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Local Angle Domain Kirchhoff Prestack Depth Migration in TI Media

Pengfei Duan* and Jiubing Cheng School of Ocean and Earth Science, State Key Laboratory of Marine Geology, Tongji University

Summary To support velocity analysis and amplitude variation with incident angle analysis in transversely isotropic media, we present a ray-based local angle domain prestack depth migration algorithm to generate common-image gathers in scattering and illumination angle domain. An efficient kinematic ray tracing system is used to construct numerical tables of traveltime, geometrical spreading factor and takeoff angle of various raypath between the image-point and source or receiver on the surface. Weighted superposition of all impulse responses according to the local angular attributes of the raypath generates scattering and illumination angle domain migrated images. Synthetic examples on the SEG/HESS VTI model and the overthrut TTI model demonstrate the validity of this algorithm Introduction In traditional prestack seismic processing, common-image gathers (CIGs) provided by common-offset or commonshot migration is commonly used for migration-based velocity analysis or amplitude variation with offset (AVO) analysis. But it has been proven that offset- and shotdomain CIGs contain unexpected kinematic and dynamic artifacts in complex areas with strong lateral velocity variations (Nolan and Symes, 1996; Xu et al., 2001). In order to overcome these artifacts, some authors (e.g., de Hoop et al., 1994; Xu et al.,1998; Brandsberg-Dahl et al.,1999) suggested constructing the CIGs in the common scattering angle domain based on ray-theory. Angledomain imaging algorithms in the frame of wave-equation migration have also been extensively studied (e.g., Prucha et al.,1999; Xie and Wu, 2002; Sava and Fomel, 2003). Compared with wave-equation migration, Kirchhoff-type migration has the advantage of efficiency and flexibility. The industry has not replaces the standard depth migration flow of iterations of migration followed by ray-based tomography with Kirchhoff as the migration engine because of the efficiency and its ability to generate CIGs that seismic tomography can be used easily. Some authors have presented various numerical implementation of raybased angle-domain Kirchhoff-type prestack depth migration (PSDM) in complex geological areas (e.g., Xu et al., 2001; Koren et al.,2002; Brandsberg-Dahl et al.,2003). Unlike conventional ray-based imaging algorithms, ray tracing is performed from the image points up to the surface where one-way “diffracted” rays are traced in all directions (including turning rays), forming a system of ray pairs for mapping the recorded surface seismic data into reflection-angle gathers. Koren et al. (2008, 2010) developed a subsurface angle domain depth imaging system for extracting full-azimuth, high-resolution angledependent reflectivity and for generating useful directional angle gathers in a continuous fashion. In this study, we focus on ray-based local angle domain PSDM implementation in VTI and TTI media. The algorithm is specifically designed for a number of seismic imaging and analysis tasks, such as seismic tomography to determine anisotropic velocity model, target-oriented highresolution reservoir imaging, and preconditioning for AVA inversion. Local angle domain: ray’s angle at image point Considering from the image point perspective, there are two wavefields at the image points, as we know, incident and reflected. Each one can be decomposed into local waves or rays, showing the direction of its propagation. The direction of the incident and scattering rays can be conventionally described by their respective polar angles. Each polar includes two components, namely dip and azimuth. Therefore, an angle domain imaging system at a given image point can be defined by a set of four scalar angles: two scattering (or reflection) angles and two illumination dip angles. The two scattering angles are the scattering opening angle (in single mode, the half-angle between source and receiver rays) and the scattering azimuth angle (azimuth of the plane containing both source and receiver rays). The scattering angles play the role, in the local depth point references, of the acquisition offset and azimuth in the surface-related references. The illumination dip angles are the angular coordinates (for instance dip and azimuth) of the illumination slowness vector, defined as the sum of the source and receiver (incidence and reflection) slowness vectors (Audebert, 2002; Koren et al.,2008; Cheng et al.,2011). Figure 1 shows the geometry and corresponding angular attributes in a 2D medium. In our angle-domain imaging algorithm, the ray tracing is performed from image points. One-way traveltime t s and take-off (phase) angle ? s of the incident ray, and one-way traveltime tr and take-off (phase) angle ? r of the scattering ray are precomputed and stored in numerical tables using an efficient ray-tracing algorithm. The true-amplitude weighting should be applied to provide an asymptotic high-frequency amplitude proportional to the reflection coefficient, with the wave propagation (geometric spreading) effects removed. With the take-off

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Local Angle Domain KPSDM in TI Media

angles, we can get average incident angle ? and dip ? using, ? 1 (1) ? ? ? ?? ? ? ?

2 2

s r

? ? ?s ?

1 ? ?s ? ?r ? 2

(2)

takes a very simple form with its right side given by group and phase velocities than complex elastic-parameter based formulas. The phase velocity V for a given type of wave in a general anisotropic medium can be calculated explicitly using the expressions given by Tsvankin (2001). Since both position vector x and unit direction vector n are the independent variables of the phase velocity function V =V ? xi , ni ? , the spatial derivatives of the phase velocity on the right side of equation 3b have no implicit chain-rule dependencies on the direction ni , and are computed simply by taking the partial derivatives of material properties appearing explicitly in phase velocity function with respect to xi . Since the angle involves the imaging system is the phase angle, so the ray tracing system is just meet our need for local angle domain PSDM. The ray tracing system is most useful for general TI media and weak orthorhombic media. Local angle domain KPSDM in TI media The imaging systems are a ray-based migration that uses the whole wavefields within a controlled aperture. Our procedure can be described as the follow. We shoot a fan of up-going rays (in all directions, including turning rays), with uniform take-off angle increment from each image point m. we directly obtain a uniform illumination at the image points from all directions, which is essential for accurate reconstruction of the image gathers. The illumination of the image points from all directions ensures that all arrivals are taken into account. Traveltimes, take-off angles and amplitude weights are selected from the precomputed tables for each ray. Kirchhoff depth migration can be implemented in two different ways, either hyperbola summation or impulseresponse superposition. Here, we used the latter to accomplish our local angle domain Kirchhoff PSDM. Once the traveltime and the angular attributes were known, we then transferred the samples of the input traces into the output traces in angle domain. Finally, the angle domain CIGs I (m , ? ) and I ( m ,? ) could be obtained with a weighted superposition of all impulse responses from all input traces, and then the migrated image I (m ) could be obtained with summation along angles. Compared with conventional common-offset migration algorithms, the only extra computation was to estimate the angular attributes. The migration formulas for the incident and dip angles dependent reflectivity are given by: ? ? (5) ?? ?

I ( m , ? ) ? ?? w ? ? ? p ( x s , xr , t ? t ( xs , xr ; m , ? )) ? d xs dxr ? ? ?t ? ? L ? ?

1 2 1 ? ? ? ? ?2 I ( m ,? ) ? ?? w ?? ? p ( xs , xr , t ? t ( xs , xr ; m ,? )) ? dxs dxr ? ? ? t ? ? L ? ?

Figure 1: Local angle characteristics of a selected ray pair at a subsurface image point.

Anisotropic ray tracing for local angle domain imaging Many approaches besides ray tracing have been developed for traveltime calculation in anisotropic media. They are, in general, more efficient and robust than the ray tracing method, but mostly limited to calculation of angular attributes of the raypath. The reason we choose ray tracing for traveltime calculation is that it can consider angle information conveniently. To overcome the difficulties of the traditional ray tracing systems (e.g., ?erven?, 1972, 2001), Zhu et al. (2005) reformulated the kinematic ray tracing systems in terms of phase velocity. The new formulas are much simpler and computationally more efficient than the traditional elastic-parameter based formulas. We summarize here only the results needed for this study. We start with the ray tracing system equations in term of phase velocity in inhomogeneous, anisotropic media: dx (3a) ?V

d?

i Gi

dpi ? ln V ?? ?xi d?

(3b)

where V is the phase velocity and VGi is the group velocity. The latter can be calculated from the former. Similar to its counterpart in isotropic media, the ray tracing system 3

(6)

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3446

Local Angle Domain KPSDM in TI Media

Where xs and xr are respectively the shot and receiver locations near the ray arrival points on the surface, and is the two-way traveltime of the t ? t s ? tr reflecting/diffracting rays. ? is the average incident angle, and ? is the dip. p is the seismic data recorded on the free surface, and I is the imaging results in the depth domain. w is the amplitude-weighting factor, L denotes the migration aperture. In mathematics, we should use 2.5D migration operator for the 2D velocity model and 2D acquisition system with 3D wave field. In 3D case, partial derivative operator in migration operator is simply the time derivative, where in 2D and 2.5D case, which is the halfderivative with respect to time. The algorithm is also targetoriented so that we can only image the target area, which provides flexibility for subsurface imaging and migration velocity analysis. Application to SEG/HESS VTI model To demonstrate its application in VTI media, we migrate the synthetic data of SEG/HESS model. Figure 2 shows the vertical P-wave velocity, anisotropy parameter ? and ? . The overall structure is not very complicated, other than having a salt body surrounded by anisotropic layers and a fault plane. The salt flanks and the fault are relatively steep. For comparison, we first migrate the synthetic data with an isotropic Kirchhoff PSDM algorithm (Figure 3a). Figure 3b shows the migrated image by our Kirchhoff PSDM algorithm in VTI media. Obviously, taking anisotropy into account in migration enhances the image accuracy and shows more focused reflected interfaces and the fault. Figure 4a shows the average incident-angle domain CIGs, in which the incident-angle ranges from 0 ° to 50 ° . Compared with the offset-domain CIGs (Figure 4b), there is less wave stretching effect and a higher signal to noise ratio in angle domain CIGs. Figure 4c shows the illumination-dip domain CIGs. On the illumination-dip domain CIGs, we see migrated reflections with concave shapes, and migrated diffractions related to the discontinued or fault events.

(a)

(b)

Figure 3. Local angle domain imaging results: (a) ignore the effects of anisotropy; (b) consider the effects of anisotropy.

(a)

(b)

Figure 2.VTI model for synthetic data: (a) vertical velocity, (b)

? and (c) ?

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3447

Local Angle Domain KPSDM in TI Media

(c)

Figure 4. Common image gathers: (a) average incident-angle domain CIGs; (b) offset domain CIGs; (c) illumination-dip domain CIGs.

Figure 5. TTI model for synthetic data: (a) vertical velocity; (b) ? ; (c) ? ; (d) ? .

Application to 2D overthrut model To demonstrate the application of local angle domain Kirchhoff PSDM in TTI media, we migrate the synthetic data of an overthrust model. It simulates a dipping, anisotropic thrust sheet, as typically seen in the Canadian Foothills and in Western China. As shown in Figure 5, the model consists of a transversely isotropic thrust sheet embedded in an isotropic background medium. The reflector below the thrust sheet represents a horizontal target-layer. This model is designed to illustrate the problems that arise when one produces seismic images beneath a dipping thrust sheet without considering the effects of its anisotropy. The widely used synthetic data generated from the model were first migrated into local angle domain but with a VTI Kirchhoff PSDM algorithm. The migrated image (Figure 6a) shows a spurious pull-up of the flat basal reflector beneath the dipping thrust. Figure 6b show the image obtained with our TTI Kirchhoff PSDM algorithm. As we see, the basal reflector has been properly flattened, and the image of the thrust is focused well. Figure 6c and Figure 6d are the CIGs in average incidentangle domain and the illumination-dip domain respectively. Since the maximum offset of the synthetic data is very small, so the incident angles are in a small range 0-25°. Conclusion We have presented an efficient implementation of local angle domain Kirchhoff PSDM algorithm for generating incident-angle domain and illumination-dip domain CIGs in TI media. The synthetic examples demonstrated that our algorithm can provide high quality migrated images in local angle domain. We expect that this local angle domain imaging algorithm can be used for target-oriented highresolution seismic imaging and ray-based tomography. Acknowledgements Thanks for supports by the National Natural Science Foundation of China (#41074083).

Figure 6. TTI data migrations: (a) VTI migration; (b) TTI migration; (c) average incident-angle CIGs; (c) illumination-dip domain CIGs.

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EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2011 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES

Cheng, J., J. Geng, H. Wang, and Z. Ma, 2011, 3D Kirchhoff prestack time migration in average illumination-azimuth and incident-angle domain for isotropic and vertical transversely isotropic media: Geophysics, 76, no. 1, S15–S27, doi:10.1190/1.3533913. Cheng, J. B., and Z. T. Ma, 2009, 3-D angle domain imaging in Kirchhoff prestack time migration: 79th Annual International Meeting, SEG, Expanded Abstracts, 2939–2942. Cerven?, V., 1972, Seismic rays and ray intensities in inhomogeneous anisotropic media: Geophysical Journal of the Royal Astronomical Society, 29, 1–13. Cerven?, V., 2001, Seismic ray theory: Cambridge University Press. Koren, Z., and D. Kosloff, 2001, Common reflection angle migration: A special issue of the JSE, in M. Tygel, ed., Seismic true amplitudes. Koren, Z., I. Ravve, A. Bartana, and D. Kosloff, 2007, Local angle domain in seismic imaging: 69th Conference and Exhibition, EAGE, Extended Abstracts. Koren, Z., I. Ravve, E. Ragoza, A. Bartana, and D. Kosloff, 2008, Full-azimuth angle domain imaging: 78th Annual International Meeting, SEG, Expanded Abstracts, 2221–2225. Koren, Z., X. Sheng, and D. Kosloff, 2002, Target-oriented common-reflection angle migration: Presented at the 72nd Annual International Meeting, SEG, Expanded Abstracts, P287. Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–1966. Tsvankin, I., 2001, Seismic signatures and analysis of reflection data in anisotropic media: Pergamon Press, Inc. Ursin, B., 2004, Parameter inversion and angle migration in anisotropic elastic media: Geophysics, 69, 1125–1142. Wu, R. S., and L. Chen, 2006, Directional illumination analysis using beamlet decomposition and propagation: Geophysics, 71, no. 4, S147–S159. Xu, S., H. Chauris, G. Lambaré, and M. Noble, 2001, Common-angle migration: A strategy for imaging complex media: Geophysics, 66, 1877–1894, doi: 10.1190/1.1487131. Zhu, T., S. H. Gray, and D. Wang, 2005, Kinematic and dynamic ray tracing in anisotropic media: Theory and application: 75th Annual International Meeting, SEG, Expanded Abstracts, 96–99.

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