Making a wavelet minimum-phase

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 9.15

The wavelet $[-0.9505,\;-0.0120,\;0.9915]$ is not minimum-phase. How would you change it to make it mimimum-phase without changing it any more than necessary? Give two methods.

Background

Minimum-phase wavelets are discussed in Sheriff and Geldart, 1995, section 9.4 and section 15.5.6a.

Solution

The transform of the wavelet is $W(z)=-0.9505-0.0120z+0.9915z^{2}$. The roots of $W(z)=0$ are 0.9652 and $-0.9531$, so to make the wavelet minimum-phase, we must change it so that the magnitudes of both roots are greater than unity. To increase both roots by the same factor, we multiply both roots by a number slightly greater than $1/0.9531=1.0492$. If we use the multiplier 1.0500 the roots become 1.0135 and $-1.00076$ and the revised equation is

${\begin{aligned}(z-1.0135)(z+1.0008)=-1.0143-0.0127z+z^{2}.\end{aligned}}$

To obtain the same value at $t=0$ as we had before, we multiply by $0.9505/1.0143=0.9371$ and the transform becomes $0.9505-0.0119z+0.9371z^{2}$ which gives the wavelet [$-0.9505$, $-0.0119$, 0.9371]. The modified wavelet is nearly the same as the original except that its “tail” has been decreased from 0.9915 to 0.9371, a 5.5% decrease.

A second method is to use a taper to reduce the tail of the wavelet. A commonly used taper is $k^{t}$ where $t$ is in milliseconds and $k$ is slightly less than unity, for example, 0.9950. Using this value for $k$ and assuming that the sampling interval is 2 ms, the values of the taper are 1.0000, $0.9950^{2}$, $0.9950^{4}$, that is, 1.0000, 0.9900, 0.9801 and we get $W(z)=-0.9505-0.9900\times 0.0120z+0.9801\times 0.9915z^{2}=-0.9505-0.0119z+0.9718z^{2}$. The roots of $W(z)=0$ are now 0.9904, $-0.9783$, both less than unity so we did not apply enough taper. We next try using $0.9850^{t}$, which reduces the wavelet to $[-0.9505,\;-0.0116,\;0.9333]$. The roots are now 1.0154 and $-1.0030$ and $W(z)=(z-1.0154)(z+1.0030)=-1.0184-0.0124z+z^{2}$. Multiplying by $0.9505/1.0184=0.9333$ makes the wavelet [0.9505, $-0.0116$, 0.9333].

Comparing the two methods, we see that the second method has changed the wavelet slightly more than the first. In practice, a wavelet will have many more than three elements and the second method will generally be more practical. To verify that the taper is large enough to achieve its objective, we have to find all of the roots.

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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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