SISMOwww.sismo.srv.br

Can we use the spectral ridges to estimate Q ?

Marcílio Castro de Matos

marcilio@matos.eng.br www.matos.eng.br

1

Attribute-Assisted Seismic Processing and Interpretation

http://geology.ou.edu/aaspi/ Signal Processing Research, Training & Consultingwww.sismo.srv.br

Summary

• Spectral ridges• Q estimation• Examples• Conclusions

SyntheticReflectivity CWT Magnitude

CWTmagnitude

0

pos

CWT MML

Spectral ridges

Introduction

Continuous Wavelet Transform (very brief review)

ICWTdec

Examples

Conclusions

3

Tim

e F

requ

ency

Reflectivity

r(t)

Source wavelet

s(t)

Noise

n(t)

Seismic data

u(t)* +

Bandlimited white spectrum

Modified from Kurt Marfurt course(Partyka et al, 1999)

Long window spectral

decomposition and the

convolutional model

White spectrum

Colored spectrum

Tim

e F

requ

ency

Reflectivity

r(t)

Source wavelet

s(t)

Noise

n(t)

Seismic data

u(t)* +

Bandlimited colored spectrum

Short window spectral

decomposition and the

convolutional model

Modified from Kurt Marfurt course(Partyka et al, 1999)

1822 Fourier book

From: http://books.google.com/

TeFtf

k

tjkkT

2.)( 0

0

Tt

t

tjkk dtetfT

F0

0

0 .).(1

Animated plot of the first five successive partial Fourier series. From wikipedia.org

f t t

(Yilmaz, 2001)

Seismic zero phase wavelet

Summation of co-sinusoids with zero phase

f t t

8

Short Time Fourier Transform – STFT

The simplest time-frequency representation

duetuhuxhtF ujx

2*;,

f t t

9

Short Time Fourier Transform – STFTAmplitude and Phase spectrum

tim

e

20 40 60 80 100 120 140 160 180 200 220

2000

2200

2400

2600

2800

3000

3200-2

-1

0

1

2

x 104

frequency

tim

e

0 50 100 150 200

2000

2200

2400

2600

2800

3000

3200

Time-frequency pattern???

f t t

Spectral ridges

Introduction

Continuous Wavelet Transform (very brief review)

ICWTdec

Examples

Conclusions

11

Continuous Wavelet Transform

0 1 2 3 4 5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

tempo

Am

plitu

de

0.02 0.04 0.06 0.08 0.1 0.12 0.14

0

2

4

6

8

10

12

14

16

18

frequencia normalizada (x )

Am

plit

ude

Cdd0 2

0

2 ˆˆ

0

dtt

s

ut

stsu 1

, 00ˆ

dtt

-5 -4 -3 -2 -1 0 1 2 3 4 5-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Wavelet Chapeu Mexicano

tempo

Am

plitu

de

(x) L2() is called a wavelet

100 200 300 400 500 600 700 800 900 1000

-0.2

0

0.2

0.4

Amostras

Am

plitu

de

Amostras

Esc

ala

100 200 300 400 500 600 700 800 900 1000

10

20

30

40

50

60

Continuous Wavelet Transform (CWT)

time

Amplitude

( ),

1( , ) ( , ) , ( )x u s

t uCWT u s Wf u s f f t dt

ss

1

( , ) ( ) s

t uWf u s f t dt f u

ss

s

t

sts 1

ˆ ˆs s s The CWT can be interpreted

as a band pass filter response at each scale s

Sca

les

Time (ms)

Time (ms)

f t t

Grossmann and Morlet introduced CWT formally

in 1984

Inverse CWT

SyntheticReflectivity CWT Magnitude Voices ICWT

CWTmagnitude

0

pos

Σ

Summary

Introduction

Continuous Wavelet Transform (very brief review)

Pseudo deconvolution (icwtdec)

Examples

Conclusions

16

Math behind…

100 200 300 400 500 600 700 800 900 1000

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

amostras

Am

plitude

20

40

60

80

100

120

Abs. and by scale Values of Ca,b Coefficients for a = 1 2 3 4 5 ...

time (or space) b

scal

es a

100 200 300 400 500 600 700 800 900 1000 1 5 913172125293337414549535761

100 200 300 400 500 600 700 800 900 1 5 913172125293337414549535761

Local Maxima Lines

Singularities detection and characterization through Continuous Wavelet Transform (CWT): Lipschitz (Hölder) Coefficients

10 20 30 40 50 60

1

2

3

s

|Wf(

u,s)

|

0 1 2 3 4 5 6

-1

-0.5

0

0.5

1

1.5

log2(s)lo

g2(|W

f(u,

s)|)

+1/2=1/2

sAsuWf 222 log2

1log,log

CWT modulus maxima line

SyntheticReflectivity CWT Magnitude

CWTmagnitude

0

pos

CWT MML

WTMMLA seismic applications

ICWT “deconvolution” workflow

SyntheticReflectivity CWT Magnitude

CWTmagnitude

0

pos

CWT MML

CWTMorlet

CWT MML

ICWTShrunken

Morlet

CWT MML voices

Σ

Relative acoustic impedance from ICWTDEC

WORKFLOW

- Re-scale seismic trace: |s(t)|<<1- Integration filter (Peacock, 1979) - High-pass filter

Summary

Introduction

Continuous Wavelet Transform (very brief review)

ICWTdec

Examples

Conclusions

24

Case 1: Synthetic seismic channel

10 ms thickness trace

Case 1: Synthetic seismic channel

30 ms thickness trace

Synthetic channel and its ICWTdec

ICWTDEC RAI

Adding random noise

Case 2: Barnet Shale

Original Seismic

ICWTdec

THINMAN™

Marble Falls

Upper Barnett Lm

Upper Barnett Sh

Forestburg

Lower Barnett Sh

Viola

ICWTdec

60Amplitude

60

0

Marble Falls

Upper Barnett Lm

Upper Barnett Sh

Forestburg

Lower Barnett Sh

Viola

THINMAN™

10Amplitude

10

0

Case 3: Pre-stack (Hampson&Russel 2D demo data set)

Twt (s)

0

0.7

Offset

ICWTdec

Twt (s)

0

0.7

Offset

RAI

Twt (s)

0

0.7

Offset

Spectral ridges Conclusions

• CWT spectral decomposition filtering process described dear generates high resolution events that correlate to major acoustic impedance changes.

• Since this broadening is a trace-by-trace independent process, laterally-consistent thin bed terminations and other truncations can be interpreted with confidence.

Q estimation

• Anelasticity and wave propagation “very brief” review• Q estimation and spectral ridges• Conclusions

Anelasticity

Berea Sandstone

Wyllie, et al, 1958

From Carl Sondergeld Rock Physics Course Notes, 2009

Anelasticity review

From Carl Sondergeld Rock Physics Course Notes, 2009

Wave equation

From Carl Sondergeld Rock Physics Course Notes, 2009

Attenuation per wavelength

From Carl Sondergeld Rock Physics Course Notes, 2009

Normal incidence anelastic reflections

From Carl Sondergeld Rock Physics Course Notes, 2009

Seismic wave behavior in absorptive media defined by v, ρ and Q.

Figure 2.4 of Seismic Absorption Estimation and Compensation by Changjun Zhang M.Sc., The University of British Columbia, 2009

• Q is inversely proportional to attenuation. The greater the Q value, the smaller the loss or attenuation!

• The phase lag Ψ is a direct measure of attenuation. The greater the phase lag, the greater the attenuation.

• Q for rock lies in the range of 10 to 200.• If Q = Q(ω) then M must also be a function of frequency!• Moduli must depend upon frequency!

Q estimation

• Anelasticity and wave propagation “very brief” review• Q estimation and spectral ridges• Conclusions

Q estimation from spectral ratio

freqAdrinal Ilyas, 2010, Estimation of q factor from seismic reflection data, MsC Curtin University

Chopra, Alexeev, and Sudhakar, TLE 2003, High-frequency restoration of surface seismic data

Q estimation from spectral ratio

• Synthetic

• Reflectivity

• CWT Magnitude

• CWT• magnitude

• 0

• pos

• CWT MML

Spectral ridges can guide Q estimation from spectral ratio

In Q computation, we need to compute the amplitude spectra of two adjacent events (Taner, 2000)

Q estimation from Peak Frequency variation

Ricker wavelet

Zhang & Ulrych, 2002, Geophysics, Estimation of Quality factors from CMP records

Q estimation from Peak Frequency variation

Zhang & Ulrych, 2002, Geophysics, Estimation of Quality factors from CMP records

Q estimation from Peak Frequency variation

Zhang_Ulrych_2007_Geophysics_Seismic absorption compensation A least squares inverse scheme

Frequency decay caused by thin-bed tuning and absorption

Figure 4.2 of Seismic Absorption Estimation and Compensation by Changjun Zhang M.Sc., The University of British Columbia, 2009

Absorption and specdecomp phase components

SyntheticReflectivity CWT Magnitude

CWTmagnitude

0

pos

CWT MML

180

-180

CWT phase

10 70

Frequency (Hz)

CWT Magnitude and Phase overlaid by spectral ridgesThe phase spectra will provide information for dispersion estimation. Attributes picked at the peak of the envelope represent the average of the wavelet attribute. That is why we pick the amplitude spectrum at the time of envelop peak for Q computation. Phase spectra is picked the same way. If we look at the phase spectra, we observe that most of the spectra of the events are horizontal, which means that these wavelets are zero phase, and their rotation angle is the phase corresponding to the envelop peak. Therefore, computation of dispersion consists of determining the phase differences at each sub-band trace and compute an average phase delay per cycle per second. Since dispersion is related to absorption, higher levels of dispersion will point to higher levels of absorption, which may indicate fracture in carbonates or unconsolidated snads in clastic environment. (Taner, 2000).

Conclusions

• CWT spectral decomposition filtering process described dear generates high resolution events that correlate to major acoustic impedance changes.

• It seems we can correlate spectral ridges with Q estimation

Acknowledgements

Attribute-Assisted Seismic Processing and Interpretation

http://geology.ou.edu/aaspi/

marcilio@matos.eng.br www.matos.eng.br

Thank you for your attention!!!

Thanks to DEVON for providing a license to one of the seismic data volume used herein.

Thanks to Carl Sondergeld

Thanks to Roderick Perez from OU for his Barnet shale interpretation.

Thanks also to PETROBRAS Reservoir Geophysics Management friends for their comments.

Signal Processing Research, Training & Consultingwww.sismo.srv.br