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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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