International Symposium on Geophysical Imaging with Localized Waves
Sanya, Hainan, 25-28 July, 2011
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
Outlines
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Introduction
Principles of 2D/3D CWT
Seismic discontinuity detection based on 2D/3DCWT
Field-data examples
Conclusions and future works
Acknowledgements
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Introductions
The consistent and reliable detection of seismic discontinuity provides
interpreters powerful means to quickly visualize and map complex geological structures. The computational cost of these methods, such as
C3 algorithm (Gersztenkorn & Marfurt, 1998) and LSE (Cohen & Coifman, 2002), will increase as analyzing window widen.
1D CWT can not properly characterize the correlated information between 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 frequently used in industry.
3D CWT has good properties such as multiscale and orientation selectivity, which has the potential to detect the seismic discontinuities
directly. So we choose 3D CWT as a novel tool to detect seismic discontinuity.
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Principles of 2D/3D CWT
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Operations on mother wavelet
Three operation on mother wavelet ψ( x) : translation, dilation, rotation
Translation
 (x  b )
b
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:translated factor
Dilation
1
1
a  (a x )
a
:dilated factor
Rotation
  ( x ) 
 :rotated operator
Use 2D Morlet as an example to illustrate three operations
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1
Principles of 2D/3D CWT
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The definition of 2D/3D CWT
f ( x ) : 2D/3D signal to be analyzed

b , a ,
: 2D/3D operated wavelet
C W T ( b , a , )  
f
b , a ,


R
n

Realizing in
Space domain
1
a
n

R
f ( x )
n


a
 1 ( x  b )  d n x
1


1
jb k
n
fˆ ( k )ˆ a ( k ) e
d k
Fast Realizing in
wavenumber domain
by using 2D/3D FFT
2D CWT: dilated factor is 1D variable, translated factor is a 2D vector,
and rotated operator only contains a dip q.
3D CWT: dilated factor is 1D variable, translated factor is a 3D vector,
and rotated operator contains a dip q and a azimuth j.
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Principles of 2D/3D CWT
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Two common-use slice/cube of 2D/3D CWT
High dimension of 2D/3D CWT coefficients
Use slice/cube to visualize
2D CWT
(i) The position slice: a and q are fixed and the slice of 2DCWT
coefficients is considered as a function of position b .
(ii) The scale-angle slice: position b is fixed and the slice of
2DCWT coefficients is considered as a function of a and q.
3D CWT
(i) The position cube: a, q and j are fixed and the cube of
3DCWT coefficients is considered as a function of position b .
(ii) The scale-angle cube: positionb is fixed and the cube of
3DCWT coefficients is considered as a function of a, q and j .
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2DCWT
2D signal to be analyzed
(contains 6 damping plane waves)

f (x) 
 
N

cn e
i kn  x
The scale-angle slice of 2DCWT
Coeffs. (modulus, in origin)
 
e
 In  x
n 1
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The position slice of 2DCWT
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Coeffs. (phase)
Principles of 2D/3D CWT
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Orientation selectivity of 2DCWT
2D signal to be analyzed
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The position slice of 2DCWT Coeffs.
(small scale , q=135º, modulus)
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discontinuity detection based on 2D/3DCWT
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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
q dis  arg m ax C W T [ f ; a sm all ,  (q ), b ]
D is ( b )  C W T [ f ; a sm all ,  (q dis ), b ]
Three dimension
q dis , j dis   arg m[q ax
,j ]
C W T [ f ; a sam ll ,  q , j  , b ]
D is ( b )  C W T [ f ; a sm all , (q dis , j dis ), b ]
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the complete procedure of our method
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.
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Field-data example 1
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A
A
B
B
C
C
Time slice of coherence
(common used software)
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Time slice of our results
(based on 3D CWT)
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Conclusions and future works
Conclusions
1. 2D/3D continuous wavelet transform is a useful tool with multiscale
properties and orientation selectivity;
2. The computation cost will not increase as the size of analyzing window enlarging by realizing high dimensional CWT in wave-number
domain through FFT algorithm;
3.The field-data examples show our method can detect seismic discontinuities more subtly comparing with commonly used methods ;
Future works
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.
2011.07
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Acknowledgements
1. We thank National Natural Science Foundation of China
(40730424, 40674064), National 863 Program (2006A09A102) 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.
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References
[1] A. Gersztenkorn, and K.J. Marfurt, “Eigenstructure-based coherence computations as an aid to 3D
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[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 Joseph 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
2011.07
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2011.07
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