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Correlation &Convolution
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Sampling of Seismic Data
• In signal processing, – sampling is the reduction of
– a continuous signal to a discrete signal.
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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
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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
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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.
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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
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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.
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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.
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Cross-correlation
• the cross-correlation can also performed – in the Fourier domain.:
• Cross-correlation = Multiplication of Amplitudes and
Subtraction of Phase spectrum.
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Mechanics of Cross-correlation
Shift one trace
Multiply, point by point
Sum
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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
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Used to Extract required Frequency
Used for band Pass Filter
Determination of Best Match between the signal
Practical Applications
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Auto-correlation• The Auto-correlation is a Cross-correlation
• of a function with itself.
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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.
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Convolution• In seismic,
– we obtain a response for a certain model
– by convolving the
– seismic signal of the
source – with the reflectivity
function.
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Mechanics of Convolution
Shift
Multiply, with other trace point by point
Sum & Plot
Reverse one trace (mirror)
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Example of a convolution
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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.
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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