# Zero-phase filtering of a minimum-phase wavelet

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 9.16

Show that the result of passing a minimum-phase signal through a zero-phase filter is mixed phase.

### Background

Zero-phase signals are discussed in Sheriff and Geldart, 1995, section 15.5.6d, where it is shown that the spectrum of a zero-phase signal comprises products of pairs of factors of the form , where can be complex. Because the imaginary part is zero, the phase is zero.

### Solution

Let be the minimum-phase signal and the zero-phase filter. Time-domain filtering is accomplished by convolution, ; in the frequency domain the result is . The factors of occur in pairs of the form , and each pair has roots , . If , then . Thus, one member of each pair of roots is not minimum-phase and consequently the filtered signal is mixed-phase.

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Making a wavelet minimum-phase | Deconvolution methods |

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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