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# Deconvolution ppt

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Lecture 13: Deconvolution, part 2 Wiener ﬁltering Deconvolution design Prewhitening Prediction distances Types of deconvolution Spiking deconvolution Predictive deconvolution Waveshaping deconvolution
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• Lecture 13:Deconvolution, part 2

Wiener filtering

Deconvolution design

Prewhitening

Prediction distances

Types of deconvolution

Spiking deconvolution

Predictive deconvolution

Waveshaping deconvolution

• The convolutional model:

x(t) is the recorded seismogram

w(t) is the source wavelet

r(t) is the earths impulse response (e.g., the reflectivity series)

n(t) is random ambient noise

The goal of deconvolution:

To remove the affect of the source wavelet and of reverberations and short period multiples in order to isolate the earths reflectivity

• Yilmaz, 2001

• Deterministic deconvolution

If the wavelet is known, we can design inverse filters to remove the effect of the source and isolate the reflectivity series of the earth

Filters with more terms provide results that are closer to the desired output

Better results are achieved if the desired output resembles the energy distribution of the input

For example, if the desired output is a spike with zero time lag, minimum phase input is required to achieve good results

• The farfield source signature of an airgun array can be recorded with a hydrophone (or modeled) and used for deterministic deconvolution

However, we usually do not really know w(t) (or what we do know does not account for all of the affects on our seismogram besides the earths reflectivity series )

Need to find a way of determining a deconvolution filter that does not require knowledge of the source wavelet

Do we know the

source wavelet?

• Revisit example of least squares filtering for minimum phase wavelet

Find the filter that has the minimum difference between the squared difference of the desired output and the actual output

Input wavelet: (1, -1/2)

Filter (a, b)

Desired output: (1,0, 0)

Sum of squared differences between desired and

actual output:

• We seek to minimize L:

Find the minima:

L

a

optimal

a

slope=0

Revisit example of least squares filtering for minimum phase wavelet

• Least squares filtering for minimum phase case expressed in matrix form

Re-arranging

• Cross-correlation of the

desired output with the

input wavelet

Auto-correlation of the

input wavelet

• Least squares filtering for maximum phase case expressed in matrix form

Re-arranging

• Cross-correlation of the

desired output with the

input wavelet

Auto-correlation of the

input wavelet

• The earths impulse response is assumed to be a white reflectivity series and thus have a flat spectrum. This means that the amplitude spectra of the seismogram is a scaled version of the amplitude series of the source wavelet.

Earths reflectivity series: a white spectrum

• Where rx, rw, and rr are the autocorrelations of the seismogram, source wavelet and reflectivity series, respectively

Where r0 is the autocorrelation of a random series, which is zero everywhere but the zero lag. Here it is the cumulative energy contained in the time series.

Key point: The autocorrelation of the seismogram is an approximation for the autocorrelation of the input wavelet

Autocorrelations and the convolutional model

• Cross-correlation of the

desired output with the

input wavelet

Auto-correlation of the

input wavelet

Approximate as the

auto-correlation of the

seismogram

Approximate as the

cross-correlation of the

desired output with the

seismogram

• The Main Message:

We can approximate the source wavelet with the seismogram because the reflectivity series of the earth is random

As a result, we can design an inverse filter if we know the seismogram and the desired output!!

• Yilmaz, 2001

• Can also demonstrate by generalizing least squares filter

Sum of squared differences between desired output (dt) and actual output (yt)

where is the lag time

• Autocorrelation of xt: rt

Cross-correlation of xt and dt: gt

• The normal equations for Wiener filter

ri: autocorrelation of the input wavelet

ai: the desired filter

gi: crosscorrelation of the desired output with the input wavelet

Robinson & Treitel, 1980

• This example demonstrates:

ri = r-i

r0 = x02+x12+x22+x32+x42

r1 = x0x1+x1x2+x2x3+x3x4

• Wiener filter

Yilmaz, 2001

• Assumptions of deconvolution

The primary reflection series is random

The source wavelet is minimum phase and is

doesn't vary though the earth (stationary).

The noise is random and is of minimal level.

The multiple period is fixed (stationary).

The data are zero offset and dip is ignored.

• Consideration in deconvolution design

Pre-whitening

Filter length (also called operator length)

Noise

Design windows

• Pre-whitening

The spectra of the spiking deconvolution operator is approximately the inverse of the amplitude spectra of the input data

If there are zeros in the original data, these are blown up by deconvolution, causing artifacts

To avoid this, add white noise to the spectra of the input spectra to stabilize deconvolution

• Pre-whitening

Yilmaz, 2001

Amplitude spectrum of input wavelet

Amplitude spectrum of inverse of input wavelet

Result of multiplying the two

• Adding a constant to the zero lag of the autocorrelation is the same as adding white noise to the spectrum

• Other Effects of Prewhitening

Pre-whitening narrows the spectrum, but does not decrease its flatness

Use a relatively small number: 0.1-1% prewhitening

Yilmaz, 2001

• Filter length

Yilmaz, 2001

• Filter length

Yilmaz, 2001

• Effects of random noise

The autocorrelation of random noise should be zero except for zero lag, where it will be a constant (e.g., akin to pre-whitening)

In practice, it effects other lags as well

The unavoidable presence of random noise in

seismic data means that only a very small amount of pre-whitening is need

• Without noise

Yilmaz, 2001

• With random noise

Yilmaz, 2001

• Design windows

To account for changes in the source wavelet with depth/time due to attenuation, etc, it is common to use windows for deconvolution, which allow you to determine different filters and apply them to different parts of the data. Considerations for design window:

It needs to be much longer than the length of the filter (rule of thumb: at least 10x the filter length)

It should avoid particularly noisy areas, multiples, etc

Ideally, merges between different windows should not occur in particular areas of interest

• Types of deconvolution

Spiking deconvolution: turn source into ideal frequency content spike

Predictive deconvolution: remove multiples and reverberations by specifying prediction distance

Waveshaping: normalize wavelets from different surveys, apply deconvolution to non-minimum phase data

Remove instrument effects

• Spiking deconvolutionPurpose: sharpen the source

Actual

Source

wavelet

Ideal

output

Filter

|H(f)|

|G(f)|

• Before

After

Yilmaz, 2001

• Bubble pulse

Befo

re

Aft

er

• In the case where the desired output is a spike, g is a spike scaled by the input wavelet

The normal equations for spiking deconvolution

• Designing spiking deconvolution operators in practice

Minimum phase or zero phase

Length

Prewhitening

Gates for the determination of an inverse filter.

Filter after deconvolution to remove artifacts

• When spiking deconvolution does not work

Yilmaz, 2001

• Predictive deconvolution

Used to remove ringy parts of source or multiples

Seeking a time-advanced form of the input series

For input series x(t), we seek x(t+) where is

the prediction lag

• A common application of predictive deconvolution:

Multiple suppression

• Main steps of predictive deconvolution

Yilmaz, 2001

• The normal equations for predictive deconvolution

In the case where the desired output is a time-advanced version of the input. is the prediction lag.

• Choosing a prediction distance or lag

Measure off of seismic record

Sometimes it is possible to simply determine the

prediction distance by examining the data

Use autocorrelation

Peaks in the autocorrelation function indicate time

delays where the two traces are most similar

http://www.xsgeo.com/course/decon.htm

• Before deconvolution

After deconvolution

• Designing prediction deconvolution operators in practice

Length

Prewhitening

Gates for the determination of an inverse filter.

Filter after deconvolution to remove artifacts

Prediction lag

• Waveshaping deconvolution: can be applied to mixed phase or maximum phase wavelets

Input wavelet

Desired output

Shaping filter

Shaping filter

Yilmaz, 2001

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