Autocorrelation

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	9
Pages	295 - 366
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 9.21a

Show that the autocorrelation $ \phi _{gg}(i-j) $ is given by

$ {\begin{aligned}\phi _{gg}(i-j)=\mathop {\sum } \limits _{t}^{}g_{t-j}g_{t-i}.\end{aligned}} $

(9.21a)

Solution

The autocorrelation of $ g_{t} $ is given by equation (9.8e):

$ {\begin{aligned}\phi _{gg}(\tau )=\mathop {\sum } \limits _{k}^{}g_{k}\ g_{k+\tau }.\end{aligned}} $

If we equate $ g_{t-j} $ in equation (9.21a) to $ g_{k} $ in equation (9.8e), we have $ k=t-j $, so equation (9.21a) becomes

$ {\begin{aligned}\mathop {\sum } \limits _{k}^{}g_{k}\ g_{k+i-j}=\phi _{gg}(i-j).\end{aligned}} $

Figure 9.21a.  Autocorrelation diagram.

Problem 9.21b

Calculate $ \phi _{ff} $ for the right triangle shown in Figure 9.21a.

$ {\begin{aligned}f(t)=0,\qquad t\leq 0,\\=(1-t)\;,\quad 0\leq t\leq 1,\\=0,\qquad t\geq 1.\end{aligned}} $

Solution

Let $ t=OA $, $ \tau =DO=BC $, $ AB=AF=1-\tau -t $, $ AG=1-t $:

$ {\begin{aligned}\phi _{ff}(\tau )={\int }_{-\infty }^{+\infty }f(t)f(t+\tau ){\rm {d}}t={\int }_{0}^{B}AG\times AF\ {\rm {d}}t\\={\int }_{}^{1-\tau }(1-t)(1-\tau -t){\rm {d}}t={\int }_{0}^{1-\tau }[(1-t)^{2}-\tau (1-t)]{\rm {d}}t\\=\left[(-1/3+t-t^{2}+t^{3}/3){\frac {\tau }{2}}-\tau t+\tau t^{2}/2\right]|_{0}^{1-\tau }\\=1/3-\tau /2+\tau ^{3}/6.\\\end{aligned}} $

Figure 9.21a shows that a displacement of $ -\tau $ gives the same result.

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Table of Contents (book)

Also in this chapter

• Fourier series
• Space-domain convolution and vibroseis acquisition
• Fourier transforms of the unit impulse and boxcar
• Extension of the sampling theorem
• Alias filters
• The convolutional model
• Water reverberation filter
• Calculating crosscorrelation and autocorrelation
• Digital calculations
• Semblance
• Convolution and correlation calculations
• Properties of minimum-phase wavelets
• Phase of composite wavelets
• Tuning and waveshape
• Making a wavelet minimum-phase
• Zero-phase filtering of a minimum-phase wavelet
• Deconvolution methods
• Calculation of inverse filters
• Inverse filter to remove ghosting and recursive filtering
• Ghosting as a notch filter
• Autocorrelation
• Wiener (least-squares) inverse filters
• Interpreting stacking velocity
• Effect of local high-velocity body
• Apparent-velocity filtering
• Complex-trace analysis
• Kirchhoff migration
• Using an upward-traveling coordinate system
• Finite-difference migration
• Effect of migration on fault interpretation
• Derivative and integral operators
• Effects of normal-moveout (NMO) removal
• Weighted least-squares

External links

find literature about Autocorrelation