Convolution and deconvolution in signal processing spectra analysis

1. Convolution and Deconvolution
By : Kareem N.Hanafi
Under supervision of Dr/ Mostafa S.Toni 1

2. 2
Introduction

3. What is a Wave ?
 A wave
is a vibration in space and time that continues in a repetitive pattern.
-Waves transfer energy from one place to another. Examples include
water waves, sound waves, light waves and seismic waves.
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4. What is the seismic waves ?
 In case of natural source .
Seismic waves :
are the waves of energy caused by the sudden
breaking of rock within the earth or an explosion .
 In case of artificial source .
Seismic waves are generated by
Dynamite , vibroseis , hummer , ……. etc.
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5. What is the seismic wave Components ?
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6. Amplitude and Wavelength
 Amplitude : the maximum displacement or distance from zero point
, also known as the rest position.
 Wavelength : the length of the wave from one peak to the next peak ,
one trough to the next trough, as well as from any point until one complete cycle has
finished.
- they are measured in meters.
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7. Time period
 Time period : the time it takes for one full wave .
- measured in sec.
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8. Frequency
 Frequency :
is the number of waves that pass by
each second.
- measured in hertz.
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9. Frequency and periodic time
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o The relation between frequency and periodic time is
Inverse relationship.
So , high frequency means short period .
low frequency means long period .

10. Waves-Phase
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o Phase :
is the position of a point with
respect to a reference.
a) Sinusoid (sine wave) - zero amplitude at
t=0 .
b) Sinusoid (cosine wave)- max amplitude at
t=0 phase position of 90° over sine wave.
c) Sinusoid – position or lags sine wave above
by 180° .

11. Zero and Minimum Phase wavelet
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o A Zero Phase wavelet is symmetric about time 0. It has energy before time 0.
o A Minimum Phase wavelet has its energy concentrated at the front end of the
pulse, but has no energy before time 0.
Phase vs. Frequency (Hz)Amplitude vs. Frequency (Hz)Time (ms) vs. Amplitude (mvolts)

12. Seismic wavelet
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o A wavelet is a wave-like oscillation with an
amplitude that begins at zero, increases,
and then decreases back to zero .
o A wavelet components are :
• Amplitude
• Polarity

13. Multiples
 Seismic energy which travels from source to interface and is then
reflected back to a detector produces a primary reflection.
 If the energy is reflected more than once in its path to the detector
then a multiple reflection is.
 Multiples produced when there are interfaces with large reflection
coefficients.
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15. Types
 According to occurrence :
1. Surface
2. Inter bed
3. Peg leg
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16. Types
 According to period :
1. Short period
2. Long period
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19. 19Sea Bottom Multiple

20. Correlation
 By correlation, we really mean the comparison of two, individual
things so as to ascertain the degree of similarity that exists between
them.
-An application of correlation outside signal theory is the comparison
of fingerprints.
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21. Correlation
Can you guess the degree of correlation of the following 2 ellipses ?
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22. Correlation
Lets rotate them. Now are they exactly the same?
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23. Correlation
 Provides a means of measuring the similarity between two waveforms.
 Correlation is used to compare two signals to see how similar they are to each
other for differing, relative positions. Essentially, one signal slides past the other
signal and their similarity is measured.
 When we correlate two waveforms that are different we call it ‘Cross-correlation”
 When a waveform is cross-correlated with itself, we call it ‘Auto-correlation’
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24. Cross-correlation
 Used in static derivation programs to compute timing shifts between
traces on land data (time shifts caused by near surface geology
variations).
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uncorrected traces pilot trace
Cross-correlation
corrected traces
compute time shifts
and correct traces

25. Autocorrelation
 Autocorrelation, also known as serial correlation, is the cross-correlation
of a signal with itself.
 Informally, it is the similarity between observations as a function of the
time lag between them.
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Convolution

27. Convolution
 Convolution is the change of a wave shape as a result of passing
it through a linear filter.
 When a signal passes through any filter (such as the earth), it is
replicated many times with different amplitudes and time delays,
by the filter.
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o A seismic trace X(t) is given by the convolution of the basic seismic
wavelet W(t) with the reflectivity series R(t) plus random noise N(t).
The Convolutional Model
X(t) = W(t) * R(t) + N(t)
Seismic trace Reflectivity series Random Noise
Basic seismic
wavelet

29. The Convolutional Model
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Note:
Acoustic impedance (Ai)= ρ * v
Where :
ρ is the density
v is the velocity

30. The Convolutional Model
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Note:
Reflection coefficient (Rc)
= (ρ2V2 − ρ1V1) / (ρ2V2 + ρ1V1) R
Where :
ρ1 = density of medium 1.
ρ2 = density of medium 2.
V1 = velocity of medium 1
V2 = velocity of medium 2.

31. The Convolutional Model
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32. The Convolutional Model
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Deconvolution

34. Deconvolution
A step in seismic signal processing used to remove the
effect of the source wavelet .
 Deconvolution is used for :
- Multiple Removal
- Noise Attenuation
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35. Deconvolutional Model
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recorded seismic trace
decon. filter
“ inverse filter “
Reflection function

36. Types of Deconvolution
 Designature .
 Predictive Deconvolution .
 Surface Consistent Deconvolution .
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Note
• Surface Consistent Deconvolution is too advanced.

37. Deconvolution Methods
 There are two methods
1. Deterministic deconvolution . ( designature )
2. Statistical deconvolution . ( Predictive Deconvolution )
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How can we determine the wavelet ?
• Wells
• Vibroseis
- We apply Deterministic Deconvolution .
• from data itself
- We apply Statistical Deconvolution .
Or

39. Deterministic Deconvolution
 where part of the seismic system is known. No random elements
are involved. For example, where the source wavelet is accurately
known we can do source signature deconvolution.
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X(t) = S(t) * W(t) * R(t) + N(t)
seismic trace Source signature system wavelet reflectivity random noise
o If we have the recorded source signature then we can do deterministic
deconvolution (‘de-signature’) to remove S(t) from the equation .
o similarly , if we have the system wavelet ,( often supplied by the manufacturer ) ,
w(t) can be removed too.
( Source wavelet )

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Is that accurate ?

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42. Statistical Deconvolution
 where no information is available about any of the components of the
convolutional model.
- Specially source wavelet.
- A statistical include :
• Trace by trace Deconvolution.
• Predictive Deconvolution.
• Spiking Deconvolution.
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43. Trace by trace Deconvolution.
 By using the autocorrelation for each trace to determine
the wavelet .
- we deconvolute each trace individually .
- But the output will be convolved with errors . ( why ? )
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44. Predictive Deconvolution
 Apply a deconvolution Filter to remove the multiple only .
 uses information from the earlier part of the seismic trace to predict
and deconvolve the latter part of the trace.
 This method is also known as ‘Gapped Deconvolution’.
 Predictive Deconvolution can be used to attenuate multiples.
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45. Spiking Deconvolution
 is a special case of predictive deconvolution where the
‘gap’ is one sample interval .
 remove the effect of wavelet only.
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