Short-time wavelet estimation

in the homomorphic domain

Roberto H. Herrera and Mirko van der Baan

University of Alberta, Edmonton, Canada

rhherrer@ualberta.ca

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