# Calculating crosscorrelation and autocorrelation

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 |

## Contents

## Problem

Four causal wavelets are given by , , , . Calculate the crosscorrelations and autocorrelations , , , , and in both the time and frequency domains.

### Background

A causal wavelet as defined in problem 5.21 has zero values when .

The *crosscorrelation* of and tells us how similar the two functions are when is shifted the amount relative to . The crosscorrelation is given by the equation

**(**)

This equation means that is displaced units to the left relative to , corresponding values multiplied, and the products summed to give . The result is the same if we move to the right units, that is,

**(**)

The functions and are closely related. Using two simple curves, it is easily shown that we can crosscorrelate by reversing and convolving the reversed function with , that is,

**(**)

(since convolution is commutative). Reversing a function in time changes the sign of , so the exponent of changes sign also, becoming and becoming the conjugate complex . The convolution theorem (equation (9.3f)) now becomes the *crosscorrelation theorem:*

**(**)

If is the same as , we get the *autocorrelation* of and equations (9.8a,d) become

**(**)

the transform relation being the *autocorrelation theorem*. Since the two functions are the same, does not depend upon the direction of displacement. The autocorrelation for equals the sum of the data elements squared, hence is called the *energy of the trace;*

**(**)

Both the autocorrelation and the crosscorrelation are often normalized; in the case of the autocorrelation, equation (9.8e) becomes

**(**)

The normalized crosscorrelation is

**(**)

### Solution

*Time-domain calculations:*

Using equation (9.8a) to calculate , i.e., is first shifted to the left; we have:

So

where marks the value at . Proceeding in the same way, we find that

To find we displace instead of and obtain , which equals .

Autocorrelations are found in the same way:

*Frequency-domain calculations*

The -transforms of , , and are , , . The conjugate complexes of these transforms are are , , . Using these transforms, we get

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Reflection field methods | Geologic interpretation of reflection data |

## 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