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1 Imain diffraction points usin the local imae matrix in prestack miration Xiaosan Zhu* 1,2, Ru-Shan Wu 1 1 Department of Earth & Planetary Sciences, University of California, Santa Cruz, CA Department of Geophysics, Pekin University, Beiji China 1871 Summary The enery anle-distribution in the local imae matrix (LIM) for a planar reflector and for a discontinuous point are different with the former exhibitin a linear enery concentration alon certain dip direction while the latter showin a scattered enery distribution. Therefore the cross-correlation value of the local imae matrix between adjacent imae points can be used to distinuish these two situations. The seismic imaes of these diffraction points may provide important information about eoloical discontinuities. Introduction Due to under- or overmiratio miration artifacts, which often manifest themselves as apparent faults or edes and are mistakenly interpreted as structural details. Diffraction points contain valuable information about the subsurface structures, such as faults, pinchouts, rouh edes, fractures, channels, salt bodies, small-sized scatters and any sudden chanes of facies. They can be used for imai inversion and interpretation of eoloical discontinuities and providin clearer identifications of eoloical discontinuities. The sinificance of diffracted waves has been reconized more than fifty years ao (Krey, 1952). The sinal amplitude from diffraction points can be extracted from the seismic section to detect local heteroeneities. A correlation procedure can be used to enhance the amplitude of the seismic sinal at the location of the diffractors on the common-diffraction-point section (D-section) (Landa et al., 1987, Landa and Keydar, 1998). However, the method needs the seismic survey data at different times for the same survey area. The difference between the reflected event from a plane specular reflector and that from a point diffractor can be use to separate specular reflections and diffraction events (Taner et al., 26). Hence we can use plane-wave destruction filters to suppress specular events to et plane-wave sections of diffractions (Fomel, 22). Several workflows are tested to enhance diffraction-like sinals and to remove ordinary reflections (Bansal and Imhof, 25). The result shows that the eienvector filter is the most efficient one for both 2D and 3D data sets. The coherence of seismic data measured by the crosscorrelation between each seismic trace and its neihborin traces has proven to be an effective method for imain eoloical discontinuities. The seismic coherence makes a clear interpretation of subtle features which may not be readily apparent in the seismic data. The coherence alorithms have developed from usin only three traces (Bahorich and Farmer, 1995) to multitrace coherence measurement which are based on the eienstructure of the covariance matrix fromed from the traces in the analysis cube (Gersztenkorm and Marfurt 1999, Marfurt et al., 2). Instead of usin the lobal Fourier transform in the seismic imai Wu et al. (2) used the Gabor-Daubechies frame (G-D frame) (Daubechies, 199; 1992) and the local cosine bases (Wu et al., 2; Wu and Che 21; 22a; 22b) to decompose the wavefield locally and derived the correspondin local propaator in the beamlet domain. A beamlet is a windowed harmonics alon spatial axes. In the case of the G-D frame, translated and modulated Gaussian window functions are used to construct frame atoms. The beamlet decomposition provides localizations in both space and wavenumber domains for the wavefield. By the beamlet imain conditio we can obtain the local imae matrix L( i, ) for each imae point durin the miration process, where i and are local incident and receivin anles, respectively. Due to different enery distribution in the local imae matrix for a diffraction point and a planar reflector, in this study, we apply the sinular value decomposition (SVD) technique and the cross-correlation method to identify and imae these diffraction points in the model. Two simple models and the SEG-EAGE salt model are used as examples to demonstrate the accessible of our approach. Method The enery distribution of local imae matrix. The local imae matrix is defined as the matrix of diffractin amplitude for incident-receivin anle pairs, hence it is the intrinsic property of the diffractin medium and is independent of the acquisition system and free from propaation effects, and it also contains the information of the local structure and the elastic properties revealed by the imae experiments at a local heteroeneity point. For a planar reflector, most of the enery in local imae matrix is distributed linearly alon certain dip direction (Fiure 1a), but for a diffraction point, the enery in the local imae matrix scatters widely in the entire matrix (Fiure 1b) because it does not have a well-defined normal direction. SEG Las Veas 28 Annual Meetin

2 Imain diffraction points usin local imae matrix in prestack miration Receivin anle (deree) Incident anle (deree) Incident anle (deree) Fiure 1. Two local imae matrices. (a) planar reflector and (b) diffraction point. (a) (b) Fiure 2. Two different representations of the local imae matrix. (a) the LIM as a function of the incident anle i and receiver anle ; (b) the LIM as a function of the dip -4 and the reflection anle n r, where = + ) / 2, = ( ) / 2. n ( i r i Data processin flow. Reconizin that the enery of a planar reflector in the LIM distributes alon a certain directio we transform the LIM from the form of L( i, ) (Fiure 2a) to the form of L( r ) (Fiure 2). In the ( r ) representation of the LIM, the enery distribution alon the same dip anle will become horizontal. The sinular values of the matrix L( r ) can be viewed as enery values alon the set of dip directions. Hence, some neihborin imae of planar reflectors will have similar sinular values and the correlation coefficient between these sets of sinular values should be lare. Lack of this correlation indicates that the imae point is a diffraction point. Therefore, we can use such a criterion to separate diffraction points from planar reflectors and finally et the imae of diffraction points. The imae amplitude of the diffraction point results from summin all the values in the correspondin LIM. In summary, the flow for separatin the enery of diffraction points from that of reflect waves is: 1) Retrieve the LIMs usin the beamlet imain technique and incorporate the acquisition aperture correction in the local anle domain; 4 8 2) Transform the local imae matrix from representation L( i, ) to L( r ) ; 3) Compute sinular values of the LIM for each point usin sinular value decomposition; 4) Select a point as a reference point O (Fiure 3) and compute the cross-correlation coefficient C (and C ) of sets of sinular values between the reference point and neihborin points Q (and Q ) in all possible directions (Fiure 3). These cross-correlation coefficicents are to be compared with the auto-correlation coefficient C OO of the reference point in the next step. 5) Make a judment by formula (1). By settin a proper threshold α, if the left side is smaller than the riht side, the reference point can be taken as a planar reflector, otherwise it will be taken as a diffraction point; 6) Take each point in oriinal model as a reference point and do the same processes as described in steps (4) and (5) and then sum up all values of local imae matrix of each diffraction point and take them as the final imae amplitude. OQ OQ ' α C OO,planar reflector C OQ C O Q (1) > α C OO,diffraction point Where ( < α <.3) and Q can be points A, B, C, D,L (Fiure 3). Fiure 3. The diaram of a reference point and its neihborin points in all possible directions. Examples Model 1. We construct a model with two hih velocity (47 km/ s ) horizontal reflectors in a backround velocity model, which has a constant vertical radient (Fiure 4). We enerated the data usin a finite difference method with a Ricker wavelet of dominant frequency 15Hz. We form imaes of the two reflectors usin different frequency bands and window lenths for the local cosine decomposition (Fiure 5, Table 1). Fiure 5(a) and (b) are two imaes of full waves after doin prestack beamlet miration by usin the backround velocity model 1. The imaes of diffraction points of the two types (dominant frequency f = 1Hz for type A and SEG Las Veas 28 Annual Meetin

3 Imain diffraction points usin local imae matrix in prestack miration f = 15Hz for type B) with different thresholds α =.1 (Fiure 5c) and α =. 12 (Fiure 5d) for all directions respectively show that the method is effective. Fiure 4. The velocity model Fiure 5. The imaes of full waves and diffraction points in model 1. ((a),(c) from type A (dominant frequency f = 1Hz )and (b), (d) from type B ( f = 15Hz )) (Table 1) Fiure 7. The imaes of full waves and diffraction points in model 2. ((a), (c) from type A (dominant frequency f = 1Hz )and (b), (d) from type B ( f = 15Hz )) (Table 1). Table 1. The computational parameters of the three models. Model Type dominant frequency window frequency bandwidth lenth Model A 1 Hz 1~2 Hz 81 m 1 & 2 B 15 Hz 1~3 Hz 51 m SEG A 1 Hz 1~2 Hz 1585 m Model B 15 Hz 1~32 Hz 85 m Model 2. This model is same with Model 1 except the two hih velocity (47 km / s ) reflectors are dippin (Fiure 6). The computational parameters are listed in Table 1 for type A and B, respectively. By usin the same workflow as the one used for model 1, the imaes of diffraction points of the two types (type A and B) with different thresholds ( α =.1 for horizontal and vertical directions and α =.15 for the rest of directions in Fiure 7c, and Fiure 6. The velocity model 2. SEG Las Veas 28 Annual Meetin

4 Imain diffraction points usin local imae matrix in prestack miration α =.15 for all directions in Fiure 7d) demonstrate the validity of the method even for a finite dippin reflector. SEG-EAGE salt model. The acquisition system of this model consists 325 shots and each shot has 176 left-handside receivers. The LIM at each point is enerated by usin shot miration and a smoothed velocity model (Fiure 8). The computational parameters are listed in Table 1. By Comparin the imaes of diffraction points ( α =. 15 for horizontal and vertical directions and α =.2 for the rest of directions in Fiure 9c and α =.2 for all directions in Fiure 9d respectively) with those of full waves (Fiure 9a, 9b), it is clear that most of Fiure 8. The smoothed velocity model of SEG-EAGE salt model x Fiure 9. The imaes of full waves and diffraction points in SEG-EAGE salt model. ((a), (c) from type A (dominant frequency f = 1Hz )and (b), (d) from type B ( f = 15Hz )). the enery of reflect wave has been removed, thouh there is still some reflected enery that cannot be completely removed due to the rouhness of some planes. Hence the boundary of the salt body can be imaed by only diffraction points and the imae is sharper with broader frequency bandwidth (Fiure 9c, 9d). Conclusions This paper proposes a method to separatin diffraction points from planar reflectors based on different enery distribution in the LIM. Throuh three numerical examples, we have demonstrated that the method is effective in obtainin imaes of diffraction points. Separatin and imain diffraction points from planar reflectors can provide valuable information about eoloical discontinuities, such as faults, pinchouts, rouh edes, fractures, salt bodies and any sudden chanes of facies. Acknowledements This work is supported by WTOPI (Wavelet Transform On Propaation and Imain for seismic exploration) Research Consortium and the DOE/Basic Enery Sciences project at University of California, Santa Cruz and China Scholarship Council. The authors thank Xiao-Bi Xie, Jun Cao, Yaofen He, Yincai zhe Hui Ya Jian Mao and Xiaofen Jia for useful comments and suestions. SEG Las Veas 28 Annual Meetin

5 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 28 SEG Technical Proram Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a hih deree of linkin to cited sources that appear on the Web. REFERENCES Bahorich, M. S., and S. L. Farmer, 1995, 3D seismic discontinuity for faults and stratiraphic features: The Leadin Ede, 14, Bansal, R., and M. G. Imhof, 25, Diffraction enhancement in prestack seismic data: Geophysics, 7, V73 V79. Daubechies, I., 199, The wavelet transform, time-frequency localization and sinal analysis: IEEE Transactions of Information Theory, 36, , Ten lectures on wavelets: Society for Industrial and Applied Mathematics. Fomel, S., 22, Applications of plane-wave destruction filters: Geophysics, 67, Gersztenkor A., and K. J. Marfurt, 1999, Eienstructure-based coherence computations as an aid to 3D structural and stratiraphic mappin: Geophysics, 64, Krey, T., 1952, The sinificance of diffraction in the investiation of faults: Geophysics, 17, Landa, E., and S. Keydar, 1998, Seismic monitorin of diffraction imaes for detection of local heteroeneities: Geophysics, 63, Landa, E., V. Shtivelma and B. Gelchinsky, 1987, A method for detection of diffracted waves on common-offset sections: Geophysical Prospecti 35, Marfurt, K. J., and R. L. Kirli 2, 3D broad-band estimates of reflector dip and amplitude: Geophysics, 65, Taner, M. T., S. Fomel, and E. Landa, 26, Separation and imain of seismic diffractions usin plane-wave decomposition: 76th Annual International Meeti SEG, Expanded Abstracts, Wu, R. S., and L. Che 21, Beamlet miration usin Gabor-Daubechies frame propaator: 63rd Annual International Conference and Exhibitio EAGE, Extended Abstracts, a, Wave propaation and imain usin Gabor-Daubechies Beamlets, in Theoretical and computational acoustics: World Scientific, b, Mappin directional illumination and acquisition-aperture efficacy by beamlet propaators: 72nd Annual International Meeti SEG, Expanded Abstracts, Wu, R. S., Y. Wa and J. H. Gao, 2, Beamlet miration based on local perturbation theory: 7th Annual International Meeti SEG, Expanded Abstracts, SEG Las Veas 28 Annual Meetin