Correlation in geophysical signal Processing
In statistics the term correlation is simply referred to the measure of association of two (2) random variables. Let's say we have random variables X and Y, which are linearly related, and the variable X is accurately known.
Then we can check how variable X is correlated (associated) with variable Y by computing the correlation coefficient (k). Since this correlation coefficient lies between -1 to +1 such as -1 ≤ k ≤ +1. If the value of our computed k is 1, then there is positive correlation (association) between variable X and Y. Also if the value of k is -1, then we can decide that there is negative correlation (association) between variable X and Y. While if the value of k is zero then, meaning that there is possibility these two variables are not related at all.
In geophysical signal processing, such as seismic processing, the term correlation is almost the same as that of statistics while the term similarity is frequently used to refer to association and the term variable should be related to geophysical signals. And the fact that investigating the similarity of various seismic signals is the one amongst tasks that seismic analyst should perfome in daily work when it comes to the issue of seismic signal processing.
Through this post you will get an idea and basic concept of why and how correlation is done in seismic signal processing. Let us together see how and why this is done.
What is correlation?
Correlation is the measure of similarity of two geophysical signals. It defines how two (2) geophysical signals quantitatively resemble each other based on predefined criteria.
Types of correlation
In geophysical signal processing, correlation can be performed based on two modes,
1. Cross - correlation
Is the type of correlation at which the similarity is checked between two (2) different signals. First you have to identify which signals you want to perform this function.
By using correlation operator (*), through cross - correlation function, such as the integral range is from - ∞ to ∞
φxy (p) = x(t) * y(t) = ∫x(t) y(t + p)dt
The Normalized cross - correlation function will look like, where p is the index variable (lag)
φxy (p) = [∫x(t) y(t + p)dt]/√(φxx (p)φyy (p)).
Let say y(t) is the part of signal that you need to identify, then x(t) is signal need to investigate
See figure 1 below where series 1 (blue line) represents x(t), series 2 (red line) represent y(t), and series 3 (black line) represent Cross correlation function (coefficient).
Figure 1: Simple cross - correlation Model between two signals ( x(t) and y(t))
2. Auto - correlation
Is the type of correlation at which the similarity is checked for the signal with itself. The Principle is the same as cross - correlation, but here only one variable (same signal) is used.
If we consider the figure 2 above, then as the signal (red line) shifted away from the very beginning of the total overlap, the two signals start to out of phase, and the autocorrelation decreasing. Since the signal is a periodic signal, the signal (red line) soon overlap with the original signal (blue line) again. However the signal (red line) shifted certain lags, as it only partially overlap with the original signal (blue line), therefore, the autocorrelation of the second peak is smaller than the first peak. As the signal (red line) shifted further, the part overlap becomes smaller, which generating the decreasing trend of the peaks.
Relationship between Correlation and Convolution
We can know a relationship between these two operations, by flipping the summation and index variables (lag). Then how can this be done?
If we make a substitute as we do with integrals.
Let consider the correlation summation:
Rxy(p)= ∑n=−∞ to ∞ x(n)y(n−p)
Then substitute n = p−m, where p is a bounded variable and m is a free variable . Then:
Rxy(p) = ∑-m=∞ to -∞ x(p−m)y(-m)
= ∑m=−∞ to ∞ y(m)x(m−p)
= ∑m=−∞ to ∞ y(m)x(−(p−m))
Rxy(p) = y(p)∗x(−p)
You can observe three things in the above derivation,
i/ The summation variable has changed from the index variable n to the dummy variable m .
ii/ The index variable has changed from n to the shift variable p .
iii/ One sequence has been flipped.
Thus, we can say that convolution can be performed by flipping one of the signals prior to correlating.
Applications of correlation in geophysical signal processing
> Extraction of reflectivity function: When the source pulse is known, then by cross - correlating it with the recorded waveform, the reflectivity function is obtained. This means that the cross - correlation function here is used as a deconvolution function.
> To identify seismic multiples: By Performing an Auto correlation, then multiples can be identified. Seismic multiples are repeated like structures seen in seismic sections due to multiple bounce of seismic waves.
> To determine relative time between two (2) signals when they are similar: This can help us to know for example in seismic signals which signal arrived first at a seismic receiver in relation to the other.
> To determine relative arrival time: When relative arrival time is determined for the same station for different similar earthquakes, then the relative location can be determined.
Performing correlation in geophysical signal processing, enables us to assess quantitatively the similarity of various signals in order to extract much information from such data.
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