# Dictionary:Autocorrelation

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(o, tō, kor, ∂ lā’ sh∂n) Correlation of a waveform with itself. The normalized autocorrelation function $\phi _{11}(\tau )$ for a continuous stationary waveform is

$\phi _{ff}(\tau )=\lim _{t_{1}\to \infty }\lim _{t_{2}\to \infty }{\frac \int _{t_{1}}^{t_{2}}f(t)f(t+\tau )dt}\int _{t_{1}}^{t_{2}}f^{2}(t)dt}}$ ,

where f(t) represents a waveform (or seismic trace) and $\tau$ is the time shift or lag. For equally sampled (digital) data the autocorrelation is

$\phi _{ff}(\tau )=\lim _{\tau \to \infty }{\frac \sum _{k}f_{k}f_{k+\tau }}\sum _{k}f_{k}^{2}}}$ .

An autocorrelation is usually evaluated only over a gate or window. The denominators in the preceding equations are the normalizing factors and sometimes are not included. The autocorrelation function is a measure of the statistical dependence of the waveform at a later time ($\tau$ ) on the present value, or the extent to which future values can be predicted from past values. The autocorrelation function contains all of the amplitude-frequency information in the original waveform but none of the phase information. An autocorrelation function is symmetrical about zero shift, that is, it is zero phase. Deconvolution operators are often based on autocorrelations; see Sheriff and Geldart (1995: 285-287, 292-403).