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

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.

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.

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.

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

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

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.

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

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

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

Forward modeled traces

Error estimation: Crosscorrelation

Error estimation: L2 norm of error

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

Inverting the Q matrix A conventional matrix inversion algorithm gets unstable for Q<70

Hale’s inversion matrix, Hale (1981) Q-1 Each column of P is the convolution inverse of the corresponding column of Q.

Hale’s inversion matrix

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

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.

Downward continuation inverse Q filter

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

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

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

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

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:

Inverse Q filtering by using pseudodifferential operators

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

Comparison for Q=50

Comparison for Q=20

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

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

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, 4737-4748 Margrave, G. F., 1998, Theory of nonstationary linear filtering in the Fourier domain with application to time-variant filtering: Geophysics, 63, 244-259. 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, 657-663.

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