# Dictionary:Autocorrelation

{{#category_index:A|autocorrelation}}
(o, tō, kor, ∂ lā’ sh∂n) Correlation of a waveform with itself. The normalized autocorrelation function **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{11} (\tau) }**
for a continuous stationary waveform is

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{ff}(\tau) = \lim_{t_1 \to \infty} \lim_{t_2 \to \infty } \frac {\displaystyle \int^{t_2}_{t_1} f(t) f(t+\tau) dt }{\displaystyle\int^{t_2}_{t_1} f^2(t)dt} }**,

where *f*(*t*) represents a waveform (or seismic trace) and **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau }**
is the time shift or lag. For equally sampled (digital) data the autocorrelation is

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{ff}(\tau) = \lim_{\tau \to \infty} \frac { \displaystyle \sum_{k} f_k f_{k+\tau} } {\displaystyle\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 (**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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).