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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