CONVOLUTION AND CORRELATION.ppt

7/29/2019 CONVOLUTION AND CORRELATION.ppt

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Correlation &Convolution



Sampling of Seismic Data

• In signal processing, – sampling is the reduction of

– a continuous signal to a discrete signal.



Sampling of Seismic Data• The output from Geophone is in analog form

– To convert it to digital form – it is sampled at regular time intervals

• The ability to correctly reconstruct a digital

signal depends upon – Frequency contents of the single

– The sampling interval

• Wrong sampling interval yield data loss



Sampling of Seismic Data• A single can be reconstructed from its

samples without loss of information• If the original signal has no frequency above

½ the sampling frequency

• For a given band limited function – the rate at which it must be sampled is called

– The Nyquist Frequency



Sampling of Seismic Data

• The Nyquist –Shannon sampling theorem states

– that perfect reconstruction of a signal is possible – when the sampling frequency is greater than

– twice the bandwidth of the signal being sampled,

• In other words,

– the sampling frequency should be more than twice

– the maximum frequency component of the signal.



Nyquist Frequency Sampling Interval (ms)

f q (Hz)

0.5 1000

1 500

2 250

4 125

8 62.5

The sampling Interval will be Half of the Nyquist Frequency





Correlation• There is a standard technique used by a large

number of processes – to analyze the similarity of selected seismic

traces.

–This technique is called "correlation".

• In other words we may define it

– as the measure of how two traces "look alike“

–or the extent to which

– one trace can be considered

– a linear function of the other.



Cross-correlation• The cross-correlation function is a measure of

– the similarity between two data sets.

– One dataset is displaced varying amounts

– relative to the other and corresponding values

– of the two sets are multiplied together

– and the products summed to give – the value of cross-correlation.



Cross-correlation

• the cross-correlation can also performed – in the Fourier domain.:

• Cross-correlation = Multiplication of Amplitudes and

Subtraction of Phase spectrum.



Mechanics of Cross-correlation

Shift one trace

Multiply, point by point

Sum



1 1 2 3 2 1 1 0 0 0

1 1 2 2

1 1 2 3

1 1 2 6

1 1 2 9

1 1 2 9

1 1 2 71 1 2 5

1 1 2 2

1 1 2 1

1 1 2 0

2

3

6

9 9

7

5

21

0

2

3

6

9 9

7

5

21

0



Used to Extract required Frequency

Used for band Pass Filter

Determination of Best Match between the signal

Practical Applications



Auto-correlation• The Auto-correlation is a Cross-correlation

• of a function with itself.



Convolution

• Convolution is a mathematical operation

– defining the change of shape of a waveform – Resulting from its passage through a filter.

The asterix denotes the convolution operator.



Convolution• In seismic,

– we obtain a response for a certain model

– by convolving the

– seismic signal of the

source – with the reflectivity

function.



Mechanics of Convolution

Shift

Multiply, with other trace point by point

Sum & Plot

Reverse one trace (mirror)



Example of a convolution



Deconvolution• The aim of deconvolution is

– the reverse of convolution – in such a way that the

– reflectivity function is reconstructed.

• In practice one obtains not the real reflectivityfunction, but it results in

– a shortening of the Signals

– Suppression of Noise – Suppression of Multiples.



Deconvolution• There are different types of deconvolution:

1. Spiking Deconvolution: – desired output function is a spike

– also whitening deconvolution

2. Predictive Deconvolution: – attempts to remove the effect of multiples

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CONVOLUTION AND CORRELATION.ppt