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# Short-time homomorphic wavelet estimation

Wavelet estimation plays an important role in many seismic processes like impedance inversion, amplitude versus offset (AVO) and full waveform inversion (FWI). Statistical methods of wavelet estimation away from well control are a desirable tool to support seismic signal processing. One of these methods based on Homomorphic analysis has long intrigued as a potentially elegant solution to the wavelet estimation problem. Yet a successful implementation has proven difficult. We propose here a method based short-time homomorphic analysis which includes elements of the classical cepstrum analysis and log spectral averaging. Our proposal increases the number of segments, thus reducing estimation variances. Results show good performance on realistic synthetic examples.

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### Short-time homomorphic wavelet estimation

1. 1. Short-time wavelet estimation in the homomorphic domain Roberto H. Herrera and Mirko van der Baan University of Alberta, Edmonton, Canada rhherrer@ualberta.ca
2. 2. Homomorphic wavelet estimation• Objective: – Introduce a variant of short-time homomorphic wavelet estimation.• Possible applications: – nonminimum phase surface consistent deconvolution. FWI, AVO.• Main problem – To find a consistent wavelet estimation method based on homomorphic analysis. SCD done in homomorphic domain.
3. 3. Wavelet estimationHow many eligible Convolutionalwavelets are there? S ( z)  W (Z ) R(Z ) Model m- roots n- roots m = 3000; % length(s(t)) Possible n = 100; % length(w(t)) solutions Example Nw  2100 How to find the n roots of the wavelet, from the m roots of the seismogram ? Combinatorial problem!!! m! * Ziolkowski, A. (2001). CSEG Recorder, 26(6), 18–28. Nw  n ! m  n ! Nw  10200
4. 4. Homomorphic wavelet estimation• Why revisit homomorphic wavelet estimation? – Homomorphic = log (spectrum) – No minimum phase assumption• Main challenges: – Phase unwrapping – Somewhat sparse reflectivity
5. 5. Homomorphic wavelet estimation• Steps (Ulrych, Geophysics, 1971): – Take log( FT( observed signal) ) • s(t) = r(t)*w(t) <=> log(S(f)) = log(R(f)) + log(W(f)) – Real part is log (amplitude spectrum) – Imaginary part is phase – Natural separation amplitude and phase spectrum – Apply phase unwrapping + deramping – Take inverse Fourier transform: ŝ(t)=FT-1(log(S)) – Apply bandpass filtering on ŝ(t) = liftering • Or simply time-domain windowing + inverse transform – Recover the wavelet w(t)
6. 6. Homomorphic wavelet estimation Math Forward Backward s(t )  w(t )  r (t ) time time FT FT IFT frequency frequency S ( f )  W ( f ) R( f ) log exp logˆ log-spectrum log-spectrumS ( f )  log(S ( f ))  log(| S ( f ) | e j arg[ S ( f )] )  log | S ( f ) |  j arg[S ( f )] IFT IFT FFT s(t )  w(t )  r (t ) ˆ ˆ ˆ quefrency quefrency
7. 7. Homomorphic wavelet estimation•Assumptions (Ulrych, Geophysics, 1971): –Somewhat sparse reflectivity –Minimum phase reflectivity •Exponential damping applied otherwise•Rationale: –Log leads to spectral whitening and wavelet shrinkage => isolation of single wavelet –Min phase reflectivity + deramping => Dominant contribution from near t=0 => emphasis on first arrival => maintains phase
8. 8. Illustration classical methodSingle echoReflectivity = 2 spikesr = [1, …, 0.9,…] a - is the amplitude of the first echo, 0.9 (forcing the reflectivity to be minimum phase) δ – is the Dirac delta function. And the echo delay is t_0 = 20 ms. Ulrych (1971)
9. 9. Illustration classical method True Wavelet Minimum Phase Ref Simulated Trace 1 1 Normalized AmplitudeTrue non-min 1 0.8 0.5 0.5phase wavelet 0.6 0 0 0.4 -0.5Min phase refl 0.2 -0.5 -1 0 -1 10 20 30 40 50 60 0 50 100 150 200 0 50 100 150 200 Time [ms] Time [ms] Time [ms] Log Amp Spectrum Wavelet Log Amp Spectrum Trace 0 Log-Amplitude -10 -10 -20 -20 Log-Spectrum -30 -30 0 50 100 150 200 250 0 50 100 150 200 250 Frequency [Hz] Deramped Phase Spectrum Wavelet Deramped Phase Spectrum Trace 80 50 Phase [degrees] 60 Phase 40 0 Spectrum 20 -50 0 0 50 100 150 200 250 0 50 100 150 200 250 Frequency [Hz] Frequency [Hz]
10. 10. Windowing Effects (Liftering) Complex cepstrum Trace + 3 Lifters 4 Normalized Amplitude 2Complex 0cepstrum -2 -4 -6 -150 -100 -50 0 50 100 150 Samples - Quefrency (segment) Estimated Wavelet LF1 Estimated Wavelet LF2 Estimated Wavelet LF3 0.2 0.2 0.2 Normalized AmplitudeEstimated 0.1 0.1 0.1wavelets 0 0 0 -0.1 -0.1 -0.1 240 260 280 240 260 280 240 260 280 Samples - Time Samples - Time Samples - Time
11. 11. Log spectral averaging• Liftering is “hopeless”• New assumption:  random reflectivity but stationary wavelet  the method becomes log-spectral averaging over many traces• Calculate the log-spectrum of many traces and average => removes reflectivity
12. 12. Log-spectrum Log-spectrum Wavelet(red) and Trace(blue) 0 Log-magnitude -5Log-spectrum -10 -15 Fundamental Period = 50 Hz -20 0 50 100 150 200 250 Frequency [Hz] Cepstrum of the Trace 5 First Rahmonic Peak at 20 ms AmplitudeComplex 0cepstrum -5 -20 0 20 40 60 80 100 120 140 Quefrency [ms]
13. 13. Our Approach– Cepstral stacking (Log-spectral averaging) • Wavelet is invariant while reflectivity is spatially non- stationary – Following the Central Limit Theorem, r(t) will tend to a mean value !!! • Requires minimum-phase reflectivity or at least strong first arrival– Averaging the log-spectrum of the STFT • Like the Welch transform in the log-spectrum domain.
14. 14. Our Approach Data IN Spectrogram TF - Overlapping segments Complex Re LOG-Spectrum ImgAmplitude LOG-Spectrum Phase Spectrum Phase unwrapping + Deramping 1/N ∑ = Average 1/N ∑ = Average EXP + IFT Estimated Wavelet
15. 15. Our Approach• Assumptions – Random reflectivity – Stationary wavelet – Nonminimum, frequency-dependent wavelet phase – Nonminimum-phase reflectivity • Deramping + Averaging of log(spectra) emphasizes main reflections => most important contribution to wavelet estimate
16. 16. Realistic example Chevron - DatasetInput data to STHWE 0.2Wavelet length (wl) = 220 ms 0.4Window length = 3 * wl 0.6Window type = Hamming50 % Overlap 0.8 1 Time (s) 1.2 1.4Comparisons with: 1.6- Original-wav 1.8- First arrival- Kurtosis Maximization (KPE). 2 Van der Baan (2008) 2.2- LSA 50 100 150 200 250 300 350 400- STHWE CDP
17. 17. Elements of comparison• Different wavelet estimates – Log spectral averaging of entire trace (LSA) – Constant-phase wavelet estimated using kurtosis maximization (= KPE) – STFT log spectral averaging (=STHWE)• Compare with true wavelet + first arrival
18. 18. Realistic example Estimated Wavelets Amplitude Spectrum 1 Org-wav Org-wav FA 10 FA 0.8 KPE-wav KPE-wav LSA-wav 8 LSA-wav STHWE-wav STHWE-wav Amp Amplitude 0.6 6 Spectrum 0.4 4Estimated 2wavelets 0.2 Normalized amplitude 0 0 20 40 60 80 100 120 0 Frequency [Hz] -0.2 Phase Spectrum 80 -0.4 60 40 Phase Phase [degrees] -0.6 20 Spectrum 0 -0.8 -20 Org-wav FA -40 KPE-wav -1 -60 LSA-wav STHWE-wav -80 -100 -50 0 50 100 5 10 15 20 25 30 35 40 45 Time [ms] Frequency [Hz]
19. 19. Real Example: Stacked section Input dataInput data to STHWE 0.5Wavelet length (wl) = 220 ms 1Window length = 3 * wlWindow type = Hamming 1.550 % Overlap Time (s) 2 2.5 3 Comparisons with: 3.5 - First arrival 4 - Kurtosis Maximization (KPE). 50 100 150 200 250 300 Van der Baan (2008) CDP - LSA - STHWE
20. 20. Real Example: Stacked section Estimated Wavelets Amplitude Spectrum 1 FA FA 7 KPE-wav KPE-wav 0.8 LSA-wav 6 LSA-wav STHWE-wav 5 STHWE-wav Amp Amplitude 0.6 4 Spectrum 3Estimated 0.4 2wavelets 1 Normalized amplitude 0.2 0 0 20 40 60 80 100 120 Frequency [Hz] 0 Phase Spectrum -0.2 80 FA KPE-wav 60 LSA-wav Phase -0.4 40 Phase [degrees] 20 STHWE-wav Spectrum -0.6 0 -20 -40 -0.8 -60 -80 -1 -100 -50 0 50 100 10 20 30 40 Time [ms] Frequency [Hz]
21. 21. DiscussionPros and cons• Wavelet could be recovered without any a priori assumption regarding the wavelet or the reflectivity.• Log spectral averaging softens the sparse reflectivity assumption by increasing the amount of traces + reduces estimation variances.• Selection window length in STFT important
22. 22. Conclusions• The short-time homomorphic wavelet estimation method provides stable results. – Comparable with the constant-phase kurtosis maximization.• Future work: nonminimum phase surface- consistent deconvolution …
23. 23. BLISS sponsorsBLind Identification of Seismic Signals (BLISS) is supported byWe also thank:- Chevron for providing the synthetic data example (D. Wilkinson)- BP for permission to use the real data example- Mauricio Sacchi for many insightful discussions

Wavelet estimation plays an important role in many seismic processes like impedance inversion, amplitude versus offset (AVO) and full waveform inversion (FWI). Statistical methods of wavelet estimation away from well control are a desirable tool to support seismic signal processing. One of these methods based on Homomorphic analysis has long intrigued as a potentially elegant solution to the wavelet estimation problem. Yet a successful implementation has proven difficult. We propose here a method based short-time homomorphic analysis which includes elements of the classical cepstrum analysis and log spectral averaging. Our proposal increases the number of segments, thus reducing estimation variances. Results show good performance on realistic synthetic examples.

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