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Elastic Green's theorem preprocessing
for on-shore internal multiple attenuation: theory and initial synthetic data tests Jing Wu* and Arthur B. Weglein May 28th, 2014 Austin, TX 1 1
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( Bahareh Boustani et al. 2013 )
Problem ( Bahareh Boustani et al. 2013 ) Ground Roll (Rayleigh Wave)
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( Bahareh Boustani et al. 2013 )
Problem Reference wave Green’s theorem Ground Roll (Rayleigh Wave) ( Bahareh Boustani et al. 2013 )
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( Bahareh Boustani et al. 2013 )
Problem Elastic Green’s theorem Reference wave Ground Roll (Rayleigh Wave) ( Bahareh Boustani et al. 2013 )
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Elastic Green’s theorem reference wave prediction
Theory of Elastic Green’s theorem reference wave prediction
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Actual medium Experiment
Reference medium Perturbation Active source ( “Passive source” ) “Source”
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Off-shore: reference medium
Air Water ( Acoustic ) F. S.
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Off-shore: reference medium
M. S. F. S. Air Water ( Acoustic )
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Off-shore: reference medium + “source”
Earth M. S. F. S. Air Water ( Acoustic )
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Off-shore: reference wave
M. S. F. S. Earth Air Water ( Acoustic ) ( Weglein and Secrest 90; Weglein 02; J. Zhang 05, 06, 07; Mayhan 12, 13; L. Tang 13 )
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Off-shore: reference wave
M. S. F. S. Earth Air Water ( Acoustic ) ( Weglein and Secrest 90; Weglein 02; J. Zhang 05, 06, 07; Mayhan 12, 13; L. Tang 13 )
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On-shore: reference medium
Elastic F. S. Air
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On-shore: reference medium
M. S. F. S. Elastic Air
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On-shore: reference medium + “source”
Elastic M. S. F. S. Air Earth
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On-shore: reference wave
M. S. F. S. Elastic Air Earth
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On-shore: reference wave
M. S. F. S. Elastic Air Earth
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On-shore: reference wave
M. S. F. S. Elastic Air Earth
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On-shore: reference wave prediction in (x,)
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On-shore: reference wave prediction in (x,)
( Stolt & Weglein 1992, 2012 )
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On-shore: reference wave prediction in (x,)
( Stolt & Weglein 1992, 2012 )
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M. S. F. S. Elastic Air
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F. S. Elastic Air M. S.
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F. S. Elastic Air M. S.
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F. S. Elastic Air M. S.
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F. S. Elastic Air M. S.
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F. S. Elastic Air M. S.
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F. S. Elastic Air M. S.
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On-shore: reference wave prediction in (kx,)
Assuming M.S. is horizontal
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On-shore: reference wave prediction in (kx,)
M. S. F. S. Elastic Air Earth
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Wavelet Estimation or
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Numerical Evaluation
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Water (acoustic) /elastic model --- OBC
Zs = -10 m O. B. 0m M.S. 1 m Elastic Water Layer P Velocity (m/s) S Velocity (m/s) Density (kg/m3) 1 1500 1000 2 1700 700 2000
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Water (acoustic) /elastic model --- OBC
Zs = -10 m O. B. 0m M.S. 1 m Elastic Water Layer P Velocity (m/s) S Velocity (m/s) Density (kg/m3) 1 1500 1000 2 1700 700 2000
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Reference wave prediction in water/earth: P wave component
Scholte wave Input data P Predicted Reference wave P0 P-P0
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Actual wavelet
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Estimated wavelet from P0
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Reference wave prediction in water/earth: S wave component
Input data S Predicted Reference wave S0 S-S0 Scholte wave
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Actual wavelet
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Estimated wavelet from S0
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Air/elastic model --- On shore
Zs = 0 m F. S. 0m M.S. 1 m Elastic Layer P Velocity (m/s) S Velocity (m/s) Density (kg/m3) 1 340 3 2 2200 1200 2000
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Reference wave prediction in air/earth: P wave component
Input data P Predicted Reference wave P0 P-P0 Rayleigh wave
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Reference wave prediction in air/earth: S wave component
Input data S Predicted Reference wave S0 S-S0 Rayleigh wave
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The elastic Green’s theorem method
Summary The elastic Green’s theorem method Predicts reference wave Estimates the wavelet Removes the ground roll without damaging the reflection data
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Discussion & Future research
Data requirements ( Weglein & Secrest 1990; Weglein, Keho & Secrest 1990; Corrigan, Weglein & Thompson 1991 )
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Discussion & Future research
Back out near surface properties (L. Tang et al.)
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Discussion & Future research
Extend to near surface with lateral variance
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Thank you Comments/Questions?
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Appendix
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For isotropic homogeneous medium
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(ux,uz) space to (P,S) space
For isotropic homogenous medium (Weglein and Stolt 1992, Zhang 2006)
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For actual medium (inhomogeneous)
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For actual medium (inhomogeneous)
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Green’s Function Reference medium Air Boundary Elastic
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Air Boundary Elastic P S
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P Air Boundary Elastic S
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Elastic Air Boundary S P
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Boundary condition At two sides of the boundary (z=0)
(Aki & Richards, 2002) Air Boundary Elastic
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The constitutive relation
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The constitutive relation
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The constitutive relation
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Air Boundary at depth 0 Elastic P S
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When z0, by using the boundary condition, The coefficients can be confirmed.
air elastic
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Air Boundary at depth 0 Elastic P S
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With boundary condition at z=0
air elastic
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Air Boundary at depth 0 Elastic S P
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With boundary condition at z=0
air elastic
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If both source and receiver are below the boundary
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Green’s theorem reference wavefield prediction derivation
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Air Earth m.s. v e.s.
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Green’s Second Identity
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Green’s Second Identity
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Reference wave prediction in (x, )
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Reference wave prediction in (kx, )
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Reference wave prediction in (kx, )
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Reciprocity of Green’s function
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Even, only real part left
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Odd, only image part left
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Appendix
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Boundary condition (acoustic/elastic)
Displacement X: viscid is low along the boundary, can be discontinuous; Z: no cavitation in the earth along the boundary, continuous. Traction Same magnitudes and opposite directions;
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At two side of the boundary (z=0)
Boundary condition Air Boundary Elastic At two side of the boundary (z=0)
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