Zero-phase filtering of a minimum-phase wavelet

Problems in Exploration Seismology and their Solutions

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

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 $(az-1)(az^{-1}-1)=(1+a^{2})-a(z+z^{-1})=(1+a^{2})-2a{\rm {\;cos\;}}(\omega \Delta )$, where $a$ can be complex. Because the imaginary part is zero, the phase is zero.

Solution

Let $\omega _{t}$ be the minimum-phase signal and $f_{t}$ the zero-phase filter. Time-domain filtering is accomplished by convolution, $\omega _{t}*f_{t}$; in the frequency domain the result is $W(z)F(z)$. The factors of $F(z)$ occur in pairs of the form $(az-1)(az^{-1}-1)$, and each pair has roots $z=a$, $1/a$. If $|a|>1$, then $|1/a|<1$. Thus, one member of each pair of roots is not minimum-phase and consequently the filtered signal is mixed-phase.

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Table of Contents (book)

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

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