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
Our main purpose is to improve Gabor deconvolution. Our motivation to work wit Q filtering, is to understand the differences between the two kinds of methods, to identify features in Q filter that can be implemented into Gabor deconvolution to improve its performance. I will show you a review of the methods to apply Q-filtering, and there is a paper in this year’s CREWES report that shows preliminary results of the comparison between Gabor deconvolution and Q filtering with respect to phase compensantion.

3. Outline Anelastic attenuation: Constant Q model
Inverse Q filter approaches
Inverting the Q matrix
Downward continuation
Pseudodifferential operators
Comparison of the methods
Conclusion
First of all I will present a very short review of the Constant Q model theory for attenuation, which is the starting point for both Gabor deconvolution and Q filter
Then I will introduce the essential ideas of three different methods to apply Q-filtering, show a few examples
And finally compare the results obtained from the three methods.

4. Attenuation mechanisms (Margrave G. F
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
A travelling way is attenuated due to different mechanisms. For each mechanisms there exist adequate methods and theories. The best possible compensation for attenuation is generally reached by combining a number of different methods.
Gabor deconvolution and Q filtering are exclusively oriented to compensate for absortion or anelastic attenuation
The Amplitude of the transmitted wave is decaying for the loss of energy reflected at each interface. In regions of high reflectivity the amplitude of the transmitted wave can rapidly approach zero. In this case, the observed transmitted wavelet will be an interference effect due to the interaction between the primary and the interbed multiples.
Interbed multiples delays can be estimated by modeling synthetic seismograms with and without interbed multiples delays from sonics and density log data.

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
Lets talk a bit about anelastic attenuation. When travelling, the kinectic energy transported by the wave is progresively converted into heat. This is a phenomenon frequency-dependent, higher frequencies are more attenueted, and the global results of the process can be summarized in waveform change, amplitude decay and phase delay.
The most common way of estimating anelastic attenuation is through the dimensionless parameter Q, defined as the fraction of energy lost during each cycle or wavelength

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
Different theories have been proposed for anelastic attenuation. However Constant Q theory is mathematically much simpler than any other NCQ models and nonlinear models. Is completely specified by two parameters Q and phase velocity at a reference frequency.
Laboratory work on the absorption in rocks shows Q independent of frequency
Velocity dispersion: high frequency plane waves travel faster than low frequency waves.
Absortion is linear far from the source and for weak attenuation
Linearity implies that convolution and Fourier transforms are valid mathematical tools to study it
Distance
Attenuation
Phase
frequency
Velocity

7. Impulse response (Q=10) With dispersion Without dispersion
(Aki and Richards, 2002)
With dispersion
Without dispersion
By using the previous equation to model the propagation of the impulse response, the three main effects of attenuation can be observed. Amplitude exponential decay due to energy absorption, broadening of the pulse because higher frequencies are more absorbed, and phase delay due to dispersion. The blue line corresponds to the impulse response without considering dispersion whereas the red line corresponds to consideration of dispersion, and is getting behind.
Dispersion in the theoretical model appears as a consequence of forcing the pulse to be causal
The time-varying phase distrorsion is often more problematic than the seismic frequency loss. Wavelet phase is what is of particular importance when the survey objective is the detailed analysis of stratigraphy, or if the processed data are to be input to inversion schemes that extract lithologic information.
Can produce time-shifts which are important when seismic data are tied to sonic-log data.

8. Nonstationary convolutional model
W matrix Q matrix r s
Fourier T attenuated signal
A very important result of the constant Q theory is the nonstationary model for a an attenuated seismic trace. For the discrete case, the Attenuated seismic trace is found to be equal to a source wavelet matrix, in which each column is a shifted version of the source, multiplied by a matrix that contains the attenuation information, in which each column is an attenuated pulse, multiplied by the reflectivity.
For the continuous case the attenuated trace in the frequency domain is equal to the Fourier transform of the wave signature multiplied by this integral which corresponds to a pseudodifferential operator on the reflectivity with symbol Alfa, where alpha is the exponential attenuation function, in which the imaginary and the real part of the exponent are related through the Hilbert transform
Pseudodifferential operator
Fourier transform source wavelet
Fourier transform attenuated signal

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
Exact inversion
Approximate inversion t 
W matrix Q matrix r s
According to our model for the attenuated trace the right order to

11. Forward modeled traces

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
A conventional matrix inversion algorithm gets unstable for Q<70

16. Hale’s inversion matrix, Hale (1981)
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)
Phase correction
Amplitude correction
Similar to Stolt F-K migration.
Hargreaves can efficiently correct for phase distortion from dispersion but neglects amplitude effect.
The amplitude compensation operator is an exponential function of the frequency, and including it in the inverse Q filter may cause instability and generate undesirable effects in the solution.
The earth Q model is assumed to be a multilayered structure.
The computational demands of this samplewise downward continuation render it impractical four routine prestack seismic processing.
It is better to overestimate Q than underestimate it according to the sensitivity analysis by Duren and Trantham.
If Q is independent of position, then the inverse filtering depends solely upon travel time and not the wave direction and an inverse Q filtering can be separated from the migration. In this situation there is no distinction between prestack and poststack inverse Q filtering, since the filter depends only on the traveltime of an event on the recorded trace, regardless of the offset.

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 pseudo- differential operators, Margrave (1998)
Stationary linear filter theory
In a stationary linear filter the
output can be obtained by
convolving an arbitrary input
with the impulse response

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

24. Inverse Q filtering by pseudodifferential operators
Nonstationary convolution and combination in the frequency domain
Nonstationary convolution
Nonstationary combination
From Margrave (1998)
These integrals are the nonstationary
extension of the convolution theorem

25. Inverse Q filtering by pseudodifferential operators
Anti-standard pseudo-differential operator
Generalized Fourier integral (direct)
Nonstationary convolution
in mixed domains
Standard Pseudo-differential operator
Generalized Fourier integral (inverse)
Nonstationary combination
in mixed domains
Forward Q filter:
Inverse Q filter:

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