Introduction to Deconvolution

Introduction to Seismic Imaging ERTH 4470/5470

Yilmaz, ch 2.1-2.5

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

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.

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.

Example of Single Channel Reflection Profile (including artifacts)

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

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= =[1 0 0 0 …] then• D S=D (W R) =(D W) R= R=R

Physical principle of convolution

Physical principle of deconvolution

Minimum Phase

• Energy of the signal wavelet W is "front loaded“ (Figs. 2-15 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 () (Fig. 2.3-2)

• Minimum phase thus becomes one of the basic assumptions of seismic processing

inverse Minimum phase output

Desired spike

Mixed phase

he = n

ae = k

• 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

Convolution Model

Mechanics of calculating the convolution

co=bo*ao+0+0c1=bo*a1+b1*ao+0c2=bo*a2+b1*a1+b2*ao

c3=bo*a3+b1*a2+b2*a1

etc

Convolution is commutative

Convolution using Fourier Transforms

• The convolution calculation in the time domain is slow

• Convolution is more conveniently done using Fourier transforms, F, since F{WR} = {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

Cr

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 auto correlation is symmetric, ie negative lags give the same value as positive lags, we usually only consider the terms for positive lags

Connection between Convolution and autocorrelograms• If RW = S, then {RR} {WW} = {RW}{RW} = S S

• Convolution of the autocorrelograms of two series is the same as the autocorrelogram of the convolution of the series

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

{RR} {WW} = {WW} = SS• This allows us to estimate the

source wavelet (W) from the initial terms of the auto-correlation of the seismogram (S)