1. On the seismic discontinuities detection in 3D wavelet domain Xiaokai Wang* and Jinghuai Gao Email: nev.s@hotmail.com jhgao@mail.xjtu.edu.cn Institute of Wave and Information, Xi’an Jiaotong University Xi'an, Shaanxi, 710049, P.R. China International Symposium on Geophysical Imaging with Localized Waves Sanya, Hainan, 25-28 July, 2011

2. XI’AN JIAOTONG UNIVERSITY 2011.07 Institute of Wave and Information, XJTU Outlines Introduction Principles of 2D/3D CWT Seismic discontinuity detection based on 2D/3DCWT Field-data examples Conclusions and future works Acknowledgements

3. XI’AN JIAOTONG UNIVERSITY 2011.07 Institute of Wave and Information, XJTU Introductions The consistent and reliable detection of seismic discontinuity provides interpreters powerful means to quickly visualize and map complex ge- ological structures. The computational cost of these methods, such as C3 algorithm (Gersztenkorn & Marfurt, 1998) and LSE (Cohen & Coif- man, 2002), will increase as analyzing window widen. 1D CWT can not properly characterize the correlated information bet- ween neighboring traces. Boucherea applied 2D CWT (Antoine, 2004) with Morlet to detect the faults in a seismogram (Bouchereau, 1997). 2D CWT has some shortages for 3D seismic data which was frequen- tly used in industry. 3D CWT has good properties such as multiscale and orientation sele- ctivity, which has the potential to detect the seismic discontinuities directly. So we choose 3D CWT as a novel tool to detect seismic disc- ontinuity.

4. XI’AN JIAOTONG UNIVERSITY 2011.07 Institute of Wave and Information, XJTU Operations on mother wavelet Translation Dilation Rotation Use 2D Morlet as an example to illustrate three operations Principles of 2D/3D CWT Three operation on mother wavelet ψ( ) : translation, dilation, rotation :translated factor :dilated factor :rotated operator

5. XI’AN JIAOTONG UNIVERSITY 2011.07 Institute of Wave and Information, XJTU The definition of 2D/3D CWT Realizing in Space domain Fast Realizing in wavenumber domain by using 2D/3D FFT : 2D/3D signal to be analyzed : 2D/3D operated wavelet 2D CWT: dilated factor is 1D variable, translated factor is a 2D vector, and rotated operator only contains a dip . 3D CWT: dilated factor is 1D variable, translated factor is a 3D vector, and rotated operator contains a dip  and a azimuth . Principles of 2D/3D CWT

6. XI’AN JIAOTONG UNIVERSITY 2011.07 Institute of Wave and Information, XJTU Two common-use slice/cube of 2D/3D CWT 2D CWT (i) The position slice: a and  are fixed and the slice of 2DCWT coefficients is considered as a function of position. (ii) The scale-angle slice: position is fixed and the slice of 2DCWT coefficients is considered as a function of a and . High dimension of 2D/3D CWT coefficients Use slice/cube to visualize 3D CWT (i) The position cube: a,  and  are fixed and the cube of 3DCWT coefficients is considered as a function of position. (ii) The scale-angle cube: position is fixed and the cube of 3DCWT coefficients is considered as a function of ,  and . Principles of 2D/3D CWT

7. 2011.07 Institute of Wave and Information, XJTU 2D signal to be analyzed (contains 6 damping plane waves) The scale-angle slice of 2DCWT Coeffs. (modulus, in origin) 2DCWT The position slice of 2DCWT Coeffs. (phase)

8. XI’AN JIAOTONG UNIVERSITY 2011.07 Institute of Wave and Information, XJTU 2D signal to be analyzed The position slice of 2DCWT Coeffs. (small scale,  =135º, modulus) Orientation selectivity of 2DCWT Principles of 2D/3D CWT

9. XI’AN JIAOTONG UNIVERSITY 2011.07 Institute of Wave and Information, XJTU discontinuity detection based on 2D/3DCWT Part of oilfield data (a), small scale 2DCWT’s modulus in position A (b) and small scale 2DCWT’s modulus in position B (c) Two dimension Three dimension

10. XI’AN JIAOTONG UNIVERSITY 2011.07 Institute of Wave and Information, XJTU We summarize the complete procedure of seismic discontinuities detection method based on 3D CWT as follows: 1.Extract the Instantaneous phase (IP) of 3D seismic data by using Hilbert transform (or 1D wavelet transform), and get the IP cube IP(x,y,t); 2. Obtain the a new cubes IP_exp(x,y,t) by using exp[j* IP(x,y,t)]. (ps: by doing this, the phase’s jump from 180º to -180º can be overcame); 3. Choose the scale and dip/azimuth searching region; 4. Do 3D CWT to IP_exp(x,y,t) and get the a series of 3D CWT coefficients (many position cubes), and obtain the modulus of these coefficients; 5. In each point, get the largest coefficients and assign the modulus as the discontinuity measure of this point. the complete procedure of our method

11. XI’AN JIAOTONG UNIVERSITY 2011.07 Institute of Wave and Information, XJTU A B A B Field-data example 1 Time slice of coherence (common used software) Time slice of our results (based on 3D CWT) C C

12. XI’AN JIAOTONG UNIVERSITY 2011.07 Institute of Wave and Information, XJTU Conclusions and future works Conclusions Future works 1.2D/3D continuous wavelet transform is a useful tool with multiscale properties and orientation selectivity; 1.The mother wavelet will effect the results, and more attention should be focused on choosing wavelets or proposing a new wavelet; 2. In order to depict more geological structure, more researches should be carried on to construct different measures in high dimensional continuous wavelet transform domain. 2. The computation cost will not increase as the size of analyzing win- dow enlarging by realizing high dimensional CWT in wave-number domain through FFT algorithm; 3.The field-data examples show our method can detect seismic disco- ntinuities more subtly comparing with commonly used methods ;

13. XI’AN JIAOTONG UNIVERSITY 2011.07 Institute of Wave and Information, XJTU Acknowledgements 1.We thank National Natural Science Foundation of China (40730424, 40674064), National 863 Program (2006A09- A102) and National Science & Technology Major Project (2008ZX05023-005-005, 2008ZX-05025-001-009) for their supports. 2. We thank Research center of China national offshore oil corporation for providing field-data. We also thank Erhua Zhang in Exploration and Development Research Institute of Daqing Oilfield Company Ltd. for the help of interpretation.

14. XI’AN JIAOTONG UNIVERSITY 2011.07 Institute of Wave and Information, XJTU References [1] A. Gersztenkorn, and K.J. Marfurt, “Eigenstructure-based coherence computations as an aid to 3D structural and stratigraphic mapping,” Geophysics, vol.64, No.5, pp.1468-1479, 1999. [2] I. Cohen, and R.R. Coifman, “Local discontinuity measures for 3D seismic data,” Geophysics, vol.67, pp.1933-1945, 2002. [3] S. Mallat, A Wavelet Tour of Signal Processing, Second Edition, Elsevier, 2003. [4] E.B. Bouchereau, “analyse d’images par transformees en ondelettes: Ph.D. Thesis,” Universite Jose- ph Fourier. [5] G., Ouillon, D., Sornette and C., Castaing, 1995, Organization of joints and faults from 1-cm to 100-km scales revealed by optimized anisotropic wavelet coefficient method and multifractal analysis: Nonlinear processes in geophysics, 2, 158-177. [6] J.P., Antoine, R. Murenzi, P., Vandergheynst and S.T., Ali, 2004, Two-Dimensional wavelets and their relatives: Cambridge University Press. [7] J.P., Antoine, and R., Murenzi, 1996, Two-dimensional directional wavelets and the scale-angle representation: Signal processing, 52, 259-281. [8] Xiaokai Wang, et.al.. 2D seismic attributes extraction based on two-dimensional continuous wavelet transform. 79th Annual Internation meeting, SEG Expanded Abstracts, pp.3650-3653, 2009. [9] Xiaokai Wang, Jinghuai Gao, Wenchao Chen, Erhua Zhang: On the method of detecting the discontinuity of seismic data via 3D wavelet transform. IGARSS 2010: 3945-3947

15. XI’AN JIAOTONG UNIVERSITY 2011.07 Institute of Wave and Information, XJTU