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Short-time homomorphic wavelet estimation
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Transcript of Short-time homomorphic wavelet estimation
Short-time wavelet estimation
in the homomorphic domain
Roberto H. Herrera and Mirko van der Baan
University of Alberta, Edmonton, Canada
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.
Wavelet estimation How many eligible
wavelets are there? ( ) ( ) ( )S z W Z R Z
n- roots m- roots
Possible
solutions
Example
m = 3000; % length(s(t))
n = 100; % length(w(t))
1002Nw
Convolutional
Model
How to find the n roots of the wavelet, from the m
roots of the seismogram ? Combinatorial
problem!!!
* Ziolkowski, A. (2001). CSEG Recorder, 26(6), 18–28. !
! !
mNw
n m n
20010Nw
Homomorphic wavelet estimation
• Why revisit homomorphic wavelet estimation?
– Homomorphic = log (spectrum)
– No minimum phase assumption
• Main challenges:
– Phase unwrapping
– Somewhat sparse reflectivity
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)
Homomorphic wavelet estimation
( ) ( ) ( )s t w t r t
( ) ( ) ( )S f W f R f
arg[ ( )]ˆ( ) log( ( )) log(| ( ) arg[ ( )]| ) log | ( ) |j S fS f S f S ff e S f j S
ˆ ˆ ˆ( ) ( ) ( )s t w t r t
time
frequency
log-spectrum
quefrency
FT
log
IFT
FT
log
IFT
time
frequency
log-spectrum
quefrency
IFT
exp
FFT
Math Forward Backward
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
Illustration classical method
Ulrych (1971)
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.
10 20 30 40 50 60
-1
-0.5
0
0.5
1
True Wavelet
Time [ms]
No
rmalized
Am
plitu
de
0 50 100 150 2000
0.2
0.4
0.6
0.8
1Minimum Phase Ref
Time [ms]0 50 100 150 200
-1
-0.5
0
0.5
1Simulated Trace
Time [ms]
0 50 100 150 200 250
-30
-20
-10
Log Amp Spectrum Wavelet
Frequency [Hz]
Lo
g-A
mp
litu
de
0 50 100 150 200 250
-30
-20
-10
0
Log Amp Spectrum Trace
0 50 100 150 200 2500
20
40
60
80
Deramped Phase Spectrum Wavelet
Frequency [Hz]
Ph
ase [
deg
rees]
0 50 100 150 200 250
-50
0
50
Deramped Phase Spectrum Trace
Frequency [Hz]
Illustration classical method True non-min
phase wavelet
Min phase refl
Log-Spectrum
Phase
Spectrum
Windowing Effects (Liftering)
Complex
cepstrum
Estimated
wavelets
-150 -100 -50 0 50 100 150-6
-4
-2
0
2
4
Complex cepstrum Trace + 3 Lifters
Samples - Quefrency (segment)
No
rmalized
Am
plitu
de
240 260 280
-0.1
0
0.1
0.2
Estimated Wavelet LF1
Samples - Time
No
rmalized
Am
plitu
de
240 260 280
-0.1
0
0.1
0.2
Estimated Wavelet LF2
Samples - Time240 260 280
-0.1
0
0.1
0.2
Estimated Wavelet LF3
Samples - Time
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
Log-spectrum
Log-
spectrum
Complex
cepstrum
0 50 100 150 200 250-20
-15
-10
-5
0
Log-spectrum Wavelet(red) and Trace(blue)
Lo
g-m
ag
nit
ud
e
Frequency [Hz]
-20 0 20 40 60 80 100 120 140-5
0
5Cepstrum of the Trace
Quefrency [ms]
Am
plitu
de
First Rahmonic Peak at 20 ms
Fundamental Period = 50 Hz
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.
Our Approach Data IN
Spectrogram TF - Overlapping segments
Complex LOG-Spectrum
Amplitude LOG-Spectrum Phase Spectrum
Phase unwrapping + Deramping
1/N ∑ = Average 1/N ∑ = Average
EXP + IFT
Estimated Wavelet
Re Img
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
Realistic example
Input data to STHWE
Wavelet length (wl) = 220 ms
Window length = 3 * wl
Window type = Hamming
50 % Overlap
Comparisons with:
- Original-wav
- First arrival
- Kurtosis Maximization (KPE).
Van der Baan (2008)
- LSA
- STHWE
Chevron - Dataset
CDP
Tim
e (
s)
50 100 150 200 250 300 350 400
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
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
Realistic example
Estimated
wavelets
Amp
Spectrum
Phase
Spectrum
-100 -50 0 50 100
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time [ms]
No
rmalized
am
plitu
de
Estimated Wavelets
Org-wav
FA
KPE-wav
LSA-wav
STHWE-wav
0 20 40 60 80 100 1200
2
4
6
8
10
Frequency [Hz]
Am
plitu
de
Amplitude Spectrum
Org-wav
FA
KPE-wav
LSA-wav
STHWE-wav
5 10 15 20 25 30 35 40 45
-80
-60
-40
-20
0
20
40
60
80
Frequency [Hz]
Ph
ase [
deg
rees]
Phase Spectrum
Org-wav
FA
KPE-wav
LSA-wav
STHWE-wav
Input data
CDP
Tim
e (
s)
50 100 150 200 250 300
0.5
1
1.5
2
2.5
3
3.5
4
Real Example: Stacked section
Input data to STHWE
Wavelet length (wl) = 220 ms
Window length = 3 * wl
Window type = Hamming
50 % Overlap
Comparisons with:
- First arrival
- Kurtosis Maximization (KPE).
Van der Baan (2008)
- LSA
- STHWE
Real Example: Stacked section
-100 -50 0 50 100-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time [ms]
No
rmalized
am
plitu
de
Estimated Wavelets
FA
KPE-wav
LSA-wav
STHWE-wav
0 20 40 60 80 100 120
0
1
2
3
4
5
6
7
Frequency [Hz]
Am
plitu
de
Amplitude Spectrum
FA
KPE-wav
LSA-wav
STHWE-wav
10 20 30 40
-80
-60
-40
-20
0
20
40
60
80
Frequency [Hz]
Ph
ase [
deg
rees]
Phase Spectrum
FA
KPE-wav
LSA-wav
STHWE-wav
Estimated
wavelets
Amp
Spectrum
Phase
Spectrum
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
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 …
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