Intro Wavelet Processing 1

Western Geophysical

© 2000 Baker Hughes Incorporated ALL RIGHTS RESERVED© 2000 Baker Hughes Incorporated ALL RIGHTS RESERVED

Introduction To Wavelet Introduction To Wavelet ProcessingProcessing

Part 1Part 1

Author: Andrew FurberAuthor: Andrew Furber

Date : 25Date : 25thth February 2000 February 2000

Western Geophysical


AcknowledgementAcknowledgement

This presentation is based on the work produced by Scott This presentation is based on the work produced by Scott Scholz Scholz (LDC presentation, July 1992)(LDC presentation, July 1992)

Western Geophysical


Topics Under DiscussionTopics Under Discussion

1.0 Introduction1.0 Introduction

2.0 Building Blocks Of A Seismic Trace2.0 Building Blocks Of A Seismic Trace

3.0 Wavelet Processing Toolkit3.0 Wavelet Processing Toolkit

4.0 Wavelet Examples4.0 Wavelet Examples

Western Geophysical


1.0 Introduction1.0 Introduction

1.1 What is a wavelet?1.1 What is a wavelet?

1.2 What is wavelet processing?1.2 What is wavelet processing?

Western Geophysical


1.1 What is a wavelet?1.1 What is a wavelet?

‘‘A wavelet is a seismic pulse usually consisting of only A wavelet is a seismic pulse usually consisting of only a few cycles’a few cycles’ (After Sheriff, 1991)(After Sheriff, 1991)

Western Geophysical


Definitely Definitely NOTNOT a wavelet! a wavelet!

Western Geophysical


Some wavelet examples:Some wavelet examples:

The wavelet is The wavelet is usually a few usually a few hundred msec hundred msec longlong

Western Geophysical


1.2 What is wavelet processing?1.2 What is wavelet processing?

‘‘Wavelet processing is a series of processes where we Wavelet processing is a series of processes where we estimate or measure the wavelet within the data and estimate or measure the wavelet within the data and try to convert it to a wavelet of constant phase (zero try to convert it to a wavelet of constant phase (zero or minimum). We may also try to make the output or minimum). We may also try to make the output wavelet as broadband as possible’wavelet as broadband as possible’

Western Geophysical


Processes which clearly apply wavelet processing:Processes which clearly apply wavelet processing:

Deconvolution; signature, spiking, surface consistent, Deconvolution; signature, spiking, surface consistent, gapped, shot averaged …Etcgapped, shot averaged …Etc

Statistical designatureStatistical designatureSpectral whiteningSpectral whitening Inverse Q compensationInverse Q compensationMonochromatic noise suppressionMonochromatic noise suppressionBandpass filteringBandpass filtering Instrument de-phaseInstrument de-phasePhase rotationPhase rotationVibroseis correlationVibroseis correlation……. Etc. Etc

Western Geophysical


Processes which also apply wavelet processing:Processes which also apply wavelet processing:

Data Sampling (and resampling)Data Sampling (and resampling)Adjacent trace summationAdjacent trace summationStackStackNMO correctionNMO correctionDMO correctionDMO correctionStatics correctionStatics correctionMigrationMigrationGainGainMutingMutingPlottingPlotting……. Etc. Etc

Western Geophysical


2.0 Building Blocks Of A Seismic 2.0 Building Blocks Of A Seismic TraceTrace

2.1 A Simplified Earth Model2.1 A Simplified Earth Model

2.2 Terms To Remember2.2 Terms To Remember

2.3 Definition Of A Seismic Trace 2.3 Definition Of A Seismic Trace

Western Geophysical


2.1 A simplified Earth model:2.1 A simplified Earth model:

Western Geophysical


A simplified Earth model (in Time):A simplified Earth model (in Time):

Western Geophysical


2.2 Terms To Remember:2.2 Terms To Remember:

VELOCITY, VVELOCITY, VDENSITY, ρDENSITY, ρ

ACOUSTIC IMPEDANCE, IACOUSTIC IMPEDANCE, I

where, I = V. ρwhere, I = V. ρ

REFLECTION COEFFICIENT, RREFLECTION COEFFICIENT, R

where, R where, R 1- 21- 2 = I = I 22 - I - I 11

I I 22 + I + I 11

Note : This definition is true only for waves travelling perpendicular Note : This definition is true only for waves travelling perpendicular to the reflecting interfacesto the reflecting interfaces

Western Geophysical


Inversion:Inversion:

If we believe that the seismic trace truly represents the If we believe that the seismic trace truly represents the reflection coefficients, we can re-arrange the reflection reflection coefficients, we can re-arrange the reflection coefficient equation such that;coefficient equation such that;

I I 22 = I = I 11 1 + R 1 + R

1 - R1 - R

This allows you to calculate the acoustic impedance for each This allows you to calculate the acoustic impedance for each rock layer.rock layer.

Western Geophysical


2.3 Definition of a seismic trace:2.3 Definition of a seismic trace:

X (t) = ω (t) * r (t) ( * X (t) = ω (t) * r (t) ( * symbol for convolution)symbol for convolution)

where, X (t) = The seismic tracewhere, X (t) = The seismic trace

ω (t) = The waveletω (t) = The wavelet

r (t) = The reflection coefficient seriesr (t) = The reflection coefficient series

This means that every spike in a reflection coefficient series is This means that every spike in a reflection coefficient series is replacedreplaced with the seismic wavelet with the seismic wavelet

This seems simple enough, but what happens when the wavelets This seems simple enough, but what happens when the wavelets overlap?overlap?

Western Geophysical


3.0 Wavelet Processing Toolkit3.0 Wavelet Processing Toolkit

3.1 Convolution3.1 Convolution

3.2 Correlation3.2 Correlation

3.3 The Fourier Transform3.3 The Fourier Transform

Western Geophysical


3.1 Convolution:3.1 Convolution:

‘‘Convolution is the superposition of linear systems (also known as Convolution is the superposition of linear systems (also known as ‘filtering’)’‘filtering’)’

A * B = CA * B = C

Convolution is a sample by sample process where one function is Convolution is a sample by sample process where one function is replaced by another:replaced by another:

1. Shift either function1. Shift either function

2. Multiply the function by the value of the sample at the reference time2. Multiply the function by the value of the sample at the reference time

3. Keep a running sum of all values at each sample location3. Keep a running sum of all values at each sample location

The order of the functions is The order of the functions is notnot important: important:

A * B is the same as B * AA * B is the same as B * A

Western Geophysical


Convolution Example:Convolution Example:

Western Geophysical


3.2 Correlation:3.2 Correlation:

‘‘Correlation is a measure of the similarity between two functions’Correlation is a measure of the similarity between two functions’

A B = CA B = C

The correlation function measures the similarity of two functions at The correlation function measures the similarity of two functions at different time shifts of different time shifts of ‘lags’‘lags’

1. Choose one function to be the fixed function. The other will be the moving 1. Choose one function to be the fixed function. The other will be the moving (or ‘(or ‘sliding’sliding’) function) function

2. Shift one function2. Shift one function

3. Multiply the corresponding sample values3. Multiply the corresponding sample values

4. Sum these products4. Sum these products

5. Place the value of the sum at the time shift or lag used in step 25. Place the value of the sum at the time shift or lag used in step 2

6. Shift the sliding function by one sample and repeat steps 3 to 66. Shift the sliding function by one sample and repeat steps 3 to 6

The order of the functions The order of the functions isis important important

A B = B A A B = B A

Western Geophysical


Correlation Example:Correlation Example:

Western Geophysical


3.3 The Fourier Transform:3.3 The Fourier Transform:

The Fourier transform measures some important characteristics of The Fourier transform measures some important characteristics of a wavelet, and makes convolution and correlation much more a wavelet, and makes convolution and correlation much more efficient on a computer.efficient on a computer.

How does it work?How does it work?

Well, any single valued function (such as a seismic trace or Well, any single valued function (such as a seismic trace or wavelet) can be synthesised from the addition of simple cosine wavelet) can be synthesised from the addition of simple cosine waves.waves.

Western Geophysical


Zero Phase Wavelet:Zero Phase Wavelet:

Western Geophysical


-90 Degree Wavelet:-90 Degree Wavelet:

Western Geophysical


Each cosine wave has only Each cosine wave has only twotwo characteristics: characteristics:

Cosine Waves:Cosine Waves:

Western Geophysical


These two characteristics may also be graphically represented These two characteristics may also be graphically represented and are known as and are known as ‘Amplitude’‘Amplitude’ and and ‘Phase’‘Phase’ spectra. spectra.

The Fourier Transform:The Fourier Transform:

Western Geophysical


Convolution and correlation are computationally easier in the Convolution and correlation are computationally easier in the frequency domain. frequency domain.

CONVOLUTIONCONVOLUTION : This can be achieved in the frequency domain by : This can be achieved in the frequency domain by simply simply MULTIPLYINGMULTIPLYING the amplitude spectra and the amplitude spectra and ADDINGADDING the phase the phase spectraspectra

CORRELATIONCORRELATION : This can be achieved in the frequency domain by : This can be achieved in the frequency domain by simply simply MULTIPLYINGMULTIPLYING the amplitude spectra and the amplitude spectra and SUBTRACTINGSUBTRACTING the the phase spectraphase spectra

Let’s see some examples: Let’s see some examples:

The Fourier Transform:The Fourier Transform:

Western Geophysical


Convolution And Correlation Examples:Convolution And Correlation Examples:

Western Geophysical


Wavelet A:Wavelet A:

Western Geophysical


Wavelet B:Wavelet B:

Western Geophysical


Convolve Wavelet A With B (A * B):Convolve Wavelet A With B (A * B):

Western Geophysical


Correlate Wavelet A With B (A B):Correlate Wavelet A With B (A B):

Western Geophysical


4.0 Wavelet Examples4.0 Wavelet Examples

4.1 Wavelet Examples And Their Fourier Transforms4.1 Wavelet Examples And Their Fourier Transforms

4.2 The ‘Shifting’ Rule4.2 The ‘Shifting’ Rule

4.3 More Wavelet Examples4.3 More Wavelet Examples

Western Geophysical


The ‘SPIKE’The ‘SPIKE’

Western Geophysical


Wavelet Example 1Wavelet Example 1

Western Geophysical



Western Geophysical



Western Geophysical


4.2 The ‘Shifting’ Rule:4.2 The ‘Shifting’ Rule:

A time shift adds slope to the phase spectrum. The time shift A time shift adds slope to the phase spectrum. The time shift and slope are defined by the following equation:and slope are defined by the following equation:

ΦΦ(Degrees)(Degrees) = (360 = (360oo)(Δf)(Δfsecsec-1-1)(Δt)(Δtsecsec))

Western Geophysical



Western Geophysical



Western Geophysical


Suggested Reading MaterialSuggested Reading Material

Yilmaz, O., 1987, Seismic Data Processing. SEG PublicationYilmaz, O., 1987, Seismic Data Processing. SEG Publication

Chapter 1; Sections 1.1 to 1.2, Pages 9 to 26Chapter 1; Sections 1.1 to 1.2, Pages 9 to 26

Chapter 2; Sections 2.1 to 2.8, Pages 83 to 153Chapter 2; Sections 2.1 to 2.8, Pages 83 to 153

Appendix B; Sections B.1 to B.6, Pages 498 to 506Appendix B; Sections B.1 to B.6, Pages 498 to 506

Documents

Intro Wavelet Processing 1