1. Introduction to Deconvolution
Yilmaz, ch
Introduction to Seismic Imaging ERTH 4470/5470

2. What we want to achieve with deconvolution
Make reflections easier to interpret - ie more like the "real" earth (Figs. 2.2, 2.3, 2.6, 2.7)
improve "spikiness" of arrivals
decrease "ringing"
But without decreasing signal relative to noise.
This is one of the main problems

5. Sources of reverberations
Airgun bubble pulse
Period depends on gun size and pressure. Use multiple guns synchronized to initial pulse to cancel bubble pulses.
Water multiples
Effect varies with water depth.
For shallow water, multiples are strong but reduce quickly with depth.
For deep water, multiple is below depth of main reflectors.
For slope depths, effect is difficult to eliminate as first (strongest) multiple arrives at main depth of interest.
Peg-leg multiples
Due to interbed multiples which can sometimes be misinterpreted as primaries.

6. Examples of acoustic pulses produced by small to large airguns
Examples of acoustic pulses produced by small to large airguns. A series of pulses follow the primary pulse due to the oscillation of the air bubble. The frequency of the bubble pulses are higher for small guns and lower for large guns. Thus by summing all the signals together, aligned at the primary pulse, we get a total signal in which the bubble pulse ringing has been reduced relative to the primary. This is referred to as a tuned airgun array.

7. Example of Single Channel Reflection Profile (including artifacts)

8. Example of strong water multiples from shallow to deep water offshore Flemish Cap

9. The Convolution/Deconvolution Operator
Convolution describes how a source wavelet (W) interacts with a set of reflectors (R) to produce the observed seismogram (S) (Fig. 6.21)
Mathematical properties
Commutative: A  B=B  A
Associative: A (B  C)=(A  B)  C
Deconvolution operator (D) is inverse
If D  W=d =[ …] then
D  S=D (W  R)
=(D  W)  R=d  R=R

10. Physical principle of convolution

11. Physical principle of deconvolution

12. Minimum Phase
Energy of the signal wavelet W is "front loaded“ (Figs to 2-18)
peak amplitude mainly occurs at the beginning of the signal.
This results in a Fourier transform of the wavelet which has a minimum phase
If W is not minimum phase, then we cannot find the operator D (ie W-1) to convert the signal W into a spike (d) (Fig )
Minimum phase thus becomes one of the basic assumptions of seismic processing

14. Desired spike
he = n
ae = k
Mixed phase
inverse
Minimum phase
output

15. Convolution Model S = W  R + N (noise) Five Main Assumptions
#1: R is composed of horizontal layers of constant velocity
#2: W is composed of a compressional plane wave at normal incidence which does not change as it travels, ie is stationary
#3: noise N = 0
#4: R is random. There is no "pattern" to the set of reflectors R
#5: W is minimum phase
Generally #3 is NOT valid
ie. there will always be some noise on our seismic records
We will need to investigate what happens when N ≠ 0
We generally do not know W

16. Mechanics of calculating the convolution
co=bo*ao+0+0
c1=bo*a1+b1*ao+0
c2=bo*a2+b1*a1+b2*ao
c3=bo*a3+b1*a2+b2*a1
etc
Convolution is commutative

17. Convolution using Fourier Transforms
The convolution calculation in the time domain is slow
Convolution is more conveniently done using Fourier transforms, F, since F{WR} = {F(W) ∙ F(R)}
We can calculate the convolution of two series by taking the Fourier transforms of the series, multiplying them together and then taking the inverse transform
Since Fourier transforms are so quick to compute, this is much faster than doing the convolution itself

19. Cross-Correlation and Auto-Correlation Functions
Cross-correlation is like convolution except that the operator series is not inverted
Cross-correlation is NOT commutative
Auto-correlation is when both series are the same. Since the auto­correlation is symmetric, ie negative lags give the same value as positive lags, we usually only consider the terms for positive lags

20. Connection between Convolution and autocorrelograms
If RW = S, then {RR}  {WW} = {RW}{RW} = S  S
Convolution of the autocorrelograms of two series is the same as the autocorrelogram of the convolution of the series

21. Use of autocorrelograms to calculate source wavelet
Following assumption #4, if R is random then R  R only has a value at t=0 lag, ie at any other lag there is no correlation of reflectors (Fig. 2-12)
In this case, the convolution of the auto-correlation of R with the auto-correlation of W is the same as the auto-correlation of the series W with some extra values that are very close to zero
{RR}  {WW} = d  {WW} = SS
This allows us to estimate the source wavelet (W) from the initial terms of the auto-correlation of the seismogram (S)