VẬT lý địa CHẤN 13 deco

Outline
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Correlation
Auto-correlation
Filter
Convolution
Deconvolution
Wiener filter


Numerical evaluation of Cross-correlation

n−τ

φ (τ ) = ∑ x y
i + τ

xy

i

i =1

xi: (i=0 ... n)
yi: (i= 0 ... n)

φxy(τ) : (-m < τ < +m)
m = max. displacement

In Fourier domain:
Cross-correlation = Multiplication of Amplitude spectrum
and Subtraction of Phase spectrum


Reynolds, 1997


Cross-correlation function


Autocorrelation
Cross-correlation of a Function with itself

n −τ

φ xx (τ ) = ∑ x i + τ x
i =1

xi = (i=0 ... n)
i

φxx (τ) = (-m < τ < +m)
m = max. displacement


Auto-correlation of two identical waveforms

Normalization of correlation
Auto-correlation

Cross-correlation

φ xx (τ )
φ xx;norm (τ ) =
φ xx (0 )
φ xy ;norm (τ ) =

φ xy (τ )

φ xx (0 )φ yy (0 )


Auto-correlation: multiples


Autocorrelation functions contain reverberations

a) A gradually decaying function indicative of short-period
reverberation
b) A function with separate side lobes indicative of long-period
reverberations: multiples


General Filter

Filter
Operator

Input function

Output function


General Filter

Filter
Operator

Deltafunction

Impulse response


Example of input response


From geology to seismogram


Convolution

Convolution

y(t) = g(t) ∗ f(t)

outputfunction
Seismic trace

Inputfunction
Source wavelet

Filterfunction
Reflectivity function


Numerical implementation of convolution
m

y k = ∑ g i f k −i
i =0

k = 0 ... m+n
gi = (i=0 ... m)
fj = (j= 0 ... n)

In Fourier domain:
Convolution = Multiplication (of Amplitudes
and Addition of Phasespectrum)


Example of Convolution

m

yk = ∑ g i f k −i

i =0

fk

f0

f1

f2

f3

2

0

0

0

-1 0

0

0 ½

0

0

0

-1
2-1 -1
2 2-1
½ ½
2-1 -1
2 ½
2-1 -1
½
½ ½
gk ½
2 2-1
2-1 -1
2 2
½
½
½ 2-1
½ ½
g
gg
gg
gg
g
g
gg
gg
gg
g
2

yk

12

021

4

10
2

0
12

-2 1

021

0

g102

gg021

gg102

ggg012

-2 1 -½ 0

ggg012

gg021

gg10

1 -½ ¼

g0

0


Convolution model of the Earth
wt
( equivalent Wavelet)

gt = kδt ∗ st ∗ nt ∗ pt ∗ e t

Impulse
of source

Near-surface
zone of source

Source effect

gt = w t ∗ e t

+ Noise

Reflectivity
of the Earth

Additional modifying
Effects (absorption, wave conversion)

+ Noise


*

=


+

=


Aim of Deconvolution
Theoretical:
Reconstruction of the Reflectivity function

Practical:
Shorting of the Signal
Suppression of Noise
Suppression of Multiples

Deconvolution
Reverse of Convolution

xt = wt ∗ e t

et = xt ∗ w-1t

=> Inverse Filtering
Problem:
w(t) is in general not known,
i.e. w

-1

(t) Can not be determined directly


Principle of Wiener filtering


Principle of Wiener-Filters

Input-Function ∗
(known)

Filter = Output-Function
(wanted)

g0

g1
…

∗

(known)

f0

g 0f 0 = y0

f1

g1 f0 + g 0 f1 = y 1
…

…

=> Solve system of equations


Wiener filter
• Spiking deconvolution:
desired output is a spike
• Predictive deconvolution:
attempts to remove the effect of multiples


After trace
Common shot-gathers

just balancing
after demultiplexing
Afterfor
spiking
deconvolution
Corrected
wavefront
divergence

Yilmaz, 1987