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.

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. 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. Wavelet estimation
How many eligible Convolutional
wavelets 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. Homomorphic wavelet estimation
• Why revisit homomorphic wavelet estimation?
– Homomorphic = log (spectrum)
– No minimum phase assumption
• Main challenges:
– Phase unwrapping
– Somewhat sparse reflectivity

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. 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
log
ˆ log-spectrum log-spectrum
S ( 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. 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. Illustration classical method
Single echo
Reflectivity = 2 spikes
r = [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. Illustration classical method
True Wavelet Minimum Phase Ref Simulated Trace
1 1
Normalized Amplitude
True non-min 1
0.8
0.5
0.5
phase wavelet 0.6
0 0
0.4
-0.5
Min 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. Windowing Effects (Liftering)
Complex cepstrum Trace + 3 Lifters
4
Normalized Amplitude
2
Complex 0
cepstrum -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 Amplitude
Estimated 0.1
0.1
0.1
wavelets
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. 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. Log-spectrum
Log-spectrum Wavelet(red) and Trace(blue)
0
Log-magnitude -5
Log-
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
Amplitude
Complex
0
cepstrum
-5
-20 0 20 40 60 80 100 120 140
Quefrency [ms]

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. Our Approach
Data IN
Spectrogram
TF - Overlapping segments
Complex
Re LOG-Spectrum Img
Amplitude LOG-Spectrum Phase Spectrum
Phase unwrapping + Deramping
1/N ∑ = Average
1/N ∑ = Average
EXP + IFT
Estimated Wavelet

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. Realistic example
Chevron - Dataset
Input data to STHWE 0.2
Wavelet length (wl) = 220 ms 0.4
Window length = 3 * wl 0.6
Window type = Hamming
50 % Overlap 0.8
1
Time (s)
1.2
1.4
Comparisons 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. 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. 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 4
Estimated
2
wavelets 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. Real Example: Stacked section
Input data
Input data to STHWE
0.5
Wavelet length (wl) = 220 ms 1
Window length = 3 * wl
Window type = Hamming 1.5
50 % 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. 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
3
Estimated 0.4 2
wavelets 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. Discussion
Pros 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. 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. BLISS sponsors
BLind Identification of Seismic Signals (BLISS)
is supported by
We 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