Compensating for attenuation by inverse Q filtering. Carlos A. Montaña Dr. Gary F. Margrave

Transcription

1 Compensating for attenuation by inverse Q filtering Carlos A. Montaña Dr. Gary F. Margrave

2 Motivation Assess and compare the different methods of applying inverse Q filter Use Q filter as a reference to assess phase restoration in Gabor deconvolution

3 Outline Anelastic attenuation: Constant Q model Inverse Q filter approaches Inverting the Q matrix Downward continuation Pseudodifferential operators Comparison of the methods Conclusion

4 Attenuation mechanisms (Margrave G. F., Methods of seismic data processing, 2002) Geometric spreading Absorption (Anelastic attenuation) Transmission losses Mode conversion Scattering Refraction at critical angles

5 Anelastic attenuation In real materials wave energy is absorbed due to internal friction Absorption is frequency dependent Absorption effects: waveform change, amplitude decay phase delay The macroscopic effect of the internal friction is summarized by Q

6 Constant Q theory (Kjartansson, 1979) Q is independent of frequency in the seismic bandwidth Absorption is linear (Q>10) Dispersion: each plane wave travels at a different velocity The Fourier transform of the impulse response is frequency Distance Attenuation Phase B( Velocity f πfx 2πifx ) = exp exp VQ V

7 Impulse response (Q=10) V ( ω 1) 1 ω1 = 1+ log V ( ω2) πq ω2 (Aki and Richards, 2002) With dispersion Without dispersion

8 Nonstationary convolutional model W matrix Q matrix τ τ r s Fourier transform attenuated signal t s( f ) t Fourier transform source wavelet 2πifτ = w( f ) αq ( f, τ ) r( τ ) e dτ τ t Pseudodifferential operator ( ωτ / 2Q ih ( ωτ / 2 )) α ( ω, τ ) = exp + Q Q

9 Outline Anelastic attenuation: Constant Q model Inverse Q filter approaches Inverting the Q matrix Downward continuation Pseudodifferential operators Comparison of the methods Conclusion

10 Q filter and spiking deconvolution s = WQr Exact inversion r = Q 1 1 W s W matrix Q matrix τ τ r s Approximate inversion r W 1 1 Q s t t τ t r = W 1 Q 1 s + [ 1 1 Q, W ]s

11 Forward modeled traces s = WQr ( W = I) s = Qr

12 Error estimation: Crosscorrelation

13 Error estimation: L2 norm of error

14 Outline Anelastic attenuation: Constant Q model Inverse Q filter approaches Inverting the Q matrix Downward continuation Pseudodifferential operators Comparison of the methods Conclusion

15 Inverting the Q matrix r r W Q 1 1 Q s 1 s A conventional matrix inversion algorithm gets unstable for Q<70

16 Hale s inversion matrix, Hale (1981) r W 1 ( PQ) 1 Ps Q -1 Each column of P is the convolution inverse of the corresponding column of Q.

17 Hale s inversion matrix

18 Outline Anelastic attenuation: Constant Q model Inverse Q filter approaches Inverting the Q matrix Downward continuation Pseudodifferential operators Comparison of the methods Conclusion

19 Downward continuation inverse Q filter, Wang (2001) 2 U ( z, ω) k U ( z, ω) = 0 U ( t + t) = 2 U ( t + t, ω) dω z 2π Phase Amplitude correction correction iωv ( ω0) ωv ( ω0) U ( t + t, ω) = U ( t, ω)exp t exp t V ( ω) 2QV ( ω) τ τ 0 τ 1 1D-Earth model Attenuated signal u(τ 0 ) u(τ 1 ) τ τ j u(τ j ) Time varying frequency limit for the low pass filter τ n u(τ n )

20 Downward continuation inverse Q filter

21 Outline Anelastic attenuation: Constant Q model Inverse Q filter approaches Inverting the Q matrix Downward continuation Pseudodifferential operators Comparison of the methods Conclusion

22 Inverse Q filtering by pseudodifferential operators, Margrave (1998) = τ τ τ d h t a t g ) ( ) ( ) ( In a stationary linear filter the output can be obtained by convolving an arbitrary input with the impulse response = ) ( ) ( ) ( t g h t a τ τ Stationary linear filter theory

23 Inverse Q filtering by pseudodifferential operators = τ τ τ τ d h t b t s ) ( ), ( ) ( = τ τ τ d h t t b t s ) ( ), ( ) ~ ( Nonstationary linear filter theory = Nonstationary convolution Nonstationary combination b(t-τ, τ) h(τ) s(t)

24 Inverse Q filtering by pseudodifferential operators Nonstationary convolution and combination in the frequency domain Nonstationary convolution S ( ω) = B( ω Ω, Ω) H ( Ω) dω From Margrave (1998) Nonstationary combination These integrals are the nonstationary extension of the convolution theorem ~ S ( ω) = B( ω Ω, ω) H ( ω) dω

25 Inverse Q filtering by pseudodifferential operators S( ω) i = β ( ω, τ ) h( τ ) e ωτ dτ 1 i = Ω t H Ω e Ω t (, ) ( ) d 2π ~ s ( t) β Ω Generalized Anti-standard Fourier pseudo-differential integral (direct) operator Nonstationary convolution in mixed domains Generalized Standard Pseudodifferential (inverse) Fourier integral operator Nonstationary combination in mixed domains Forward Q filter: Inverse Q filter: ( ωτ / 2Q ih ( ωτ / 2 )) β ( ω, τ ) = α ( ω, τ ) = exp + Q β ( ω, τ ) α Q 1 Q ( ω, τ ) = exp ( ωτ / 2Q ih ( ωτ / 2Q) )

26 Inverse Q filtering by using pseudodifferential operators

27 Outline Anelastic attenuation: Constant Q model Inverse Q filter approaches Inverting the Q matrix Downward continuation Pseudodifferential operators Comparison of the methods Conclusion

28 Comparison for Q=50

29 Comparison for Q=20

30 Conclusions Amplitude recovery Phase recovery Numerical stability Computat. cost Hale s matrix inversion A+ (Q>40) D (Q<40) A+ (Q>40) D (Q<40) A+ (Q>40) D (Q<40) C Downward continuation C A- A+ A+ Pseudodifferential operator C+ A A+ A

31 Future work Consider as variables Uncertainty in Q estimation Variations of Q with depth Noise Improve amplitude recovery and efficiency in pseudodifferential operator Q filtering

32 References Aki, K., and Richards, P. G., 2002, Quantitative seismology, theory and methods. Hale, D., 1981, An Inverse Q filter: SEP report 26. Kjartansson, E., 1979, Constant Q wave propagation and attenuation: J. Geophysics. Res., 84, Margrave, G. F., 1998, Theory of nonstationary linear filtering in the Fourier domain with application to time-variant filtering: Geophysics, 63, Montana, C. A., and Margrave, G. F., 2004, Phase correction in Gabor deconvolution: CREWES Report, 16 Wang, Y.,2002, An stable and efficient approach of inverse Q filtering: Geophysics, 67,

33 Acknowledgements CREWES sponsors POTSI sponsors Geology and Geophysics Department MITACS NSERC CSEG