de convolution

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Presentation title - Geoscience Training Centre

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BASIC SEISMIC PROCESSING BASIC SEISMIC PROCESSING

DECONVOLUTION DECONVOLUTION

PART 06PART 06

Presentation title - Page 2 Geoscience Training Centre

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Purpose of DeconvolutionPurpose of Deconvolution

Deconvolution attempts to undo unwanted convolution processes

applied to the seismic wavelet as it passes through the seismic system

Ultimate purpose:

to extract the reflectivity seriesto extract the reflectivity series

Deconvolution improves temporal resolution by compressing the

wavelet

This can include attenuation of multiplesThis can include attenuation of multiples

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Convolutional Model of the EarthConvolutional Model of the Earth

Deconvolution makes some assumptions about the seismic system:

Vertically propagating downVertically propagating down--going plane wavegoing plane wave

earth made of horizontal layers, constant Vearth made of horizontal layers, constant V

compression (P) wave at normal incidence, no shear wavescompression (P) wave at normal incidence, no shear waves

Source signature does not change: stationarity Source signature does not change: stationarity

Despite the fact that non of these criteria are not satisfied,

deconvolution is a remarkably robust process


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Reflection coefficient Wavelet Result

Principle of Superposition Principle of Superposition -- 11

Reflection coefficient for a single boundary represented by a spike:size indicates degree of impedance contrastsize indicates degree of impedance contrastcan be positive or negativecan be positive or negative

Source wavelet replicates itself:

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Principle of Superposition Principle of Superposition -- 22

Earth reflectivity series is a series of reflection coefficients

Convolution of reflection coefficients with wavelet Convolution of reflection coefficients with wavelet

*

=


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ResolutionResolution

To properly identify all boundaries wavelet must be ‘removed’

Reverse the effect of the convolutionReverse the effect of the convolution

Inverse FilterInverse Filter

DECONVOLUTIONDECONVOLUTION

3 boundaries How many boundaries

If reflections are closer (in time) than the wavelet length then

interference occurs and these events cannot be uniquely identified

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Convolution in the Time Domain ...Convolution in the Time Domain ...

The convolution of the reflectivity series with the source wavelet can be

expressed as:

x(t) = w(t) * e(t)x(t) = w(t) * e(t)

where:where:

xx is recorded traceis recorded trace

ee is reflectivity time seriesis reflectivity time series

ww is seismic wavelet (source signature in this model)is seismic wavelet (source signature in this model)

noise - free model


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… Multiplication in the Frequency Domain… Multiplication in the Frequency Domain

assume reflectivity is randomamplitude spectra multiplied

Amplitude spectrumof wavelet

X

Amplitude spectrum of reflectivity

(white)

Amplitude spectrumof trace

=

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Inverse FilteringInverse Filtering

From

xx((tt) = ) = ww((tt) * ) * ee((tt))

to remove w (t)

derive operator derive operator gg((tt) such that ) such that gg((tt) * ) * ww((tt) = spike) = spike

gg((tt) is mathematical inverse of ) is mathematical inverse of ww((tt) = ) = w w --11((tt))

Inverse filterInverse filter = = w w --11((tt))


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DeconvolutionDeconvolution TypesTypes

Deconvolution falls into two main categories:

Statistical Statistical DeconvolutionDeconvolution• Operators are derived statistically from the seismic data

Deterministic Deterministic DeconvolutionDeconvolution• Operators are derived from known or modelled functions

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Statistical Deconvolution: Application CasesStatistical Deconvolution: Application Cases

Wavelet estimated directly from the seismic data trace(s)

zerozero--lag spike ( ‘spiking deconvolution’ )lag spike ( ‘spiking deconvolution’ )

spike at arbitrary lagspike at arbitrary lag

time advanced form of the input (‘predictive’)time advanced form of the input (‘predictive’)

zerozero--phasephase

any desired arbitrary shape (‘shaping’)any desired arbitrary shape (‘shaping’)


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Spiking Deconvolution Spiking Deconvolution -- 11

White Noise (pre-whitening)

constant added to zeroconstant added to zero--lag of autocorrelationlag of autocorrelation

prevents amplification of noise at edges of frequency spectrumprevents amplification of noise at edges of frequency spectrum

Autocorrelation of trace is that of the wavelet

Assumes stationarity

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Spiking Deconvolution Spiking Deconvolution -- 22

X

=Wavelet Operator

Spiked output

prewhitening

amplitude

frequency

amplitude

frequency

amplitude

frequency


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PrePre--Whitening Whitening -- 11

Amplitude Spectrum - No White Noise

0

1

2

3

4

5

6

7

8

9

10

0 25 50 75 100 125

Inverse of Amplitude Spectrum - No White Noise

0

2

4

6

8

10

12

0 25 50 75 100 125

Amplitude Spectrum with White Noise

0

2

4

6

8

10

12

0 25 50 75 100 125

Inverse of Amplitude Spectrum with White Noise

0

1

2

3

4

5

6

7

8

9

10

0 25 50 75 100 125

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Effect of White Noise in Spiking Effect of White Noise in Spiking DeconvolutionDeconvolution

Added white noise

(% of zero lag)

0.1

No deconvolution

1.0

50.0

100.0

(Operator length 80ms)


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Effect of Operator Length in Spiking Effect of Operator Length in Spiking DeconvolutionDeconvolution

40

80

160

No deconvolution

(Prewhitening 1%)

Total Operator Length

(ms)

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Operator Length Operator Length -- 11

Inspect autocorrelation of input trace

Use first transient zone

Input Wavelet Autocorrelation


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Predictive Deconvolution Predictive Deconvolution -- Principles Principles -- 11

Predictive deconvolution presupposes that if an event can be predicted, it can not be a primary - the reflection series we have already defined as being random, i.e. non-predictable

aim to remove the predictable components

Unpredictable:Reflection Series

t 2t3t

4t

Predictable:Reverberations

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Predictive Deconvolution Predictive Deconvolution -- Principles Principles -- 22

Two methods:

PREDICTION filterPREDICTION filter• Predicts time advanced form of input series• Subtraction from input

PREDICTION ERROR filterPREDICTION ERROR filter• Convolve with input


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Effect of Gap in Effect of Gap in PredicitvePredicitve DeconvolutionDeconvolutionTotal Operator Length

(ms)

Gap

(ms)

88 8

96 16

104 24

120 40

No deconvolution

(Prewhitening 1%)

80 0(4)

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MultiMulti--channel Deconvolution channel Deconvolution -- 11

Calculate an operator from a group of traces

Preserves continuityPreserves continuity

Avoids deconvolving the geologyAvoids deconvolving the geology

signal well estimated but the noise is not well attenuatedsignal well estimated but the noise is not well attenuated


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DECMC ExampleDECMC Example

spiking DECON

multi -channel DECON

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Frequency Domain Frequency Domain DeconvolutionDeconvolution

Input Trace

Compute Autocorrelation

Compute W-L Operator

Fourier TransformAmplitude Spectrum Phase Spectrum

Amplitude Spectrum Phase Spectrum

Multiply AddInverse Fourier

Transform to get output trace

Fourier Transform


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Zero Phase Zero Phase DeconvolutionDeconvolution

Input Trace

Compute Autocorrelation

Compute W-L Operator

Fourier TransformAmplitude Spectrum Set Phase = 0

Amplitude Spectrum Phase Spectrum

Multiply AddInverse Fourier

Transform to get output trace

Fourier Transform

de convolution

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