Autocorrelation

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

Problem 9.21a

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

 {\displaystyle {\begin{aligned}\phi _{gg}(i-j)=\mathop {\sum } \limits _{t}^{}g_{t-j}g_{t-i}.\end{aligned}}} (9.21a)

Solution

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

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

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

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

{\displaystyle {\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 ${\displaystyle t=OA}$, ${\displaystyle \tau =DO=BC}$, ${\displaystyle AB=AF=1-\tau -t}$, ${\displaystyle AG=1-t}$:

{\displaystyle {\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 ${\displaystyle -\tau }$ gives the same result.