Deconvolution(Ta3520)

Deconvolution – p. 1/14

Convolutional model of seismic dataSeismogram :

x(t) = s(t) ∗ g(t)

x(t) = seismogram

s(t) = seismic wavelet

g(t) = earth response (Green’s function)

Deconvolution – p. 2/14

Inverse filter: deconvolutionDesired: undo effect of source s(t):

f(t) ∗ s(t) = δ(t)

Apply f(t) to seismogram x(t):

f(t) ∗ x(t) = f(t) ∗ s(t) ∗ g(t)

= δ(t) ∗ g(t)

= g(t)

Neutralizing effect of s(t) is called Deconvolution

Deconvolution – p. 3/14

Inverse filter in frequency domainIn frequency domain:

X(f) = S(f)G(f)

Inverse of S(f):

F (f) =1

S(f)

Apply to spectrum of seismogram X(f):

X(f)

S(f)= G(f)

Deconvolution – p. 4/14

Deconvolution in presence of noiseSignal model in presence of noise:

x(t) = s(t) ∗ g(t) + n(t)

In frequency domain:

X(f) = S(f)G(f) + N(f)

Deconvolution (dividing by S(f)):

X(f)

S(f)= G(f) +

N(f)

S(f)

If S(f) is small for certain frequencies, this blows up thenoise.

Deconvolution – p. 5/14

Deconvolution in presence of noise

frequency

signal spectrum

noise spectrum

frequency

signal spectrum

noise spectrum

a) Before deconvolution

1.0

b) After deconvolution

Deconvolution – p. 6/14

Deconvolution in presence of noiseTherefore, stabilize division:

Apply filter:

F (f) =S∗(f)

S(f)S∗(f) + ε2

If |S(f)| >> ε : F (f) = 1S(f)

If |S(f)| << ε : F (f) = S∗(f)ε2

≈ 0

Choose ε as, e.g., : ε = α × max(|S(f)|) withα ≈ 0.01 − 0.1

Deconvolution – p. 7/14

Deconvolution without noise

0 20 40 60 80 100 1200

2

0 20 40 60 80 100 1200

2

4

0 20 40 60 80 100 1200

1

Deconvolution – p. 8/14

Deconvolution in presence of noise

0 20 40 60 80 100 1200

2

0 20 40 60 80 100 1200

2

4

0 20 40 60 80 100 1200

1

Deconvolution – p. 9/14

Deconvolution to desired outputDeconvolution problem:

f(t) ∗ s(t) = d(t)

where d(t) is desired output (previously δ(t))

Deconvolution – p. 10/14

Deconvolution to desired output

frequency

signal spectrum S(ω)

smoothed spectrum D(ω)

smoothed signal d(t)original signal s(t)

time

Deconvolution – p. 11/14

Deconvolution to desired outputThen, output after deconvolution:

F (f)X(f) = D(f)G(f)

So filter:F (f) =

D(f)

S(f)

Or, stabilized,:

F (f) =D(f)S∗(f)

S(f)S∗(f) + ε2

Deconvolution – p. 12/14

Deconvolution: field data

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

time

(s)

500 1000 1500 2000 2500offset (m)

Deconvolution – p. 13/14

Deconvolution field data

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

time

(s)

500 1000 1500 2000 2500offset (m)

Deconvolution – p. 14/14