and Gray, 2001;Zhu et al., 2007). However, the ray-based wave propagators are under the high-frequency assumption, which have several drawbacks, e.g., ray shadow areas, the caustics and traveltime interpretation etc. ...
Conference Paper
Full-text available
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 sufficient smooth velocity models, which limits the developments of ray-based methods for the fact that practical 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 operator for transversely isotropic with a vertical symmetry axis (VTI) media, which considers both characteristics of band-limited wave propagation and local anisotropy. Frequency-dependent traveltimes are computed with the wavelength-dependent smoothed model 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 method by using the wavelength-dependent Fresnel beam propagator for VTI media, which can generate angle domain common imaging point gathers (ADCIGs) conveniently. Numerical examples are included to demonstrate the effectiveness of our approach, which shows its potential application for subsequent velocity estimation via ADCIGs.
... Meantime, it has been extended to complex topography (Gray 2005;Yue et al. 2010Yue et al. , 2012. In addition, taking into account the complexity of subsurface media, GBM has been applied to elastic media (Casasanta and Gray 2015;) and anisotropic media (Alkhalifah 1995;Zhu et al. 2007;Protasov 2015;Li et al. 2018). For the true amplitude GBM, one branch is proposed by Gray and Bleistein (2009). ...
Article
Full-text available
For large-scale 3D seismic data, target-oriented reservoir imaging is more attractive than conventional full-volume migration, in terms of computation efficiency. Gaussian beam migration (GBM) is one of the most robust depth imaging method, which not only keeps the advantages of ray methods, such as high efficiency and flexibility, but also allows us to solve caustics and multipathing problems. But conventional Gaussian beam migration requires slant stack for prestack data, and ray tracing from beam center location to subsurface, which is not easy to be directly applied for target-oriented imaging. In this paper, we modify the conventional Gaussian beam migration scheme, by shooting rays from subsurface image points to receivers to implement wavefield back-propagation. This modification helps us to achieve a better subsurface illumination in complex structure and allows simple implementation for target reservoir imaging. Significantly, compared with the wavefield-based GBM, our method does not reconstruct the subsurface snapshots, which has higher efficiency. But the proposed method is not as efficient as the conventional Gaussian beam migration. Synthetic and field data examples demonstrate the validity and the target-oriented imaging capability of our method.
... The Gaussian beam method has been used for high-frequency wave field calculations (Červený, 2001;Červený et al., 1982;Popov, 1982) as well as for migration imaging (Hill, 1990(Hill, , 2001Gray, 2005;Gray and Bleistein, 2009). Much work has been carried out to improve the Gaussian beam method (Alkhalifah, 1995;Červený, 1985;George et al., 1987;Klimes, 1984;Kravtsov and Berczynski, 2007;Nowack, 2003;Nowack and Kainkaryam 2011;Popov et al., 2010;Tanushev, 2008;Zhu et al., 2007). However, the beam is calculated using the ray-centered coordinate system and it is necessary to calculate the distance of the imaging point to the nearest point on the ray or to carry out coordinate transformation (Červený and Pšenčik, 2010;Klimes, 1994). ...
Preprint
We present the Eulerian Gaussian beam method in anisotropic media. We derive kinematic and dynamic ray tracing equations based on the level set theory and Eulerian theory using the anisotropic eikonal equation. Compared with the traditional anisotropic Gaussian beam method using ray-centered coordinates, the anisotropic Eulerian Gaussian beam method derived in this work has the following three advantages: (1) it can handle the problem of calculating the distance from the imaging point to the beam point more easily; (2) it allows the travel time and amplitude to be distributed uniformly within the actual computational domain without interpolation; (3) it can handle late arrivals, both theoretically and in calculations, due entirely to ray tracing in the phase space.
... Gaussian beam migration (GBM) (Hill, 1990(Hill, , 2001Hale, 1992;Gray and Bleistein, 2009) is a ray-based depth migration method whose accuracy rivals that of wave-equation based migrations and whose efficiency rivals that of KPSDM. The GBM can be extended to general transversely isotropic (TI) case via the anisotropic ray tracing system (Alkhalifah, 1995;Zhu et al., 2007;Han et al., 2014;Liu et al., 2015) and generalized to viscoacoustic media by introducing complex traveltimes to the ray tracing procedure (Keers et al., 2001;Bai et al., 2016;Liu et al., 2017). On the other hand, the Gaussian beam can be used as wave propagator in inversion imaging. ...
Article
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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.
... The Gaussian beam method for the research of wave-propagation phenomena in rather complicated geophysical models are broadly applied in seismic depth imaging owing to its effectiveness and not so time-consuming numerical procedures with low memory requirements (Hill, 1990(Hill, , 2001Hale, 1992;Gray et al., 2005Gray et al., , 2009. To date, it has been effectively extended to complex media, including elastic and anisotropic models (Protasov et al., 2012;Alkhalifah, 1995;Zhu et al., 2007), which are used to obtain a better final pre-stack depth image based on the algorithm in frequency domain. Adhering to the basic framework of Gaussian beam migration (GBM), in addition, numerous novel seismic beam techniques have been investigated (Nowack, 2008(Nowack, , 2011Huang et al. 2015), which provide several excellent alternatives to enhance the imaging accuracy or computational efficiency of beam migration algorithms in frequency domain. ...
... Alkhalifah (1995) extended the Gaussian beam migration technique to anisotropic media and analyzed the anisotropic parameters. Zhu et al. (2007) reformulated the ray tracing systems in terms of phase velocity and extended pre-stack Gaussian beam migration to apply to general transversely isotropic (TI) media. Han et al. (2014) introduced a Gaussian beam prestack depth migration method for the converted PS-wave in TI media. ...
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We study problems associated with seismic data decomposition and migration imaging. We first represent the seismic data utilizing Gaussian beam basis functions, which have nonzero curvature, and then consider the sparse decomposition technique. The sparse decomposition problem is an l0-norm constrained minimization problem. In solving the l0-norm minimization, a polynomial Radon transform is performed to achieve sparsity, and a fast gradient descent method is used to calculate the waveform functions. The waveform functions can subsequently be used for sparse Gaussian beam migration. Compared with traditional sparse Gaussian beam methods, the seismic data can be properly reconstructed employing fewer Gaussian beams with nonzero initial curvature. The migration approach described in this paper is more efficient than the traditional sparse Gaussian beam migration.
... An essential element of traveltime-based GBM is to form local plane-wave gathers using a slant-stack transformation of the data, which are then projected to the image domain according to the traveltime relationship between the source-side beam and receiver-side beam. Subsequently, this migration scheme was extended to complex irregular topographic conditions (Gray, 2005;Yue et al., 2012;Yang et al., 2014), true-amplitude imaging (Gray and Bleistein, 2009;Popov et al., 2010), anisotropic media (Alkhalifah, 1995;Zhu et al., 2007;Han et al., 2014;Protasov, 2015), least-squares Gaussian beam migration (Hu et al., 2015(Hu et al., , 2016Yang et al., 2018a), as well as elastic media (Protasov and Tcheverda, 2012;Yang et al., 2018b). In addition, within the GBM framework, numerous novel seismic beam techniques have been investigated. ...
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In the Gaussian beam (GB) method, initial beam parameters are principal factors influencing the accuracy and computational efficiency of seismic depth imaging. Various optimized beam parameter strategies for Gaussian beam migration (GBM) have been proposed to improve imaging quality as well as computational efficiency, while optimized space-time Gaussian beam schemes for seismic migration have still not been fully investigated. In this paper, an optimized space-time Gaussian beam approach with dynamic parameter control for seismic depth imaging is developed. We first provide an expression for dynamic beam parameter by taking in account the effect of velocity field variation on the beam forming. Based on dynamic beam parameters, the new space-time adaptive Gaussian beam generated by an arbitrary source wavelet is obtained, which can adaptively calculate the beam width to make the seismic beam energy better focused in the central ray neighborhood. Then, the forward wavefield is constructed in two-dimensional (2D) acoustic media by space-time adaptive Gaussian beam for the implementation of migration. Adhering to the framework of conventional space-time Gaussian beam method, we perform the up-going ray tracing from subsurface imaging points to the receiver surface to compute the asymptotic Green function for the construction of the backward wavefield. Numerical experiments demonstrate that the new presented approach has a superior accuracy for seismic depth imaging in both shallow and deep regions compared to the conventional space-time Gaussian beam migration scheme.
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Riemannian spaces are described by nonorthogonal curvilinear coordinates. We generalize one-way wave-field extrapolation to semiorthogonal Riemannian co-ordinate systems that include, but are not limited to, ray coordinate systems. We obtain a one-way wave-field extrapolation method that can be used for waves propagating in arbitrary directions, in contrast to down-ward continuation, which is used for waves propagating mainly in the vertical direction. Ray coordinate systems can be initiated in many different ways; for example, from point sources or from plane waves incident at vari-ous angles. Since wavefield propagation happens mostly along the extrapolation direction, we can use inexpen-sive finite-difference or mixed-domain extrapolators to achieve high angle accuracy. The main applications of our method include imaging of steeply dipping or over-turning reflections.
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Gaussian beams concentrated close to rays of high-frequency seismic body waves prop-agating in an inhomogeneous anisotropic layered structure are studied. The amplitude profiles of the Gaussian beam along the plane perpendicular to the ray and along the plane perpendicular to the slowness vector are Gaussian. The Gaussian profile is controlled by the 2 × 2 complex-valued matrix M of the second derivatives of the travel-time field at any point of the ray. The matrix M can be simply determined at any point of the ray if the ray-propagator matrix along the ray is known and if the value of M is specified at a selected point of the ray. The ray-propagator matrix can be determined by dynamic ray tracing along the ray. In inhomogeneous anisotropic medium, the dynamic ray tracing can be performed alternatively in several coordinate systems: in global Cartesian coordinates, in ray-centred coordinates and in wavefront orthonormal coordinates. In addition, also simplified dynamic ray tracing in global Cartesian coordinates can be used, which reduces the number of equations of the dynamic ray tracing system. The derived expressions for the Gaussian beams are applicable to general 3-D inhomogeneous layered structures of ar-bitrary anisotropy (specified by upto 21 independent position-dependent elastic moduli). Possible simplification of the procedure are outlined.
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Due to imaging problems of a seismic wavefield in complex geological regions by the use of zero order ray theory as Green function, we report on a Gaussian-Beam (GB) type operator in the kernel of a finite-offset true-amplitude prestack depth migration Kirchhoff operator. This op-erator is a weighted GB superposistion integral constrained to the pro-jected Fresnel zone of a seismic experiment. The current migration operator is applied to a simple synthetic example and is compared to the conventional Kirchhoff-PSDM in a complex synthetic geological model of the Solimões Basin (Brazil). The depth images in all the cases showed an increase in horizontal resolution and a reduction of migration artifacts. INTRODUCTION
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Reflection tomography in the migrated domain can help reconstruct heterogeneous, anisotropic velocity fields needed for accurate depth imaging of complex geologic structures. The presence of anisotropy, however, increases the uncertainty in velocity analysis and typically requires a priori constraints on the model parameters. Here, we develop a 2D P-wave tomographic algorithm for heterogeneous transversely isotropic media with a tilted symmetry axis (TTI) and investigate the conditions necessary for stable estimation of the symmetry-direction velocity V-P0 and the anisotropy parameters epsilon and delta. The model is divided into rectangular cells, and the parameters V-P0, epsilon, delta, and the tilt nu of the symmetry axis are defined at the grid points. To increase the stability of the inversion, the symmetry axis is set orthogonal to the imaged reflectors, with the tilt interpolated inside each layer. The iterative migration velocity analysis involves efficient linearized parameter updating designed to minimize the residual moveout in image gathers for all available reflection events. The moveout equation in the depth-migrated domain includes a nonhyperbolic term that describes long-offset data, which are particularly sensitive to epsilon. Synthetic tests for models with a "quasi-factorized" TTI syncline (i.e., epsilon and delta are constant inside the anisotropic layer) and a TTI thrust sheet demonstrate that stable parameter estimation requires either strong smoothness constraints or additional information from walkaway VSP (vertical seismic profiling) traveltimes. If the model is quasifactorized with a linear spatial variation of V-P0, it may be possible to obtain the interval TTI parameters just from long-spread reflection data.
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Kirchhoff-type prestack depth migration is the method most popular for outputting offset gathers for velocity-model updating because of its flexibility and efficiency. However, conventional implementations of Kirchhoff migration use only single arrivals. This limits its ability to image complex structures such as subsalt areas. We use the beam methodology to develop a multiarrival Kirchhoff beam migration. The theory and algorithm of our beam migration are analogs to Gaussian beam migration, but we focus on attaining kinematic accuracy and implementation efficiency. The input wavefield of every common offset panel is decomposed into local plane waves at beam centers on the acquisition surface by local slant stacking. Each plane wave contributes a potential single-arrival in Kirchhoff migration. In this way, our method is able to handle multiarrivals caused by model complexity and, therefore, to overcome the limitation of conventional single-arrival Kirchhoff migration. The choice of the width of the beam is critical to the implementation of beam migration. We provide a formula for optimal beam width that achieves both accuracy and efficiency when the velocity model is reasonably smooth. The resulting structural imaging in subsalt and other structurally complex areas is of better quality than that from single-arrival Kirchhoff migration.
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Accurate velocity model building is a key in achieving high quality subsalt images. To reach this goal, we propose a target oriented and efficient velocity analysis methodology. Our methodology comprises of two steps. In the first step the surface data is downward extrapolated to a new datum plane using a beam based redatuming algorithm. In the second step we switch to a Kirchhoff or fast beam migration algorithm to perform the actual velocity analysis. The redatuming process has the advantage that it not only regularizes the subsurface data, but also reduces the data to a shorter recording times and a narrower offset range. Most importantly the resulting wave field at the new datum plane is generally much simpler in nature as supra salt sediment and top salt reflected as well as refracted arrivals have been removed - hence revealing the weaker subsalt events. A fast migration method can therefore be used to scan the subsalt velocity to determine the best image quality. As the beam based redatuming converts the irregular surface data into regular data at the datum level, it can also be used as a regularization tool.
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We develop beamlet propagation and imaging using Gabor-Daubechies (G-D) frame decomposition based on local perturbation theory and apply it to target-oriented prestack depth migration. The method is formulated with local background velocities and local perturbations in wavefield extrapolation. The localized propagators and phase-correction operators are obtained analytically or semianalytically by one-way operator decomposition and screen approximation in the beamlet and space-beamlet mixed domain. Beamlet wavefields have superior localiza-tion properties in both local space and direction (wave-number) over Gaussian beams in the sense that localiza-tions are not limited within short propagation distances in either homogeneous or heterogeneous media. Com-parisons of the prestack depth-migrated images for the 2D SEG-EAGE salt model and the Marmousi model indicate that, for seismic-wave propagation and imaging in complex structural environments, the G-D beamlet propa-gator has higher accuracy and better wide-angle properties than do global propagators. Target-oriented prestack G-D beamlet migration is performed by means of local-angle-domain imaging and controlled superposition of common-dip-angle images based on the local directivity features of the target structures. This considers the spatial and di-rection localizations of beamlets. As a numerical exam-ple, we process the 2D SEG-EAGE salt-model prestack data. The results show that the proposed migration method has considerable advantages in suppressing noise and en-hancing structural features. Image quality for subsalt struc-tures, especially for steep faults, is improved through structure-based superposition of common-angle images. This demonstrates the potential and capability of beamlet migration in target-oriented seismic imaging.
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Stereotomography was proposed 10 years ago for estimating velocity macromodels from seismic reflection data. Initially, the goal was to retain the advantages of standard traveltime tomography while providing an alternative to difficult interpretive traveltime picking. Stereotomography relies on the concept of locally coherent events characterized by their local slopes in the prestack data cube. Currently, stereotomography has been developed in two and three dimensions, and precious experience has been gained. The expected advantages have been demonstrated fully (in particular, the efficiency and reliability of the semiautomatic stereotomographic picking strategies), and further studies have increased the method's potential and flexibility. For example, stereotomographic picking can now be done in either the prestack or poststack domain, in either the time (migrated or unmigrated) or depth domain. It appears that the theoretical frame of stereotomography can reconcile, very satisfactorily and efficiently, most methods proposed for velocity-macromodel estimation for depth imaging. Moreover, an extension of the method to full-waveform inversion already exists and opens the way for very interesting developments. © 2008 Society of Exploration Geophysicists. All rights reserved.
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Based on anisotropic kinematic and dynamic ray tracing systems, a P-wave Gaussian beam prestack depth migration method for VTI media is introduced in this paper. The imaging principle of anisotropic Gaussian beam prestack depth migration is presented on the basis of Gaussian beam prestack depth migration algorithm in isotropic media. For horizontally layered VTI media model, the influence of anisotropic parameters on Gaussian beam prestack depth migration is illustrated. Tests of multi-layer shale model and SEG/Hess model show that the proposed method is an accurate and efficient anisotropic prestack depth migration approach in VTI media.
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In this paper we discuss an efficient way to apply Gaussian Packet method to data representation and seismic imaging. Similar to Gaussian Beam method, wavefield radiating from a seismic source as a set of Gaussian Packets can represent synthetic seismic data, and recorded wavefield at the surface can be expressed and downward continued by a set of Gaussian Packets at surface as well. The evolution of Gaussian Packet is determined by parameter of central frequency, local time, local space and ray emergent angle, and the shape of Gaussian Packet is determined by its initial value. Each Gaussian Packet is directly related to the ray emergent angle, and propagates along a central ray. Therefore, representation of seismic data using Gaussian Packets provides the local time slope and location information at certain central frequency, while summation of Gaussian Packets' evolutions constructs the corresponding propagated wavefield. These properties also make seismic imaging using Gaussian Packets easily be understood and implemented. The method becomes efficient because 1) to represent seismic data, Gaussian Packets with given initial value can be used and inner product can be applied to obtain the useful information of seismic data; 2) with proper initial value, only Gaussian Packets with few central frequencies are needed for representing propagated seismic wavefield. Numerical examples on impulse responses and a 4layer zero-offset data are calculated to demonstrate the valid of the method.
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We have developed a common-reflection-point (CRP)-based kinematic migration velocity analysis for 2D P-wave reflection data to estimate the four transversely isotropic (TI) parameters V-Po, delta, and epsilon, and the tilt angle phi of the symmetry axis in a TI medium. In each iteration, the tomographic parameter was updated alternately with prestack anisotropic ray-based migration. Iterations initially used layer stripping to reduce the number of degrees of freedom; after convergence was reached, a couple of more iterations over all parameters and all CRPs ensured global interlayer coupling and parameter interaction. The TI symmetry axis orientation was constrained to be locally perpendicular to the reflectors. The V-Po dominated the inversion, and so it was weighted less than delta and epsilon in the parameter updates. Estimates of delta and epsilon were influenced if the error in phi was >5 degrees; estimates of V-Po were also influenced if the error in phi was >10 degrees. Examples included data for a simple model with a homogeneous TI layer whose dips allowed recovery of all anisotropy parameters from noise-free data, and a more realistic model (the BP tilted transversely isotropic (TTI) model) for which only V-Po, delta, and phi were recoverable. The adequacy of the traveltimes predicted by the inverted anisotropic models was tested by comparing migrated images and common image gathers, with those produced using the known velocity models.
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A series of ocean-bottom cable (OBC) surveys has been conducted in the Bohai Sea in China in recent years to overcome difficulties experienced with streamer surveys in shallow water, such as strong currents, missing near offsets, and obstacles. The main challenges in OBC data imaging include steeply dipping structures, serious multiples in the shallowwater environment, large lateral velocity variations, fault shadow effects, and low signal-to-noise ratio (S/N). To obtain optimal images, advanced processing technologies have been developed and applied to OBC data which involve effective PZ summation and shallow-water demultiple, a high-fidelity beam migration in the wide-azimuth domain, and accurate velocity-model building in 3D tilted-transverse-isotropy (TTI) media. The PZ summation and shallow-water demultiple methods aim to effectively eliminate shallow-water ghosts to achieve broadband seismic data. Furthermore, high-fidelity controlled-beam migration (CBM) and TTI velocity-model updates greatly enhance steep dip imaging, improve S/N, and reduce turnaround time. Through the combination of these technologies, OBC data processing provides high-quality images with well-defined steeply dipping structures to reduce exploration risk in the Bohai area.
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As a kind of improved ray migration method, Gaussian beam migration method has got more and more attention with the advantages of high imaging precision and computation efficiency. Through modifying kinematic and dynamic ray tracing equations and introducing underground medium anisotropy parameters, we developed a prestack Gaussian beam depth migration method in common-shot domain for anisotropic media based on the algorithm of conventional Gaussian beam migration. Then we used simple anisotropic sub-sag model and the international standard anisotropic Hess model to verify the method and drew the conclusion that anisotropic parameters have a great influence on the large offset information of seismic data. Meanwhile for the exploration area where the stratum anisotropy can't be ignored, Gaussian beam migration method based on anisotropic media can be utilized to characterized the complex subsurface structures more accurately.
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Angle-domain prestack depth migration based on anisotropic ray theory can provide strong support to the structural imaging in the depth domain and angle-gather-based seismic tomography. The Gaussian beam prestack depth migration has shown its great advantages in accuracy of complex structural imaging, computational efficiency and flexibility. In this paper, we focus on implementing Gaussian beam perstack depth migration in the framework of local angle domain imaging. In order to improve the efficiency and practicability of proposed algorithm, we discuss a kind of phase velocity based kinematic and dynamic ray tracing systems, which is evolved from the classic ray tracing equation of anisotropic media. And we propose a more economical program for constructing anisotropic Gaussian beams. Combined with the principle of seismic local angle domain imaging, we discuss an angle parameter calculation method which is suitable for Gaussian beam migration. Numerical tests on the standard theoretical anisotropic models show that compared with local angle domain Kirchhoff migration, our approach has a higher imaging precision and better anti-noise ability. It also can be used as an imaging tool for complex geological structures and efficient migration engine for anisotropic migration velocity analysis and model construction.
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Neglecting anisotropy in seismic imaging may result in remarkable positional errors and focusing problem, especially for long-offset and wide-azimuth seismic data. To meet the demands of seismic imaging and anisotropic velocity analysis in laterally heterogeneous transversely isotropic (TI) media, we present an angle-domain imaging approach for prestack depth migration based on two advanced anisotropic ray tracing algorithms. It not only outputs migrated section and offset-domain common image gathers, but also obtains incident angles domain and illumination angles domain imaging results according to the local angular characteristics at a subsurface image point based on extended superposition of the impulse responses. In order to efficiently calculate traveltimes and local angular attributes in TI medium, we discuss and compare two improved ray tracing systems. One is based on phase velocity, which is evolved from the classic ray tracing equation of anisotropic media. Another system is derived from qP wave equation with acoustic approximation for VTI medium, and is extended to tackle TTI medium through coordinate rotation. Numerical examples on the standard theoretical anisotropic models show that our approach can be used as imaging tool for complex geological structures and efficient migration engine for anisotropic migration velocity analysis and model building.
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Elastic-wave migration is a desirable technique because it can image the structure of the earth more accurately. We develop a new elastic Gaussian beam migration method with 3D three component (3D-3C) seismic data that focuses on a complex PS-converted wave. Based on the elastic-wave equations and complete boundary conditions, we derive effective work formulas for an accurate multimode wave downward continuation for the free-space, ocean-bottom, and free-surface models. We separate the PS-wave into linear-polarized P-S1 and P-S2 waves to simplify the expression and derivation of the migration. To image the vectorial wave directly and solve the reverse-polarity issue, we use the crosscorrelations of P-wave divergence and PS-wave curl operators as the 3D P- and PS-imaging conditions, and we develop a unit vector to define the rotation direction of the PS-wave. With our approach, 3D-3C multimode waves are automatically decomposed to P- and PS-waves during the migration without the need for prior data separation, which not only reduces the crosstalk noise caused by inaccurate multimode wave decomposition but also decreases the processing cost. Applications of this method to two 3D-3C synthetic examples indicate successful PS-wave migration. Also, confirming that the PS-image can be constructed by summing the P-S1 and P-S2 images and is independent of the choice of the ray-centered local coordinates validates this method.
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Subsurface rocks (e.g., shale) may induce seismic anisotropy, such as transverse isotropy (TI). Traveltime computation is an essential component of depth imaging and tomography in TI media. It is natural to compute the traveltime using the wavefront marching method. However, tracking the 3D wavefront is expensive, especially in anisotropic media. Besides, the wavefront marching method usually computes the traveltime using the eikonal equation. However, the anisotropic eikonal equation is highly nonlinear and it is challenging to solve. To address these issues, we present a layer‐by‐layer wavefront marching method to compute the P‐wave traveltime in 3D TI media. To simplify the wavefront tracking, it uses the traveltime of the previous depth as the boundary condition to compute that of the next depth based on the wavefront marching. A strategy of traveltime computation is designed to guarantee the causality of wave propagation. To avoid solving the nonlinear eikonal equation, it updates traveltime along the expanding wavefront by Fermat's principle. To compute the traveltime using Fermat's principle, an approximate group velocity with high accuracy in TI media is adopted to describe the ray propagation. Numerical examples on 3D VTI and TTI models show that the proposed method computes the traveltime with high accuracy. It can find applications in modeling and depth migration. This article is protected by copyright. All rights reserved
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Processing pressure shear (PS)-wave data is more challenging than PP-wave data because of the asymmetry of the source-to-receiver ray paths, particularly for irregular surfaces and sparse acquisition. Gaussian beam summation (GBS) migration is an effective method for imaging seismic data from irregular surfaces. In this letter, we introduce a converted PS-wave migration method for irregular surfaces using GBS, in which a scalar wavefield is used for wavefield propagation imaging. Cross correlation imaging is performed using forward-continued source wavefields calculated with the P-wave and reverse-continued wavefields calculated using the S-waves from the receivers. The corresponding Green's function is constructed as a superposition integral of the Gaussian beams emitted from the source and the receivers on the irregular surfaces, respectively. Numerical tests demonstrate that the method is a flexible and effective alternative for accurate imaging of converted PS-wave data from irregular surfaces.
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Seismic wave imaging in complex media requires an accurate wavefield simulation method that can accurately describe the wave propagation in realistic media. Reverse time depth migration is the preferred method for seismic wave imaging in complex media. Although it is relatively expensive, its imaging accuracy is usually better than migrations based on the ray method. Migration of primary reflection data requires a wave propagation simulation method that can accurately describe primary reflected/scattered wave energy and incorporate anisotropy. Accordingly, we propose the simulation of wave propagation in tilted transversely isotropic media using a 15° one-way wave equation in a ray-centred coordinate system, combining the flexibility of ray theory and accuracy of wave theory. We use this equation to describe the propagation of body waves in a single ray tube, a “beam”. The wavefield along the beam, guided by its central raypath, has an angle limit defined only by the ray angle; therefore wave propagation in complex and steeply dipping media can be simulated with a 15° one-way wave equation. Numerical experiments show that the simulation results for beam propagation using the 15° equation in the ray-centred coordinate system have good accuracy. For prestack depth migration in tilted transversely isotropic media, we built a beam imaging method using this propagator, and this migration method yielded accurate images with greater efficiency than RTM. This article is protected by copyright. All rights reserved
Article
In areas with a complex surface, the acquisition and processing of seismic data is a great challenge. Although elevation-static corrections can be used to eliminate the influences of topography, the distortions of seismic wavefields caused by simple vertical time shifts still greatly degrade the quality of the migrated images. Ray-based migration methods which can extrapolate and image the wavefields directly from the rugged topography are efficient ways to solve the problems mentioned above. In this paper, we carry out a study of prestack Gaussian beam depth migration under complex surface conditions. We modify the slant stack formula in order to contain the information of surface elevations and get an improved method with more accuracy by compositing local plane-wave components directly from the complex surface. First, we introduce the basic rules and computational procedures of conventional Gaussian beam migration. Then, we give the original method of Gaussian beam migration under complex surface conditions and an improved method in this paper. Finally, we validate the effectiveness of the improved method with trials of model and real data. Keywordscomplex surface-local plane-wave-Gaussian beam migration
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Summary We investigate the Gaussian beam migration of common- shot and common-receiver data. The imaging of common- shot data is useful for seismic data where the receiver coordinates are well sampled, but the source coordinates are less well sampled. By reciprocity, this approach can also be applied to common-receiver data, such as from OBS experiments where the source locations are dense but the receiver locations are sparse. Since Gaussian beam migration uses smoothed, localized windowing of the data, it can provide more stable results for the inversion of sparsely sampled data. In order to test the Gaussian beam imaging, pre-stack data sets have been computed using the finite difference method for slices obtained from the SEG/EAGE salt model which has significant lateral velocity variations. Prior to tracing of the beams, smoothing of the velocity was performed. The phase times were then corrected using first-order perturbation theory. A receiver spacing of 20 m was used with an offset range of 5000 m, and the shot spacing was allowed to vary. Comparing the Gaussian beam migration images with the true model, most features were well imaged. In particular, features beneath the salt were imaged including steeply dipping events and a lower flat reflector. The Gaussian beam images also compare well with results from other migration methods.
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Kinematic and dynamic raytracing in inhomogeneous, anisotropic media has been traditionally formulated in terms of elastic parameters. Such a formulation is inefficient for computation as it requires evaluating complicated right-hand-side functions and solving an eigenvalue problem at each ray step. It also requires that a medium be specified with elastic parameters. This is inconsistent with the common practice in seismic data processing where anisotropy is usually described with Thomsen (1986) parameters. This inconsistency may result in ambiguity in specifying the elastic parameters. To overcome these difficulties, we have reformulated the kinematic and dynamic raytracing systems in terms of phase velocity. The new formulation is much simpler and computationally more efficient than the previous elastic-parameter based formulations. Solution of the eigenvalue problem at each ray step is no longer required. As the medium for raytracing is now specified with phase velocity, the possible ambiguity in specifying elastic parameters is also eliminated. The kinematic and dynamic raytracing systems developed in this study have been used to implement Gaussian beam depth migration in anisotropic media. Numerical results show that our formulation is efficient and accurate and has greatly speeded up the depth migration in anisotropic media.
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Most bulk elastic media are weakly anisotropic. The equations governing weak anisotropy are much simpler than those governing strong anisotropy, and they are much easier to grasp intuitively. These equations indicate that a certain anisotropic parameter (denoted delta ) controls most anisotropic phenomena of importance in exploration geophysics, some of which are nonnegligible even when the anisotropy is weak. The critical parameter alpha is an awkward combination of elastic parameters, a combination which is totally independent of horizontal velocity and which may be either positive or negative in natural contexts.-Author
Article
In a transversely isotropic medium, pre-stack time migration becomes necessary due to the lack of information on the vertical velocity in the P-wave reflection data. Here we describe the theory and computational details of a fast traveltime computation algorithm in the offset-two way time domain that is valid in weakly heterogeneous, anisotropic media. Our algorithm makes use of two interval parameters, namely, the elliptic velocity (small angle moveout velocity) and anisotropy parameter κ (equivalent to η) estimated by interactive moveout analysis in the τ-p domain. Our travel time formulation is essentially a perturbation approach in which we compute the vertical delay time in a laterally homogeneous background medium to which perturbations are applied to account for lateral heterogeneity. Although this approach is valid in weakly heterogeneous media we do not compute head wave paths, which are often computed by a finite difference solution of Eikonal equation for first arrivals. The computed traveltimes are then applied to perform pre-stack time migration using the Split-Step Fourier migration technique. We demonstrate the applicability of our algorithm with application to a 2D seismic line from the Gulf of Mexico.
Article
Gaussian beam migration is a depth migration method whose accuracy rivals that of migration by wavefield extrapolation - so-called "wave-equation migration" - and whose efficiency rivals that of Kirchhoff migration. This migration method can image complicated geologic structures, including very steep dips, in areas where the seismic velocity varies rapidly. However, applications of prestack Gaussian beam migration either have been limited to common-offset common-azimuth data volumes, and thus are inflexible, or suffer from multiarrival inaccuracies in a common-shot implementation. In order to optimize both the flexibility and accuracy of Gaussian beam migration, I present a common-shot implementation that handles multipathing in a natural way. This allows the migration of data sets that can include a variety of azimuths, and it allows a simplified treatment of near-surface issues. Application of this method to model data typical of Canadian Foothills structures and to model data that includes a complicated salt body demonstrates the accuracy and versatility of the migration. © 2005 Society of Exploration Geophysicists. All rights reserved.
Article
An approach for calculating first-arrival traveltimes in a transversely isotropic medium is developed and has the advantage of avoiding shadow zones while still being computationally fast. Also, it works with an arbitrary velocity grid that may have discontinuities. The method is based on Fermat's principle. The traveltime for each point in the grid is calculated several times using previously calculated traveltimes at surrounding grid points until the minimum time is found. Different ranges of propagation angle are covered in each traveltime calculation such that at the end of the process all propagation angles are covered. This guarantees that the first-arrival traveltime for a specific grid point is correctly calculated. The resulting algorithm is fully vectorizable. The method is robust and can accurately determine first-arrival traveltimes in heterogeneous media. Traveltimes are compared to finite-difference modeling of transversely isotropic media and are found to be in excellent agreement. An application to prestack migration is used to illustrate the usefulness of the method.
Article
Kirchhoff migration is the most popular method of three-dimensional prestack depth migration because of its flexibility and efficiency. Its effectiveness can become limited, however, when complex velocity structure causes multipathing of seismic energy. An alternative is Gaussian beam migration, which is an extension of Kirchhoff migration that overcomes many of the problems caused by multipathing. Unlike first-arrival and most-energetic-arrival methods, which retain only one traveltime, this alternative method retains most arrivals by the superposition of Gaussian beams. This paper presents a prestack Gaussian beam migration method that operates on common-offset gathers. The method is efficient because the computation of beam superposition isolates summations that do not depend on the seismic data and evaluates these integrals by considering their saddle points. Gaussian beam migration of the two-dimensional Marmousi test data set demonstrates the method's effectiveness for structural imaging in a case where there is multipathing of seismic energy.
Article
Gaussian beam migration (GBM), as it is implemented today, efficiently handles isotropic inhomogeneous media. The approach is based o the solution of the wave equation in ray-centered coordinates. Here the author extends the method to work for 2-D migration in generally anisotropic inhomogeneous media. Extension of the Gaussian-beam method from isotropic to anisotropic media involves modification of the kinematics and dynamics in the required ray tracing. While the accuracy of the paraxial expansion for anisotropic media is comparable to that for isotropic media, ray tracing is anisotropic media is much slower than that in isotropic media. However, because ray tracing is just a small portion of the computation in GBM, the increased computational effort in general anisotropic GBM is typically only about 40%. Application of this method to synthetic examples shows successful migration in inhomogeneous, transversely isotropic media for reflector dips up to and beyond 90°. Further applications to synthetic data of layered anisotropic media show the importance of applying the proper smoothing to the velocity field used in the migration. Also, tests with synthetic data show that the quality of anisotropic migration of steep events in a medium with velocity increasing with depth is much more sensitive to the Thomsen anisotropy parameter ε than to the parameter δ. Thus, a good estimate of ε is needed to apply anisotropic migration with confidence.
Article
Seismic anisotropy in dipping shales causes imaging and positioning problems for underlying structures. We developed an anisotropic depth-migration approach for P-wave seismic data in transversely isotropic (TI) media with a tilted axis of symmetry normal to bedding. We added anisotropic and dip parameters to the depth-imaging velocity model and used prestack depth-migrated image gathers in a diagnostic manner to refine the anisotropic velocity model. The apparent position of structures below dipping anisotropic overburden changes considerably between isotropic and anisotropic migrations. The ray-tracing algorithm used in a 2-D prestack Kirchhoff depth migration was modified to calculate traveltimes in the presence of TI media with a tilted symmetry axis. The resulting anisotropic depth-migration algorithm was applied to physical-model seismic data and field seismic data from the Canadian Rocky Mountain Thrust and Fold Belt. The anisotropic depth migrations offer significant improvements in positioning and reflector continuity over those obtained using isotropic algorithms.
Article
We develop the theory of localized slant-stack migration in the common-offset domain. We use three asymptotic solutions to implement the theory: Gaussian beam, Maslov, and coherent state solutions. We find that all three techniques give improvements over Kirchhoff migration in areas of multipathing and caustics, but amplitude issues remain.
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A scaled physical model was constructed to investigate the magnitudes of imaging errors incurred by the use of isotropic processing code when there is seismic velocity anisotropy present in the dipping overburden. The model consists of a block of transversely isotropic (TI) phenolic material with the TI axis of symmetry dipping at an angle of 45°. Its scaled thickness is 1500 m, and it is intended to simulate the dipping clastic sequences found in many fold-thrust belts. A piece of isotropic Plexiglas, affixed to the underside of the anisotropic block, has a step function in it to simulate a target reef edge or fault. The anisotropy parameters of the material are δ = 0.1 and ε = 0.24. On zero-offset data the imaged position of the target is shifted laterally 320 m in the updip direction of the beds, whereas on time-and depth-migrated multichannel sections the shift is 300 m. The lateral shift is offset dependent, with the amount of shift in any common-midpoint gather decreasing from 320 m on the near offsets to 280 m on the far offsets. Prestack depth-migration velocity analysis based upon obtaining consistent depth images in the common-offset domain results in the base of the anisotropic section being imaged 50 m (about 3%) too deep.
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This paper describes a zero-offset depth migration method that employs Gaussian beam downward continuation of the recorded wave field. The Gaussian-beam migration method has advantages for imaging complex structures. Like finite-difference migration, it is especially compatible with lateral variations in velocity, but Gaussian beam migration can image steeply dipping reflectors and will not produce unwanted reflections from structure in the velocity model. Unlike other raypath methods. Gaussian beam migration has guaranteed regular behavior at caustics and shadows. In addition, the method determines the beam spacing that ensures efficient, accurate calculations. The images produced by Gaussian beam migration are usually stable with respect to changes in beam parameters. -from Author
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SummaryA ray series expansion for seismic body waves propagating in inhomogeneous anisotropic media is studied. Methods for calculation of rays and amplitude coefficients of the ray series are suggested. A seismic ray is described by a system of ordinary differential equations of first order which can be solved by standard numerical techniques. Another system of ordinary differential equations can be used to compute amplitude coefficients. The method may be applied to general anisotropic media in which the elastic parameters are arbitrary continuous functions of all three co-ordinates.
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Historically, seismic migration has been the practice (science, technology, and craft) of collapsing diffraction events on unmigrated records to points, thereby moving ("migrating") reflection events to their proper locations, creating a true image of structures within the earth. Over the years, the scope of migration has broadened. What began as a structural imaging tool is evolving into a tool for velocity estimation and attribute analysis, making detailed use of the amplitude and phase information in the migrated image. With its expanded scope, migration has moved from the final step of the seismic acquisition and processing flow to a more central one, with links to both the processes preceding and following it. In this paper, we describe the mechanics of migration (the algorithms) as well as some of the problems related to it, such as algorithmic accuracy and efficiency, and velocity estimation. We also describe its relationship with other processes, such as seismic modeling. Our approach is tutori