GEOPHYS 6001 KfUPM Advanced Geophysical Data Analysis Problem Set Need help with Geophysics homework as you can see in the attachment files the homework an

GEOPHYS 6001 KfUPM Advanced Geophysical Data Analysis Problem Set Need help with Geophysics homework as you can see in the attachment files the homework and lectures if needed . GEOPHYS 6001
Advanced Geophysical Data Analysis
Contents
Lecture 19
 Cross correlation
Kelly Liu
 Auto correlation
 Correlation coefficient
 Convolutional Neural Network
References: Yilmaz, “Seismic data analysis”
Kearey et al., “An introduction to geophysical exploration”
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Cross-correlation
 In data analysis, it often requires measurement of the similarity or time
alignment of two traces.
 Correlation is another time-domain operation that is used to make such
measurements.
φab (τ ) = ∑ ak bk +τ
k
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Cross-correlation
Exercise 1:
φab (τ ) = ∑ ak bk +τ
φab (τ ) =k∑ ak bk +τ
a={1,2,3,2,1}
b={1,-1,1,-1,2}
Xcorrelation of a with b
∴φab (0) = ∑ akk bk = a0b0 + a1b1 + a2b2 + a3b3 + a4b4 + a5b5 = 1 − 2 + 3 − 2 + 2 = 2
φab (1) = ∑ ak bk +1 = a0b1 + a1b2 + a2b3 + a3b4 = −1 + 2 − 3 + 4 = 2
φab (2) = ∑ ak bk + 2 = a0b2 + a1b3 + a2b4 = 1 − 2 + 6 = 5
φab (3) = ∑ ak bk +3 = a0b3 + a1b4 = −1 + 4 = 3
φab (4) = ∑ ak bk + 4 = a0b4 = 2
-2, -1, 0, 1, 2, 3, 4
= a-3
φab (−1) = ∑ ak bk −1-4,
1b0 + a2 b1 + a3b2 + a4 b3 = 2 − 3 + 2 − 1 = 0
φab (−2) = ∑ ak bk − 2 = a2b0 + a3b1 + a4b2 = 3 − 2 + 1 = 2
φab (−3) = ∑ ak bk −3 = a3b0 + a4b1 = 2 − 1 = 1
φab (−4) = ∑ ak bk − 4 = a4b0 = 1
φab (τ ) = {1,1,2,0,2,2,5,3,2}
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Cross-correlation
Exercise 1:
φab (τ ) = ∑ ak bk +τ
a={1,2,3,2,1}
b={1,-1,1,-1,2}
Xcorrelation of a with b
k
∴φab (0) = ∑ ak bk = a0b0 + a1b1 + a2b2 + a3b3 + a4b4 + a5b5 = 1 − 2 + 3 − 2 + 2 = 2
φab (1) = ∑ ak bk +1 = a0b1 + a1b2 + a2b3 + a3b4 = −1 + 2 − 3 + 4 = 2
φab (2) = ∑ ak bk + 2 = a0b2 + a1b3 + a2b4 = 1 − 2 + 6 = 5
φab (3) = ∑ ak bk +3 = a0b3 + a1b4 = −1 + 4 = 3
φab (4) = ∑ ak bk + 4 = a0b4 = 2
φab (−1) = ∑ ak bk −1 = a1b0 + a2b1 + a3b2 + a4b3 = 2 − 3 + 2 − 1 = 0
φab (−2) = ∑ ak bk − 2 = a2b0 + a3b1 + a4b2 = 3 − 2 + 1 = 2
φab (−3) = ∑ ak bk −3 = a3b0 + a4b1 = 2 − 1 = 1
φab (−4) = ∑ ak bk − 4 = a4b0 = 1
φab (τ ) = {1,1,2,0,2,2,5,3,2}
Length: 5+5-1=9
Compare with convolution, correlation don’t need to reverse the second sequence.
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Auto-correlation
φaa (τ ) = ∑ ak ak +τ
k
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Cross-correlation
Exercise 2:
Wavelet 1: (2,-1,1,0,0)
Wavelet 2: (0,0,2,-1,1)
 Xcorrelation of w1 with w2
Two wavelets are identical in shape, wavelet 2 is
shifted by two samples with respect to wavelet 1.
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Cross-correlation
Exercise 2:
Wavelet 1: (2,-1,1,0,0)
Wavelet 2: (0,0,2,-1,1)
 Xcorrelation of w1 with w2
It measures how much two time series resemble each other.
If wavelet 2 were shifted two samples back in time, then
these two wavelets would have the maximum similarity.
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Cross-correlation
Exercise 3:
Wavelet 1: (2,-1,1,0,0)
Wavelet 2: (0,0,2,-1,1)
 Xcorrelation of w2 with w1
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Cross-correlation
Exercise 3:
Wavelet 1: (2,-1,1,0,0)
Wavelet 2: (0,0,2,-1,1)
 Xcorrelation of w2 with w1
If wavelet 1 were shifted by two samples forward in time,
these two wavelets would have maximum similarity.
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Cross-correlation
φ ab (τ ) = φba (−τ )
Unlike convolution, cross correlation is not commutative
—the output depends on which array is fixed and which is moved.
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Auto-correlation
Exercise 4:
Wavelet 1: (2,-1,1,0,0)
Wavelet 2: (0,0,2,-1,1)
 Autocorrelation of W1
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Auto-correlation
Exercise 4:
Wavelet 1: (2,-1,1,0,0)
Wavelet 2: (0,0,2,-1,1)
 Autocorrelation of W1
 Maximum correlation occurs at zero lag.
 The autocorrelation function is symmetric.
 Only one side of the autocorrelation needs to be computed.
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Convolution
&
Correlation
Correlation is a
convolution without
reversing the moving
array.
Fig. 1.1-19. The frequency-domain description of convolution and correlation.
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Convolution
&
Correlation
Convolution
Y=|Y|e iφy
=|X1|.|X2| ei(φx1+φx2)
Corss-Correlation
Y=|Y|e iφy
=|X1|.|X2| ei(φx1-φx2)
Fig. 1.1-19. The frequency-domain description of convolution and correlation.
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Convolution & Correlation
 Both convolution and correlation produce an output with a spectral bandwidth that
is common to both the input time series. The immediate example is the band-pass
filtering process.
 Phases are additive in case of convolution and subtractive in case of correlation.
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Convolution & Correlation
 Auto-correlation is symmetric with respect to zero lag. The output series is
zero-phase.
 The autocorrelation function contains all the amplitude information of the
original waveform but none of the phase information. The original phase
relationships being replaced by a zero phase spectrum.
 The autocorrelation function and the power spectrum form a Fourier pair.
φ xx (τ ) < − > x( f )
2
Power spectrum
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Cross-correlation
As a measure of similarity, cross-correlation has been used widely at various
stages of data processing.
For instance,
 Traces in a CMP gather are cross correlated with a pilot trace to compute
residual statics shifts.
 The fundamental basis for computing velocity spectra is cross correlation.
 The constituent elements of the Wiener filter are cross correlation of the
desired output waveform with the input wavelet, and autocorrelation of the
input wavelet.
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Auto-correlation
Sketch the auto correlation
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Haton et al., “Seismic Data Processing”
Autocorrelation
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Haton et al., “Seismic Data Processing”
Auto-correlation
+
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Haton et al., “Seismic Data Processing”
Auto-correlation
+
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Haton et al., “Seismic Data Processing”
Cross-correlation
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Haton et al., “Seismic Data Processing”
Crosscorrelation
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Haton et al., “Seismic Data Processing”
Autocorrelation
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Haton et al., “Seismic Data Processing”
Autocorrelation
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Haton et al., “Seismic Data Processing”
Correlation Coefficient
Or Linear correlation coefficient
∑ ( x − x )( y
γ=
i
i
i
− y)
∑ (x − x) ∑ ( y
i
x
y
2
i
i
.
Note that
i
− y)
2
| γ |≤ 1
Is the mean of X
Is the mean of Y
γ =1
γ = −1
means complete positive correlation.
means complete negative correlation.
Numerical recipes in Fortran 77; P630
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Correlation Coefficient
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http://www.okstate.edu/artsci/botany/bisc3034/lnotes/corr.jpg
Cross-correlation
Sweep signal
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Robinson and Coruh, “Basic Exploration Geophysics”
Cross-correlation
Correlogram
Normally, a correlogram is a plot of the sample autocorrelations versus the time lags.
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Kearey et al., An introduction to Geophysical Exploration
Cross-correlation
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CNN
Convolutional Neural Network
The convolution is performed on the input data with the use of a filter to
produce a feature map.
In the convolution operation, the filter (the
green square) is sliding over the input (the
blue square) and the sum of the
convolution goes into the feature map (the
red square).
2D convolution. Good for image processing.
Convolution or cross-correlation?
= cross-correlation.
https://www.freecodecamp.org/news/an-intuitive-guide-to-convolutional-neural-networks-260c2de0a050/
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Summary




Cross correlation
Auto correlation
Correlation coefficient
Convolutional Neural Network
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Geophys 6001: Advanced Geophysical Data Analysis
Assignment 8; 10 points
Two time series are given by Xt=[1, 9, 2, 1, 4], Yt=[2, -5, 3, 8, 1]. Calculate the
cross-, auto- correlations, and the correlation coefficient. You may use Matlab for
the calculations. Explain your results.
Note:
 Study Lecture 19
 xcorr – cross correlation in Matlab
 corrcoef – correlation coefficient in Matalb

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